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,-
(c) Analytical model
Analytical model for distortional beam buckling
These beam models are, of course, directly analogous to the column model shown in Fig. 4. The only significant difference lies only in the stiffness of the rotational spring and the necessary modifications to the design expressions for the rotational spring stiffness k^ and the buckling lengtih X are given in Hancock's paper. This then leads to similar equations for the critical stress for distortional buckling. When comparing the results given by Hancock's original expressions and GBT for the distortional buckling of channel section beams (Davies and Jiang 1996b), it was found that good results were
13 obtained for small web depths but that, for deeper sections, Hancock's expression could become unsafe due to the neglect of web local buckling and the assumption of fixity at the bottom (tension) end of the web. In order to improve the accuracy of the design expressions, modifications were proposed by introducing a reduction coefficient into the equation for k^ and by assuming the tension end of the web to be pinned. It was then shown that the modified design expressions gave an excellent estimate of the critical bending moment which was somewhat better than that given by Hancock's original expressions. Distortional buckling of restrained beams Unrestrained beams are, of course, relatively rare. It is more usual for cold-formed section beams to be at least partly restrained by the floor or roof that they support. An important practical case is that of a purlin supporting a lightweight roof. In this, and other similar cases, the purlin receives both lateral and torsional restraint to one flange. It is an important feature of GBT that such restraints can be incorporated into the member properties (Davies at al. 1994). Consider a channel section with the upper flange restrained in position laterally and restrained torsionally by a spring which can have varying stiffness c^. This is a typical representation of a purlin restrained by sheeting. Fig. 7 shows the orthogonal modes of deformation, computed with the aid of GBT, for a typical value of c^.
^
Figure 7
Orthogonal deformation modes for a restrained purlin
The presence of the restraints modifies the orthogonal deformation modes from those shown in Fig. 5. Mode 1 is axial strain which is not relevant for a purlin in bending and the remaining modes are each associated with a buckling mode. Mode 3 is global buckling by bending about the major axis. Due to the existence of the restraints, the mode of buckling by bending about the minor axis of the section is eliminated and the mode of pure torsion about the shear centre is turned into a combination of torsion and distortion (mode 2). Modes 4 and 5 are the distortional modes. Under downward load, such a purlin usually fails in distortional buckling with a combination of modes 4 and 5. However, for the important case of uplift load, there is a transition from distortional buckling to the combined torsional-distortional mode as the span increases. The span at which this transition takes place depends on the amount of restraint from the cladding and, if this is sufficiently great, it may not occur at all. Evidently, the behaviour is more complex than for an unrestrained section and there are two distortional modes to consider in design. Lack of space precludes a full discussion of this topic but it is sufficient to say that Pekoz and Soroushian (1982) have described an adequate model for the combined torsional-distortional mode 2 and that Hancock's distortion model in Fig. 5 can be readily modified further (Davies and Jiang 1996c) to include the additional torsional restraint from the cladding. Both of these models must be considered and the more critical of the two gives a satisfactory design expression for a restrained purlin.
14 Direct strength design of cold-formed sections The foregoing paragraphs demonstrate that the available models for local and distortional buckling design are far from simple and have significant limitations. The current trend is to increase the complexity of section shapes by rolling in additional features such as intermediate flange stiffeners, web stiffeners and compound lips. This increases the complexity of the required mathematical models and, consequently, directs attention towards second-order analytical techniques such as the finite strip method or GBT which consider the complete cross-section. The "direct strength method" (Schafer and Pekoz 1998) recognises this trend and proposes a formal design procedure based on elastic bucking solutions for the complete cross-section. This procedure recognises that, for economic design, it is necessary to take advantage of the postbuckling strength. It therefore takes the effective width equations (3) and (4) and applies these to the complete cross-section. The method has been initially expressed in terms of flexural members but it is clearly also applicable to members under axial compression. Thus, the design moment capacity is given by: M^ = pMy
where:
if A. =
if A, > 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 (P, =7r^EI„ iV is Euler's load), The cubic interpolation ftmctions may be used with good accuracy in comparison with the stability ftmction for stiffness formulation. Fig. 1 shows a plot of the percent error in the element stiffness matrix terms percent error = 100 ^^%^"f^-' [ksJ
(1)
in which [ k j and [ k j are the element stiffness matrices derived from the cubic and stability interpolation ftmctions, respectively.
21 It can be seen from Fig. 1 that if |P / P j < 0.4, the largest error in any of the terms of [kj is less than one percent. If the geometric stiffness matrix [kgj is used for cases of |P / P j > 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
^ <^'
U
3 elements
2 elonents/
J
e^ ''
o o o — -
r
0.0
/
\ / /*
[^
/''
.
1
1 1
1.4
1 py
Tip displacement w/L
Figure 3 Load-displacement behaviour of a cantilever
*P
-t^ 0.0 I 1 1 1 1 1 1 1 «0.00 0.02 0.04 006 0.08 010 012 0.14 0.16 Lateral displacement V/L
Fig.4 Lateral buckling behaviour of a beam
24
ends restrained against rotations about the X- and Z-axes. The section properties for the beam are: E = 71,240 MPa, G = 27,190 MPa, A = 18 mm^ ly = 0.54 mm\ t = 1,350 mm\ J = 2.16 m m \ and span length L = 200 mm. The theoretical lateral-torsional buckling moment can be predicted as Mcr = VEIGJ /L = 1493.3 N-mm. In the geometrical nonlinear analysis, a disturbing torsional moment of 0.0 IM is applied at mid span of the beam to initiate the displacement and twist. The beam is modelled as two, four and eight proposed elements and, for the purpose of comparison, by the cubic elements. Since the coupling terms between the lateral, flexural and torsional displacements are included in deriving the stiffness matrices of both elements, the lateral-torsional instability can be predicted. As shown Fig. 4, the lateral-torsional buckling load can be obtained accurately by modelling the beam as eight elements. The lateral-torsional buckling loads predicted by modelling the beam as two and four elements exceeds the theoretical load by seven and ten percent respectively. Since the axial force and the flexural deformations of the beam are small in the pre-buckling range, the differences between the load-displacement curves obtained by the proposed element and the cubic element are negligible. However, the differences become obvious in the far post-buckling range because the bowing effect in the beam becomes large. To fully capture the nonlinear behaviour in the entire range of loading, the proposed element is recommended.
HEXAGONAL FRAME Figure. 5 shows a hexagonal space frame with member properties given as: Young's modulus E = 3032 MPa, shear modulus G = 1096, cross-sectional area A = 3.187 cm^, torsional constant J = 1.378 cm"^, and moments of inertia ly = t = 0.832 cm"*. The frame is used as a benchmark problem to investigate the accuracy of the proposed element for analysing the nonlinear behaviour of space frames. Each jframe member is modelled by one fi*ame element. The load-displacement curve obtained by the proposed method is found to be consistent v^th that given by Chan and Zhou (1994) as shown in Fig. 6. Chan and Zhou's approach is based on point-wise equilibrium polynomial element in which the element forces are directly updated from the member basic deformations. Although their force recovery method can be used successfully for geometrical nonlinear analysis, it is limited by its usage in the path-dependent plastic hinge analysis. In inelastic analysis, the incremental forces at the plastic hinge must be calculated to keep the state of forces on the plastic strength surface. In the proposed analysis, the element incremental forces are updated based on the element natural deformations (Gattass and Abel, 1987). The spatial rigid body movement does not contribute to force increment. It serves to rotate the initial forces acting on the element from the previous configuration to the current configuration. If each member is modelled as one or two cubic elements (i.e, cubic interpolation functions) ignoring the bowing effects, discrepancy in the load-deflection curves is quite obvious (see Fig. 6). The discrepancy becomes small when four cubic elements are used for each member. This example demonstrates the capability of the proposed element in capturing the nonlinear behaviour of space frames with fewer elements per member.
SIX STOREY SPACE FAME Fig. 7 shows a six-storey steel building frame to be analysed using the proposed advanced plastic hinge method. A36 steel is used for all sections. USFOS, a computer program for the progressive collapse analysis of steel offshore structures (S^eide et al., 1994), is used to verify the results firom the proposed analysis program. In USFOS, beams and colunms are modelled as the inelastic beam-column elements based on an Updated Lagrangian approach. In the plastic hinge analysis, every member is modelled as one inelastic frame element. The effect of member shear deformation is considered by adjusting the flexural stiffness. The gravity loads are applied at the columns of every storey and are equivalent to a uniform floor load of 9.6 kN/m . The
25
wind loads are simulated by applying point loads of 53.376 kN in the Y-direction at every beamcolumn joints of the front elevation. The loads are proportionally applied until the frame collapses. A total of 20 plastic hinges are detected at the frame's limit load. The limit strength of the frame reaches at a load ratio of 1.005 in this study, as compared with 0.995 from USFOS. The difference between the two limit loads is less than 1%. Fig. 7 shows the sequence of plastic hinge formation and the deformed shape of frame at its limit strength. Plastic hinges initially occur in the beams of axis 1. With the increase of loads, plastic hinges spread in columns and beams of other axes. As the loads and structure are asymmetrical, torsional forces are induced and the frame deforms in a twisting mode. Due to the change of the structural plan starting from the fourth storey, severe torsional effect is induced and more plastic hinges occur in the beams and columns of the fourth storey. The frame collapses when three hinges form at the tops of three columns of the fourth storey. The load-displacement curves along the global X and Y-axes plotted at node A are shown in Fig. 8. The results generated by both analyses compare well. CONCLUSIONS The methodology of nonlinear analysis of space frames based on the plastic hinge approach is presented in this paper. The presentation focuses on the frame element formulation and the plasticity model. The proposed method considers the member's P-6, initial out-of-straightness and bowing effects, and the elastic coupling effects between axial, flexural and torsional displacements. The analysis method is particularly suitable for space frame structures in which the members are slender or subjected to high axial force. The study of the six-storey space frame illustrates that practical and reliable tools are now available for the static collapse analysis of space frame structures. The direct second-order plastic hinge analysis provides better insight into the structural behaviour up to failure by observing the load-displacement characteristic of the structure and the sequence of hinge formation in the frame. It is particularly usefiil for flexible and non-symmetrical structures. The evaluation of system's limit-state will provide a more uniform safety level compared to the conventional approach based on first member failure. Moreover, design economy can be realised if higher proportion of the inelastic reserved strength built into the system can be mobilised. It should be noted that the accuracy of plastic hinge analysis is reasonable only for cases where spread-of-plasticity if not significant and where material stress-stain law is essentially elastic-plastic. Otherwise the spread-of-plasticity approach must be used. Research and development on refined plastic hinge analysis of space frames, which allow for gradual plastification of yield hinge, is currently being carried out (Liew and Tang, 1998). Future research work will focus on modelling the inelastic lateral-torsional behaviour of beam-columns. This would eventually eliminate the need of semi-empirical interaction equations for checking the lateral-torsional stability of beamcolumns. ACKNOWLEDGEMENT The work is funded by the research grants (RP 940661 & RP 960648) made available by the National University of Singapore. REFERENCES Chan, S.L., and Zhou, Z.H. (1994), Pointwise equilibrium polynomial element for nonlinear analysis of frames, J. Struct Eng., ASCE, 120(6), 1703-1717. Chen, W.F., and Toma, S. ed. (1994), Advanced Analysis of Steel Frames: Theory, Software, and Applications, Boca Raton, FL: CRC Press. Chen, W.F., Goto, Y., and Liew, J.Y.R. (1996), Stability Design of Semi-Rigid Frames, John Wiley & Sons Inc., New York.
26 Gattas, M., and Abel, J.F. (1987), Equilibrium considerations of the Updated Lagrangian formulation of beam-columns with natural concepts. Int. J. Numer. Metk Eng., 24,2119-2141. Liew, J.Y.R., Punniyakotty, N.M., and Shanmugam, N.E. (1997a), Advanced analysis and design of spatial structures, J. Construct. Steel Res., 42(1), 21-48. Liew, J.Y.R., Chen, H., Yu, C.H., Shanmugam, N.E., and Tang, L.K. (1997b), Second-order inelastic analysis of three-dimensional core-braced frames, Research Report No. CE024/97, Department of Civil Engineering, National University of Singapore. Liew, J.Y.R., Yu, C.H., Ng, Y.H., and Shanmugam, N.E. (1997c), Testing of semi-rigid unbraced frames for calibration of second-order inelastic analysis, J. Construct. Steel Res., Elsevier, UK, 41(2/3), 159-195. Liew J.Y.R., and Tang, L.K. (1998), Nonlinear refined plastic hinge analysis of space frame structures. Research Report No. CE027/98, Department of Civil Engineering, National University of Singapore. Oran, C. (1973), Tangent stiffiiess in space frames, J. Struct. Div., ASCE, 99, 987-1001. Orbison, J.G. (1982), Nonlinear static analysis of three-dimensional steel frames. Report No. 82-6, Department of Structural Engineering, Cornell University, Ithaca, New York. 80
Chan and Zhou (1994) • Proposed element (1 element) Cubic element
70 60
X
50
a* T3 CO
O
H4
40 30 20
0.0
.Fig. 5 Hexogonal frame
0.5
1.0
1.5 2.0 2.5 3.0 3.5 Vertical deflection (in)
4.0
4
Fig. 6 Load-vertical deflection curves of hexagonal frame 1.2 I
1 o
0.000 •
Sequence of plastic hinge formation
Fig. 7 Deformed shape of six-storey space frame at limiting strength
0.004
0.008
Displacements u/H and v/H at node A Fig. 8 Load-displacement curves of six-storey space frame
0.012
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
27
LOCAL, DISTORTIONAL, FLEXURAL AND FLEXURAL-TORSIONAL BUCKLING OF THIN-WALLED COLUMNS B. Young^ and K. J. R. Rasmussen^ ^ School of Civil and Structural Engineering, Nanyang Technological University, Singapore 639798 (Formerly, Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia) ^ Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia
ABSTRACT The paper describes an overall bifurcation analysis of locally buckled columns. The overall flexural and flexural-torsional bifurcation loads are calculated using the tangent rigidities of the locally buckled cross-section. An elastic non-linear finite strip local buckling analysis is used to determine the tangent rigidities. The columns are assumed to be geometrically perfect in the overall mode but may include imperfections in the local mode. In addition, the elastic distortional buckling loads and the local buckling loads are determined using a finite strip buckling analysis. The bifurcation loads and the distortional buckling loads are compared with tests of fixed-ended lipped channel columns conducted at the University of Sydney. The tests involved pure local buckling and distortional buckling as well as overall flexural buckling and flexural-torsional buckling. The failure modes observed in the tests are also compared with the failure modes predicted by the analysis. The bifurcation loads and the predicted failure modes are shown to be in good agreement with the tests.
KEYWORDS Bifurcation analysis, Distortional buckling. Finite strip buckling analysis. Fixed-ended columns, Flexural buckling, Flexural-torsional buckling. Local buckling. Tangent rigidities.
INTRODUCTION The use of thin-walled structures has increased rapidly in recent years, particularly those involving high strength steel structural members. Thin-walled members can be used economically for building construction and many other structural applications. When compared with thicker hot-rolled members, they provide a substantial increase in strength to weight ratio and ease of construction. The manufacturing process of thin-walled members usually involves roll-forming or brake-pressing of coils or steel sheets respectively to produce a wide range of cross-sectional shapes.
28 Thin-walled members are prove to local buckling in their ultimate limit states. One of the effects of local buckling is to reduce the member stiffnesses against overall flexure and torsion, thus reducing the overall buckling loads. Bifurcation in an overall mode can be calculated by using the effective stiffnesses of the locally buckled cross-section to estimate the member strength. For members buckling in an overall flexural mode after local buckling, the analysis simply substitutes the reduced flexural rigidity of the locally buckled member into the Euler formula. This method has been used by Bijlaard and Fisher (1953), and Hancock (1981) to determine the flexural buckling loads of locally buckled doubly symmetric columns. A similar approach can be used for flexural-torsional buckling, in which case the flexural and warping torsion rigidities of the locally buckled member are required. The rigidities of the locally buckled member are assumed to be constant along the length. They are determined by applying a small curvature or twist to a length of member equal to the local buckle halfwavelength and dividing the resulting moment (or bi-moment) by the applied curvature (or twist). The overall bifurcation analysis of locally buckled thin-walled members was formally described by Rasmussen (1997), and Young and Rasmussen (1997). The analysis was applied to singly symmetric columns and doubly symmetric cross-sections in combined compression and bending, deriving an explicit expression for the overall bifurcation load. The members were assumed to be geometrically perfect in the overall mode but may include imperfections in the local mode. The bifurcation analysis described in Young and Rasmussen (1997) is applied in the present paper to fixed-ended lipped channel columns, in which the tangent rigidities of a locally buckled section were obtained by using an elastic non-linear finite strip method. In addition, the elastic local buckling load was obtained using a finite strip buckling analysis (Hancock, 1978), and the elastic distortional buckling load was calculated according to Lau and Hancock (1987). The purpose of this paper is to compare the loads and failure modes predicted by these analyses with tests of fixed-ended lipped channel columns conducted at the University of Sydney (Young and Rasmussen, 1998). Tests were performed over a range of lengths which involved pure local buckling, distortional buckling as well as overall flexural buckling and flexural-torsional buckling.
FLEXURAL AND FLEXURAL-TORSIONAL BIFURCATION ANALYSIS The overall bifurcation analysis of locally buckled singly symmetric columns is described in this section. The analysis applies to cross-sections composed of thin plates for which the post-local buckling behaviour is governed by von Karman's non-linear plate equations. A simple model is used which assumes that the locally buckled cross-section consists of an assembly of narrow strips, the stiffnesses of which vary around the cross-section as functions of the extent of local buckling. This simple model allows the effect of local buckling, which is to cause a geometrical loss of stiffness to be considered as a material effect in the overall bifurcation analysis. In the bifurcation analysis of the locally buckled member, the actual locally buckled cross-section, as shown in Fig. la, is replaced by an undistorted cross-section, as shown in Fig. lb, for which the tangent stiffness {Ei) varies around the cross-section. This procedure allows the formulation of bifurcation of thin-walled members with undistorted cross-section to be used by changing only the stress-strain relations, as detailed in Rasmussen (1997).
29
. /
^ '
J (a) Actual cross-section
(b) Model
Figure 1: Model of locally buckled cross-section As derived in Young and Rasmussen (1997), the overall flexural and flexural-torsional bifurcation loads of a locally buckled fixed-ended singly symmetric column are obtained from the determinantal equation,
(£4),
0
-(Y)^^^^'
0 0
0
=0
'2n
(1)
Y^^^^^^^^^^cNxs -X^Nxl-X^W where Xc is the critical load factor for flexural and flexural-torsional buckling, L is the specimen length, Xs is the jc-coordinate of the shear centre, N and W are the applied axial force and Wagner stress resultant respectively, G is the shear modulus, and J is the torsion constant. In Eqn. 1, the first two rows pertain to buckling displacements (w^, Ub) in the longitudinal z-axis and major jc-axis directions, and the last two rows pertain to buckling displacements (v^) in the minor >'-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 (<100 mm).
CONCLUSIONS An experimental investigation into the buckling behaviour of the cold-formed steel wall frame systems lined with plasterboard under axial compression has been described in this paper. It included a series of twenty frill-scale tests on unlined and lined wall frames and another series of twenty stub column tests. Appropriate effective length factors have been recommended for unlined and both sides lined wall frames within the AS4600 and AISI provisions. The inadequacy of the current design methods in some cases has also been identified. Further investigations using finite element analysis and testing are needed to develop improved behavioural models for wall studs with one or both sides lining. For the commonly used lined wall frames, any improvement to local buckling behaviour can be ignored. ACKNOWLEDGEMENT The authors wish to thank AusAid for providing a scholarship to the first author, Mr Ross Morris from GSR for donating the lining material and QUT's Physical Infrastructure Centre and School of Civil Engineering for providing other materials and test facilities. REFERENCES American Iron and Steel Institute. (1996). Specification for the Design of Cold-formed Steel Structural Members, Washington, USA American Iron and Steel Institute. (1986). Specification for the Design of Cold-formed Steel Structural Members, Washington, USA CSR Plasterboard. (1990). Do it yourself Gyprock Plasterboard Installation Manual, Sydney Miller, T.H. and Pekoz, T.A (1993) Behaviour of Cold-formed Steel Wall Stud AssembUes. Journal of Structural Engineering ASCE 119:2, 641-651 Miller, T.H. and Pekoz, T.A. (1994). Behaviour of Gypsum Sheathed Cold-formed Steel Wall Studs. Journal of Structural Engineering ASCE 120:5,1644-1650. Rondo Building Systems (RBS). (1993). Design Manual for Steel studs systems in non-cyclonic areas. Simaan, A. and Pekoz, T.A. (1976). Diaphragm Braced Members and Design of Wall Studs. Journal of Structural Division, ASCE Proceedings 102:1, 77-93. Standards AustraUa (SA). (1996). AS 4600, Cold-formed Steel Structures, Sydney Telue, Y.K. and Mahendran, M. (1997). Behaviour and Design of Plasterboard Lined Cold-formed Steel Stud Walls under Axial Compression, Research Monograph No.97-4, QUT, Brisbane Young, B. and Rasmussen, K.J.R. (1998a) Tests of Cold-formed Channel Columns, Proc. of l4^ Int. Specialty Conference on Cold-formed Steel Structures, St.Louis, pp. 239-264. Young, B. and Rasmussen, K.J.R. (1998b) Behaviour of Locally Buckled Singly Symmetric Columns, Proc. ofl4^ Int. Specialty Conference on Cold-formed Steel Structures, St.Louis, pp. 219-238.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
45
COMPRESSION TESTS OF THIN-WALLED CHANNELS WITH SLOPING EDGE STIFFENERS B. Young^ and G. J. Hancock^ ^ School of Civil and Structural Engineering, Nanyang Technological University, Singapore 639798 (Formerly, Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia) ^ Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia
ABSTRACT The paper describes a series of tests on cold-formed channels with sloping edge stiffeners compressed between fixed ends. The current Australian/New Zealand Standard and the American Specification for cold-formed steel structures provide for the calculation of the strength of channels with sloping edge stiffeners. However, the provisions are based mainly on tests of sections having edge stiffeners at right angle to the flanges. The provisions for sloping edge stiffeners are determined based on intuition without rigorous experimental evidence. Hence, it is important to obtain test data for sections with edge stiffeners not at right angle to the flanges. The purpose of this paper is to provide test data for channel sections in compression with sloping edge stiffeners. A series of tests was performed on channel sections brake-pressed from high strength steel sheets. The edge stiffeners of the channels are sloped at different angles, both outwards and inwards. The distortional buckling mode was observed in all tests. The test strengths are compared with the design strengths predicted by the Australian/New Zealand Standard and the American Specification for cold-formed steel structures.
KEYWORDS Channel column. Design strength, Distortional buckling. Edge stiffener. Fixed-ended, Experimental investigation. High strength steel. Test strength.
INTRODUCTION Edge stiffeners are commonly used in thin-walled sections to provide a continuous support along a longitudinal edge of the flange to enhance the buckling stress. The edge stiffener can be easily brakepressed or roll-formed on the free edge of an unstiffened plate. Therefore, thin-walled sections having edge stiffeners can lead to an economic design as a result of enhancing the buckling stress of a section. The current rules in the Australian/New Zealand Standard (1996) and the American Specification (1996) for cold-formed steel structures for designing an edge stiffened element are based mainly on
46 tests performed by Desmond et al. (1981). The tests were conducted on channels with edge stiffeners at right angle to the flanges. However, the current standard and specification also allow for sloping edge stiffeners. This provision appears to be based on intuition without any supporting tests. Therefore, it is important to obtain test data for sections with sloping edge stiffeners. The purpose of this paper is to present a series of tests of cold-formed channel sections in compression with sloping edge stiffeners. In the past, compression tests of thin-walled channel sections with sloping edge stiffeners have not been performed. Therefore, the test strengths of such sections are not known. This paper provides the test strengths of channels with sloping edge stiffeners. In addition, a comparison of the test strengths with the design strengths predicted by the Australian/New Zealand Standard (1996) and the American Specification (1996) for cold-formed steel structures are also presented in this paper.
EXPERIMENTAL INVESTIGATION Test Specimens Tests were performed on channels with sloping edge stiffeners compressed between fixed ends. The specimens were brake-pressed from high strength zinc-coated Grade G450 structural steel sheets having nominal yield stress of 450MPa and specified according to the Australian Standard AS 1397 (1993). The test program comprised six series with different cross-sections, referred to as ST15, ST19, ST24, LT15, LT19 and LT24 according to their flange width and thickness. The nominal flange width was either 50mm or 100mm, where the first letter of the test series label "S" or "L" refer to "Small" (50mm) or "Large" (100mm) flange width respectively. T15, T19 and T24 specify the three nominal thicknesses of 1.5mm, 1.9mm and 2.4mm respectively. A nominal width of the web of 100mm and a nominal width of the lip (edge stiffener) of 12mm was used for all channels. The edge stiffeners were sloped at different angles specified from 30° to 150° (30° < 0 < 150°) measured from the plane of the flanges as shown in Fig. 1. The measured cross-section dimensions are given in Tables 1-6 using the nomenclature defined in Fig. 1. In the tables, the first four letters of the specimen labels indicate the test series and following by the nominal angle of the sloping edge stiffeners. The nominal angle of 150° was not achieved, and it was measured as approximately 140°. The nominal length of each specimen was 1500mm as supplied from the manufacturer in an uncut length. Both ends of each specimen were milled flat by an electronic milling machine to an accuracy of 0.01mm to ensure full contact between specimen and end bearings. The largest value of ley/ry ratios for Series ST 15, ST 19 and ST24 was approximately 45, while that for Series LT15, LT19 and LT24 was approximately 20. The parameter ley is the effective length for buckling about the minor y-axis which assumed equal to half of the column length (L) for fixed-ended columns, and ry is the radius of gyration about the y-axis. Effective Thickness The steel sheets used in the test program were finished with a thin layer of zinc coating for corrosion protection purposes. From a structural point of view, the zinc coating is much softer than steel, and so it is reasonable to assume that the layer of zinc coating does not carry any load (Young and Rasmussen 1998a, b). Hence, the effective thickness of the steel sheet is the base metal thickness (without the zinc coating).
47
In Tables 1-6, the base metal thickness (t*) was measured by removing the zinc coating by acid etching. The thicknesses of the zinc coatings were measured as 50|Lim for Series ST 15 and LT15, and 40^m for Series ST19, ST24, LT19 and LT24.
^
% !
h-t
B^ B^
B^
(a) Outwards edge stiffeners
(b) Inwards edge stiffeners
Figure 1: Definition of symbols TABLE 1 MEASURED SPECIMEN DIMENSIONS AND EXPERIMENTAL ULTIMATE LOADS FOR SERIES S T 1 5
Specimen
Web
Flanges
Lips
Thickness
Internal Radius
Angle
Length
Area
Exp. Ult. Load
fiw
Bf
Bi
t
t*
rt
e
L
A
A^Exp
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
(Degree)
(nrni)
(mm^)
ST15A30
100.4
50.3
10.6
1.55
1.50
1.0
32.0
1504.1
325
(kN) 76.0
ST15A45
100.9
50.3
10.6
1.55
1.50
1.0
46.2
1503.0
325
81.3
ST15A60
98.8
51.4
10.7
1.54
1.49
1.0
61.1
1503.5
322
83.4
ST15A90
98.9
49.5
10.7
1.55
1.50
1.0
89.5
1503.7
315
97.3
ST15A120
99.7
49.8 .
10.8
1.55
1.50
1.0
119.8
1503.8
320
102.2
ST15A135 ST15A150
99.7 99.3
49.8 49.8
10.9 10.8
1.53 1.54
1.48
1.0 1.0
134.0 140.5
1504.2
318 320
90.4 97.3
Note: 1 in. = 25.4mm;
1.49
1502.9
1 kip = 4.45 kN
* Base metal thickness
TABLE 2 MEASURED SPECIMEN DIMENSIONS AND EXPERIMENTAL ULTIMATE LOADS FOR SERIES ST 19
Specimen
Web
Flanges
Lips
Thickness
Internal Radius
Angle
Length
Area
Exp. Ult. Load A^Exp
Bw
Bf
Bi
t
t*
ri
d
L
A
(nmi)
(mm)
(mm)
(mm)
(mm)
(nrni)
(Degree)
(nmi)
(mm^)
(kN)
ST19A30
99.4
51.2
10.5
1.92
1.88
1.0
31.2
1503.8
406
117.5
ST19A45
99.0
51.4
10.8
1.94
1.90
1.0
46.6
1503.6
410
126.6
ST19A60
99.2
51.1
10.7
1.94
1.90
1.0
60.7
1503.8
408
139.1
ST19A90
100.3
49.2
10.5
1.93
1.89
1.0
89.1
1503.9
395
144.8
ST19A120
100.1
49.6
11.6
1.93
1.89
1.0
118.9
1504.0
403
155.5
ST19A135
99.6
49.6
11.7
1.93
1.89
1.0
133.8
1503.8
405
152.8
ST19A150
100.2
49.4
11.6
1.94
1.90
1.0
139.6
1504.3
408
154.2
Note: 1 in. = 25.4mm; * Base metal thickness
1 kip = 4.45 kN
48 TABLES MEASURED SPECIMEN DIMENSIONS AND EXPERIMENTAL ULTIMATE LOADS FOR SERIES S T 2 4 Specimen
Web Bw
(mm) ST24A30 100.0 ST24A45 101.1 ST24A60 100.6 ST24A90 100.4 101.4 ST24A120 ST24A135 99.7 ST24A150 100.5 Note: 1 in. = 25.4mm;
Flanges
Lips
Bf
Bi
(mm) (mm) 50.8 11.8 50.6 • 11.6 51.2 11.5 12.0 49.9 11.8 50.1 12.1 50.3 50.3 11.9 1 kip = 4.45 kN
Thickness
Internal Radius
Angle
Length
Area
Exp. Ult. Load ^Exp (kN) 155.9 180.7 198.5 194.0 198.6 197.2 195.5
t
t*
r,
e
L
A
(nrai) 2.40 2.44 2.42 2.44 2.41 2.43 2.43
(mm) 2.36 2.40 2.38 2.40 2.37 2.39 2.39
(mm)
(Degree) 30.1 45.3 61.8 90.3 119.1 135.9 139.8
(mm) 1485.0 1481.7 1502.9 1479.1 1477.9 1478.2 1482.5
(mm^)
1.0 1.0 1.0 1.0 1.0 1.0 1.0
513 520 514 506 507 513 514
* Base metal thickness
TABLE4 MEASURED SPECIMEN DIMENSIONS AND EXPERIMENTAL ULTIMATE LOADS FOR SERIES L T l 5 Specimen
Web By,
(mm) LT15A30 100.7 LT15A45 99.8 99.7 LT15A60 LT15A90 99.9 LT15A120 99.3 99.0 LT15A135 99.4 LT15A150 Note: 1 in. = 25.4mm; * Base metal thickness
Flanges
Lips
Bf
Bi
(mm) (nmi) 100.4 10.5 100.8 10.7 100.7 10.7 10.1 99.6 100.0 10.8 100.1 10.6 99.7 10.8 1 kip = 4.45 kN
Thickness
Internal Radius
Angle
Length
Area
Exp. Ult. Load A^Exp (kN) 70.4 71.5 75.9 74.2 80.2 79.0 76.8
t
t*
r,
e
L
A
(nmi) 1.55 1.53 1.56 1.53 1.56 1.55 1.54
(mm) 1.50 1.48 1.51 1.48 1.51 1.50 1.49
(mm)
(Degree) 30.6 45.9 61.1 88.8 119.8 134.8 138.3
(mm) 1503.7 1503.6 1503.3 1503.6 1503.4 1503.3 1503.7
(mm^)
1.0 1.0 1.0 1.0 1.0 1.0 1.0
476 469 477 459 473 471 468
TABLE 5 MEASURED SPECIMEN DIMENSIONS AND EXPERIMENTAL ULTIMATE LOADS FOR SERIES L T l 9 Specimen
Web fiw
(nmi) 98.5 LT19A30 99.9 LT19A45 99.6 LT19A60 LT19A90 99.9 99.9 LT19A120 99.7 LT19A135 99.6 LT19A150 Note: 1 in. = 25.4mm; ' Base metal thickness
Flanges
Lips
Bf
Bi
(nmi) (nrni) 10.4 101.6 10.7 100.6 10.5 101.0 99.7 10.5 11.7 99.8 99.4 11.4 99.5 11.5 1 kip = 4.45 kN
Thickness
Internal Radius
Angle
Length
Area
Exp. Ult. Load
t
t*
r,
e
L
A
A^Exp
(mm) 1.94 1.94 1.94 1.94 1.93 1.93 1.94
(mm) 1.90 1.90 1.90 1.90 1.89 1.89 1.90
(nrni)
(Degree) 31.1 46.4 60.8 89.8 119.0 133.6 139.7
(mm) 1503.5 1503.1 1503.0 1503.5 1503.5 1503.8 1503.9
(mm^)
(kN) 99.0 107.6 115.8 113.2 127.2 120.0 129.5
1.0 1.0 1.0 1.0 1.0 1.0 1.0
600 599 597 588 593 592 596
49 TABLE 6 MEASURED SPECIMEN DIMENSIONS AND EXPERIMENTAL ULTIMATE LOADS FOR SERIES L T 2 4 Specimen
Flanges-
Web Bw
(mm) LT24A30 99.0 99.4 LT24A45 LT24A60 99.2 LT24A90 100.4 LT24A120 99.8 100.3 LT24A135 LT24A150 100.7 Note: 1 in. = 25.4mm; * Base metal thickness
Thickness
Lips
Bi B, (mm) (mm) 11.3 101.7 11.2 101.7 11.6 101.7 12.0 100.1 12.4 100.4 11.8 100.3 100.4 11.8 1 kip = 4.45 kN
/
t*
Internal Radius r,
(mm) 2.43 2.42 2.42 2.45 2.43 2.42 2.44
(mm) 2.39 2.38 2.38 2.41 2.39 2.38 2.40
(mm) 1.0 1.0 1.0 1.0 1.0 1.0 1.0
Angle
Length
Area
e (Degree) 28.6 43.7 60.4 89.9 119.6 135.2 139.8
L (nrni) 1502.7 1503.7 1503.6 1502.5 1484.0 1491.8 1484.9
A (mm^) 758 753 752 750 751 749 757
Exp. Ult. Load A^Exp
(kN) 127.9 137.4 149.0 161.7 176.9 167.7 166.8
Material Properties The tensile properties of each series of specimens was determined by tensile coupon tests. Longitudinal coupons were taken from the centre of the web plate of the finished specimens belonging to the same batches as the column test and so could be expected to have nearly the same material properties. The coupons were prepared and tested according to Australian Standard AS 1391 (1991) using 12.5mm wide coupons and a gauge length of 50mm. The coupons were tested in a 300kN capacity MTS Sintech 65/G displacement controlled testing machine using friction grips to apply loading. A calibrated extensometer of 50mm gauge length was used to measure the longitudinal strain. A data acquisition system was used to record the load and the gauge length extensions at regular intervals during the tests. The static load was obtained by pausing the applied straining for one minute near the 0.2% tensile proof stress and the ultimate tensile strength. Table 7 summarises the material properties determined from the coupon tests. The table contains the nominal and measured static 0.2% tensile proof stress (00.2), the static tensile strength (Cu) and the elongation after fracture (e„) based on a gauge length of 50mm. The 0.2% proof stresses were used as the corresponding yield stresses in calculating the design strengths of the columns. TABLE 7 NOMINAL AND MEASURED MATERIAL PROPERTIES Test Series
Nominal Ob.2
(MPa) 450 ST15 450 ST19 ST24 450 LT15 450 450 LT19 LT24 450 Note: 1 ksi = 6.89 MPa
Measured Ob.2
0"„
Su
(MPa) 515 505 415 520 510 420
(MPa) 555 535 505 560 535 505
(%) 12 12 26 13 13 26
Test Rig and Operation The SOOkN capacity MTS Sintech 65/G displacement controlled testing machine was also used to apply compressive axial force to the specimens. Displacement control was used to drive the actuator at
50
a constant speed of 0.2 mm/min. The use of displacement control allowed the tests to be continued into the post-ultimate range. A data acquisition system was used to record the load and the readings of displacement transducers at regular intervals during the tests. The static load was recorded by pausing for one minute near the ultimate load. This allowed the stress relaxation associated with plastic straining to take place. The load was applied at the upper end through a rigid end plate which was restrained against rotation. A spherical bearing was used at the lower end support. After the specimen was positioned on the spherical bearing, the ram of the actuator was moved slowly toward the specimen until the upper end plate and the lower bearing were in full contact with the ends of the specimen having a small initial load of approximately IkN. This procedure would eliminate any possible gaps at the ends of the specimen, since the spherical bearing was free to rotate in any directions. The spherical bearing was then retrained from twist and rotations by using horizontal and vertical bolts respectively. Hence, the spherical bearing became a fixed-ended bearing. The fixed-ended bearing was considered to restrain both minor and major axis rotations as well as twist rotations and warping. The displacement transducers were positioned at mid-length and 200mm away from mid-length of the specimens as well as positioned at the loading rigid end plate to measure the axial shortening of the specimens.
TEST RESULTS The experimental ultimate loads (A^Exp) obtained from the tests are given in Tables 1-6. The distortional buckling mode was observed near the ultimate load during testing. It should be noted that for each test series, the maximum load was obtained when the angle of the edge stiffeners sloped at 120° (inwards edge stiffeners), except for Series LT19. The test results are also plotted against the angle (9) of edge stiffeners in Figs 2-7 for Series ST15, ST19, ST24, LT15, LT19 and LT24 respectively.
COMPARISON OF TEST STRENGTHS WITH DESIGN STRENGTHS The fixed-ended column test strengths (Afexp) of channels with sloping edge stiffeners are compared in Figs 2-7, with the unfactored design strengths predicted using the Australian/New Zealand Standard (1996) and the American Specification (1996) for cold-formed steel structures. The Australian/ New Zealand Standard (AS/NZS 4600) was adopted from the American Iron and Steel Institute (AISI) Specification. In the compression member rules, the AS/NZS 4600 Standard includes a separate check for distortional buckling of singly-symmetric sections as specified in Clause 3.4.6. However, the AISI Specification does not have a separate check for distortional buckling. The design strengths were calculated using the measured cross-section dimensions and the measured material properties as detailed in Tables 1-6 and 7 respectively. The base metal thickness was used in the calculation. The elastic distortional buckling stresses (/od) were obtained from a rational elastic buckling analysis (Papangelis and Hancock 1995). Hence, the distortional buckling loads were calculated according to Clause 3.4.6 of the AS/NZS 4600 Standard. It is noted that the dimensional limits of flange flat width to thickness ratio is approximately 65 for Series LT15. This value is higher than the maximum value of 60 specified in Clause 2.1.3.1 of the AS/NZS 4600 Standard and Section Bl.l of the AISI Specification. Hence, the test Series LT15 does not strictly comply with the requirements of the AS/NZS 4600 Standard and the AISI Specification.
51 200
200
180
180 h
^ 160
^ 160
I 140
. 140
i 120
120
' 100
100
I 80
I 60 • Tests — AS/NZS4600 • • • AISI
i 40 20 0
80 60 40 20
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
-AS/NZS4600 •AISI 0
Angle of edge stiffeners, 0 (Degree)
Figure 2: Comparison of test strengths with design strengths for Series ST 15
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Angle of edge stiffeners, 9 (Degree)
Figure 3: Comparison of test strengths with design strengths for Series ST 19 200 180 ^ 160 . 140
S 140 ^ 120 '^ 100 2 80
120 100 80 60
- Tests — AS/NZS4600 --AISI 0
40 20
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Angle of edge stiffeners, 0 (Degree)
Figure 4: Comparison of test strengths with design strengths for Series ST24 200 180 ^ 160 h . 140 , 120 100 80 60 40 20 0
Tests -AS/NZS4600 •AISI
Angle of edge stiffeners, 0 (Degree)
Figure 5: Comparison of test strengths with design strengths for Series LT15
• 160 h
• Tests — AS/NZS4600 ••AISI 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Angle of edge stiffeners, 0 (Degree)
Figure 6: Comparison of test strengths with design strengths for Series LT19
• 140 h 120 100 80 60 40 20
-AS/NZS4600 •AISI 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Angle of edge stiffeners, 0 (Degree)
Figure 7: Comparison of test strengths with design strengths for Series LT24
For Series ST 15, ST 19 and ST24 (channels with small flange width), the design strengths predicted by the AS/NZS 4600 Standard and the AISI Specification are conservative for all channels with outwards (6 < 90°) and inwards (6 > 90°) edge stiffeners, as shown in Figs 2, 3 and 4 respectively. In general, the conservatism increases as the angle (6) increases. For Series LT15, having the most slender series with a flange flat width to thickness ratio of approximately 65, the design strengths predicted by the AS/NZS 4600 Standard are generally conservative for all channels with outwards and inwards edge stiffeners. However, the design
52
strengths predicted by the AISI Specification are unconservative for all channels, as shown in Fig. 5. For Series LT19 and LT24 (channels with large flange width), the design strengths are closely predicted by the AS/NZS 4600 Standard for channels with outwards edge stiffeners, and conservatively predicted for channels with inwards edge stiffeners. However, the design strengths predicted by the AISI Specification are unconservative for channels with outwards edge stiffeners, but closely predicted for channels with inwards edge stiffeners, as shown in Figs 6 and 7. The unconservatism of the AISI Specification is a result of the fact that it does not contain design rules specifically for distortional buckling which always control for Series LT15, LT19 and LT24. CONCLUSIONS A test program on cold-formed channels with sloping edge stiffeners has been presented. The axial compression tests were performed using fixed-ended support conditions. The test specimens having a nominal yield stress of 450MPa and the edge stiffeners of the channels were sloped at different angles for both outwards and inwards. A comparison of the test strengths and the design strengths obtained using the Australian/New Ziealand Standard (1996) and the American Specification (1996) for cold-formed steel structures has been presented. It is shown that the design strengths predicted by the standard and specification are conservative for all channels with sloping outwards and inwards edge stiffeners for Series ST 15, ST 19 and ST24, where the flange flat width (b) to thickness (t) ratio of approximately 30, 25 and 20 respectively. For Series LT15 having a b/t ratio of approximately 65, the design strengths predicted by the AS/NZS 4600 Standard are generally conservative for all channels, but unconservatively predicted by the AISI Specification. For Series LT19 and LT24 having the b/t ratios of approximately 50 and 40 respectively, the design strengths predicted by the AS/NZS 4600 Standard are generally conservative for all channels with outwards and inwards edge stiffeners, but unconservatively predicted by the AISI Specification for channels with outwards edge stiffeners, except for channels with inwards edge stiffeners. ACKNOWLEDGMENTS Funding was provided by BHP Steel for the purchase of the test specimens. REFERENCES American Iron and Steel Institute (1996). Specification for the Design of Cold-Formed Steel Structural Members, AISI, Washington, DC. Australian Standard (1991). Methodsfor Tensile Testing of Metals, AS 1391, Standards Association of Australia, Sydney, Australia. Australian Standard (1993). Steel Sheet and Strip - Hot-dipped zinc-coated or aluminium/zinc-coated, AS 1397, Standards Association of Australia, Sydney, Australia. Australian/New Zealand Standard (1996). Cold-Formed Steel Structures, AS/NZS 4600:1996, Standards Australia, Sydney, Australia. Desmond T.P., Pekoz T. and Winter G. (1981). Edge Stiffeners for Thin-walled Members. Journal of Structural Engineering, ASCE, 107:2, 329-353. Papangelis J.P. and Hancock G.J. (1995). Computer Analysis of Thin-walled Structural Members. Computers and Structures, 56:1, 157-176. Young B. and Rasmussen K.J.R. (1998a). Tests of Fixed-ended Plain Channel Columns. Journal of Structural Engineering, ASCE, 124:2, 131-139. Young B. and Rasmussen K.J.R. (1998b). Design of Lipped Channel Columns. Journal of Structural Engineering, ASCE, 124:2, 140-148.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
53
NUMERICAL ANALYSES ON HIGH CAPACITY STEEL GUARD FENCES SUBJECTED TO VEHICLE COLLISION IMPACT Y. Itoh', M. Mori^and C. Liu^ ' Center for Integrated Research in Science and Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603,Japan ^Department of Civil Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan
ABSTRACT Recently, there has been an increasing requirement to develop a new type of roadside guard fences to improve their capacities required due to the increases of the traffic speed, the large-scale vehicles and heavy trucks, and the height of the center of gravity of trucks. However, it is very difficult to test the performances of full-scale guard fences in the field because of the huge consumption in time and cost. In this study, finite element models are developed for both the trucks and the guard fences to reenact their behaviors. The feasibility of these models is demonstrated in an experimental case of collisions of the heavy truck with guard fences, and good agreements are obtained.
KEYWORDS Dynamic analysis, Energy absorption, FEM, Guard fence, Heavy truck, Vehicle collision impact
INTRODUCTION With the improvement of the road network and the vehicle capacities, the vehicles have taken a more important role in the freight transport. In Japan, the change of the allowable weight of trucks from 20 tf to 25 tf from November 1994 increases the percentage of heavy trucks and the height of the gravity center of the trucks. Accordingly, from both the function and safety viewpoints, these changes challenge the conventional transportation infrastructures such as roads, bridges, and guard fences. Therefore, a research effort is needed to recognize the capacities of the guard fences due to the collision impact of heavy trucks. Several approaches in this field have been carried out such as on the simulation of the impact load characteristics based on the experimental results of automotive vehicles (Miyamoto et al. 1991), and the impact response analysis for the shock softening type guard fences (Kobayashi et al. 1994). Further study was also carried out on the impact simulation between vehicles and roadside safety hardware (Wekezer et al. 1993) and a finite element computer simulation for the
54
vehicle impact with a roadside crash cushion (Miller and Carney 1997). Because of the huge consumption of time and cost, it is difficult in the field to measure the collision performances of the full-scale guard fences for various cases. In this research, by taking the advantages of both computer software and hardware, the collision impact process between the heavy trucks and the guard fences is simulated inside the laboratory based on the presented numerical calculation models for both the heavy trucks and guard fences. A nonlinear, dynamic, three-dimensional finite-element code LS-DYNA3D is capable for simulating the vehicle impact into the guard fences on an enhanced Workstation (Hallquist 1991). The analysis results are further compared with the in-situ experimental results to demonstrate the feasibility of the approach presented in this research.
NUMERICAL ANALYSIS MODELS Analytical Model of Guard Fences This research focuses on the collision impact of heavy trucks with a high speed onto the guard fences at the two sides of roads and bridges. The angle between the truck movement direction and the guard fence plane is an important parameter to determine the impact force and displacement in addition to the truck speed, the truck weight, the height of the gravity center of the truck, the guard fence, the curb, and others. Fig. 1 shows the basic collision analysis components in this research. 250 80km/h
Figure 1: Collision features
Figure 2: Analysis model of guard fence (mm)
The analytical model for the bridge guard fences is modified according to the structural model used in the experiment carried out in the Public Work Research Institute of Japan (1992). Fig. 2 shows the FEM model of the guard fence in the cross section and three-dimensions. The fence column is made of the H-type steel whose web and flange are 150 mm wide and 9 mm thick, and 150 mm wide and 9 mm thick, respectively. Both the main beam and sub-beam are of round cross sections. The pipe diameter and slab thickness of the main beam are 165 mm and 7 mm, respectively. The pipe diameter and slab thickness of the steel sub-beam are 140 mm and 4 mm, respectively. The Young's modulus of steel is 206 GPa, and the Young's modulus of concrete is 24.36 GPa. The Possoin's ratios of steel and concrete are 0.3 and 1/6, respectively. The shear moduli of steel and concrete are 15.85 MPa and 10.44 MPa, respectively. The yield stress and initial strain hardening of steel are 235 MPa and 4.12 Gpa, respectively. The strain hardening of steel starts from 0.0014. The concrete volume modulus is 12.18 GPa. The concrete compressive and tensile strengths are 23.52 MPa and 2.29 MPa, respectively. The steel is assumed to be an isotropic elasto-plastic material following the Von-Mises yielding condition. The strain hardening and strain velocity are taken into consideration the stress-strain relationship. The concrete in the curb is assumed as a general elasto-plastic material. This means the concrete is in the general elasto-plastic condition while the concrete in the compressive side reaches the yield point and only the cut-off stress is available once the tensile stress increases to the tensile strength. The boundary condition at the concrete curb is considered as a fixed end.
55 Analytical Model of Trucks In previous research, the trucks are modeled in various manners (Zaouk et al. 1996). The trucks whose weight is 25 tf is studied by modeling the truck frame, engine, driving room, cargo, tiers and so on. The structure of the 25 tf truck is similar to the 20 tf truck except the strengthened frame and the loading capacity of the vehicle axles. Fig. 3 shows the model of the ladder-type truck frame whose two side members are of channel sections so that some facilities such as the fuel tanks and pipelines can be attached inside the side members. The thickness of the side member is 8 mm, and the yield stress is 295 MPa. The general elasto-plastic stress-strain relationship is adopted. The solid element with the same shape and volume is modeled for the engine and the transmission, and their weights are adjusted according to the practical vehicles. The tiers, wheels, and gears of a truck influence its behaviors during the collision impact significantly. The connection of the tier and the wheel is assumed to be a rotation joint so that the movement of the wheel can be simulated. A constant value of 0.45 is used for the friction coefficient between the tier and the road pavement. The driving room and other small portions are also modeled for the purpose of the numerical calculation.
Figure 3: Truck structural model
Figure 4: Truck FEM model
Fig. 4 shows the presented FEM model of a truck used in this research. In this model, the numbers of nodes and elements are 3532 and 3904, respectively. The Young's modulus of steel is 206 GPa, while that of aluminum is 70 GPa. The Possoin's ratios of steel and aluminum are 0.30 and 0.34, respectively. The shear modulus is 15.85 MPa. In the case of guard fences, the steel is assumed to be an isotropic elasto-plastic material following the Von-Mises yielding condition, and the stress-strain relationship is perfectly elasto-plastic. The aluminum used for the cargo body is assumed to follow a multi-piece linear stress-strain relationship as shown in Fig. 5 (Itoh et al. 1998).
^
g CO
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—1
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-r
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d o &
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.1 10-2
1
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Strain v e l o c i t y e
Figure 6: Steel yield stress with strain velocity
56 PARAMETRIC STUDY Effects of Strain Hardening and Strain Velocity Parametric study is first carried out to check the effects of the strain hardening and strain velocity on the displacement of the guard fence components. It is assumed that the strain hardening starts from 0.0014 and the initial strain hardening modulus is 4.01 GPa (2% of the Young's Modulus). On the other hand, the yield stress usually increases with the increase of the strain velocity. The scaling relation of the yield stress shown in Fig. 6 is used in this research to investigate the effects of the strain velocity. Fig. 7 shows the displacement of a column with time in four combined cases by considering the strain hardening and strain velocity or not. In this calculation, the truck weight, collision speed and collision angle are 14 tf, 80 km/h and 15^, respectively. According to the displacement tracks as shown in this figure, the effects of the strain hardening and strain velocity on the maximum response displacement and the residual displacement are very obvious. The displacements follow the similar tracks with time if one of the strain hardening and the strain velocity is considered and the other is eliminated. It should also be noticed that at about 0.5 second after the collision impact the displacement increases rapidly within a very short time at all cases. The experimental results are very close to the results obtained by considering the strain hardening and the strain velocity simultaneously. Therefore, these two factors will be taken into account in the following part of this paper. ' PIO (1-2-8 model) • Pin (1-2-8 model) PIO (1-2-8 model)
Experimental maximum displacement
With hardening/Wilh velocity
Experimental maximum displac Experimental residual displac
IM 0.5
Time (sec) Figure 7: Effects of strain hardening and velocity
Figure 8: Effects of mesh sizes
Effects of Mesh Sizes Further study is carried out to determine the appropriate mesh sizes by following the tracks of the displacement of the bridge guard fence with time. The calculation results are compared with the experimental values by adjusting the mesh sizes of the column web, the column flange, and the horizontal beam pipe. Three cases, 1-2-8 model, 4-4-16 model and 8-8-32 model, are studied. The three numbers of each model represent the classified portions of the column web, column flange and beam pipe, respectively. The calculation results are shown in Fig. 8 in addition to the detected values from the actual experiment for the case when the truck weight, collision speed and collision angle are 14 tf, 80 km/h and 15^, respectively. This figure shows the displacement of only one column PIO whose position can be recognized from Fig. 9. According to the displacement curves in Fig. 8, the residual response displacements in cases of 4-4-16 model and 8-8-32 model are very large at 0.5 second after the collision impact (about 40%). The tracks in these two cases are almost same within the first 0.5 second. The final displacement is about 10% less than the maximum value in all cases. Compared to the maximum and residual displacements from the experiment, the 4-4-16 model contributes very good agreements. Therefore, this model will be adopted in the following analysis.
57
Pll
PIO
Figure 9: Codes of columns and beams
Figure 10: Movement track of a collision truck
IMPACT ANALYSIS OF BRIDGE GUARD FENCES Displacement of Bridge Guard Fence Further research is carried out to demonstrate the presented models by comparing the calculated values with the actual experimental results in the case of collisions between the heavy truck and guard fences. In both the experiment and calculation, the impact speed is 80km/h and the impact angle is 15^.The weight of the truck is 14 tf. Fig. 10 shows the moving track of a truck. The front of the truck first touches the guard fence and the front wheel runs onto the curb very soon and impacts the guard fence. Then, the truck frame inclines, the direction of movement is changed suddenly, and the rear wheel runs onto the curb. Finally, the rear wheel of the truck touches the guard fence and the sloped angle of the back truck frame increases so that the truck frame shifts the moving direction and leave the bridge guard fence. Fig. 11 shows the performances of both trucks and guard fences at several collision conditions in detail. The graphs from the previous experiment and the simulation in this research are compared.
(e) 0.8 s
(c) 0.4 s Figure 11: Impact performance of a truck
(0 1.0 s
58
The responses of several fence column tops in the form of displacement are shown in Fig. 12 in terms of different types of lines. The maximum and residual response displacements of the column PIO are 95 mm and 85 mm, respectively. In the practical vehicle experiment, these two values are 97 mm and 84 mm, respectively. It is obvious that the calculation results are quite near to the experimental results. Figs. 13 and 14 show the displacement curves of several main beams and sub-beams with time, respectively. The calculation value of the main beam BIO at the central section is 99 mm, about 30% higher than the detected value of 76 mm from the practical experiment. However, the calculated displacement value of 105 mm of the sub-beam B9 at the central section is less about 20% than the experimental value of 130 mm. —J-BS '
150
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Figure 12: Displacement responses of columns
Experimental residual displacement
1
1
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Figure 13: Displacement responses of main beams
Energy Absorption of Bridge Guard Fence Using the FEM model of the bridge guard fences, the horizontal force in the vertical direction to the guard fence plane is calculated and compared with the practical experimental results as shown in Fig. 15. It is obvious that the calculated values and the experimental results are rather consistent within the first 0.1 second or after 0.2 second. However, the differences are quite large from 0.1 second to 0.2 second. This may result from the fact that the strain gauge is used to detect the shearing strain while the experimental horizontal force is determined, and the relationship between the load and the shearing force is modeled according to the static loading experimental results. Further, after 0.6 second in both the experiment and the simulation, the forces are stable. This may represent that the dynamic impact response between the trucks and the guard fences last about 0.6 second. I — - L B8 '
'
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Time (sec)
Time (sec)
Figure 14: Displacement responses of sub-beams
-
1
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i,"
Figure 15: Horizontal force onto columns
59 The detail of energy absorption among the guard fence components with time is shown in Fig. 16. About 65%, 30%, and 5% of the impact energy are absorbed by the main beam, the sub-beam and the fence column, respectively. This percentage distribution is related to the assumption that the truck collision impact starts from the main beam. Figs. 17, 18, and 19 show the energy absorption of several specific components of the columns, beams, and sub beams, respectively. - I "
'••—
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.
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Figure 17: Energy absorption of columns
Figure 16: Energy distribution
In Fig. 17, it can be seen that the column PIO absorbs more energy than other columns. About 50% of energy is absorbed by this column although P9 and P8 are the two nearest columns to the collision impact point as pointed out above in Fig. 9. At about 0.5 second after the collision impact, the energy consumption of both PIO and Pll increases suddenly within a short time. In the case of the main beam as shown in Fig. 18, the beam BIO absorbs about 50% of the energy. However, as shown in Fig. 19, the sub-beam B9 absorbs more than 50% of the energy absorbed by all sub-beams. It should be noticed that the energy absorption is consistent with the displacement curves as shown in Figs. 12, 13, and 14. For example, in both the displacement and the energy absorption, PIO, BIO and B9 take the most important effects in columns, main beams, and sub-beams, respectively. Further, the order of columns PIO, P l l , P9, P12 and P8 according to the decrease of displacements is similar with the order PIO, P9, P l l , PI2, and P8 in absorbing the internal energy. The main beam and the sub-beams have the similar characters of orders. This means that a column or beam that has a relatively large displacement may absorb relatively more energy than other columns or beams. 1
1
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.
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~1-
60 CONCLUSIONS In this research, the numerical analysis models are prepared for simulating the behaviors of both trucks and bridge guard fences due to the collision impact. The usefulness of these models is demonstrated by comparing with the practical test results. The main conclusions of this research are as follows: 1) It is possible to simulate the collision process and to visualize the performances of bridge guard fences due to the collision impact of heavy trucks based on the FEM models for trucks and guard fences. 2) The performances of heavy trucks during the collision impact obtained from this research are very similar to the actual experimental results. 3) Based on the comparative study with the practical experimental results, the dynamic horizontal response forces of bridge guard fences subjected to the collision impact of heavy trucks can be accurately reenacted using the presented FEM models. 4) A high percentage of the collision impact energy is absorbed by the main beam of the guard fence, and the displacement of the main beam is also relatively large. The collapse of the main fence beam can effectively absorb the impact energy from the collision impact of a heavy truck.
REFERENCES Hallquist J. (1991). LS-DYNA3D Theoretical Manual, Livermore Software Technology Corporation, LSTC Report 1018, University of California. Itoh, Y., Ohno, T. and Mori, M. (1998). Numerical Analysis on Behavior of Steel Columns subjected to Vehicle Collection Impact. Journal of Structural Engineering, JSCE, 40A, 1531-1542 (in Japanese). Kobayashi K., Okuda M., Ishikawa N. and Hiruma Y. (1994). Impact Response Analysis for the Model Test of the Shock Softening Type Precast Concrete Guardfense. Journal of Structural Engineering, JSCE, 44A, 1531-1542 (in Japanese). Miller, R and Carney, J. (1997). Computer Simulations of Roadside Crash Cushion Impacts. Journal of Transportation Engineering, ASCE, 123:5, 370-376. Miyamoto A., King M. and Masui H. (1991). Numerical Modeling of Impact Load Characteristics Acting on Structures. Journal of Structural Engineering, JSCE, 37A, 1531-1542 (in Japanese). Public Works Research Institute (1992). A Study on the Steel Guard Fences, Research Report No. 74, Tsukuba (in Japanese). Wekezer, J., Oskard, M., Logan, R. and Zywicz, E. (1993). Vehicle Impact Simulation. Journal of Transportation Engineering, ASCE, 119:4, 598-617. Zaouk A. , Bedewi N. , Kan C. and Schinke H. (1996). Evaluation of a Multi-purpose Pick-up Truck Model Using Full Scale Crush Data with Application to Highway Barrier Impacts, 29th International Symposium on Automotive Technology & Automation, Florence, Italy.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
61
ON THE USE OF THE BUCKLING LENGTH CONCEPT IN THE DESIGN OR SAFETY CHECKING OF STEEL PLANE FRAMES A. Baptista^, D. Camotim^, J. P. Muzeau^ and N. Silvestre^ 1 Laboratorio Nacional de Engenharia Civil (LNEC), Av. do Brasil, 101, 1700-066 Lisboa, Portugal 2 Civil Engineering Department, Technical University of Lisbon, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal 3 LERMES, CUST, Universite Blaise Pascal de Clermont-Ferrand, BP 206, 63714 Aubiere Cedex, France
ABSTRACT This work deals with the use of the buckling length concept in the design/safety checking of steel plane frames, namely in (i) the determination of elastic critical loads and (ii) the verification of the compressed members resistance. The aim of the paper is to draw attention to several limitations and ambiguities related to the common use of the BL concept and to provide a contribution to shed some light on the issues involved. A number of simple illustrative examples are presented and discussed. Their results are physically interpreted and used to suggest a few guidelines, concerning the convenience/validity of the BL concept and the definition of alternative approaches.
KEYWORDS Elastic stability, buckling length, plane frames, leaning columns, member resistance, amplification factor.
INTRODUCTION The concept of "buckling length" (BL), also termed "effective length", was introduced by Jasinsky in 1893 (Timoshenko, 1953) to quantify the boundary conditions influence on the elastic critical load of uniform columns, i.e., prismatic columns under uniform compression (since Jasinsky only looked at columns with restrained ends, the original designation was "reduced length"). In this context, the BL is physically interpreted as the distance between adjacent points of contraflexure in the column buckling mode shape (eventually "extended"). Later, the simplicity and appeal of the BL concept led to its extension to compression members belonging to frames and incorporation in design/safety checking procedures still prescribed by the existing codes. In the case of Eurocode 3 (EC3, 1992), the design/safety checking of steel plane frames with respect to in-plane buckling phenomena may or must use the BL concept (i) to evaluate elastic critical loads, (ii) to define the member geometrical imperfections and (iii) to verify the resistance of compressed frame members.
62 Unfortunately, however, using the BL concept to solve problems other than isolated uniform columns (e.g., frames) may (i) lack a physical meaning and (ii) lead to unsafe/uneconomical designs, due to inappropriate interpretations. It is, therefore, important (i) to identify such problems and (ii) to discuss the advantages/drawbacks of using the BL concept. The objective of this paper consists of drawing attention to several limitations and ambiguities related to the common use of the BL concept, mainly in the context of the design/safety checking of plane frames with respect to in-plane buckling phenomena, and providing a contribution to shed some light on the issues involved. For this purpose, a number of simple illustrative examples are presented and discussed. On the basis of these examples, a few guidelines, concerning the convenience/validity of the BL concept and the definition of alternative approaches, are suggested and commented. The paper deals first with the use of the BL concept to perform linear stability analyses (determine critical loads), after which procedures related to the verification of a compressed member resistance are addressed (including the definition of the initial geometrical imperfections).
LINEAR STABILITY ANALYSIS Isolated Members - Column Stability As mentioned earlier, the BL concept was originally introduced to take into account the influence of the boundary conditions on the value of the critical buckling load of an uniform column. The fact that it represents the distance between adjacent points of contraflexure in the column buckling mode justifies the classical BL definition: "it is the length of a ficticious simply supported uniform column that has the same critical buckling load as the real column under consideration". The underlined word, which is often omitted from the definition, explicitly indicates that the usual BL concept cannot be directly applied to non-uniform columns (varying cross-section and/or compression), such as the ones shown in figure 1 (ECCS, 1976). X
X /T777T7
//l/h/
Figure 1: Non-uniform columns It is obviously not practical to define BL for non-uniform columns, as they would depend on the particular combination of column profile and axial force diagram and, therefore, vary from case to case (Baptista, 1998). In such situations, the best solution is probably to perform an accurate linear stability analysis of the column. However, alternative approaches may be undertaken to estimate the column critical load parameter value X-cr using the BL concept, namely: (i) to calculate an (eventually too) lower bound (>-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 0<(le/L)
l, in unbraced frames. It is still worth noticing that, in unbraced frames, a difference in the (j) values of columns located at the same storev has no consequences on the ^cr value, as it is known that only the total amount of axial force acting on all the columns is relevant (it may always be "distributed" to obtain uniform ^ values, before calculating Xcr). (iii) use of the relation (le/L)j = ae/L)i V(EI)j(NL')i/(EI)i(NL')j
(2)
to determine the BL of the "non-critical members" ("destabilised members"). The non-critical (le/L) values are always larger than the critical ones and (le/L)>l 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)
where A and Wei are the cross-section area and elastic modulus, CQ is the peak amplitude of the member initial geometrical imperfection, fy is the steel yield stress and C^ is a coefficient which takes into account the shape of the bending moment diagram and depends on the ratio (Nsd/Ncr). The value of CQ also depends on Ncr, through the non-dimensionai slendemess A,=(Afy/Ncr)^ ^ One observes, therefore, that Ncr influences the member resistance through (i) the moment amplification factors Cni(l-Nsd/Ncr)~^ (primary moments) and (1-Nsd/Ncr)"^ (imperfection moments) and (ii) the imperfection amplitude CQ. Since Ncr is usually estimated by means of the BL, assuming each member as critical (recall the widespread belief that taking (\Q/L)=\ is always conservative in braced frames), a number of issues need clarification. Before discussing them in some detail, which requires a comparison with the results of a genuine second-order analysis, it is important to mention that: (i) in the moment amplification factors, Ncr rnust be taken as N(A.cr) - axial force at the frame critical state (the BL may be obtained from (1)), which means that (Nsd/Ncr)=(A.sd/^cr) for all members. Assuming the member as critical guarantees conservative results only for the critical member(s). (ii) in order to calculate the value of CQ, it makes no sense to use the BL derived from (1), as this would imply extremely large imperfection amplitudes for weakly compressed member. Therefore, it is correct to use the BL derived under the assumption that each member is critical.
Moment Amplification Factors Bracedframes Figure 5 presents an illustrative example of the second-order behaviour of a "geometrically perfect" braced frame (L=5m and EI=21000 kNm^). Figures 5(a)-(b) show the first-order deformed configurations produced by the two sets of loads indicated (cases A and B). Both lead to the axial force distribution presented in figure 5(c), corresponding to 25% of the frame critical load (>-cr^l062 kN and the columns are the critical members) - mode shape also sketched. A comparison between the (i) amplified first-order and (ii) second-order ("exact") column and beam maximum displacements and moments shows that: 2655 kN 375 kNV
2655 kN
1655 kN V
V
Nl
fTlym
(a)
1655 kN
^1
167 kN/
Case A 40 \r><\ir^ kN/m
400 kN/m \k
2655 kN (10?.)
2655 kN I (lOX)
\A\
266 kN
CaseB /TTrrT?
40 kN/m
/Trrm
(\>\ /7T?n7 (b)
nT/TT? /_\ /TTrrn
Figure 5: Illustrative example - second-order effects in braced frames (i) as expected, the amplification of the first-order displacements, using (l-X,sd/^cr)~^=l-33, leads to the exact results, in case A, and to conservative results (5n/5i=1.28 and 1.14, respectively for the
67 columns and beam), in case B. This is due to the fact that, unlike in case B, the deformed configuration of case A is identical to the mode shape, (ii) the amplification of the first-order moments, using Cni(l-A.sd/^cr)"^"l-33Cni, with Cm^O.99(A); 0.75(B) and Cm=l.06(A); 0.97(B), respectively for the columns and beam, leads to either exact (columns in A) or conservative (beam in A - Mn/Mi=1.33, columns in B - Mn/Mi=0.91 and beam in B - M„/Mi=1.09) results, (iii) assuming each member as critical, one has Nsd/Ncr=0.25 (columns) and 0.017 (beam). These values lead to the same results in the columns (critical members "by far") and to unconservative results in the beam (1.017 vs. 1.33, incase A, and 1.015 vs.1.09, in case B). Unbracedframes Figure 6 presents an illustrative example of the second-order behaviour of a "geometrically perfect" unbraced frame (L=5m and EI=21000 kNm^). Figure 6(a) shows the first-order deformed shape produced by the indicated set of loads, which comprises a sway and a non-sway components (figures 6(ai) and (a2)). Both components are associated to the axial force distribution presented in figure 6(b), corresponding to 25% (6.3%) of the frame sway (non-sway) critical load (A.^^i.=276 kN, X,^^=1095 kN and column CI is the critical member) - the two mode shapes are also sketched. A comparison between the (i) amplified firstorder and (ii) second-order column and beam maximum displacements and moments shows that: 490 kN
A 231 kN 100 kN/m 1
/ L_jt
]L
k
/
f 19kN
^ ^
y
i 50 kN CI
440 kN 740 kN 25 kN
^2
/ 25 kN
flSlkN
100 kN/m SC A
25 kN
JL_
690 kN ,(10^)
69 kN
«* ^ J H . . . . H ' *J\""':
25 kN
f\
7 —
h •
^ + /rPm
( a i ) /7M77
/jhm
(ai)
/zAn
/7M77 ( b )
/7A77
Figure 6: Illustrative example - second-order effects in unbraced frames (i) the sum of the first-order displacements non-sway component with an amplified sway component, using (l-Xsd/^cr)~^=l-33, leads to the exact results. This is due to the fact that the sway and nonsway components are, respectively, identical to and very different from the critical mode shape. (ii) the design moments are the sum of the first-order non-sway moments with the amplified sway moments, using (l-A<sd/^cr)^^l-33. Amplifying these moments by Cx^{\-X^^I}^^)-^=\.061Cxn, with Cni=0.937, slightly conservative results are obtained (Mn/Mi=1.17 vs. 1.14, in column C2, and M^P/M^J5'=1.09, in column CI - Mi«0). (iii) assuming each member as critical, one has Nsd/Ncr==0.063 (column CI) and 0.0063 (column C2). These values lead to the same results in both column CI (critical member) and column C2.
Geometrical Imperfections In order to define the amplitude CQ and shape of the geometrical imperfection to be used in the verification of a compressed member resistance, attention should be paid to the stability behaviour of the isolated member and to the deformed shape induced by the primary moments (it may be similar to a member higher buckling mode). Although it is clear that the value of CQ must depend on the isolated member BL (i.e., on its X value), it is questionable whether the common suggestion of assuming the member simply supported (Villette, 1997) always leads to conservative moments. It seems that such a procedure may significantly underestimate the second-order end moments in members with high end rotational restraint (the simply supported assumption implies that the end moments are unaffected by the presence of the geometrical imperfection). However, further studies are required to clarify this matter.
CONCLUDING REMARKS The use of the buckUng length (BL) concept in the design/safety checking of steel plane frames, namely in (i) the determination of elastic critical loads and (ii) the verification of the compressed members resistance, was discussed. Attention was drawn to several limitations/ambiguities and the analysis of a number of simple examples was used to illustrate and shed some light on the issues involved. It was recalled that the BL concept is not useful for isolated non-uniform members (varying cross-section and/or compression). Concerning uniform frame members, the determination of their exact BL (distance between adjacent points of contraflexure) requires the knowledge of the frame stability (critical load >Lcr). The individual (isolated) members BL may be used to determine lower bounds of A-cr, the quality of which, however, depends on the frame "regularity" or on the use of complicated methods. Moreover, the BL concept is not able to handle effectively frames displaying "weak" members, such as leaning columns. Except for very "regular" frames, it seems preferable to perform a frame exact linear stability analysis. In the context of a compressed member resistance verification, the BL may be used to define (i) moment amplification factors and (ii) geometrical imperfection amplitudes and shapes. It was argued that the BL should be obtained from frame stability, in the first case, and from the isolated member stability, in the second. The analysis of a simple "geometrically perfect" braced frame showed that the individually determined BL may lead to unconservative results in the non-critical members, even if le=L. Such a situation did not occur for a similar unbraced frame. In any case, it seems that the use of the BL concept should be replaced by second-order elastic analyses, much more accurate and nowadays easily available. Concerning the definition of the appropriate geometrical imperfections, further studies are required. REFERENCES Baptista A. (1998). EC3 Design of Tapered Elements. Proceedings JPEE 98 (in Portuguese), 301-310. Barreto V. and Camotim D. (1998). Computer-Aided Design of Structural Steel Plane Frames According to Eurocode 3. Journal of Constructional Steel Research 46:1-3, 367-368 (full paper in CD-ROM - # 80) Chen W.F. and Lui E. (1991). Stability Design of Steel Frames, CRC Press, Boca Raton, USA. Cheong-Siat-Moy F. (1986). The K-factor Paradox. Journal of Structural Engineering (ASCE) 112:8, 1747-1760. Discussion and Closure 114:9, 2139-2150 Cheong-Siat-Moy F. (1996). Multiple K-factors of Leaning Columns. Engineering Structures 19:1,50-54. Comite Europeen de Normalization - CEN (1992). ENVl993-1-1 Eurocode 3: Design of Steel Structures, Part 1.1: General Rules and Rules for Buildings, Brussels, Belgium. European Convention for Constructional Steelwork (ECCS) (1976). Manual on Stability of Steel Structures, publ. 22, Brussels, Belgium. Gon9alves R. (1999). MASc. Thesis in progress (in Portuguese), Lisbon, Portugal. Hellesland J. and Bjorhovde R. (1996). Improved Frame Stability Analysis with Effective Lengths. Journal of Structural Engineering (ASCE) 122:11, 1275-1283. Johnston B.G. (Ed.) (1976). Guide to Stability Design Criteria for Metal Structures (^^ edition), John Wiley & Sons, New York, USA. Picard A., Beaulieu D. and Kennedy J.L. (1992). Longueur de Flambement des Elements en Compression (in french). Revue Construction Metallique 1992:2, 3-15. Timoshenko S. (1953). History of Strength of Materials, McGraw-Hill, New York, USA. Villette M. (1997). Considerations sur le Flambement, Proposition de Revision de L'Eurocode 3(in french). Revue Construction Metallique 1997:3, 15-38. Wood R. (1974). Multi-Storey Frame Column Effective Lengths. The Structural Engineer 52:1, 235-244.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
69
The Elasto/Plastic Behaviour and Load Capacity of a Riveted Aluminium/Steel Combined Member in Bending B. D. Dunnel M. Macdonald^ G. T. Taylor^ J. Rhodes^ 1 Department of Energy and Environmental Technology, Glasgow Caledonian University, Glasgow, UK ^Department of Engineering, Glasgow Caledonian University, Glasgow, UK ^Department of Mechanical Engineering, Strathclyde University, Glasgow, UK
ABSTRACT The combined member is becoming popular for light structural applications. It is formed from an extruded aluminium box section into which is inserted a cold formed plain steel channel. Design analysis is complicated by the mix of materials, the combination of the open and closed profiles and the altered mechanical properties of the cold formed section. Analysis of the structural member is performed to British Standards [Ref 1,2]. A complex in-depth finite element study of the individual members is performed. Permutations of parametric studies of the combined members are expanded in great detail and an elasto/plastic^uckling investigation is attempted. A final study of the elasto/plastic behaviour is attempted and the shortcoming and difficulties are elucidated. The work is concluded with a comparison of equivalent results from the FE analysis, BS calculations and experimental work performed. From the results obtained a factor is applied to the beam equivalent FE model in order that these analyses may be used in compliance with the relevant British Standards.
KEYWORDS Aluminium, Steel, Combined Members, British Standards, Load Capacity, Finite Element Analysis, Design Analysis, Elasto/plastic.
INTRODUCTION Increasing use is being made of light structural frameworks of combined sections. Southside Engineering Ltd. of Glasgow has had much success with this class of structure. The main structural unit is the Riveted Combined Alu/Steel Beam. It is used as the structural framework for long walkways, bus stations and similar structures.
70
This structural unit is comprised of an extruded aluminium box section, into which is inserted a cold formed steel plain channel section. The two components are riveted together, non-structurally at discrete points along the mating web face to form the combined section. Initial design work using the Finite Element Method, modelled the complete structure using simple beam equivalents for the structural units and discrete plates for the interconnecting skin. (Figure 1) Differences in the production dimensions of the cold rolled steel channel (Figure 3/Table 1) and validation of the modelling of the structure required that an in-depth analysis of the basic structural unit was required. A comprehensive analysis by Dunne [Ref 3] using the FE method was performed and this was backed up by in-depth testing of component materials by Macdonald [Ref 4] and Macdonald et al [Ref 5].
MODEL DEVELOPMENT FE studies of the structural components begins with a rigorous Convergence Study (Figure 2/Graph 1 and 2) to generate accurate models of each of the component units. A graded mesh is produced for each of the components (Figure 4), with each model located at its relevant position in 3D space as defined by a standard right handed Cartesian axis. The two single models are then combined to provide an FE model of the combined section. Various models of requisite boundary conditions are generated to determine that a valid modelling of these conditions is achieved. (Figure 6) Obviously with this number of model variations there is a requirement for a consistent method for the evaluation and presentation of results. Spreadsheet applications for the reduction and presentation of these results were developed. In the majority of cases displacements and stresses around the cu"cumference of both members were presented simultaneously for direct comparison. These graphs present resuhs that are logically related to their physical location within the individual members.(Graph 3)
PARAMETER STUDIES A suitable mesh that adequately models the combined members has been generated. There are however some variation in the plain channel section due to the cold forming process which require investigation. In addition the most effective use of the tie rivets, although non-structural , require some investigation. Parameter studies of both of these phenomena were investigated and two series incorporating the variations are presented. Permutations of both sets of variables are included to provide a complete analysis of all possibilities. Series 1 is a FIT parameter study investigating variation in the web breadth and flange depth of the steel channel section which result in problems regarding the FIT of the channel within the aluminium box section. Investigation in this section also check the validity of treating the open channel as adequately restramed and therefore subject to pure bending about its major axis for treatment by British Standards. (Graphs 4 and 5/Figure 7) Series 2 studies some alternative arrangements for the RIVET attachments which are applied as an aid in manufacturing, but do have some bearing upon how the combmed section will behave in the manufactured article. Of the many variations possible in applying the rivets for manufacture only three cases are presented; sans Rivet, a single row of rivets at mid-web and a double row of rivets at the web quarters. The confiising stress patterns around the level of load application can be clearly seen in the sans Rivet graph. Distribution of the load in the twin rivet model is much smoother and the effectiveness of the rivets can be seen in the twin stress peaks. (Graphs 6 and 7/Figures 8 and 9)
71 POST ELASTIC ANALYSIS When a specimen is loaded beyond the elastic limit some deformation becomes permanent. This plastic deformation prescribe the material ductility. The yield point and plastic deformation pathway may be determined from tensile tests in the laboratory. In these models the onset of plasticity occurs incrementally at various locations i.e. comer stress concentrations and due to shear forces at rivets. In order to identify the locations and elements most likely to yield a series of increasing load increments was analysed. From this series three sets of elements subject to yielding were identified. Elements below 115 mm, elements connected to rivet points and elements above 1.1 metres. This last set are peculiar to the model and are partially due to stress concentrations generated by the point loads applied. Although the PLASTIC models are run in the Snug FIT configuration, checks are made of the Loose FIT models to ensure that torsional twisting is adequately restrained by the box section. The FE models and in particular the LOAD extension models identify plastic fronts beginning at the comers of the members. The PLASTIC models bear out this effect. Plasticity begins at the comers and develops as twin plastic fronts at each of the flanges. When the plastic fronts coincide the fixll section behaves as a plastic hinge. Plastic models m the Loose FIT configuration sans rivets show quite clearly that torsion in the channel section becomes a major factor once the box section has become fiilly plastic. Local Buckling of the channel section flanges is rapid and comprehensive. (Figure 10)
Summary FE analysis of the combined members in a variety of configurations has produced a bandwidth of values which may be applied with confidence to the beam equivalent results as used in stmctural analysis and complying with British Standards requirements. However Post elastic analysis will require much deeper attention especially regarding the sufficiency of lateral restraint within the manufactured stmcture. (Tables 2 to 6)
References 1. 2. 3. 4. 5.
BS5950, ''Structural Use of Steelwork in Building - Part 5: Code of Practice for the Design of Cold Formed Sections'". 1990. BS8118, ''The Structural Use ofAluminium''. 1990. Dunne, B.D., "The Design and Analysis of Light Structures with Combined Aluminium/Steel Sections". Draft PhD Thesis, Glasgow Caledonian University. Febmary 1999. Macdonald, M, "Bending of a Thin-Walled Combined Section Beam". MSc Thesis, University of Strathclyde. September 1993. Taylor, G.T., Macdonald, M. and Rhodes,!., "The Design Analysis of Light Structures with Combined Aluminium/Steel Sections". Ex. Thin-Walled Structures, 1997.
72
Figure 1: Schematic Plate and Beam FE Model of Complete Structure
^iiiiiiii ^iiliiiiiillii iiiiiiiiiiiii!
iffliiiiii
k*+
Figure 2: Increasing Mesh Densities for Convergence Study
Convergence Study - Max. Displacement vs Mesh Density - Aluminiam Box Section 9.165
I ^ 9.16 I I 9.155 1 " " 9.15 ^
9.145 0
2000 4000 6000 8000 Mesh Density (# Elements)
10000
Ux-Max Graph 1: Convergence Displacement Results - Aluminium Box Section
73
CopiYergence Study - Max Displacement vs Mesh Density - Steel Channel g
9.492 9.49
U ^ S
9.484 9.482 9.48
I ^ 9.438 I I 9.486 4000
2000
6000
8000
Mesh Density (# Elements) -X
Ux-Max
Graph 2: Convergence Displacement Results - Steel Plain Channel Lerel - 0 aetre - Nodul poiltloa ••
0.04
+
0.033
I'"
i
0.025 Nod* PoritioB I
0.02
3 0.015
0.01
0.02
0.09
0.04
O.OS
0.06
0.07
0.08
0.09
O.I
X-Axis Coordlaate ( H e t r e i )
Graph 3: Plot Order for Results Presentation A^vrag* ttrvit AUag LcMftk - TcailU riaag* - r i T l r* riT4
(9 30000000 3000 &
23000000
0.1
0.2
0.3
0.4
0.5
0.4
0.7
i.9
1
1.1
Laagth ( « )
Graph 4: Comparison of Average Flange Stress Along Length - FIT1/FIT4 With Rivet
74 Artrnft
Streif AUag Lcagth - Tcaille Strvti - FITS vt FITS
.<SOOOOOO
$7 30000000
I 20( ISOOOOOO 10000000
0.2
0.3
0.4
0.5
O.ff
0.7
0.8
0.9
Length ( B ) -FIT5A1
F I T S St
—-
FITS Al
F I T S St
Graph 5: Comparison of Average Flange Stress Along Length - FIT5/FIT8 sans Rivet
Stress Distribntion hj Surface - sans Rivet C onfignratioii
SIGcT Al SIGcM AL SIGcB Al SlGcT St SIGeM St SlGeB St
Node Position
Graph 6: Stress Distribution About Sections @ 1.2246 metre Level (sans Rivet) stress DistribBtioB by Sarface - Twin Rfvet CeafigaratioB
SfGcT AI SIGcM Al - SIGeB Al SIGeT St SIGcM St SIGoB St
Node PositioB
Graph 7: Stress Distribution About Sections @ 1.2246 metre Level (Rivet)
75
Figure 3: General Arrangement of Combined Section
Table 1: Dimension Parameters for Steel Channel
Figure 5: Displaced Channel Showing Twist
Figure 4: Stress Distribution About Box Section
Figure 6: Schematic Boundary Conditions
76
Figure 7: Stress Distribution - Alu/Steel Snug (Left) Then Alu/Steel Loose (Right)
Figure 9:Stress Distribution in Twin Rivet Model - Alu to Left, Steel to Right
Figure 8: Standard and Modified Mesh for RIVET Study
Figure 10: Displaced PLASTIC Compression Flange. Alu @ Left Showing Post Elastic Torsion & A Zoom View of the Mating Steel Flange (Right) Showing Local Buckling
77 Load Capacity (kN) Model Name 8.1728 Full Snug 8.0577 Loose Web 8.0222 Loose Flange 7.9106 Full Loose Table 2: Summary of Load Capacities Extrapolated from Flexure Formulation Load Capacity (N) Model Name 7669.60 Snug Al 7498.69 Loose Web Al 7538.35 Loose Flange Al 7373.23 Loose Al 8271.14 Snug St 8086.82 Loose Web St 8129.60 Loose Flange St 7951.52 Loosest Table 3: Summary of Load Capacities Extrapolated From Flexure Formulation - Combined Member as Structural Monolith L^ Beam £q. 1 Model Name L^Shear^late 1 7.887712 8.252347 COMBSNUG 7.780561 8.149319 COMBLOWB 7.747671 8.099674 COMBLOFL 7.644437 7.998599 | LCOMBLOOS Table 4: Extrapolated Load Capacities of Single Members From FE Analysis L^ Beam Eq. 1 Model Name L, Shear/Plate 7.887712 8.252347 COMBSNUG COMBLOWB 7.780561 8.149319 7.747671 COMBLOFL 8.099674 7.644437 7.998599 | 1COMBLOOS Table 5: Extrapolated Load Capacities of Combined Members From FE Analysis [ 1 1
Model Snug FIT Loose Web Loose Flange Loose FIT
Load Capacity (kN) 8.525 8.408 8.367 8.254
Table 6: Load Capacities From British Standards
1
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79
SHEAR BUCKLING OF BEAM WITH SCAFOLD WEB Jacek Tasarek Institute of Structural Engineering, Poznan Technical University, Poznan, POL
ABSTRACT The effect of shear buckling and stif&iess characteristic due to pure shear has been considered in the analysis of steel I beams composed of traditional flanges and the non flat folded plate web. Lideed shear buckling can limit the load capacity of high depth web girders and stiffiiess characteristic due to shear can increase calculated deflection. Some existing analytical formulae can predict more or less precisely shear buckling stress of the folded plate wed. However it's still uncertain how shape of fold influences on the behavior of web under shear and overall stif&iess of the beam. A preliminary analysis of the problem and laboratory experiments on beams of limited length indicates the necessity offiirtherinvestigations. The computational analysis of the experimentally tested models with the use of finite element methods gives the possibility to establish the influence of different parameters of the model on the results of test.
KEYWORDS Steel beam, web, corrugated sheet, thin walled structure, buckling, shear.
INTRODUCTION In order to check validity of theoretical results on the limit stress of corrugated plate web of I steel beam in shear, tests were made on a specially designed steel beam. Flanges of the beam were made with flats D 10x100mm and web with rectangular corrugated sheet (Figure 1). The pitch of corrugation, c, was 400mm There were two different length models. The long one consisted of five complete corrugations and short model had two complete corrugations and some small extra fraction of corrugation . The sheet nominal thickness was t=2mm and t=2.33mm according to measurements, the corrugation high, h, was equal to 50mm. The through and crest with, ao, was equal 200mm. The depth of web measured between flanges, b, was 1000mm. The web was connected to the flanges by one side fillet weld of throat, L5mm thick. From the corrugated sheet six standard tension specimens were cut in order to determine yield stress and Young modulus. The yield stress was determined to be 325MPa as a mean value from tests and Young modulus was estunated as 205GPa. There is no sufficient space available to present and discuss some experiments done with the long models.
80 TEST SETUP AND PROCEDURE The laboratory stand (Figure 2) was in a shape structural steel rectangular frame fixed by anchors to the concrete floor and from one side to the concrete wall. Upper and lower beams of the frame were built of 1200HEA profile. Stanchions welded to the bottom beam were built up members. The one from concrete wall side, close to hydraulic jack, consists of I200HEA profile strengthened by U200 profile. The second stanchion was made with two parallel-situated I200HEA profiles. The upper beam was connected to the stanchions by end plate bolted connections using six high-tension MIO bolts. That way it was possible to lower upper beam of the stand and to test smaller depth beams.
Figure 1: Model dimensions
Figure 2: Test setup
The lower flange of tested model was fixed to the bottom beam of stand by bolts MIO situated m pairs symmetrically on both sides of the web in a distance of 200mm along the beam. The upper flange of model was kept in a horizontal and vertical straight position by pairs of roller bearings connected to the upper beam of the stand. The load was applied at one end of upper flange along the centerline of flange by mean of hydraulic jack of SOOOkN capacity coupled with hydraulic manometer to measure load. Longitudinal displacements were measured by three inductive deflection gauges, two at upper and one at lower flange. Additionally fifteen inductive and dial type gauges measured perpendicular deflection of web and upper flange. The course of experiment was recorded by two hi^-resolution video cameras in order to check displacements by mean of photogrametric methods. In any test the load was applied in small increments and all displacements gauges were read at each load level.
THEORETICAL ANALYSIS hi order to prepare laboratory experiments and to fmd out importance of different parameters of test there were done some theoretical analysis using finite element method and some existing analytical formulae described corrugated web shear critical stress. Model of computational analysis. The method of finite elements was used to provide numerical analysis. The physical and geometrical nonlinearities are assumed. The geometry of construction part under testing allows preparing regular element mesh. Quadrilateral shell finite elements with constant thickness are applied. Material behavior is modeled by elastic-plastic law with isotropic hardening, described by the
81 following parameters: Young modulus, yield stress and hardening parameter. The construction is loaded m iie kinematics way. The displacement is applied for the whole edge of flange end. The multi-point constrain (MPC) is defined to have equal displacement for all nodes of the edge and also to be able to get total reaction force, representing experiment loading mechanism, in this place. The boundary conditions of analyzed problem are formulated as supports (equality conditions) and contact (inequality conditions). There were several computational models prepared and calculated to show the influence of various parameters on structural behavior of construction part. First the geometrical dimensions (thickness of the web) was changed. Next in geometrical modifications the imperfection forfi-eevertical edge of the web was analyzed. Local effects and total change on edge length were introduced. Another group of parameterization was done for material parameters, concerning mainly yield stress. The most important modifications for correct analysis results concentrate on boundary conditions. The defmition with only supports gives not satisfactory structural behavior. The contact type constrains allow to observe real effects, taking place during experiment. They are important for roller provides on the top flange and in the area under bottom flange where tension in bolts occurs. The decision on modeling the reality of boundary conditions becomes the most influencing factor in analyzed problem.
Analytical formulae Four analytical formulae described value of critical shear stress in corrugated web beams were taken into consideration: • From Timoshenko & Gere (1961) r. =4A:
VAA
(1)
A/AA^ b I where: k parameter can be determined according to value 0 = -^—^— and p = —yD^D^
A
fi-om table 1: TABLE 1 k PARAMETER VALUE 11/teta= 0 1 beta 0 0,2 0,4 0.6 0,8
1 '
k 8 8,1 8.5 9 10 11.5
1/teta= 0,2 1 beta k 1 9,2 0 9,4 0,2 9.8 0,4 10.8 0,6 0,8 11.9 1 13.8|
From :Eastley McFarland (1969,1975) r„=36
VA^
(2)
From:fflavacek(1968) r„=41
V^
(3)
82 •
From:. Dukarskij J.M (1970) , n'D.a
(4)
Plate stififtiess can be evaluated from equations: Et'c ^ EJ^ ^
A = 12(1-v^K
'
c
'
Et'c, 12(1-v')c
(5)
Comparison of analytical and computational results The results of computation as a reaction/displacement curves and calculated from
Figure 3: Comparison computational results with analytical estimation. analytical formulae values of critical reactions was presented on Figure 3 .The critical values of reaction calculated from considered formulae, presented by dashed lines are in a good agreement with values of limit stress from computation. However they are mostly too conservative. The parameters used in the analysis can change the value of limit reaction in a range of 15 % and the stiffiiess of model within very small limits only.
COMPARISON LABORATORY TEST RESULTS WTH COMPUTATIONS The computational results for data taken from experiments are represented on the Fig. 4 by the deformed shape with the contours of displacements perpendicular to web surface. The same shape of deformation shown on the Figure 5 and Figure 6 was obtained in the experiment The experimental and computational results for the loaded top flange end in the experiment cases as reaction versus displacement were plotted for comparison on Fig. 7.
83
Figure 6: Deformed shape of model from laboratory test.
REACTION [kN] 200.00 —1
C0MP.1 ^t-ATEST2
1 0.00
4.00
'
\ 8.00
'
\ 12.00
'
1 16.00
'
1 20.00
DISPLACEMENT[mm]
Figure 7: Reaction -displacement curves computed and obtained from tests
84
Figure 4: Deformed shape of model from computation
Figure 5: Deformed shape of model from laboratory test.
85 It can be seen from analysis that the loads limits obtained from tests and from computation are in a reasonable good agreement; however the shape of. buckled web differs from that expected on the basis of theory. There are also differences between calculated and obtained in tests stif&iess of beam measured by slope of reaction displacement curves to displacement axis. Models seem to be less stiff then it was predicted in the computation. It means that some extra factors have inflicted on the results. It is supposed that some initial shape imperfections of the models, their too short lengths, small eccentricities of applied loads particularly in test land deformations of laboratory stand disturbed the results.
CONCLUSIONS The results of investigations presented in the article in this paper constitute a part of preliminary studies on the problems connected with buckling behavior of corrugated web steel beams in shear. The analytical formulae for critical shear stress of corrugated webs considered in the paper can be applied for estimation critical stress for short beams, although the concern long beams. Computational analysis of the beams in shear makes it possible to estimate the importance of different parameters of the model on results of experiments. It is very usefiil in better understanding and explanation of the test results. The test results comparison of short models indicates that several additional factors beside those analyzed in the computations and which importance was impossible to predict in advance, can influence on the results of laboratory experiments. Further detailed analysis of presented preliminary investigations make it possible to planfiirtherexperiments more precisely.
References Timoshenko S.P. and Gere J.M.(1961), Theory of Elastic Stability, Mc Grew Hill Book Company Easley I.T. Mc Farland D.E. (1969^ Buckling of Light Gauge Corrugated Metal Diaphragm. Journal of Structural Divison ASCE v.95 No. St7, 1497-1516 Easley I.T.(1975) Buckling Formulas for Corrugated Shear Diaphragms. Journal of Structural Division ASCE No. St7, 1403-1417 Hlavacek V. (1972) The Effect of Support Conditions on the Stiffiiess of Corrugated Sheets Subjected to Shear. Acta Technica CSAV No 2 , 209-236 Dukarskij J.M.(1970), "Issledovanie raboty gofrirovannogo lista na sdwig. Aluminievyje Konstrukcji. Isdatielstwo po Stroitielstwu ,21-41.
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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,,<1
FlangeAVeb': (/,,)^
k=
FlangeAVeb': (/,,)^
k = {2-{b/hf'}4{b/hf
mdd/b<0.6
U25min[4,{05Ce,+'^^L+'^X^/hf}
={2-{h/bf'}4
\f h/b>\
\fh/b<\
local buckling of the flange and web when the web is under flexure ^ local buckling of the flange and web when the web is in pure compression.
91 Local buckling solutions for both the element model and the semi-empirical interaction model are given in Table 1. For an edge stiffened member (e.g., a lipped channel) h = web height, b = flange width, and d = lip length. All of the k values are in terms of the critical buckling stress of the flange, where: b't ' Several of the elements are subjected to a stress gradient, which is defined in terms of ^,
^ =/ , - / 2 Where,/i and/2 are defined as the stresses at the opposite edges of the element. For the web,/i is at the web/compression flange juncture. For the lip, f\ is at the lip/compression flange juncture. Compression stresses are positive. Distortional Buckling Prediction of distortional buckling by hand methods is markedly more involved than local buckling. The key issue is related to the rotational restraint at the juncture of the flange and the web. Consider an isolated edge stiffened flange in which the restraint at the web/flange juncture is first i; modeled as simply supported, then fixed - finite strip results of such a model are given in Figure 2. The plate buckling coefficient and hence the buckling < i stress are markedly more sensitive to rotational restraint for distortional buckling than for local buckling. Hence, distortional buckling requires careful treatment of the rotational restraint at the web/flange juncture and local buckling in most cases does not. < 0
I
10
KM)
Closed-form prediction of the distortional buckling r:or ^ ^r> ^ ^- 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.
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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' -
\
I + O-210
a'
(7)
{El lE'y^a^- <^l)+(^l> a'
«3
V (np
1 irp\r^2
-,2\j.,2
1
^12=^1+^2+3^3-4^33
When X is a reference direction, then a^ =1, (E** = Ex** and
a = CTJ, = (TJO + E^s^ ).
Summarising the above considerations we state that for a strain hardening orthotropic material not only initial yield stresses but the parameters Ex^ , Ey^ , E''e=45° and G^xy are to be determined as well. They can be evaluated in unixial tensile tests and the pure shear test.
103 4. ENERGY OF PLASTIC DEFORMATION The following two postulates are taken in order to obtain an effective approximate solution of the problem [4], The radius of curvature of a yield strip is approximated as p = nt/p while the maximum (boundary) bending strain s^ is expressed as 8^ = s^ = p/2n. A multiple n of wall thickness of the beam can be determined in an experimental way or by means of minimalization procedure and P is an angle of relative rotation between two adjacent walls of the global plastic hinge. The analysis was limited to the linear stress distribution when stresses vary from the value which equals an initial yield stress Co at the neutral beam axis up to the value corresponding to an effective stress o^^r in a boundary layer. Such a stress distribution leads to the evaluation of the fully plastic moment at a yield line. Thus, the plastic moment capacity at yield lines perpendicular to principal du-ections of orthotropy is expressed as
^^ 4 12« ^ ^ ^ where i=l,2(x,y). The plastic moment capacity at a yield line the situation of which is determined by an angle y with respect to the reference direction x is
W ^Zl^^ZllL '^
12
(8b)
6
^ ^
where ayo is an initial yield stress for the direction y given by (3), while a^ is an effective stress expressed by (4) and the effective stress in x direction takes form: a,=c7,,+£;^/ where
(9)
e^ = f ^ cos^ y
Finally, the total energy of plastic deformation absorbed at the global plastic hinge during a relative rotation of two beam parts and corresponding to the rotation angle 9 is expressed as W(e) = I l j m ' p d p + i r , ( m ; p , P o „ p , ) + i;n,(inJp,Po„P,) J
0
k
(10)
1
The first component of the right side of the expression (10) is the energy absorbed at the rotation along stationary yield lines while two next components express the energy absorbed at both travelling and non-stationary yield lines [2]. A length of any yield line Ij as well as angles Pj of rotation of two adjacent walls of the global plastic hinge along a stationary yield line and all other functional parameters of Fk and 0.\ have to be expressed in terms of the angle 6. The total energy of plastic deformation (10) can be calculated using a numerical procedure only- by means of the incremental method - for subsequent increments of the angle AG which correspond to the increments of Ap. A bending moment at the global plastic hinge is calculated using a numerical differentiation procedure of the energy with respect to the angle 0. At each step of the numerical procedure the following limit conditions have to be fulfilled - at yield lines perpendicular to principal directions
104 (11a)
i=l,2 (x,y)
Ojo + EfEi" < a„„,i
where auit,i is an ultimate tensile (or compression) stress for the i* principal direction which can be estimated in the uniaxial tensile test; - at yield lines inclined at an angle y with respect to the reference direction x ^y^a„,,,
(lib)
where auit,yis an ultimate stress for an arbitrary direction determined by the angle y. This stress can be calculated in the similar way as the effective stress given by (4).
5. NUMEMCAL RESULTS On the basis of the procedure presented above a computer programme has been elaborated and numerical calculations have been carried out for beams made of structural steel and aluminium alloys mainly. The plastic mechanism taken into consideration, typical for a rectangular boxsection beam, is shown in Fig. 3 a while the numerical verification of its geometry in the initial stage of failure, performed usingfiniteelement method, is presented in Fig. 3b.
b)
a)
Figure 3: Plastic mechanism of failure, a) - theoretical model, b) - FEM verification In the next two diagrams the structural behaviour of the box-section beam made of mild structural steel is presented. Material parameters of the beam, established in three tensile tests and the pure shear test were as follows o„. =232<*> cJuit,! = 322 Ex =2.03-10^ \EJ' =1.73-10^ (*) all values in MPa
ao2 cJuiu Ey EyP
= 226 =313 = 1.99-10^ =1.54-10^
CJo,45° CJult,45° E450
E^y
= 222 = 313 = 2.04-10^ = 1.63-10^
To = 1 4 1
1
Gxy'' = 0.57-10^
The behaviour of the beam in the stage of failure is shown in Fig.4. Both bending moment capacity and energy of plastic deformation at the global plastic hinge in terms of the rotation angle at that hinge are presented. Continuous curves represent the isotropic material with average material parameters while dotted lines correspond to the orthotropic material of properties given above. The behaviour of the same beam in the whole range of loading from zero up to and beyond the uhimate load as well as in the initial stage of failure is presented in Fig. 5. Two curves without
105 markers represent failure paths for „average isotropic" and orthotropic material, respectively, while lines with markers show pre- and post-buckling paths evaluated using post-buckling nonlinear analysis based on the asymptotic Koiter method. The results concerning the post-buckling analysis are taken from Kolakowski [5,6] under the author's kind permission. The horizontal marked levels represent the upper bound estimation of the load carrying capacity for the „average isotropic" and orthotropic material, respectively.
M/tn
[MN/mm]
Energy [J]
Figure 4: Failure behaviour of the box-section beam a=l 10 m, b=140 mm, t=lmm. M [kNm]
Figure 5: Structural behaviour of the box-section beam. Dimensions as above. Exemplary diagram of bending moment capacity in the stage of failure in terms of the yield stress ratio Ooi/doi for a material displaying strong orthotropic properties ( the presented range of cJo2/croi corresponds to material parameters of some aluminium alloys as well as of non-metallic materials) is shown in Fig.6.
106
^
[A ,
n
1.S M[kNmn
X ^
»
/> ^—
Mbl]
Figure 6: Bending moment capacity in the phase of failure. Square section beam: a= 100mm, t=1.25mm; Mbi-e = 0.1°,Mb2-e = 5°
/ / 1
1,6
2
CJ02/O0I
6. FBVAL CONCLUSIONS In the case of a structure made of mild structural steel the factor of orthotropy does not influence the collapse behaviour significantly (Fig.4). However, material orthotropic properties play more significant part in the initial stage of failure (Fig. 5) and influence the maximum load at the highest point of the load-deformation diagram. The upper bound estimation of the load-carrying capacity is about 11 % lower when taking into account orthotropic properties of the material than for the „average isotropic" material. Generally, orthotropic properties of the investigated structural material are more influential in the phase of failure than in pre- and post-buckling state, since sheet metals display distinct anisotropic properties in the plastic range of deformation. Finally it should be mentioned that this investigation provided a general solution of the problem but did not go into sufficient depths to provide any comprehensive information about the influence of anisotropic plastic properties of sheet metals after manufacturing processes on the collapse structural behaviour. A substantially more exhaustive material investigations and subsequent theoretical calculations carried out on the basis of the experimental results would be of aid in the estimation of this influence. However, a researcher faces here a very complex problem since depending on the history of deformation during the manufacturing process, anisotropic plastic properties of sheet metals may be different. 7. REFERENCES [1] Murray N.W. (1986), Introduction to the theory of thin-walled structures. Clarendon Press, Oxford [2] Kotdko M.(1996),Ultimate load and postfailure behaviour of box-section beams under pure bending, Engineering Transactions, 44:2, pp229-251, PWN-Warszawa [3] Hill R. (1950), Mathematical theory of plasticity, Oxford University Press [4] Kotelko M.(1998), Selected problems of collapse behaviour analysis of structural members builtfi-omstrain-hardening material, Thin-Walled Structures, 30:1-4 [5] Manevich A.,Kolakowski Z. (1996), Influence of local postbuckling behaviour on bending of thin-walled beams, Thin-Walled Structures., 25: 3, pp 219-230 [6] Kotelko M., Kolakowski Z. (1998), Lower and upper bound estimation of thin-walled composite beams, Proc. of/w/. Conf on Lightweigth Structures in Civil Engineering, Warsaw, pp 325-330 [7] Szczepinski W. (1993), On deformation-induced anisotropy of sheet metals. Archives of Mechanics, 48:1, pp 3-38, PWN-Warszawa
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
107
ELASTIC WEB CRIPPLING OF THIN-WALLED COLD FORMED STEEL MEMBERS M.C.M.Bakker\ H.H.Snijder^'^ and J.G.M.Kerstens^ * Faculty of Architecture, Building and Planning, Eindhoven University of Technology, the Netherlands ^ Holland Railconsult, Utrecht, the Netherlands
ABSTRACT In this paper, first two different analytical web crippling stiffiiess models are described, namely a beam-on-elastic-foundation model and an energy model. Then a description is given of the finite element model, which was used to check the accuracy of these models. These finite element simulations have been validated against experimental results. Subsequently the parameter study is described, which was carried out to compare the results of the beam-on-elastic foundation model and the energy model with the results of finite element simulations. Based on these results the analytical models have been optimized by adding empirical correction factors. Finally conclusions and recommendations for fiirther research are given.
KEYWORDS Web crippling, thin-walled, cold-formed, steel, elastic foundation, energy method, Raleigh-Ritz.
INTRODUCTION When a thin-walled cold-formed member is subjected to a concentrated load, its web may cripple due to the high local intensity of the load. This type of localized failure is denoted as web crippling. Web crippling may occur under various loading conditions and in various types of members. The research was confined to web crippling of interior supports of continuous members consisting of cold-formed hat or deck sections with unstiffened web and flange elements. In this paper only models for deck sections will be discussed. The existing web crippling prediction formulae are based on curve fitting of test results rather than on a physical understanding of the occurring failure modes. These formulae appear to give inconsistent and sometimes unsafe results (Bakker & Pekoz, 1985). Moreover, they have a limited and often not well described range of applicability. The research described in this paper is directed to the fiirther development of an analytical web crippling model (Bakker & Stark, 1994). According to this model (see Figure 1) the load-web crippling deformation behavior can be described by means of a linear elastic curve until the formation of a spatial plastic mechanism, and thereafter by a rigid-plastic mechanism curve. Here the web crippling deformation is defined to be the decrease in the height of the web of the member at the load application. The object of the current research (Vaessen, 1995) was to develop an analytical model describing the linear elastic load- web crippling deformation behavior.
108 This curve is fully described by its slope, the web crippling stiffhess k^j^ . The behavior of the spatial plastic mechanism has been described elsewhere (Bakker & Stark ,1994).
linear elastic curve
rigid plasHc mechanism curve y r c t a n k^hw —•Ahw l/2bbf
l/2bbf Figure 1: Web crippling
BEAM-ON-ELASTIC-FOUNDATION MODEL According to the beam-on-elastic-foundation model (as proposed by Tsai, 1987) the load bearing plate is supported by a beam (formed by the top flange and a part ab^ of the web) on a continuous elastic foundation (formed by infinitely small slices of the member, supported near the bottom flange comer radius). Note that in this idealization the top flange and the part ab^ of the web are used twice (see Figure 2). Furthermore note that the beam-on-elastic-foundation model only describes the local web crippling deformation of the member underneath the concentrated load, and not the overall deflection of the member due to the concentrated load. To determine the foundation stiffhess k^ the slice of the cross-section is idealized to a portal fi*ame with a width dx (see Figure 2). The foundation stiffness k^ is then determined as: (1) where w ^ is the vertical displacement of point A. This displacement can be calculated using Castigliano's law. For a deck section this results in: 1 _ b,fb„(3btf + 2 b J + 3r,^^sine,cose„(b,f + b j ( b ^ -r^^^ sin9 J + bt(2btf + b j k,
Et(b,,(3btf+2bJ + b „ ( 2 b t f + b J ) Tj^tf sin^ e„b|,f(ri.tf sine„(6btf -9r,.tf sine„) + b„(6btf -8r,.tf sin9„)) ^,Et^(b^,(3btf+2bJ + b„(2btf+b J ) bX^ sin^ e„(4r,^ sine„(btf - b J + Sb^b^ -61^^^ sin'0„)
y,,Et'i\,(2h^+2hJ
+ K(2\
+b J )
(2)
109 To determine the bending stiffness EI of the beam, the second moment of area of the top flange and a part ab^ of the web (see Figure 2) is calculated, resulting in: g^b^+t^
Ei = J ? ^ + Etab, 12
cos29,(a^b^-t^) ^ tab,sin2e,^
12
^g^b^sine^+tab ^ + Ebtft|
12 Etgb„
bjf + 2 g b ^
\2
b^gb^sine^ +tbtf btf+2ab^ J
(3)
In this calculation the rounding of the comers has been neglected.
btf
4 njtfsine,, n^ine„
btf
o
1/2qa i^1/2q ^* ,'f]^-=^.A'\ elastic >/
-+
vVoc^w
bPa^
^ 1/2 F
\i^ foundation ,\ d x
-t
•!/2bbf
4-
/M
t / \t
1/2 F
M lu I ^ Mb
-span
-f
Figure 2: Beam-on-elastic-foundation model Once the beam and foundation properties are determined, the deflections of the beam under various loading conditions can be calculated. To determine the web crippling stiffness, the member is idealized to a finite beam loaded by two concentrated forces X F • The web crippling stiffness can then be calculated as:
Vl024EIkJ AhYv;boef
where:
WM
l + f,(pLJ-Xf,(P(L,^ - L , J ) + Ke-'*'^""-"^' and f,(px) = e"P^(cospx + sinpx)
(4)
(5)
ENERGY MODEL The energy model is based upon the Raleigh-Ritz method. This method provides a direct way for obtaining approximate displacement solutions of mechanical systems, and hence a direct way to obtain approximate stiffness expressions. The Raleigh-Ritz method is based upon the principle of the stationary potential energy and the use of assumed kinematically admissable deformation functions (trial functions) with unknown parameters. Substituting these deformation functions in the potential
no energy functional and subsequently requiring this function to be stationary, results in a set of linear equations from which the unknown deformation parameters can be solved.
1/4 F
1//.F
Figure 3: Assumed deformations in energy model The assumed deformations (see Figure 3) are based on the mode of deformation observed in finite element simulations. The member is ideahzed to a structure composed of the top flange plate and two web plates. The bottom flanges are not included, because their deformation variations are negligibly small. The web and top flange deformations do not extend over the entire length of the member, but only over an effective length L^^. The variations in in-plane deformations are very small. Thus it is assumed that the contribution of stretching to the potential energy of the plates may be neglected. The trial functions, describing the out-of-plane deformations w^^ and w^ in the top flange and web plates respectively, are chosen as: Wtf(x,y) = Wtf(x)w^f(y) and w^(x,y) = w^(x)w^(y)
(6)
where the deformation in the width direction is described by: (7)
W,,(x) = Wj(l + C i ^ t f ^ + C 2 ; t f ^ ) Dtf Dtf
x^
x'
(8)
W w W = W2(l + Ci^^—- + C 2 . ^ — ) b^ b^
and the deformation in the length direction by: 16
Wtf(y) = w „ ( y ) = — ( y ^ - X ^ f ) ^
and:
Wtf(y) = w „ ( y ) = 0
(9)
|y|>%Lef
(10)
Ill Lgf, Wj and W2 are unknown coefficients to be determined by minimizing the potential energy. Assuming that the web and top flange plates are rigidly connected to each other, the coefficients ^i;tf' ^2;tf' ^1 w ^^^ ^2;w ^^^ ^^ determinedfi*omthe compatibility requirements between webs and flanges, resulting in: 2b^^-5nb„b.,-3nbt nbl+5nb,b^-4b^ ^itf ~ (11) c,.«- = %nb„(b„+btf) X.nb„(b„+bJ 2nbl-5b„b^-3b^ %b,f(b„+bj
b^tf+5b„b„-4nb:
(12)
To limit the number of unknown parameters in the potential energy function, it is furthermore assumed that the maximum deflection w, of the top flange can be expressed as a function of the maximum deflection W2 of the web: Wj =nw2. (13) The factor n is calculated from the ratio between the mid-web and mid-top flange deflection in the portal frame model used to determine the foundation stiffiiess for the beam-on-elastic-foundation model (see Figure 2). For deck sections: c, + c . 3bt(b,,-r,,,sin0J(3b,,+bJ
(14)
with:
c, =4r,^^sin^e^(bjb^ +2(b,, + b j ) + 3 b , , b j
(15)
and:
C2 =-3bff(3b,fr.,,sine, + b , ( b , +2(b,f +r.tf sin0^)))
(16)
Using Eqns. 6 to 16 the potential energy of the member can be determined. The resulting expression is so complex that the unknown coefficients L^^ and W2 can only be determined (by requiring the potential energy to be stationary) after neglecting some terms in the potential energy function. Once these coefficients are known, the web crippling stiffness can be calculated as:
Ah^;energy
Wtf(XL,b.ri.tfSine^)
nw2(XLib,ri.tf sin0^)
(17)
This results in: ^Ahw;energy
where:
and:
D=
Et^ 12(1-v^)
Db^f^f 1575(L^,-L^,f)*(b^^+b»
c,+c^
b^tf(34bl +504L;) + btn^(17b^^ +252L^,) 14b^^+7btn^
(18)
(19)
112 C3 = b ^ ( b t ^ f ( 8 6 4 0 + 2400n) + 5248Ltfb, +btf(L',,(1536^f +3936bt +2400btn))) + bJfbt(b,(3150 + 15750n') + 7875btfn') + bt(b'tfL',,(2400n'-768n + 8800) + 6656btfL>') (20) c^ =bt(Ufn'(768L',f +2400b'tf) + b'tfn(7875bJfn-8960L',f)-2100b^
+4096b^btfL',fn')
+ hi (hi (8400n(btf + b , ) - 5250btf) + L^f b'^ (b^n(3200n - 9984) - btf (4608n - 6400)))
(21)
C5 =b,sin'e^ri^,f(sine^r.^,f -btf)(b^(btf(4sin0^r.,, - b t f ) - b , n ( b , +5sine^ri.tf)~4sin'O^r-^^) + b , sinO^r. tfn(sine,r. J b , +5b J - b ^ b ^ ) )
(22)
FINITE ELEMENT MODEL The finite element model (see Figure 4) has been set up to simulate web crippling tests, which are performed as three-point bending tests (see Figure 1). In these tests the members are loaded by a very stiff load-bearing plate. The top flange of the member is not attached to the load-bearing plate, and due to the downward curling of the top flange underneath the load-bearing plate, only four contact points will remain after initial loading. Each of these contact points transmits a concentrated load XF • Using symmetry, only one quarter of the member needs to be modeled. The finite element model has been developed within the finite element program ANSYS (version 5.0A), using quadratic eight-node iso-parametric shell elements. This element has both bending and membrane capabilities. With respect to the boundary conditions a distinction is made between the finite element models for hat sections (used to validate the finite element model against test results) and first generation deck panels (used in the parameter study to verify the accuracy of the analytical models). In both models the web crippling deformation is calculated as w^ - w^ (see Figure 4). Hence the web crippling stif&iess is determined
boundary conditions: hat and deck sections 2:v = 9 , = ( p , = 0 deck sections only 4:v = 9 , = ( p , = 0
U
Figure 4: Finite element model
113 The finite element model has been validated against experimental tests (three-point bending tests on small span members, Bakker & Stark (1994)). From these simulations it was found that the model predicted the web crippling stiffness reasonably well.
PARAMETER STUDY To validate the analytical web crippling models, their resuhs are compared to the resuhs of finite element simulations. The influence of the following parameters (see Figure 1) has been studied: span length Lgp3„, web angle 6^, width of bottom flange b^^, width of web b^ , width of top flange b^^, interior comer radius between the web and bottom flange rj.^f (which is taken equal to the interior comer radius between the web and top flange r-.^^), plate thickness t and length of load-bearing plate Lj,,. The influence of each parameter on the web crippling stiffness is investigated starting from three different basic members. These three basic members have the following geometrical dimensions (in mm) in common: L^p^^ = 1560,0^ = 90% b^^ = 60, b^ = 60, t = 0.7 en L^^ = 100. The comer radii vary: for basic member 1 r^ t^ = rj ^f = 1, for basic member 2 r-.^^ = r-.^^f = 5, and for basic member 3 ^i;tf = ^i;bf = 10 • For each basic member one parameter at a time is varied, while keeping the other parameters at their initial value. The following parameter values (in mm) have been used: L^p,^: 520, 1040, 1560, 2080, 2600
L,^: 25, 50, 100, 150, 200, 250, 300
0 , : 50°, 60°, 70°, 80°, 90°
t: 0.5, 0.7, 0.9, 1.1, 1.3, 1.5
\ , and b,,: 20, 40, 60, 80, 100, 120
b , : 30, 50, 65, 80,100
Besides, in the basic members the interior comer radii r-.^ = T-.^^ are varied from 1 to 10 mm (in steps of 1 mm). The total program included 106 simulations. In all simulations the modulus of elasticity E is taken equal to 210000 N/mm^ and Poisson's ratio v equal to 0.3. RESULTS According to the finite element simulations, the web crippling stiffness increases with decreasing comer radius, width of web, width of top flange and web angle, and increases with increasing plate thickness and width of bottom flange. The web crippling stiffness is independent of the span length, since the deformations do not extend over the entire span length. The web crippling stiffhess is most sensitive to the comer radius and the wall thickness. For short load-bearing plates the web crippling stiffhess increases, for larger load bearing-plates the web crippHng stiffhess remains constant with increasing length of the load-bearing plate. This is caused by the fact that for a large length of the load-bearing plate, the four concentrated loads used to model the load bearing-plate result in four separate dents in the member and the distance between these separate dents does not influence the web crippling stiffhess. The finite element model is a valid representation of a web crippling test, provided there are only four contact points between the member and the load-bearing plate. This is tme for members with a small comer radius, where the web crippling deformation is very small compared to the global beam deflection. However, this may not be tme for members with a large comer radius and a large length of the load-bearing plate. Then, additional contact points may be present. Adjusting the finite element model by including these additional contact points will result in larger web crippling stif&iesses.
114 For a having a constant value, the beam-on-elastic-foundation model qualitatively predicts the correct influence of all parameters (compared with the finite element model) except for the width of web and the width of the bottom flange (which have however only a small influence on the web crippling stiffness). To improve the quantitative performance, the beam-on-elastic-foundation model has been adjusted by determining the factor a (used in Eqn. 3) to be a function of the comer radius: a = 0.118rj^,f.
(24)
This results in a mean value kAhw;boef/^Ahw;FEM ^^ 0.9967 and a standard deviation of 0.1750 . The energy model qualitatively predicts the correct influence of all parameters (compared with the finite element model), except for the length of the load-bearing plate. In the energy model the web crippling stiffness keeps increasing with increasing length of the load-bearing plate. To improve the quantitative performance, the energy model has been improved by multiplication by an empirically derived correction factor, taking L,^ = 100 irrespective of the actual length of the load bearing-plate: 0.92 kAhw;energy;improved = Y ^^^^•,.n.r,y(K
= 1^0 m m )
With Y = ^^^^^'^,,3 •
(25)
This results in a mean value k AH^ ;energy;improved / ^ AH^ ;FEM ^^ ^-^^^ ^ ^^^ ^ standard deviation of 0.1386. CONCLUSIONS AND RECOMMENDATIONS Both the beam-on-elasfic-foundation model and the energy model give reasonable results for the web crippling stifftiess of thin-walled-cold formed steel members when compared to the validated finite element model. The energy model is slightly more accurate, but the beam-on-elastic-foundation model results in simpler equations. For reliable web crippling stiffness results for members with large comer radii, the modeling of the load-bearing plate in the finite element model needs to be refined. Then this modified finite element model should be used to determine whether the beam-on-elastic foundation model and the energy model need further modifications. ACKNOWLEDGEMENT The research described in this paper has been carried out by M.J.Vaessen in a research project to obtain his master's degree. REFERENCES Bakker, M.C.M. and Pekoz, T. (1985). Comparison and evaluation of web crippling prediction formulas, EUT-report 86-B-Ol, Eindhoven University of Technology, Eindhoven, the Netherlands. Bakker, M.C.M. and Stark, J.W.B. (1994). Theoretical and Experimental Research on Web Crippling of Cold-Formed Flexural Steel Members. Thin-walled structures 18:4, 261-290. Tsai, Y.M. (1987). Comportement sur appuis de toles minces formees a froid, These no.689, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland. Vaessen, M.J. (1995). On the elastic web crippling stiffness of thin-walled cold-formed steel members, master's thesis Faculty of Architecture, Building and Planning , Eindhoven University of Technology, Eindhoven, the Netherlands.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
115
ELASTIC BUCKLING OF THIN-WALLED MEMBERS WITH CORRUGATED ELEMENTS J.P. Papangelis and G.J. Hancock Centre for Advanced Structural Engineering, Department of Civil Engineering, University of Sydney, NSW 2006, Australia
ABSTRACT Although corrugated steel sheets are used extensively in many building applications, very little research has been done on the buckling of thin-walled members with corrugated elements. Research has been limited to the study of beams with trapezoidally corrugated webs and flanges of flat plate. This paper describes a finite strip buckling analysis for investigating the buckling behaviour of cold-formed C- and Z-section purlins with corrugations. The analysis is performed by the computer program THIN-WALL, which can predict the local, distortional, flexural and flexural-torsional buckling modes of a section and draw the shapes of these modes on the screen. Examples are given of a C-section in compression and a Z-section in bending.
KEYWORDS Buckling, thin-walled, cold-formed, corrugated, C-section, Z-section, purlin, finite-strip, computer, program.
INTRODUCTION Corrugated steel sheets are used in building construction in a variety of different applications. They are frequently used for roofing and siding in buildings, drainage pipes, flooring systems, and silos and tanks. Studies of corrugated plates have mainly focused on corrugated shear diaphragms and I-section beams with corrugated webs. It appears that no research has been done on the effect of corrugations on the behaviour of thin-walled structural members such as C- and Z-section purlins. This paper describes a finite strip method for analysing the buckling modes of thin-walled structural members composed of corrugated elements. The analysis is used to investigate the buckling behaviour of C- and Z-section purlins with and without corrugations. The structural behaviour of corrugated plates is quite different to that of flat plates. By the nature of their form, corrugated plates have flexural and membrane properties which are different parallel to and perpendicular to the corrugations. In structural analysis, a corrugated plate can be treated as an
116 equivalent flat plate of uniform thickness, but with different flexural and membrane stiffnesses in the two directions. This assumption has proven to be quite valid for analysing shell and plated structures in which the size of the corrugations are small compared with the overall dimensions of the structure. Early studies of corrugated plates were done by Blodgett (1934) and Wolford (1954), who developed formulas for the section properties of corrugated sheets. Comparisons of theory and tests on the flexibility, strength and buckling of shear diaphragms composed of corrugated sheets have been done by many researchers including Bryan and El-Dakhakhni (1968), Easley (1975) and Davies and Lawson (1978). Equivalent flexural, membrane and shear properties of corrugated plates were first developed by Timoshenko and Woinowsky-Kreiger (1959) for corrugations in the form of a sine wave. Theoretical formulas for the equivalent properties of corrugated plates have also been developed by Marzouk and Abdel-Sayed (1973), Trahair et al (1983) and Briassoulis (1986). Zhang and Rotter (1988) tested the validity of these theories for cylindrically curved shells. The behaviour of beams with trapezoidally corrugated webs and flanges of flat plate has been studied by a few researchers. Sherman and Fisher (1971) conducted tests to determine the minimum amount of web connection which would maintain the strength and stiffness of the webs under static load. The efficiency of beams with unstiffened and stiffened flat webs and corrugated webs was studied by Roth well (1985). Luo and Edlund (1996) studied the shear capacity of plate girders with corrugated webs by using a non-linear finite element method. Elgaaly et al (1996) studied the shear strength of beams with corrugated webs by comparing test results with a nonlinear finite element analysis. Elgaaly et al (1997) also studied the bending strength of beams with corrugated webs and concluded that, for design purposes, the moment capacity can be based on the moment capacity of the flanges alone. A theoretical and experimental investigation of the flexural-torsional buckling of beams with corrugated webs has been done by Lindner (1990). More recently, sections with trapezoidally corrugated webs and cold-formed rectangular hollow section flanges (University of Sydney 1997) have been used in the construction of large portal frame buildings with clear spans up to 60 m. Cold-formed C- and Z-section purlins are mainly used in building construction to support roof sheeting. These purlins are frequently required to carry high bending moments caused by wind load acting on the sheeting. Purlins in bending undergo complex buckling deformations which must be taken into account when calculating the design capacity of the purlin. The types of buckling modes that occur in purlins include local, distortional, flexural and flexural-torsional buckling. However, it appears that no research has been done on the effect of corrugations on the buckling behaviour of C- and Z-section purlins. The effect of corrugations is likely to be most pronounced for the local and distortional buckling modes because of the plate bending action inherent in these modes. In this paper, a cross-section analysis and finite-strip buckling analysis for thin-walled members with corrugated elements is described. The corrugations are in the form of a sine-wave and are aligned transversely to the longitudinal axis of the member. The effect of the corrugations on the cross-section analysis is taken into account by reducing the elastic modulus. In the finite strip buckling analysis, the effect of the corrugations is allowed for by altering the property matrix of the strip. The analysis is performed by the computer program THIN-WALL (1998), which has been developed to run on a micro-computer under Windows 95/98/NT. This program is used to investigate the buckling behaviour of C- and Z-section purlins with corrugated elements and to compare the results with those obtained for purlins without corrugations.
117 CORRUGATION THEORY A finite strip oriented in the z-direction with its corrugations oriented transversely is shown in Figure 1. The thickness of the sheet is t, the wavelength of the corrugations is b and the crest to valley depth is d. The major effect of the corrugations is to increase the transverse (x-direction) flexural stiffness and to reduce the longitudinal (z-direction) membrane stiffness. Other variations are minor and are mainly a result of the greater area of material on corrugated sheeting compared with flat sheeting. The equivalent membrane and flexural properties per unit projected width are given by (Trahair et al 1983) tx = t(l+7cV/4b^) tz = 2tV3d^ txz = t/(l+7tMV4b^)
(la) (lb) (Ic)
Ix = (tVl2)/(l+7iV/4b^) Iz = (dV8)(l+7rV/8b^) J = (tV3)(l+7i^d^/4b^)
(2a) (2b) (2c)
where I is the second moment of area and J is the torsion constant. y
Figure 1: Finite strip corrugated transversely
CROSS-SECTION ANALYSIS The cross-section analysis performed by THIN-WALL is described by Papangelis and Hancock (1995). In this analysis, a general matrix method is used to analyse the section properties and stresses in thinwalled cross-sections of any shape. The cross-section is subdivided into an assemblage of rectangular elements, with the ends of the elements intersecting at nodes. The analysis calculates the section properties, warping displacements, and the longitudinal and shear stresses caused by axial force, flexure and torsion. The longitudinal stresses are used to perform a finite strip buckling analysis of the section. The effect of the corrugations on longitudinal stress is allowed for in the cross-section analysis by reducing the elastic modulus of the corrugated elements to E(2/3)(t/d)^ as given by Eqn. lb, where E is the actual elastic modulus of the material.
118 FINITE STRIP BUCKLING ANALYSIS The finite strip method used in THIN-WALL is the same as that presented by Cheung (1976). The finite strip buckling analysis can be represented in matrix format by (3)
[K]{D}-MG]{D}=0
where [K] and [G] are the stiffness and stability matrices of the thin-walled member and X is the load factor against buckling under the initially assumed applied longitudinal stress used to assemble the matrix [G]. The values of X for which the determinant of the coefficients of {D} in Eqn. 3 vanishes are called the eigenvalues. The corresponding values of {D} are called the eigenvectors which are the buckling modes. The plate theory and displacement fields used to derive the stiffness matrix [K] are the same as those presented by Cheung. The plate theory used by Cheung is the orthotropic theory derived by Timoshenko and Woinowsky-Krieger (1959). The method for the calculation of the stability matrix [G] from the potential energy is described by Cheung. The stability matrix for flexural displacements was first presented by Przemieniecki (1973) and the stability matrix for membrane displacements was first developed by Plank and Wittrick (1974). In the finite strip buckling analysis THIN-WALL, the structure being analysed may be a folded plate system, a stiffened plate or a thin-walled structural member but it must be uniform in thickness in the longitudinal direction and simply supported at its ends. The longitudinal edges may be simply supported, clamped or free along the full length of the plate system. The analysis can be done for a number of different buckle half-wavelengths of the structure and the load factor and buckled shape are output for each length. Figure 2 shows the main window of THIN-WALL displaying a Z-section undergoing distortional buckling. ••ioixri
Be m wm-fsm-mmmMm^mmmMmmmm rOfim- - ' - • ' " - ' -
'-- -
,r—=====
"3Jr~3 °t/|g| H'''! ^I«^
'M M
i: I
M M
3!
M 3
ll i|
f^^'-^yjf,.''/-Xf,
Figure 2: Distortional buckling of Z-section
119 FINITE STRIP BUCKLING ANALYSIS FOR CORRUGATED PLATES The effect of corrugations in a plate can be accounted for in the finite strip buckling analysis by altering the terms in the property matrix [D] of the strip. The property matrix is used to derive the stiffness matrix [K]. The property matrix relates the bending moments Mx, Mz and torsional moments Mxz per unit length of plate to the curvatures px, pz and twist pxz of the plate, and the membrane normal stresses Ox, cjz and shear stresses Xxz to the axial strains 8x, 8z and shear strain yxz in the plate. It is represented in matrix notation by {a} = [D]{8}
(4)
[a] = {Mx Mz Mxz CTxCTzTxz}^
(5)
in which
(6)
{8} = {px pz Pxz Sx Ez Yxz}^
The property matrix for an isotropic strip is given by Et^' 12(l-v') vEt^ 12(1-v')
vEt" 12(l-v') Et^ 12(1-v')
0
0
0
0
0
0
0
0
0
0
Gt^ 12
0
0
0
E
vE
0
0
[D] =
0
0
(7)
0 0
0
G
where v is Poisson's ratio and G is the shear modulus. By the use of Eqns. 1 and 2, the property matrix for a corrugated strip is assumed to be given by
[D] =
Edna, 8
0
0
0
0
0
0
Et^ 12ai
0
0
0
0
0
0
0
0
0
0
0
Gt^t 12 0
Ettj
0
0
0
0
0
0 2Et' 3d^
0
0
0
0
(8)
0 a,
where ai = l+7i^d^/4b^ and a i = l+7r^d^/8b^. The property matrix used for the corrugations is assumed to be diagonal thus ignoring Poisson's ratio effects in corrugated sheeting.
120 EXAMPLES To illustrate the effect of corrugations on the buckling capacity of cold-formed purlins, examples of a C-section in compression and a Z-section in bending were analysed with THIN-WALL. The sections are of standard size with an overall depth of 152 mm, flange width 64 nmi, lip depth 15 mm, comer radius 5 mm, thickness 1.2 mm and a yield stress of 450 MPa. The webs of the sections have transverse corrugations of wavelength b = 20 mm and a depth d which varies from 0-5 mm. Figures 3 and 4 show graphs of dimensionless buckling capacity versus buckle half-wavelength for the two examples, where Nc and Ny are the buckling and yield loads, and Mc and My are the buckling and yield moments. The graphs generally show two distinct minimum points. The first minimum occurs at low lengths and represents local buckling of the section. This buckling mode consists of deformation of the web, flange and lip elements without movement of the line junctions between the flange and web and the flange and lip stiffener. The second minimum occurs at intermediate lengths and represents distortional buckling since movement of the line junction between the flange and lip stiffener occurs without a rigid body rotation or translation of the cross-section, as shown in Figure 2. At long lengths, the C-section buckles in a flexural mode while the Z-section buckles in a flexural-torsional mode. The effect of the transverse corrugations in the web is to increase the local and distortional buckling capacities of the purlins and to reduce the flexural and flexural-torsional buckling capacities. The increase in capacity is more pronounced for the C-section in compression, where the local buckling capacity has increased by over 300% and the distortional buckling capacity by over 100%. The reason for this increase is because these buckling modes involve out-of-plane bending of the web, which has an increased transverse flexural stiffness because of the corrugations. The slight reduction in the flexural and flexural-torsional buckling capacities is due to the lower out-of-plane flexural stiffness of the purlin because of the transversely corrugated web.
' ' - • " •
d=0 1 d=1M — d = 2 n d = 3 11
If.
III /?' •••' \
M rv < K\\\'
•a 1.4
O)
\
o c
m (0 0)
iJ c o •<5
c
d=4 M
\
(0
0.8
\
41
\ \ i '
In
III
r
0.6
/
d=5 N
m\
14f f
\
W\
0)
E 0.4
rfu
0.2
10
100
1000
Buckle Half-Wavelength (mm)
Figure 3: C-section in compression
\ V 10000
121
100
1000
10000
Buckle Half-Wavelength (mm)
Figure 4: Z-section in bending
CONCLUSIONS A finite strip buckling analysis has been described for the elastic buckling of thin-walled members with corrugated elements. The analysis involves alteration of the property matrix in a conventional finite strip analysis to account for the corrugations in a strip. The analysis is performed by the computer program THIN-WALL, which can predict the local, distortional, flexural and flexural-torsional buckling modes of a section. The program was used to investigate the buckling behaviour of cold-formed C- and Z-section purlins with transverse corrugations in the web. The effect of the corrugations is to increase the transverse flexural stiffness and to reduce the longitudinal membrane stiffness of the web. This results in a significant increase in the local and distortional buckling capacities of the purlins and a slight reduction in the flexural and flexural-torsional buckling capacities.
REFERENCES Blodgett H.B. (1934). Moment of Inertia of Corrugated Sheets. Civil Engineering 4:9,492-493. Briassoulis D. (1986). Equivalent Orthotropic Properties of Corrugated Sheets. Computers and Structures 23:2, 129-138. Bryan E.R. and El-Dakhakhni W.M. (1968). Shear Flexibility and Strength of Corrugated Decks. Journal of the Structural Division, ASCE 94:11, 2549-2580. Cheung Y.K. (1976). Finite Strip Method in Structural Analysis, Pergamon Press, New York.
122 Davies J.M. and Lawson R.M. (1978). The Shear Deformation of Profiled Metal Sheeting. International Journal for Numerical Methods in Engineering 12:10, 1507-1541. Easley J.T. (1975). Buckling Formulas for Corrugated Metal Shear Diaphragms. Journal of the Structural Division, ASCE 101:1, 1403-1417. Elgaaly M., Hamilton R.W. and Seshadri A. (1996). Shear Strength of Beams with Corrugated Webs. Journal of Structural Engineering, ASCE 122:4, 390-398. Elgaaly M., Seshadri A. and Hamilton R.W. (1997). Bending Strength of Steel Beams with Corrugated Webs. Journal of Structural Engineering, ASCE 123:6, 772-782. Lindner J. (1990). Lateral-Torsional Buckling of Beams with Trapezoidally Corrugated Webs. Proceedings, 4th International Colloquium on Stability of Steel Structures, Budapest, Hungary, 79-82. Luo R. and Edlund B. (1996). Shear Capacity of Plate Girders with Trapezoidally Corrugated Webs. Thin-Walled Structures 26:1, 19-44. Marzouk O.A. and Abdel-Sayed G. (1973). Linear Theory of Orthotropic Cylindrical Shells. Journal of the Structural Division, ASCE 99:11,2287-2306. Papangelis J.P. and Hancock G.J. (1995). Computer Analysis of Thin-Walled Structural Members. Computers and Structures 56:1, 157-176. Plank R.J. and Wittrick W.H. (1974). Buckling Under Combined Loading of Thin, Flat-Walled Structures by a Complex Finite Strip Method. International Journal for Numerical Methods in Engineering 8, 323-329. Przemieniecki J.S. (1973). Finite Element Structural Analysis of Local Instability. American Institute of Aeronautics and Astronautics Journal 11, 33-39. Rothwell A. (1985). On the Efficiency of Stiffened Panels, with Application to Shear Web Design. Aspects of the Analysis of Plate Structures, Edited by Dawe D.J., Horsington R.W., Kamtekar A.G. and Little G.H., Clarendon Press, Oxford, 105-125. Sherman D. and Fisher J. (1971). Beams with Corrugated Webs. Proceedings, 1st Specialty Conference on Cold-Formed Steel Structures, St Louis, USA, 198-204. THIN-WALL (1998). Cross-Section Analysis and Finite Strip Buckling Analysis of Thin-Walled Structures, Centre for Advanced Structural Engineering, University of Sydney. Timoshenko S.P. and Woinowsky-Krieger S. (1959). Theory of Plates and Shells, 2nd Edition, McGraw-Hill, USA. Trahair N.S., Abel A., Ansourian P., Irvine H.M. and Rotter J.M. (1983). Structural Design of Steel Bins for Bulk Solids, Australian Institute of Steel Construction, Sydney. University of Sydney (1997). Design Bending and Shear Capacities of Industrial Light Beams. Investigation Report SI 113, Centre for Advanced Structural Engineering. Wolford D.S. (1954). Sectional Properties of Corrugated Sheets Determined by Formula. Civil Engineering 24:2, 59-60. Zhang Q. and Rotter J.M. (1988). Equivalent Orthotropic Properties of Cylindrical Corrugated Shells. Proceedings, 11th Australasian Conference on the Mechanics of Structures and Materials, Auckland, New Zealand, 417-421.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
123
COMPRESSION BEHAVIOUR OF PERFORATED STEEL WALL STUDS Jyrki Kesti and Pentti Makelainen Laboratory of Steel Structures, Helsinki University of Technology, P.O.Box 2100, FE^-02015 HUT, Finland
ABSTRACT Li this paper, the compression behaviour of web-perforated steel wall studs is studied. The series of compression tests on studs with a length of 800mm were performed between fixed ends. The web perforation decreases distortional buckling strength of the section. The tests showed, however, that even a small restraint given by the perforated web has an influence on compression capacity. An analytical prediction for compression capacity was performed. The elastic distortional buckling strength was determined by replacing the perforated web part with plain plate of the same bending stiffness and the ultimate capacity was determined by using design curves given in design recommendations. The comparison showed that the method used gives reasonable results for Csections but overestimates the compression capacity of sigma-sections. KEYWORDS Cold-formed steel, wall stud, perforation, compression, test, distortional buckling INTRODUCTION Web-perforated steel wall studs are especially used in the Nordic countries as structural components in small housing. The slotted thermal stud offers a considerable improvement in thermal performance over the solid steel stud. The wall structure consists of web-perforated C- or sigma-sections as studs and U-sections as tracks and, e.g. gypsum wallboards attached to the stud flanges. The sections investigated in this paper are shown in Figure 1. Both stud types had six rows of slots. The perforation reduces the bending stiffness of the web causing decreased distortional buckling strength. The aim of this paper is to investigate the local and distortional buckling behaviour of the perforated steel stud. A series of column tests were performed for the perforated studs. Reference tests were also carried out using only the flange parts of the stud to see the restraint behaviour of the perforated web with respect to distortional buckling mode. The ultimate compression capacity is determined using design codes, such as Eurocode 3 (1996) and the Australian and New Zealand Standard (1996).
124
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Mi Figure 1: Web-perforated sigma-section and web-stiffened C-section (Dimensions in mm) COMPRESSION TESTS Test Specimens The tests were performed on sigma-sections and web-stiffened C-sections, whose webs were perforated. 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 cut away. The latter arrangement was used to investigate the influence of the perforated web on the compression resistance of the section. The sections were labelled as CS-1.3-# for sigma-sections with a nominal thickness of 1.3mm and CC-1.2-# and CC-1.5-# for web-stiffened C-sections with a thickness of 1.2mm and 1.5mm, respectively. For the code, #, W was used for whole sections and F for sections with only flange parts. The ordinal 1 or 2 was also added if there were two identical tests. The dimensions of the specimens using the nomenclature defined in Figure 2 are shown in Tables 1 and 2. The mid-line dimensions are the averages of the measured values at both ends and of two identical specimens. TABLE 1 MEASURED SPECIMEN DIMENSIONS FOR SIGMA-SECTIONS b c e a 1 Area A | bs [mm] [mm] [mm] [mm] [mm] [mm] [mm^] CS-1.3-W 145.4 12.4 51.9 33.5 13.8 9.0 278.7 12.9 33.7 51.1 15.5 11.9 1 283.0 1 1 CS-1.3-F 1 146.0
1^
TABLE 2 MEASURED SPECIMEN DIMENSIONS FOR WEB-STEFFENED C-SECTIONS Area A a d h b e f c [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm^l 302.7 CC-L2-W 16.2 22.4 173.6 49.5 23.8 9.3 11.9 12.1 300.7 173.5 23.0 CC-1.2-F 8.9 22.6 49.5 16.1 376.2 10.7 16.4 8.5 22.7 CC-1.5-W 173.8 49.5 23.0 11.3 1 373.3 22.2 CC-1.5-F 8.3 22.7 173.5 49.8 15.6
Figure 2: Definition of symbols for sigma-sections and web-stiffened C-sections
125 Material Properties The material of the test specimens was S350GD+Z250 (EN 10147) with a nominal yield strength of 350N/mm^. The material properties of each series were determined by tensile coupon tests. Four longitudinal coupons were tested for each series. The coupons were cut out from the centre of the flange plates of the finished specimens. The coupons were prepared and tested according to the EN10002-1 standard. The mean values of the test results are shown in Table 3. TABLE 3 MEASURED MATERIAL PROPERTIES Tensile Core Yield stress Modulus of A; strength fu Thickness elasticity E fy [N/mm^] [%] [N/mm^] [N/mm^] [mm] 481 13.2 CS-1.3 1.25 379 203266 CC-1.2 386 490 1.15 200455 15.1 492 380 14.3 CC-1.5 1.47 204167 Ag : Percentage total elongation at maximum force minus elastic elongation A : Percentage total elongation at fracture minus elastic elongation
A* [%] 19.8 23.4 23.1
Column Test Arrangement All the test specimens were tested in fixed-end conditions. The fixed-end conditions were arranged by casting each end of the specimen in concrete. The free length of each specimen between the concrete blocks was 800mm, when minor axis flexural buckling is not yet critical. The columns with concrete blocks were centred in a 500kN hydraulic testing machine equipped with a lockable plate at one end. The lockable plate allowed the ends of the specimen and the loading plates to bed-in, thus ensuring full contact between the end bearing and test specimen. The loading rate used was 4kN/min., corresponding to about 10.5 - 13.5N/mm^ stress. The set-up for the column tests is shown in Figure 3. The displacements of the specimens were measured using linear displacement transducers around the sections. One displacement transducer measured the axial shortening of the specimen. gid end platten Concrete block
Lateral displacement measurement
Concrete block Lockable end platten
Figure 3: Test set-up for column tests Column Test Results Both flange parts of the test specimens, whose perforated part were removed, independently failed in the flexural-torsional buckling mode. A significant post-buckling reserve of strength was observed, especially in the buckling mode of the CS-1.3-F specimens. The failure loads of these specimens are
126 shown in Table 4. The failure loads for whole specimens are shown in Table 5. These sections failed in the distortional buckling mode, which was naturally almost similar to the buckling mode failure of the specimens whose perforated parts were removed. In the case of web-stiffened C-sections, CC-1.2-W and CC-1.5-W, one specimen failed such that the lips buckled inwards and the other identical specimen such that lips buckled outwards. In the latter case, the capacity was considerably lower. The restraint of the perforated web with respect to distortional buckling mode was found to have some importance. The failure loads of the whole section CS1.3-W section was approximately 12% higher than those of the CS1.3-F sections without the web part. In the case of web-stiffened C-sections, CC1.2 and CC1.5, the difference was about 10% with the lips of the whole section failing outwards and 27% with the lips failing inwards. Nonlinear FE analyses were also performed using the NISA application [1996] to gain a better understanding of the behaviour of perforated sections. Full details of the FE analysis are not given here due to the space limitation. The sections were modelled completely, including perforations and an initial imperfection magnitude of IVIOOO was used in the analysis. The initial imperfection was given by applying a small force to the tip of the stiffener. The force was applied both inwards and outwards for sections CC-1.2 and CC-1.5, leading to different ultimate strength. The FE analysis results shown in Tables 4 and 5 indicate good correlation with the test results. TABLE 4
1
1
TEST RESULTS FOR "FLANGE PART" SECTIONS Test 1 Failure T FEM Lips failure [kN] mode direction specimen load[kN] CS-L3-F-1 49.7 53.1 inward CS-L3-F-2 52.0 53.1 inward inw.+outw. CC-L2-F 58.0 61.1 inward | CC-L5-F 1 76.2 1 76.1
TABLE 5 TEST RESULTS FOR WHOLE SECTIONS Lips failure Test specimen 1 Failure 1 FEM mode direction load[kN] [kN] inward CS-1.3-W-1 55.7 1 59.6 inward CS-1.3-W-2 58.2 59.6 outward CC-L2-W-1 64.4 66.9 inward CC-1.2-W-2 73.5 76.7 CC-1.5-W-1 96.2 inward 98.9 outward CC-L5-W-2 1 83.1 1 87.1
ANALYTICAL METHODS Elastic Buckling Stress Because of perforation of the web, the transverse bending stiffness of the section is very low and the section is very sensitive to distortional buckling under compression. In the distortional mode of buckling, the edge-stiffened flange elements of the section deform by rotation of the flange about the flange-web junction. The distortional buckling mode occurs at longer wavelengths than local buckling. The distortional buckling modes are shown for the perforated sigma-section in compression in Figure 4. When the stiffness of the web area is very low, the buckling mode of each flange part approaches pure flexural-torsional buckling.
127
Figure 4: Asymmetric and two symmetric distortional buckling modes for sigma-sections The distortional buckling stresses for the sections were determined using the GBT computer program written by Davies and Jiang (1995), which is based on the generalized beam theory (GBT). A unique feature is that GBT can separate and combine individual buckling modes and their associated load components. In the analysis, the perforated web part was modelled as a plain plate with reduced thickness corresponding to the same bending stiffness of perforated web part. The bending stiffness of the perforated part, determined by FEM, is 4% for CS-section and 6% for CC-sections of stiffness of the plain plate. The buckling stress versus column length under the pin-ended boundary conditions is shown in Figure 5 for the CS-1.3 section (using dimensions of the whole test specimen CS-1.3-W) and in Figure 7 for the CC-1.2 section. Figures 6 and 8 show buckling behaviour for the fixed-end condition as used in the tests. Local buckling modes were omitted in these figures. Three curves are presented in each figure. The lowest curve is for purely the flange part. The second curve is for sections with reduced thickness in the web and the highest curve is for comparison for the full section without perforation. OUU ] !s 500 E E 2 400
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Figure 7: Elastic buckling stress for section CC-1.2, pin-ended condition
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Figure 6: Elastic buckling stress for section CS-1.3, fixed-ended condition R.600 E E 500
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128 Design Recommendations Cuitent design recommendations do not include the design of perforated sections. In this research, the elastic buckling stresses of perforated sections were determined using numerical methods and the ultimate compression resistance was determined using the different design methods presented in design recommendations and literature. Eurocode 3 (1996) includes several alternative "column curves" for different buckling modes and cross-section types. The design strength in torsional and torsional-flexural buckling is usually based on curve b or c with a = 0.34 or a = 0.49, correspondingly. Distortional buckling is taken account of by reducing the thickness of the stiffeners, not the whole section. The reduction is based on curve ao with a = 0.13. In each case, the local buckling should also be considered by using the effective width approach. In this research, the perforated web was ignored and the plate element between the perforated part and plain part was assumed as an edge element when effective widths were determined. In the Australian and New Zealand Standard (1996), the distortional buckling strength is checked by using separate design curves. Another "direct" design method for distortional buckling has been proposed by Schafer and Pekoz (1998). In their paper, the method was used for flexural members. This method has been presented as an alternative to the effective width approach. In the Schafer model, the reduction factor applies to the entire section instead of to separate plain elements. The well-known Winter approach has been used with the following reduction: p = l when A<0.673 p = (l-0.22/A)/A
when A > 0.673
where
/„=min[(/„)^,«,(/„X The term (fcr)f is local buckling stress based on a buckling coefficient value of k=4.0 and (fcr)d is distortional buckling stress. The reduction factor for distortional buckling stress is as follows: R^ = 1 when Xd < 0.673 1.17 R. = -^ + 0.3 when 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
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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
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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
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1200 1 [mm]
Figure 4: Experimental results related to specimens with and without perforations
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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
(3)
>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
This Page Intentionally Left Blank
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
|
—©—
1 o G
•—— *——«— B—
glOQ>
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
/
/
/
/
J
1 ^— \
\
\
! \
\
\ X
A
1 1 I^Euler . 1 1 Li 2Li 3Li 4Li L Figure 4: Dependence of the critical buckling stress of an I-section on the column length ^
A geometric imperfection with the shape of the first plate buckling mode is used with an amplitude of bf/125 at the free flange end. Further calculations show the influence of residual stress to be negligible. Making use of the symmetry in longitudinal direction, two extremes for the loaded edge are investigated. In the first case the edge can be freely deformed in longitudinal direction and the applied load follows as a slack load bundle. In the other case the edge is constrained to remain straight. Figure 5 shows the strong influence of this boundaiy condition on the ultimate load Nu in dependence of the non-dimensional plate buckling slendemess X. Figure 6 demonstrates the corresponding lines of equal normal stress ax in plate midsurface along the length. The tensile stress is anchored at the free flange ends in the case of straight edges. Of course, the condition of equilibrium is only retained by forces through the constraint equation. In our tests (see Table 1) the solid end plates fulfil this condition. They are capable of keeping the balance between the outer load pressure and the resulting internal tensile stresses, which only occur at the area of load application. The test results can also be seen in Figure 5. Longer columns exhibit local buckling patterns that consist of series of half-wave segments. At the transitions from one segment to the other, the continuity has to be retained and the tensile stress of each half-wave can be anchored in its direct neighbours. Contrary to this, continuity cannot be retained with the edges that are free to be deformed. Thus, a column of a length of two half-waves would have a wedge-shaped gap in the middle of the flanges which is not possible, see Figure 7. A model with deformable loaded edges cannot reflect the behaviour of a longer column sufficiently well.
160
0.5
1.0
1.5
2.0 /I
=4fy'^cr
Figure 5: Dependence of the cross-section capacity on the loaded edge conditions It has to be considered that the effective widths equations were derived from stub column tests, (Kalyanaraman et al. (1977)). The stub columns were tested between end plates which have not been welded onto the ends of the sections. Consequently, the tensile stress cannot be anchored at the column ends for these stub columns. COMPRESSIVE STRESS lELD STRESS
COMPRESSIVE STRESS ^YIELD STRESS
TENSILE STRESS
STRAIGHT LOADED EDGE
DEFORMABLE LOADED EDGE
Figure 6: ax-normal stress distribution at different loaded edge conditions Also the FEM-model with deformable edges does not reflect a stub-column test because the load follows the edge deformation. The load is not a slack load bundle in the test machine. If tensile strains are present, a gap will open between the column and the end plate. This has only little influence on columns where the web is more slender than the flanges. In the case where the flanges are more slender than the web a flexural buckling-type failure occurs at the outer edges of the flanges. This explains why the FEM-results remain below the Winter curve in Figure 5.
161
Figure 7: Deformation pattern at different loaded edge conditions Fischer et al. (1996) and Brune (1998) confirmed the values for the effective widths with an extensive FEM parameter study on single plates with straight loaded edges. But they did not take into account the continuity. In these analyses a loaded edge of a three-sided simply supported plate is kept straight under constant central loading ("static loading"), but allowed to rotate. If two plates were joined together, there will again evolve a gap, this time enclosed by straight lines. In addition Fischer et al. (1996) investigated the loading caused by the edge displacement ("geometric loading"). The integral about the non-linear stress distribution always results in a normal force plus a bending moment for the three-sided simply supported plate. However, Figure 8 shows the moments of the right and left flangehalves to neutralize. Thus, in tiiis case static and geometrical loading merge into one another.
rjj
u[nnm]
r1
q[^ J/mm]
Oy M=0 L
•
actio
Me ^ reactio
equd c ictio
Figure 8: Static and geometrical load application in the flanges of an I-section The stub column tests of Chick and Rasmussen (1995) can also confirm the effect described above. Two welded I-sections fabricated from the same steel plate were tested. Table 2 shows the ultimate test load to be 15 percent higher in case of solid plates welded to the flanges at the ends compared to those columns without the plates. The evaluation of the Winter formula" Eqn. (1) shows good correlation with the ultimate load without the plates. However, the load-bearing capacity with plates is in sufficient agreement with the FEM-calculation of the cross-section capacity considering of a straight loaded edge. TABLE 3 STUB COLUMN-TESTS WITH AND WITHOUT END PLATES (CHICK AND RASMUSSEN (1995))
Specimen 800-SC2
|800^sc3
b
h
t
fy
fmml
fmml
fmml
fN/mm^l
240
240
5
435
End Plates ?
No Yes
Nu,tcst
Nu.Wintcr
Nu.FEM
rkNl
rkNl
fkNl
911,5
1066
964 1095
162 CONCLUSION Thin-walled I-profiles are found to buckle locally in a distinct half-wave pattern. The half-wave length is the length corresponding to the first minimum of the critical buckling stress. The free flange ends show tensile stresses in the post-critical range. The tensile stresses have to be anchored at the ends of the member. In tests and in practical applications this can be achieved with stiff end plates. Effective widths equations are derived from stub column tests, where no anchorage of the tensile stress is possible because of the test rig. Thus, a gap will open instead of building up tensile strain. If the length of a column is a multiple of a stub column, this effect will not occur. This is the reason for the deviation of the pure cross-section capacity as determined v^th the effective widths from the test member capacity including overall stability influence. In view of the tensile stress anchorage a FEMcalculation can reflect the load-carrying capacity much better. The tests on short columns show only little effect on the stress distribution and the buckling pattern due to measured imperfections. Thus, there is hardly no centroid shift caused by these measured imperfections. A design concept that takes into account an unintentional eccentricity also for short columns, therefore may underestimate the load-carrying capacity of thin-walled I-section columns. ACKNOWLEDGEMENTS The authors thank the Deutsche Forschungsgemeinschaft DFG for fiinding. We are grateful to the Salzgitter AG for the generous gift of steel plates. REFERENCES AISI (1968). Specification for the Design of Cold-formed Steel Struct. Members, Washington, USA AISI (1986). Specification for the Design of Cold-formed Steel Struct. Members. Washington, USA Brune B. (1998). Die dreiseitig gelagerte Platte in der Methode der wirksamen Breiten. Stahlbau 67:11,851-863. Chick CO. and Rasmussen K.J.R. (1995). Research Report No. R717: Tests of Thin-walled I-sections in Combined Compression and Minor Axis Bending, Part II - Proportional Loading Tests. School of Civil and Mining Engineering, University of Sydney, Australia DIN 18 800-2(1990). Stahlbauten - Stabilitdtsfdlle, Knicken von Staben und Stabwerken. BeuthVerlag, Berlin, Germany EC 3 Part 1.3 (1996), ENV 1993-1-3 General Rules, Supplementary Rules for Cold-formed Thin Gauge Members and Sheeting, CEN, Brussels, Belgium Fischer M. and Konowalczyk R. (1988). Traglastversuche an langsgestauchten unversteiften dreiseitig gelagerten Rechteckplatten. Stahlbau 57:5, 135-141. Fischer M., Zhu J., Priebe J. (1996). The Method of Effective Width used for Bars with Thin-walled Cross-sections - Remarks on Insufficiencies and Improvements. Proceedings of the 2^^ Intern. Conf of Coupled Instabilities in Metal Structures, Imperial College Press, London, UK Kalyanaraman V., Pekoz T., Winter G. (1977). Unstiffened Compression Elements. Journal of Structural Division, ASCE 103:9, 1833-1848. Kalyanaraman V., Pekoz T. (1978). Analytical Study of Unstiffened Elements. Journal of Structural Division, ASCE 104:9, 1507-1524. Lindner J. and Rusch A. (1998). Stutzen mit diinnwandigen Querschnittsteilen im Bereich geringer Schlankheiten; I. Zwischenbericht DFG-Forschungsvorhaben Li-351-15/1, Fachgebiet Stahlbau, TU Berlin, Germany
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
163
ULTIMATE LOAD OF ORTHOTROPIC THIN-WALLED BEAM-COLUMNS Slawomir Kedziora, Katarzyna Kowal-Michalska, Zbigniew Kolakowski Department of Strength of Materials and Structures, Technical University of Lodz, PL - 90-924 Lodz, ul. Stefanowskiego 1/15, Poland
ABSTRACT The investigation is concerned with open and closed cross-section columns under the eccentric compression. The asymptotic Koiter's method is employed in the form of the numerical transition matrix method for the elastic range. The numerical analysis allows to investigate the influence of the material orthotropy of columns upon all buckling modes from global (flexural, flexural-torsional, lateral, distortional and their combinations) to local and upon the uncoupled post-buckling state. In the solution obtained the transformation of buckling modes with the increase of load and shear lag phenomenon is included. For short columns the post-buckling behaviour of walls in the elasto-plastic range is analysed on the basis of Hill Yield Criterion and Prandtl-Reuss equations. As the result of analytical-numerical solution the ultimate load of a considered column can be found out.
KEYWORDS thin-walled structures, orthotropy, non-linear stability, elastic and elasto-plastic range
INTRODUCTION Thin-walled structures made of fibrous composites find more and more frequent use in civil engineering designs (e.g. vessels), aircraft stififeners as well as in the construction of space vehicles. The increasing popularity of fibrous composites as structural materials can be attributed to the many reasons, e.g.: • high strength-to-weight and stiffiiess-to-weight ratios, • high resistance of some fibrous composites to aggressive chemical compounds and to corrosions. Generally, fibrous composites are non-homogenous and anisotropic. However, fibres are often laid in composite matrices in one or in two perpendicular directions; in such cases the fibrous composite is usually considered as an orthotropic material with a selected orthotropy factor.
164 In the present paper the analysis will be restricted to composites with orthotropic matrix, the fibres being evenly distributed across the structural component (plate, beam wall) along two directions perpendicular to each other but parallel to the principal orthotropy directions of the matrix. Such an orthotropic material with a given orthotropy factor will fiirther be deah with in macro-scale as a homogenous. The macrostructure of composite will be characterised by its mechanical and strength properties in the elastic and plastic range. Thin-walled composite structures, especially columns and beams may have many buckling modes and are able to sustain load after local buckling. The determination of their load carrying capacity requires consideration of the nonlinear analysis of stability in the elastic-plastic range. In the present paper the post-buckling behaviour of thin-walled composite structures in the elastic range being under eccentric compression is examined on the basis of Koiter's method. The study is based on the numerical method of the transition matrix (by Unger (1969)) using Godunov's orthogonalization (Biderman (1977)). The most important advantage of this method is that it enables to describe a complete range of behaviour of the thin-walled structures from all global (flexural, flexuraltorsional, lateral, distortional and their combinations) to local stability for uncoupled buckling analysis. In the solution obtained, the co-operation between all the walls of structures being taken into account, the transformation of buckling modes with the increase of load, the shear lag phenomenon and also the effect of cross-sectional distortions are included. Further the obtained elastic post-buckling solution is employed to investigate the behaviour of the orthotropic columns in the elasto-plastic range. The solution of non-linear elasto-plastic problem is reached by an incremental and iterative procedure on the basis of Rayleigh-Ritz variational principle involving Hill's Yield Criterion and plastic flow theory. As a result of numerical calculations the ultimate load of orthotropic columns of different geometry and material properties can be found out.
STRUCTURAL PROBLEM EV THE ELASTIC RANGE The considerations concern a prismatic beam-column, of the length /, whose flat walls are treated as thin-walled homogenous orthotropic rectangular plates. These rectangular plates, of principal axes of orthotropy parallel to their edges, are connected along their longitudinal edges and form a beamcolumn. A plate model is adopted for the beam-columns (Fig.l). For the i-th plate component more precise geometrical relationships are assumed in order to enable the consideration of both out-of-plane and inplane bending of each plate:
^"=V.y+l(W?,y+U^ ^^Zy
= Yixy = Ui,y + ^i,^ + Wi ,,Wi y.
Physical relationships for the i-th wall are formulated in the following way:
(1)
165
1-TliVf Miy = ^ A ( V i K i , + K i y ) , *'
1-TliV?
Nixy = GihiY™y = 2Gihi8:^,
(2)
M i ^ = D^Ki^.
where: E _
'
V »y _
E:
»yx
V:
Zi
'
Vi+i
Fig. 1: A segment of a structure cross-section with local coordinates systems The differential equilibrium equations resultingfromthe virtual work principle and corresponding to expressions (1) for the i-th plate can be written as follows: NiX,X + N i ^ y +(NiyUi,y)y = 0,
Nixy,x+Niy,y+(Ni,Vi,J^=0, (NixWi, J,^ +(NiyWi^y) +(Ni^Wi J
(3) +(Ni^Wi y)^ +Mi, „, +Miy yy +2Mi^ ,^ = 0.
The solution of these equations for each plate should satisfy kinematic and static continuity conditions at the junctions of adjacent plates (Kolakowski et al. (1998)) and the boundary conditions referring to the free support of the structure at its both ends. The non-linear elastic problem is solved by Byskov and Hutchinson asymptotic method (1977). Displacement fields U, and sectional force fields, N, are expanded in power series in the buckling mode amplitude, ^ (divided by thickness of the first component plate):
166
The expression for the total elastic potential energy corresponding to the non-linear equation of equilibrium for uncoupled buckling mode with regard to the imperfection of buckling mode with amplitude C, has the following form:
Ve=
-a^>i^/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—
0.6 0.5
a*
0.4 0.3
,
•
Tl-1
•
T|—U.J
0.2 0.1
0.5
1.5
Fig. 2: Influence of elastic orthotropy factor on the L-S curves for a cubic column of b/h =80 At the beginning the numerical calculations for cubic column uniformly compressed have been performed. As a result the full L-S (load versus shortening) curves presenting average stress a*=aav/cJo as a function of nondimensional strain S*=SxE/ao are obtained (Fig.2). The ratio width to thickness of a component plate is equal to 80 and the material properties are taken fi-om the works of Owen and Figueiros (1983): in the elastic range: E=Ex = r|Ey; Ee = Ey= 30000 MPa; G=10000MPa; Vyx=0,3; in the plastic range: EXP= Ey^ =EeP=3000 MPa; G^ =1000 MPa; initial yield limits: ao=axo=ayo =aeo=30 MPa; TO=17,32 MPa; The results of numerical calculations for orthotropic columns of different geometry will be presented during the Conference.
ACKNOWLEDGMENTS The paper has been carried out in the range of research project supported by National Research Committee (KBNNo. PB 251/T07/97/12).
References 1 Biderman, B. L. (1977). Mechanics of thin-walled structures - Statics; Moscou, Mashinostroenie, pp.488, An Russian/. 2 Byskov, E. and Hutchinson, J. W. (1977). Mode interaction in axially stiffened cylindrical shells; AIAAJ., 15,7,pp.941-948.
170 3 Camotim, D. and Prola, I. C. (1996). On the stability of thin-walled columns with Z, S and sigma sections; Proc. of Second Int. Conf on Coupled Instability in Metal Structures, Imperial Press College, pp. 149-156. 4 Dubina, D. (1996). Coupled instabilities in bar members - general report; Proc. of Second Int. Conf on Coupled Instability in Metal Structures, Imperial Press College, 119-132. 5 Graves-Smith T.R. (1968). The ultimate strength of thin-walled colunms of arbitrary length, ThinWalled Steel Structures, Crosby Lockwood 6 Gradzki, R. and Kowal-Michalska K. (1988) Collapse behaviour of plates. Thin-walled Structures, 6 7 Hill, R. (1948). A theory of yielding and plastic flow of anisotropic metals, Proc.R.Soc.Lon. Ser.A, 193,281. 8 Koiter, W. T. (1976). General theory of mode interaction in stiffened plate and shell structures; WTHD Report 590, Delft, pp.41. 9 Kolakowski, Z. and Krolak, M. (1995). Interactive elastic buckling of thin-walled closed orthotropic beam-columns; Engineering Transactions, 43, 4, pp.571-590. 10 Kolakowski, Z., Krolak, M., Kowal-Michalska, K. (1998). Modal interactive buckling of thinwalled composite beam-columns regarding distortional deformations. Int. Journal of Eng. Sci., (to be published) 11 Krolak, M. and Kolakowski, Z. (1995). Interactive elastic buckling of thin-walled open orthotropic beam-columns; Engineering Transactions, 43, 4, pp.591-602. 12 Little, G.H. (1977), Rapid analysis of plate collapse by live energy minimisation. Int. J.Mech. Sci., 19 13 Owen, D.J.R., Figueiras., J.A. (1983), Elasto-plastic analysis of anisotropic plates and shells by the semiloof element, Int.J.for Num.Meth.in Engng., 19. 14 Owen, D.J.R., Figueiras., J.A. (1983), Anisotropic elasto-plastic finite element analysis of thick and thin plates and shells. Int. J. for Num. Meth. in Engng., 19. 15Unger, B. (1969). Elastisches Kippen von beliebig gelagerten und aufgehangten Durchlauftragem mit einfachsymmetrischen, in Tragerachse veranderlichem Querschnitt und einer Abwandlung des Reduktionsverfahrens als Losungsmethode; Dissertation D17, Darmstadt.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
171
COMBINED AXIAL LOAD AND VARYING BENDING MOMENT IN BEAM COLUMNS Jim Rhodes Department of Mechanical Engineering, University of Strathclyde Glasgov^Gl IXJ, Scotland
ABSTRACT The interaction equations used in some design specifications to deal with the behaviour of beam columns under combined bending and compression are outlined in this paper, and some comparison is made between the predictions of each code. It is found that there can be substantial differences between the interaction diagrams produced on the basis of the equations used in different specifications, particularly in the case of relatively slender members under substantially varying moment. The results of a short series of tests on small lipped channel section columns under various degrees of eccentric loading are reported, and these are compared with the predictions of different design codes for this type of loading. The predictions of the AISI Cold Formed Steel Specification, the British Standards for cold formed steel and for hot rolled steel and EC3: Part 1.3 were compared with the experimental results, and all of these would appear to be reasonably accurate on the whole in relation to the experimental results obtained despite showing substantial differences in the interaction curves.
KEYWORDS Beam Columns, bending, axial load, interaction, buckling, eccentricity, channel sections, failure.
INTRODUCTION The behaviour of beam columns has been subjected to a substantial degree of research, as illustrated in Refs. (1) and (2). The interaction of bending moment and axial compressive loading has been investigated by a number of researchers, and a wide variety of different interaction equations have been derived to take account of the complexities which arise when beam columns of various cross sections, with varying degrees of compactness, are subjected to combinations of axial load and moments which may vary to any specified degree along the length of the member. As part of a calibration exercise between the Eurocode, EC3: Part 1.3 (3), against the British cold formed steel specification (4) the rules dealing with the interaction of axial loading and bending
172 moment on columns were compared and found to give substantially different results in some cases. The differences seemed to be greatest in cases when the moment varies along the column. The interaction formulae used in the British code were largely taken from those of the AISI specification (5), and if the safety factors etc. are discarded then there is not a great deal of difference in these two codes. The Eurocode interaction formulae were taken from the corresponding rules in the parent Eurocode 3, Part 1.1 (6), and had been initially set up for hot rolled steel columns. Examination of other design specifications, such as the European Recommendations and the British Aluminium code showed that these used yet different interaction formulae. Perhaps of prime interest to the writer was the fact that the British Standard for hot rolled steelwork (7) had also different interaction equations. The formulae given in the AISI specification, The British cold-formed steel specification, the British hot rolled steel specification and the Eurocode are now considered and compared
DESIGN FORMULAE The interaction formulae used in the different specifications are subject to different conditions with regard to load and safety factors etc. The Eurocode and the British standards, being limit analysis based, calculate load capacities without safety factors, although the Eurocode uses a material factor of 1.1 rather than unity as is the case with the British code. The AISI specification, on the other hand, has safety factors incorporated. Thus to perform a meaningfiil comparison of the different codes, the safety factors, load and material factors must be taken into consideration. In the investigation discussed here it was decided that all factors would be taken out of the formulae used, so that all specification formulae would be taken, with the appropriate modifications where required, to be appUcable directly to the evaluation of the ultimate strength of the structural member under consideration. The members tested were chosen to have a cross section which was fiilly effective at failure, under either compression or bending, so that the effects of local buckling could be eliminated, and the cross sections assumed to be frilly effective. In such a case there may be considered to be no neutral axis shift due to local buckling, and if the member is bent about one axis only, and compressed axially, the interaction formula to be satisfied may be vmtten as foliows:AISI(1996) In this specification the interaction equations applying to the problem of a beam column with applied axial load P and Bending moment varying along the length from a maximum value of Mi to a value of M2 as shown in Figure 1 are (with safety factors discarded) :-
p + — ^, M, Pc MS -P/PE) where
and
Q = 0.6 + 0.4 M,MJM, IM,
< 1 bui hut
P — + —i- < < 1 p
Q > 0.4
(1) (2)
(3)
173 Here PE is the Euler buckling load, equal to n^Ell 1}, Mc is the moment capacity in the absence of axial force, Pc is the axial load capacity in the absence of moment, but including any reduction induced by the possibility of Euler buckling. Pes is the stub column capacity and Cm is an equivalent uniform moment factor.
Figure 1. Primary moment variation along a column BS 5950: Part 5 The interaction equations used in this specification are based to a large extent on those of the AISI specification, but with some minor differences. Here:P
M
Pc
QMc(l-P/P,)
(4)
< 1
r w 1^
where
but
5 - 1.05 - ^ + 0.3
Q<2.3
1
and Pes
(5)
(6)
Mc
Note that Eqn (4) covers the situation when buckling governs the failure, while Eqn (6) covers the situation which arises when excessive stresses at some point promote failure. ECS: Part U The interaction equation in this code may be written, for the columns investigated:ATM
Z Pes
< 1
^c
(7)
where x is a reduction factor for overall buckling such that zPcs - Pc ^^<^ the factor K is obtained as follows :K = \
JLL xPcs
but K < 1.5
(8)
174 In the above expression, the factor jj, is dependent on the column slendemess and the moment distribution along the column. This factor may be obtained from the following expression:{2/3-4)
but
y^ = 1.8 - 0.7
ju < 0.9
^
(9)
(10)
Note that M2 may be positive or negative, and that the factor |LI may also be negative, and is indeed negative in all cases where single curvature exists along the column. BS 5950: Part 1 In this specification, for the case combined axial load and bending about the minor axis the relevant "more exact" interaction equation is:m M 1+
0.5 P <1
(11)
M. where
m = 0.57 + 0.33 ML + 0.1
and
Ml
ML < 1 Mr
but
m > 0.43
(12)
(13)
It should be mentioned here that the evaluation of Po M : and Pes varies from one code to another. In the beam columns considered here it was assumed that the moment capacities evaluated by each code were the same, although there are actually small differences, and the column capacities were determined on the basis of the individual specifications. In the case of the moment capacities, as all codes allow elasto-plastic behaviour for the cross section considered the plastic capacity used in the British codes was taken for all four specifications. With respect to the equivalent moment factors it can be seen that m and Cm are approximately equal for a given moment distribution, and are close to the inverse of C^. All of these are approximate, and none take into account the effects of axial load on the equivalent moment factor. This is discussed in Ref(l)
CODE COMPARISONS Figure 2 shows interaction curves based the four different design codes for stocky beam columns (Length to radius of gyration ratio =10) with uniform moment along the member. It is interesting to observe that for these short members the interaction curves based on EC3 and more substantially, BS 5950:Part 1 are somewhat concave, thus suggesting that the interaction is more damaging than would
175 Uniform moment M2=Mi -BS5950:Part5 EC3: Part 1.3 -AISI -BS5950: Parti
P/Pc
Figure 2. Interaction curves from different design codes for a short beam column, L/r =10 be given by simple linear interaction expressions. For moment which varies along the member, as shown in Figures 1, then in the case of stocky members such as these all codes give rather similar interaction curves. For the British and AISI codes the governing equations in such cases are the linear equations covering the maximum stress at a single point. — BS5950:Part5 - - EC3: Part 1.3 AISI ^ BS5950: Part 1
^v
1 0.80.60.40.20-
1
1
1
0.08
0.12
0.2
P/Pc
Figure 3. Interaction curves for beam column with L/r=180, M2=Mi 1.2— BS 5950:Part 5 - EC3: Part 1.3 —AISI ^ BS5950: Part 1
0.08
0.12 P/Pc
Figure 4. Interaction curves for beam column with L/r=180, M2=0
176 More interesting comparisons arise in the examination of more slender members. Figures 3, 4 and 5 show the interaction curves for members with L/r = 180. Figure 3 shows the uniform moment case, and it can be seen that all curves suggest that a linear interaction approximation would be nonconservative. Figure 4, for members in which the bending moment varies linearly from zero at one end to a maximum at the other, suggests that the different codes give substantially different interaction behaviour for the varying moment case. Here the EC3 curve gives slightly greater moments than AISI/BS 5950:Part 5 for large axial forces, but substantially lower moments for small axial forces. The BS 5950:Part 1 moments for any axial load are substantially greater than the other codes. 1.2 1 —
BS5950:Part5 EC3:Part1.3 AISI
—
0
0.04
0.08
P/Pc
0.12
0.16
BS5950: Part 1
0.2
Figure 5. Interaction curve for beam column with L/r = 180, M2 = -Mi In the case of frill moment reversal along the member, as shown in Figure 5, the interaction curve from the EC3 rules suggests significantly greater moment capacity in the presence of high axial forces than does any of the other codes. Indeed values of axial load can be selected for which EC3 suggests a moment capacity greater than 5 times that given by BS 5950:Part 5. These results are somewhat alarming. It seems that there are widely divergent interaction expressions in use to deal with beam column behaviour. To check this out a research program was instituted, in which the intention is to examine the interaction behaviour thoroughly. So far, only a rather short series of experiments has been carried out to examine interaction behaviour in the presence of varying moments, but frirther research is in progress at the present time. The initial test series was reported in ref. (8), and the main results are summarised here.
EXPERIMENTS ON MOMENT-AXIAL FORCE INTERACTION. Thirty tests to failure were carried out on small channel cross section specimens under eccentrically applied loading. The specimens were of length varying in increments of 100 mm from 100 mm to 500 mm. The two end blocks through which loading was applied added another 25 mm to the overall length, and therefore the overall length between load points varied from 125 nmi to 525 mm. The radius of gyration of the specimens with respect to the minor axis was 2.05 mm, so that the slendemess ratios under examination varied from 61 to 256. Special clamping blocks were made up to fit the ends of the specimens. The specimens were fitted into the blocks which were then tightened by bolts to fix the ends securely into the blocks. The clamping blocks were bolted to loading blocks which had serrated outer edges, v^th serrations at 3mm
177 pitch as shown in Figure 6. The loading was applied through knife edged vee blocks as shown in the figure. By selection of appropriate serrations the degree of eccentricity of load at each end of the column could be specified. In practice during tests the column arrangement was aligned so that the loading points were as close to being in the same vertical line as possible, but this was not essential to produce the specified load variation, as transverse loading was automatically induced at the load points to ensure equilibrium by producing the moment variation specified by the end eccentricities. The tests were carried out in a Tinius Olsen electro-mechanical testing machine.
Tinius Olsen loading head
33
FY
t=1.034
Close up view Of load points.
21
4r
3 mm spacing of grooves
Material and Section Properties Area = 50.5 mm^ Imin = 212mm Yield Stress = 339 N/mm^ Mc = 23.63 Nm
Section A- A
Diagrammatic view of test rig Figure 6. Test details Six different loading conditions were examined. These were to some extent dictated by the layout of the loading blocks and could be specified by the eccentricities ei and QI Under each loading condition five column tests were carried out, with overall lengths of 125, 225, 325, 425 and 525 mm. The eccentricity ei considered in examination of the tests was modified to take account of the fact that under varying moment the maximum moment actually suffered by the column was Pei* as shown in Figure 2, due to the fixity within the clamping blocks. The modified eccentricity can be obtained simply from the geometry of the moment diagram as:The failure loads obtained from the tests are tabulated below, and the comparisons with the various codes are shown in Figure 7:TABLE 1. TEST FAILURE LOADS Loading Condition 1 2 3 4 5 6
ei
62
mm
mm
3 9 15 15 15 9
3 -3 3 15 -9 9
125 mm Length 3.83 kN 3.24 kN 1.94 kN 1.50 kN 2.94 kN 1.83 kN
225 mm Length 2.73 kN 2.66 kN 1.50 kN 1.16kN 2.01 kN 1.66 kN
325 mm Length 1.76 kN 2.23 kN 1.15kN 0.91 kN 1.65 kN 1.27 kN
425 mm Length 1.34 kN 1.50 kN 0.86 kN 0.71 kN 1.22 kN 1.00 kN
525 mm Length 0.94 kN 1.08 kN 0.71 kN 0.61 kN 0.94 kN 0.79 kN
178 —
ui 1.4
o o
BS5950:Part5
° Load type 1
O 1.2
° Load type 2 » Load type 3 * Load type 4 • Load type 5 A Load type 6 0
200
400
lu 1.4 Q
-AISI Load type 1 Load type 2 Load type 3 Load type 4 Load type 5 Load type 6
o
O 1.2 hZ
I 1 uj 0.8 a. X ^ 0.6
600
400
600
Length of member in mm lu 1.4 Q O O 1.2 Im 1
-
UJ 0.8
p
^
_
><
0.6
i
1
«
200
-EC3:Part1.3 Load type 1 Load type 2 Load type 3 Load type 4 Load type 5 Load type 6
9
°
•
X
"
i
2
400
2
600
— BS 5950: P a r t i
lU 1.4 Q O O 1.2
o Load type 1 ° Load type 2 « Load type 3 • Load type 4
UJ 0.8
• Load type 5
^ 0.6 L
' 400
Load type 6
600
Figure 7. Comparison of failure loads with predicted loads from different codes The comparisons shown indicate that BS 5950:Part 5 and the AISI code give similar results, with the scatter reducing as the length increases. BS 5950 :Part 1 and EC3: Part 1.3 have somewhat similar overall accuracy for short members, but there is a slight tendency for non-conservatism for longer members
CONCLUSIONS The brief experimental investigation reported here indicated that the codes examined give realistic evaluations of the load capacities of columns subjected to varying moment along their lengths. This investigation, however, did not go into sufficient depths to provide any comprehensive calibration of the design code predictions, and a substantially more exhaustive investigation should be carried out to provide assurance of the safety and accuracy of the relevant formulae.
REFERENCES 1. Chen W.F and Lui E.M. (1991) Stability of Steel Frames. CRC Press 2. Liew R.J.Y., White D.W. and Chen W.F. (1992). Beam Columns. Constructional Steel Design. Eds. Dowling J.P, Harding J.E. and Bjorhovde R. 105-132 3. CEN ENV 1993-1-3:1996. Eurocode S.Design of steel structures-Part LS.General Rulessupplementary rules for coldformed thin gauge members and sheeting 4. British Standards Institution (1987). BS 5950: Structural use of steelwork in building. Part 5. Code of Practice for the design of cold-formed sections. 5. American Iron and Steel Institute. (1996). Specification for the Design of Cold-Formed Steel Structural Members. 6. British Standards Institution (1985). BS 5 950 .Structural use of steelwork in building. Part 1. Code of Practice for design in simple and continuous construction: hot rolled sections. 7. CEN ENV 1993-1-1;1992: Eurocode 3: Design of steel structures-Part I.LGeneral Rules and rules for buildings. 8. Rhodes J.(1998). Column under loads of varying eccentricity. Proc. 14^^ International Specialty Conference on Cold-Formed steel Structures.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
179
ANALYSIS OF THIN-WALLED STEEL BEAM-COLUMNS
A. M. S. Freitas' and F. G. F. Bueno'
^ Department of Civil Engineering, Federal University of Ouro Preto Campus Morro do Cruzeiro, Ouro Preto - MG - Brazil, 35400-000
ABSTRACT
The main goal of this research is to investigate the ultimate limit state of opened thin-walled beamcolumns at monosymmetric sections. This paper presents the analysis of a set of experimental results of eccentrically compressed thin-walled unstiflfened channel cold-formed steel members. Experimental results from the authors were adopted. As the flexural-torsional buckling mode has to be carefiilly considered in the case of eccentricity compressed members, two ways to evaluate this critical buckling load were tested: a simplified one that use the interaction equation and the exact theoretical value. The experimental results are compared with AISI-LRFD.
KEYWORDS Steel Structures, structural stability, metal structures, thin-walled, cold formed, beam-column.
180 INTRODUCTION
In this paper design approaches of beam-columns of cold-formed steel profiles are discuss. The interaction equations generally adopted to analyze those structural elements have been researched for several decades. As cold-formed members are frequently thin-walled sections, stability problems as local buckling introduces one more non-linearity. Several researches to simplify the prescriptions and introduce the non-linear effects are available. Therefore, an experimental program was developed by the authors to analyze the behavior of beam-colunms, constituted by cold-formed steel profiles with unstiflFened channel cross section, U. The paper discuss the non-linear effects and the approach designs are compared with AISI-LRFD prescription.
CODE PRESCRIPTION One of the most complex evaluations of cold-formed steel profiles are beam-columns members. Codes as AISI (American Iron and Steel Institute), and Eurocode 3 - Part. 1.3, adopted the interaction Eqn. 1.
<|)oPn <|).M„,(1-P/P^)
<|),M,,(1-P/P^)
P - axial load applied Mx, My - bending moments applied Pn - design axial strength for concentric loads compressed members MjK, Mny - bending strength of the member Cmx, Cmy" cocfificient that depends on the bending distribution 1-P/Pkx, 1-P/Pky - amplification factor for consideration of non linear effect P-6 Pkx, Pky - Euler critical load.
The analysis of these structural members have to consider the influence of the local buckling, the Euler buckling, flexural-torsional buckling, lateral buckling, the strength such a colunm and such a beam, and non linear effects.
181 STABILITY OF STEEL PROFILES
Members composed of cold-formed steel profile are sensitive to the local buckling, torsional and flexural-torsional buckling. The predominant mode in the analysis of these profiles depends on the geometric characteristics of the cross section, the length of the member, the slender and the boundary conditions. Monosymmetric section under eccentric axial compressed load in relation the main axis of inertia, are govern by the flexural-torsional mode of instability. Table 1 Batista (1987) Critical load for double eccentric load the solution of the Eqn. 2, roots are the critical loads of the column, as show in Table 1.
(^kx - ^ ^ \ - ^ ^ ^ k t - ^)^« ^ ^^y^] - (^kx - ^>^yP' - (^ky - ^^^^ + ^o>'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
This Page Intentionally Left Blank
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). <0 JF2C
(1)
where F is the support reaction and FR the support reaction resistance and an - NS/AF2 the compressive stress and fpzc the compression strength of the face, which is against the supporting structure. The compression strength fF2c is determmed from the bending tests of smgle span sandwich panels. The expression (2) for the support reaction resistance FR has been presented in the European Recommendations for sandwich panels (ECCS&CIB 1993).
193 F,=U{L,
(2)
+ tje)B
To verify the design model given in Eqn. 1, comparisons have been made using tests results from the laboratories from Germany and Finland (Bemer 1995, Martikainen&Hassinen 1996) (Figure 4). The most test results, 92 results for polyurethane and 13 for rock-wool cored sandwich panels, represent the ultimate load of the three point loaded sandwich beams, which are also called to be simulated support reaction tests, while the triangular bending moment distribution is similar to that at an intermediate support of a continuous multi-span sandwich beam. In addition to the results from simulated support reaction tests, there are eleven results from the two-span tests on rock-wool cored sandwich panels and four results on two-span polyurethane core sandwich panels shown in Figure 1 included in the comparison. Test results for continuous two-span panels represent the first failure mode of the panel, which is in two cases a shear failure and may in other cases be the core crushing failure in the case of narrow support width or the buckling failure of the compressed face for large support widths. In the Figure 4 the coefficient TI describing the distribution of the support pressure in
the core has been chosen to be n = 0.5 for the polyurethane and r| = 0.4 for the rock-wool cored sandwich panels. Of2 has been calculated assuming the loads and support reactions to be exposed to the panel along a line located on a mid-lines of the loading beams and the supports. The comparison in Figure 4 shows some scattering in test results, the reasons for which may be the failures initiated from the bonding failures and from other imperfections in addition to the core crushing and face buckling failures. All the results base on the tests having a synmietric loading at the support. However, on the basis of the comparisons with a relative large number of test results the equations (1) and (2) can be recommended for the design work in practice. The equation gives a smooth interaction between thefriUcompression strength fF2c and the support reaction resistance FR. 1.2
1
m
^
^
mm
11 «" * P^Ofc t ^ ^k<*. Mi
0.8
^H
>;
i
i >
0.6
CD
\p]
0.4 T
lu) "
(SO
\ j p
1 — 1 ^ pu, simulated - M l - rw,simulated
0.2-4-
m \
E3 '• rw, two-span test -G^~ pu, two-span test
T 4~
Design curve (
1 — = = ? = ^-4
0.2
\
^
.H
(---—"H
V-"
0.4 0.6 0.8 1 Relative support strength [-/-]
1
—1
1.2
i " —
1.4
Figure 4: Relative support reaction resistance (F/FR) versus compression strength of the face (aF2/fF2c)Comparison between experimental results and design curve given in Eqn. 1.
194 REMAINING BENDING RESISTANCE The ultimate load-bearing capacity of a multi-span sandwich beam is evaluated on the basis of a shear or bending failure in a span of the structure, if the structure is resistant enough to the required rotations at the intermediate supports. The remaining bending resistance of a thin-faced sandwich beam can be evaluated from (3)
fF2cM^F2e{0)
M,S,pl
O^,=0-0,=0.
(4)
-('^"•'^sX
where Ikcpi 1« the eOfflpfSgiiSft itfength of th# fee© alter the tirst iailure at die intermediate support, AF2 the cross-sectional area of the face and e the distance between the centroids of the faces, which may be reduced because of the crushing of the core on the support. 0ei and 0pi are the elastic and inelastic components of the rotation at the intermediate support. 1 Tfs:
n1 vCs*-**^^ ^ J 1 \
0.8 1
Test9 L,=200mm
Figure 5: a) Remaining bending resistance of two polyurethane foam sandwich panels tested in two-span panel tests. Tensile strains in upper face, when the width of the supporting structure is b) large (Ls = 200 mm) or c) narrow (Ls = 70 mm).
"""^^
0.6]
Test 10 L, = 70 mm
0.4 1J
1
0.2! 0
1
0.01
1
0.02
^-
0.03
0.04
Plastic rotation B^ [rad]
-200 -200
400
0
100
200
Distancefromthe mid-line of the mid-support [mm] Support plate
-100
-35
0
35
100
Distancefromthe mid-line of the mid-si^pport [mm] ^^^^^^^^^— Support plate
Experimental investigations on two-span sandwich panels with a polyurethane foam core shown in Fig. 1, have resulted in high inelastic bending resistances after the first failure (Figure 5). And further, the remaining bending resistance seems to be independent on the support width in this case, it means, independent on the first failure mode on the support. Bending failures were exceeded in the spans of the test panels after a relatively small inelastic rotation of about 2° on the mid-support. Axial strains in the upper face, which is loaded by tensile stresses, remain elastic, while the yield strain for the steel material of the face is 8y = 1.77E-3 (Figure 5b,c). The strain values may include strains caused by additional internal stresses in the post-ultimate loading phase, when the compressed face is pressed against the edges of the supporting structure.
195 The value of the remaining bending resistance depends of the properties of the core when it is loaded with large compression strains. The core made of structural rock-wool material has a clear compression strength value resulted in from the buckling of the fibres. With strains larger than the compression strain, the transverse support provided by the core to the buckled face is reduced, which leads to smaller relative remainmg bending moment resistance compared to panels with a core made of structural foams. However, the value of the remaining bending resistance on the intermediate supports is relatively large compared to the resistance causing the first failure, the serviceability limit state failure, and it can be utilised in the design in practice for plastic foam and rock-wool cored sandwich panels.
MATEMAL MODELS Analyses of the load-bearing capacity of sandwich panels have shown, that the use of the traditional isotropic elastic-plastic material models and von Mises type failure criterion for the core made of a structural foam or of a mineral wool leads to failure modes and loads, which have not been observed in the tests. Obvious reasons for the differences in the analyses and tests are the inadequate modelling of the stress-strain behaviour of the core materials. a) nu
Tension
100
X /
50 0
7
-50
/
/
-100 -150 .
""""^ 1
Compression (
1
\
,
j
^ _
1
-0.1 -0.08 -0.06 -0.04 -0.02 Strain [-/-]
20
H . _ _
— — H
0
40 60 80 Shear strength [kPa]
1
1
'
0.02 0.04
100
0.05 0,1 0.15 Compressive/Shear strain [-/-]
0.2
120
Figure 6: a) Stress-strain curves for a polyurethane foam (p = 39.7 kg/m^) when loaded a) by compressive and tensile stresses and b) by simultaneous compressive and shear stresses, c) experimental strength values from compression-shear tests and d) a test specimen. Figure 6a shows experimental results for a polyurethane foam loaded by repeated tensile and compressive stresses. The results show the material being brittle in tension but havmg a large plastic
196 capacity after the relative yield stress when loaded by compressive stresses. If, in addition to the compressive stress, the foam is loaded simultaneously also by shear stresses, a clear yield stress value with a softening phase can be observed in the compression strain-stress curve (Figure 6b). The strength values determined from the conditions (fcc , fcv) = [max{fcc,max , CTCCCSCC - 0.10)}, TC ] or (fcc, fcv) = [cFcc, fcv,max] have been drawn in Figure 6c. TC and acc in the conditions are the shear and compressive stresses, which correspond the maximum compressive or shear stress value in the same tests. The results show that the polyurethane foam has in reality a higher capacity in a stress state composed of simultaneous compressive and shear stresses than could be expected on the basis of the von Mises type yield criterion. A scattering in the results at large shear stress values results in from the brittle failures in the bonds between the steel face sheet and foam, when the specimens were loaded by large shear stresses.
CONCLUSIONS In the design of contmuous multi-span sandwich panels, the limit loads have to be determmed separately for the serviceability and ultimate limit state. For the evaluation of the serviceability limit state load, an interaction model has been developed and verified using test results on sandwich panels with polyurethane foam and rock-wool core layers. The ultimate limit load can be evaluated more accurately, if the inelastic bending resistance at intermediate supports is taken into account. The remaining inelastic bending resistance depends on the properties of the core and may have relatively large values. At intermediate supports the core layer is loaded by compressive and shear stresses simultaneously. For numerical analyses the special stress-strain behaviour of plastic foams and mineral wools under simultaneous compressive and shear stresses has to be modelled accurately in order to get results, which correspond the load-bearing capacities recorded in experiments.
REFERENCES Bemer, K. (1995). Erarbeitung vollstandiger Bemessungsgrundlagen im Rahmen bautechnischer Zulassimgen fiir Sandwichbauteile (Development of complete design basis for the type approval of sandwich panels). Part 2. Fachhochschule Rheinland-Pfalz. Research Report. (In German) Chong, K.P. & Hartsock, J.A. (1974). Flexural Wrinkling in Foam-Filled Sandwich Panels. Journal of the Engineermg Mechanics Division, 100(1974),EM1, 95 -110. European Convention for Constructional Steelwork (ECCS) & International Council for Building Research, Studies and Documentation (CIB) (1993). Preliminary European Recommendations for Sandwich Panels with Additional Recommendations for Panels with Mineral Wool Core Material. CIB Report, Publication 148. Hassinen, P. & Martikainen, L. (1994). Analysis and design of continuous sandwich beams. Proc. of Twelft International Specialty Conference on Cold-Formed Steel Structures held in St.Louis, Missouri, U.S.A, October 18-19 1994. pp. 523-538. Hassinen, P., Martikainen, L. and Bemer, K. (1997). On the Design and Analysis of Continuous Sandwich Panels. Thin-Walled Structures 29:1-4,129-139. Martikainen, L. & Hassinen, P. (1996). Load-bearing capacity of continuous sandwich-panels. Helsinki University of Technology, Department of Structural Engineering. Report 135.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
197
THE CREEP AND RELAXATION IN SANDWICH PANELS WITH THE VISCOELASTIC CORES P. Rappl, J. Kurzyca^ and W. Szostak^ 1 Institute of Structural Engineering, Poznan University of Technology, Poland ^Metalplast - Obomiki, ul. Lukowska 7/9, 64-600 Obomiki, Poland
ABSTRACT The paper deals with cylindrical bending of sandwich panels including the time dependent creep and relaxation of the core. The sandwich panel consists of two thin and flat faces separated by a thick core of a low density material. The core material is described by the standard rheological model. An experimental test on a simply supported panel, subjected to uniformly distributed load has been carried out to obtain the constants of the rheological model. The relaxation phenomenon in a two-span continuous panel with no external load and with the forced and fixed displacement of the central support is demonstrated. It has been shown that the standard rheological model of the core can be correctly and effectively appUed to the problems of creep and relaxation of sandwich panels. The discussion of other rheological models and a numerical example are given.
KEYWORDS sandwich panel, creep, relaxation, standard rheological model
1. VISCOELASTIC MODELLING OF THE CORE In this paper, the strain - shear stress correspondence, y - x, in the core has been described by the standard rheological model, where y denotes the shear strain and T denotes the shear stress in the core. This model consists of a spring and a dashpot in parallel, with another spring in series. The constants of the model related to the core material are: G^ - the instantaneous shear modulus, G - the retarded shear modulus, ;; - the viscosity constant. Figure 1. The spring G^ deforms instantly under a load. The spring G is restrained by the dashpot ?] so that the strain y can occour only gradually, and at a reducing rate. For the standard rheological model the strain shear stress relationship, y -T, is given by the equation: \ G„+Gdt
G„+G
G„+Gdt
198 2. THE CONSTITUTIVE EQUATIONS Let's consider a sandwich panel with two thin faces separated by a core of thickness a. The thicknesses of the upper and lower faces are ti, t2 respectively. The distance between centroids of the faces is d, and the width of the panel is Z>, 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
for -^
' 0<x
(12) ^ ^ (13) ^ ^
where:
GoG
G^^^^
Ti j
The displacement at the midspan equals:
^
Pl^ 6B
PI 2AG{t)
The time dependent shear modulus G(0, given by the formula (14), varies from the value G^^ at ^ = 0 to the value G^ = G^GIiG^ +G) at / = 00.
201 The instantaneous bearing reaction P^, at the central support of the two-span panel (Figure 7) calculated from (15) with ^ = 0, G(0) = G^, and w(l,0) = 6 equals:
Po= ,^2
6ABG0 - ^-^^-
(1^)
Hence, it is evident, that the instantaneous shear strains y(x, 0) and deflections w{x,0) in the two span panel have been defined by the formulae (12) and (13), where / = 0, P= P^ and G(0) = G^j. If time t tends to infinity, for P remaining constant, the displacement w(lj) increases and for / = 00, and according to (15), it equals: Pl^ W(1,OQ) =
^'
where:
^
PI +
-—,
(17)
^ ^
2AG'
6B
°'-^-
Thus, it is obvious, that the force P^ being constant and satisfying the equahty 5=
p i^
PI
6B
lAG^
makes the displacement w(/,oo) equal 5. The force P^ equals: P . - ^ ^ ^ ^ 5. l{rAGoG + 3{Go+G)B)
(19)
Consequently, in the two - span panel, the asymptotic shear strains y(x,oo) and the asymptotic deflections w(x,oo), by the formulae (12) and (13), with ^ = 00, P= P^ and G(oo) = G^ have been defined.
6. NUMERICAL EXAMPLE Let's consider the two - span continuous sandwich panel ISOTHERM SC 80 as it has taken in the unit 4 of this paper. Assume / = 2000 mm. The forced displacement of the central support equals 5. The values 5 and x should be taken in millimetres in the sequel. According to the formulae (12) and (13), where t = 0, P = Po and G(0) = G^ the following instantaneous values have been calculated: P^ = 147,8 5 [N] y(x,0) = 2,39-10"^5
for
0<x<0,5/
w(x,0) = (63,06-10"^x-3,27-10~^^x^)5[/wm]
for
0<x<0,5l.
202
Using the formulae (12), (13) and (14), where / = oo, P = P^md asymptotic values have been obtained:
G(oo) = G^ the following
P^= 120,6 b[N] G{t)-
1 0,118-0,134 exp(-0,217/) 0^ = 2,39 N/mm^
y(x,oo) = 2,87-10~^6 for
0<x<0,5l
w(x,oo) = {60,66-10'^ X-2,67-10'^^ x^)S [mm] for
0<x<0,5L
The effect of the relaxation in the two - span sandwich panel ISOTHERM SC 80, subjected to the forced displacement 5 of the central support has been shown in Figure 7. The plot of the time dependent shear modulus G{t) has been shown in Figure 8.
7. CONCLUDING REMARKS The physical model of the constructive material should accurately describe its basic real features. In the real constructive material, when applying loads, instantaneous elastic deformations have occurred. At a given fixed load, deformations increase due to creeping effect, and after a period of time they become constant. Similarly, the initial deformations cause immediate inner forces in the material. At given fixed deformations, inner forces will decrease because of relaxation and again, after a period of time, will remain constant. Creeping deformations cannot increase infinitely, and as the result of relaxation inner forces cannot disappear. The standard rheological model is the simplest physical model describing properly the real material features mentioned above. Other rheological models used so far, such as Maxwell, Kelvin-Voigt and Burger models, do not fit properly to the basic features of the real materials. The Maxwell model consists of a spring and a dashpot in series. Its response to an apphed load is producing an instantaneous elastic deflection followed by a time dependent creep deflection, increasing iofinitely. For constant deformations complete relaxation of the inner forces is observed, Stamm & White (1974). The Kelvin-Voigt model consists of a spring and a dashpot in parallel. This model being perfectly stiff agaiast suddenly apphed load, does not show instantaneous deformations of the real materials. The Biirger model, Shenoi ... (1995), consists of a Maxwell and a Kelvin-Voigt models in series. Its response to an apphed load is similar to the Maxwell model.
REFERENCES 1. Allen H.G. (1969). Analysis andDesing of Structural Sandwich Panels, Pergamon Press, Oxford. 2. Shenoi R.A., Clark S.D., Allen H.G., Hicks LA. (1995). Steady State Creep Behaviour of FRP Foam Cored Sandwich Beams, pp 789-799 in: Third International Conference on Sandwich Construction, University of Southampton, UK. 3. Stamm K., White H. (1974). Sandwichkonstruktionen, Berechnung, Fertigung, Ausfuhrung, Springer - Verlag, Wien.
203
g= 13.97 doN/m ^
p=77.99 doN/m
i;,l,i/iiiiUiM,M,M,M,i|^,l,M,^,i/i,MJ,M,t,t,i,llM,M,i,WMI
\i/w(x,t)
instantaneous deflection due to the load p equals w (0) = 11.94 mm
Go - instantaneous shear modulus G- retarded shear modulus r| - viscosity constant
T - shear stress y - shear strain
Figure 1. The core material modeling by the standard rheological model
long term deflection due to the load g+p estimated as 2.8 mm
Figure 3. The experimental creep testing of the panel ISOTHERM SC 80
t, Ti=34.4 Nday/mm^ o
b
T *
Go=3.52N/mm^ ^AAAAAA/^
—
^
—
G=7.46 N/mm^
U
Figure 2. The cross-section of a sandwich Figure 5. The standard rheological model of panel the core in the panel ISOTHERM SC80 3,5 3
w(t)-w(0) y(t) [mm] 2.8mm
j;(0 = 2.8(l-exp(-0.2170)
w(0-w(0)
2,5 2 1.5 1 0,5
t [days] w(t)-w(0) experimental creep deflections y(t) standard model approximation Figure 4. The experimental creep deflections and the standard model approximation
204
X
I
/i^ ^1 I :[
1
s
r
/
(N
»*»»
W
\I \ \ \\ \
\
1 ^1\
CIH
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^-»^
/'-N
V
7^
13
g
u. §
& 1
i
<§
1
•§ C/3
^ a>
\
fS 2?
1, E
c
T-H
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s—^
Tiro
o
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00
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O
1—t
II
^
V—''
1 \/ o
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o
ON
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U)
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c> O tf > i O
V^i-^^C-4V r 1 r lO CO
(0
o o d
:
CM
"*
o
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f^
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00
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1 E
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
205
IMPACT TESTS ON STEEL AND ALUMINIUM ROAD SIDE COLUMNS J. Valtonen* and K. Laakso^ ^Department of Civil and Environmental Engineering, Helsinki University of Technology, P.O.B 2100,02015 HUT, Finland
ABSTRACT Thousands of people die every year when hitting different road side objects. Lighting columns are very common objects. That is why it has been cost effective to develop safer products. While developing these products, one may not forget other important functional properties for example durability against wind loads and loads caused by road maintenance. Passive safety classification of road side objects is based on the results of full-scale impact tests. The speed alternatives in the impact tests vary from 35 to 100 km/h. The test is covered by two or three high-speed cameras and couple of video cameras. The most important results are measured by accelerometers placed in the vehicle. Helsinki University of Technology has made nearly 150 impact tests and tested mainly lighting columns made of steel: - profile plate structures - lattice structures - pipe structures Also aluminium columns have been tested. Nowadays it is no more enough that the column is frangible or detachable, but also at least some energy absorbing properties are needed. One of the main results of the whole amount of impact tests is, that steel and aluminium have proved to be suitable materials when the column should act safely at the same time absorb the kinetic energy of the encroaching vehicle.
KEYWORDS impact test, road side safety, traffic accident, lighting column, break-away column, energy absorbing
206 INTRODUCTION There are annually ca. 500 road traffic fatalities in Finland, which means ca. 100 fatalities per one million inhabitants. That is a typical value for Scandinavia. About 150 of the fatalities in Finland take place in two lane main roads, and about 20 of them are single accidents. Ten years ago the number of single accidents was higher, ca. 30. Some common accidents are shown in table 1. TABLE 1. TYPES OF SINGLE ACCIDENTS IN THE YEARS 1 9 8 8 - 9 0 ON RURAL MAIN ROADS IN FINLAND.
Objects Rigid lighting column Bridge piers Intersection, culvert Safety barrier Forest (tree or stone) Ditch or slope Rock cut
Fatalies/3years 10 8 21 6 19 29 8
Fatal accidents density per year 0,1 /1000 columns 7 /1000 piers 0,07 / intersections 1,3 / 1000 km safety barrier 0,4 /1000 km roadside in forest 0,4 / 1000 km road-side with ditch 2,3 /1000 km road-side rock cut
One main reason why the number of fatalities in single accidents in two lane main roads in Finland has decreased from 30 to 20 are the measures done during the 90's: Existing lighting columns have been modified to break-away columns, and different kinds of break-away columns have been used in new installations. More safety barriers have been installed in front of bridge supports.
Also the following methods are considered for future use: • • • •
Rehabilitation of existing safety barriers to fulfil modem requirements Reshaping side ditches and slopes Installation of safety barriers in front of rock cuts Systematic inspection of existing roads and for example replacing, removing or modifying dangerous road side objects
Some studies are going on right now in Finland to find out the effects of these methods on fatal and injury accidents.
LIGHTING COLUMNS Break-away lighting columns can divide in two categories: energy absorbing and non-energy absorbing. Energy absorbing structures slow the vehicle considerably and the risk of secondary accidents with structures, trees, pedestrians and other road users can be reduced. Non-energy absorbing colunms have weak section, slip-base or other detachment mechanism near ground level. At the impact the column come loose the foundation.
207
At the impact the cross section of energy absorbing column deforms and the column bends under the vehicle. Typically energy absorbing columns are thin wall steel or aluminium tubes with reinforcement steel bars, or columns are lattice structures. In the year 1992, when the Finnish Road Administration ( FinnRA ) had decided to prefer break-away lighting columns, the Helsinki University of Technology was contacted. At the beginning the idea was to make six crash tests in a hall with a big pendulum. The pendulum tests had not even started, when it was found out, that at the same time a wood company was developing a wooden hollow break-away lighting column and had to build a crash test track for that about 130 km north of Helsinki. At that time it proved to be an ideal place to make crash tests. The test vehicle was guided by timbers and drawn by a Range Rover. The area was quite large, so that the Range Rover started behind the tested column and it had the same direction and speed as the impacting vehicle. In the first place the test speed was only 35 km/h. At first the acceleration measurements were done with equipment originally built for roughness measurements and it was placed on the rear axle of a car. During the first year more than ten different columns were tested and always two tests were made for one column type. The test procedure was very simple, for example there were not many restrictions for the test vehicle. The test vehicles were of course of the same weight but very many car makes were used. To convince the clients ( = column producers ) about the repeatability of the tests, cars that were out of the ordinary were not used for example if the engine was in the very front or in the rear. Through experience, also the scale of the test speed could be increased, in the autumn even 65 km/h was carried out. The results of the first year of crash testing were good. Several good or at least satisfying columns got an approval from FinnRA. At that time FinnRA decided to use only break-away columns where the speed limit exceeds 50 km/h and when the columns are not behind a safety barrier. The price difference between rigid columns and new safe ones is very small. It is also rather inexpensive to modify existing columns into break-aways. Methods to modify wood poles and steel columns were developed in the beginning of the 90's. Nowadays most of the existing lighting columns along the Finnish main roads are already modified. The modification started in 1992. The first year of crash testing also showed that the embankment material and the compaction of the soil influence the performance of some slip-base columns. That is why the Helsinki University of Technology showed two crash tests for installation contractors and traffic safety engineers so that they could see with their own eyes the importance of careful installation. In the year 1993 some difficulties occurred in the shape of a patent disagreement. When the companies saw each other's columns during testing nobody was sure who had invented and what. That was the main reason to look for a neutral test track. Fortunately a closed factory with a paved yard was found and it was only 50 km from Helsinki At that time even higher test speeds were needed it was noticed that the whole construction of the test track should be redesigned. The basic idea for the track was found in United States. First the vehicle was tried to guide with a wire, but the impact point was very seldom in the right place. Soon a rail was found to be more suitable for guiding and a truck was used for drawing. The truck goes to the opposite direction compared to the test vehicle and the wire is fourfold so when the speed of the truck is 25 km/h, the test vehicle is going 100 km/h. Even a 120 km/h test speed is possible to reach.
208 In the year 1993 Working Group 10 of CEN TC226 began to write a European Standard for the impact tests of narrow obstacles. It was important to have a delegate of our own in the group so that it could be possible to react rapidly to the directions where the test procedure was heading to. Very soon it was for example noticed that some high-speed cameras and more sensors for acceleration measurements are needed. Also columns with overhead cables were tested just to see if the overhead cables have an influence to the performance of different kinds of columns. The cable was tightened by the same strain as the cable of ten columns on a row along the road.
SAFETY BARRIERS In the same year also crash testing of safety barriers started. After a couple of tests it was realised that the old safety barrier type used in Finland had relatively strong posts compared to the rail. Since then the barrier has been modified. Now the posts and the bolts between the post and the rail are weaker than earlier, which leads to better performance. The design has been used on new installations since 1995. Even though there are not many severe accidents caused by the old types of safety barriers, the modification of the barrier seems to be cost-effective, since the price of the barrier did not rise and the barrier still can stand the loads caused by snow removal. The problem with that traditional barrier is that a small car can easily hit other cars after impact with the barrier. It might be cost-effective to modify the barriers that are already installed along roads, by making the posts and bolts weaker and by lifting the post and the rail up in order to get the same height and performance as in the new designs. FinnRA might start modifying the existing barriers if the ongoing studies show it to be a competitive method for preventing severe accidents. The ends of the barriers may cause roll over accidents as well. These accidents may be prevented by using energy-absorbing end treatments instead of ramped-down ones. The energy absorbing end treatments are quite expensive, and that is why the economy of using them should be carefully evaluated. Until now there is only one end treatment of the energy absorbing kind in Finland. In the year 1998 it is intended to continue the crash tests with safety barriers by testing the influence of the post frequency in the performance of the barrier. It is possible that the frequency used in Finland will change from 4 to 2 m on the basis of those tests, especially in cases where the working width of the barrier may not be wide. Crash cushions are very expensive and only two of those have been installed in Finland, one of them has already saved a young car-driver's life. In the year 1994 there was an idea to create a cheap wooden crash cushion and eight crash tests were made with slightly different versions of wooden crash cushions. The work ended hopelessly, because not even one single test gave satisfying results. Still a low-cost crash cushion is badly needed but it certainly will require much work and plenty of tests. Also the markets may be quite limited, so that there may be no sense in investing much money on such efforts. It is also intended that crash testing of bridge barriers will start in the year 1998. It will be a new subject for Helsinki University of Technology, but FinnRA has made that kind of tests in the year 1964! The whole arrangements will need modification so that a 16-ton truck as a test vehicle could be possible.
209 CONCLUSIONS After about 160 tests it seems that same mistakes has been done as everywhere else when developing the test track and the test procedure but not much have been written about those kinds of things. For example safer barriers have already existed, but still nearly every country has to do the developing work on their own. All the time during these six years of crash testing the purpose of the job has been to help producers develop safer products. It has been of great importance to get also foreign clients. Most important is that approximately 20 people have survived when hitting lighting columns after crash testing activities started in Finland. Finland produces several excellent lighting column types so also the export of those products has increased due to cost-effective testing possibilities.
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
211
VIBRATION OF IMPERFECT SLENDER WEB J ^ Ravinger Department of Structural Mechanics, Faculty of Civil Engineering Slovak University of Technology, Radlinskeho 11, 813 68 Bratislava, Slovakia
ABSTRACT Von-Karman theory has been used for the description of the post-buckling behaviour of slender web with geometrical imperfections and residual stresses. Using Hamilton's principle in incremental form the problem of the free vibration has been established.
KEYWORDS Stability, dynamic, post-buckling behaviour, vibration, imperfections, residual stresses.
steel slender web, initial geometrical
EWRODUCTION The slender web as a main part of the thin-walled structure has significant post-buckling reserves and for a description of them it is necessary to accept a geometrical non-linear theory. The problem of the vibration of a slender web as a non-linear system was formulated by Bolotin (1964). One partial problem which is still focused on by researchers is the influence of initial imperfections on the vibration of colunms and slender webs. Burgreen (1951) formulated the problem of vibration of an imperfect colimm in the begirming of the fifties, but this is still a subject of research (e.g. EHshakoff et.al.(1991), Wedel-Heinen (1991)).It is difficult to say who first formulated the problem of vibration of an imperfect slender web. At present time, a number of theoretical works as well as the results from experiments have been published.
THEORY The total description is used, and according to the von Karman theory for large deformations of the plates, the strain vector is
212 W ,x
8=i
w^
1 2
v..
r-z< w„ 2w^
(1)
"..+^.x where u, V are the functions of in-plain displacements, w is the function of plate displacements (the indices denote the partial derivatives; and for slender web we can suppose w»u,v). Assuming a linear elastic material, the relationship between stresses and strains can be written as follows (2)
a = D ( 8 - 8 o ) + a^, where G =<
D=
E,v
}
l-v^
is the stress vector,
1
V
V
1
a ^ - the vector of residual stresses,
- the elasticity matrix,
1-v - Young's modulus, Poisson's ratio, respectively, W^O,x
Wn
w^o,y y — zi w,o,yy ^^0,.^0,y
- the vector of initial strains.
2w,0,xy
We consider the inertia forces acting in the direction of the plate displacements only. Then the inertia force per unit volume is
Pm
P ^.2 '
where p is the mass per unit volume,
t -the time.
Using Hamilton's principle we obtain the system of conditional equations K^aD+KcaD+(K^+K^^-K^^Ja^+K^a,=P^+P^„
(3a)
K,^aD+K^a5=P,+P,o,
(3b)
where K^
is the mass matrix,
K^
- the damping matrix, K ^
- the linear stif&iess matrix
of the web, K^^ - the non-linear stiffiiess matrix, the interaction between the in-plane and the plate displacements parameters.
213 K ^ - the non-linear stiffiiess matrix, the interaction between the plate and the in-plane displacements parameters, K^^ - the geometrically non-lmear stif&iess matrix of the plate, K^^Q - the matrix of the increase of the bending stif&iess of the plate due to the initial displacements, nonlinear part, K ^ - the linear stif&iess matrix of the plate, P^Q- the vector of the transformed in-plane internal forces due to the initial displacements, P^Q the vector of the transformed plate internal forces due to the initial displacements, P^ ofthe transformed external load of the plate, P^
- the vector
- the vector of the transformed external in-plane
load of the web, a ^, a ^ - the plate, in-plane displacement parameters, respectively.
THE PROBLEM OF THE FREE VIBRATION Taking out the inertia forces and the damping effects we have conditional equations describing the static post-buckling behaviour of a slender web or large displacements of plates. The problem can be solved by Newton-Raphson iterative procedure. The natural frequencies can be evaluated from the equation
det K ^ - ( » ' K ,
(4)
=0
where CO is the natural frequency of the thin-walled panel (slender web) taking into account the effect of the extemal load including the initial imperfections, K^^. is the incremental stiffiiess matrix. We can write K^^^ = J - incremental stiffiiess matrix is equal to the Jacobian of the NewtonRaphson iteration of the system of non-linear algebraic equations. The properties of the Jacobian characterise stable and unstable branches of the static solution.
INFLUENCE OF THE RESIDUAL STRESSES We have investigated a thin-walled panel with h=2.505 mm thick web manually arc-welded with the residual stresses (Fig. 1)
^ _ . ; : _ # oi%»««:a?. J <^wdy-0
<^<3C0i,Joywdx-0
Fig. 1 Distribution of the residual stresses in the slender web
214 The comparison of theoretical and experimental results is arranged in the Fig. 2. l*^*>*>MPo
h=2505mm cr, = 7.4A N mm"'
K=8.86
R^=2.2A
cr>0.38
(j,= 861.5 s
Fig.2 Vibration of thin-walled panel with residual stresses for different level of the load CONCLUSION The theory and the experiments have proved the sensitivity of the circular frequency of free vibration on the level of the load and different types of initial imperfections. This knowledge can be used as an inverse idea. Measuring of the natural frequencies can give us a picture of the stresses and imperfections in a thin-walled structure. This idea represents a base for a non-destructive method for the evaluation of the properties of thin-walled structure. The natural frequency is sensitive even to residual stresses and it is very important detail. References: Bolotin, V. v.: Dynamic Stability of Elastic Systems. GITL, Moscow, 1956 (in Russian, English translation by Holden-Day, New York, 1964). Burgreen, D.: Free vibration of pin-ended column with constant distance between pin-ends. J. Appl. Afec/za«., 18(1951)135-9. Elishakofif, I. - Birman, V. - Singer, J. > Influence of initial imperfections on non-lmear free vibration oiQldiSiich^rs, ActaMechanica, 55 (1985), 191-202. Wedel-Heinen, J.: Vibration of geometrically imperfect beam and shell structures. Int. J. Solids & Structures, 1, (1991) 29-47. Hui, D.: Effects of geometric imperfections on large amplitude vibrations of rectangular plates with histeresis damping. J. Appl. Mechan. 55 (1984) 216-20. Ilanko, s. - Dickinson, S. M.: On naturalfrequenciesof geometrically imperfect simply supported plates under uni-axial compressive loading. J. Appl Mechan. 58 (1991) 1082-4. Vohnir, A. C: Non-Linear Dynamic of Plates and Shells. Nauka, Moscow 1972 (in Russian) Ravinger, J.: Vibration of an imperfect thin-walled panel. Part 1 : Theory and Illustrative Examples. Part 2: Numerical results and experiment. Thin-Walled Structures 19 (1994) 1-36. Ravinger, J. - §volik, J.: Parametric resonance of geometrically imperfect slender web. Acta Technica 05^^,3,(1993)343-56.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
215
A CRACKED MODEL FOR FATIGUE DAMAGE DETECTION AND EVALUATION IN STEEL BEAM BRIDGES ^ E. P. Deus, ^ W. S. Venturini and ^ U. Peil ^Department of Mechanical Engineering, Federal University of Ceara Fortaleza-CE, Bloco 714 - Campus do PICI - 60455-760, BR ^Department of Structural Engineering, USP- Sao Carlos School of Engineering Sao Carlos-SP, Av. Dr. Carlos Botelho 1465 -13560-250, BR 3 Institute for Steel Structures, Technical University of Braunschweig Braunschweig, BeethovenstraPe 51, 38106, DE
ABSTRACT The most of the non-destructive techniques used nowadays are based on the local inspections. This paper presents a simple non-destructive method to identify damage occurrence due to fatigue in steel beam-bridges. For simplicity and according to the practical experience related to this problem, only the opening fracture mode is considered to develop the proposed technique. In order to identify cracks and evaluate their sizes a fracture mechanic model was used to derive reduced beam stiffness. The natural frequency changes are directly related to moment of inertia variation and consequently to a reduction in the flexural stiffness of a steel beam. Thus, the results obtained from dynamic analysis can also be properly adopted to find cracked cross sections, their places and sizes. Numerical examples show that this proposed simplified approach can lead to satisfactory results when compared with solutions obtained by using another well established computer code.
KEYWORDS Steel Bridges, Fatigue, Fracture Mechanics, Inverse Techniques, Structural Damage, Crack Detection.
INTRODUCTION Some of the most important failure modes for welded bridge structures is related with strength failure, in light of fracture due to overload or excessive plastic deformation; serviceability failure, such as vibration under service-load condition; and failures due to fatigue. Fatigue and fracture is a principal failure mode for steel structures, and it is still less understood than any other modes. The design factors are usually subjected to considerable uncertainties and, in theory, it is appropriated to make engineering decisions on the basis of a qualified system analysis.
216 The problem of safety is produced by a number of parameters such as increase of actions, deterioration of components, higher degree of capacity utilization, and negligence conditions during construction and supervision. There are several well-known international codes available covering the subjects: security, maintenance, inspection and rehabilitation of bridges structures, BrinckerhofiF (1993). The most elementary method to identify defects in steel bridges is Visual Inspection. Other nondestructive testing measures can be used to inspect and to discover cracks such as dye penetrant. X-ray, magnetic particle (magnaflux), and ultrasonics. Any of those examination methods requires the presence of a specialist to follow the inspection. The dynamic performance of cracked steel beams was investigated by numerical and experimental methods. The studies have indicated that increase in damage reflects at the same time decrease in the natural frequency of a structure. The changes in the natural frequencies are directly related to moment of inertia variation and consequently to a reduction in the flexural stiffness of a steel beam, Ostachowicz & Krawczuk (1991). The development of a non-destructive technique employing eigenvalue variation is used to identify a crack as well as to evaluate its grown size (inverse technique). Then, using fracture mechanics concepts, the cracked beam stiffness is approached here (Figure 1).
Figure 1: Method to Identify a Fracture in a Steel Beam Bridge. The steel beam model is based on finite element method and it is introduced to simulate crack openings due fatigue. This method can help in previous detection of cracks in vibrating elements before they grow and cause structural collapse. When a crack is found by inspection, it is important to predict further growth in order to decide the convenient time to repair the structure. Previously, a fatigue crack growth prediction based on the linear elastic fracture mechanics (LEFM) is needed. In addition, the crack behaviour during an overload has to be known in order to prevent brittle or ductile fractures. The credit of the cracked beam model is in its fiinctional application for crack identification in steel bridges and its efficacy is demonstrated by a comparison with numerical simulation results.
ELASTIC CRACKED MODEL The physical model considered is of a simply supported uniform beam (see Figure 2). Suppose the beam of length L exhibiting initially a simple crack (Mode I) at the mid-point. The beam, characterised by a constant cross section A and moment of inertia lo, is loaded by two moments M applied at its ends. The crack is simulated here by an equivalent rotational spring of constant k (Deus,1997).
f\—jr—1^
^A^,
Figure 2: Representation of pure bending model.
217
As was previously stated, deterioration of a structure can alter the flexural stiffness. According to the Saint-Venant's principle, the stress field is affected only in the vicinity of the crack, represented here by ^(a), being a the crack length. Thus, one can assume that the value elastic modulus x inertia product, EI, at any beam cross section is given by: rEIJl-(t>(x,a)] ^^^^'^^^k
for for
0<x<5(a) x>5(a)
(^)
where
I„=d^^
and
l' = I o ( l - j ]
(2a,b)
The complementary energy U* of the beam taking into account only one half can be computed as indicated: u* = u : + : ^ [ 0 , + 2i-'(K - 1 ) 0 , + i-^(K -1)^03]
(3)
where Uo* is the complementary energy in the uncracked situation, Me= K. Mi (K = constant) and: 0. =£^^*^x*-'Ti(x,a)dx
(4)
with Ti(x,a) =
u:=^^
(5)
' 2 k Comparing the second term of equation (4) with equation (5) one obtains: k=- ^ ^
(6)
r(x,a) where r(x,a) is given by the relation: r(x,a) = [01 + 21"'(K -1)02 + r'(K -1)^03]
(7)
218 For the whole beam, the spring coefficient k is obtained from (see figure 3):
A^
I
;
c^c^ M.= k^((|)-0^)
^i='<2(4'2-4>)
Figures: Formulation of Equivalent Stiffness.
^
Mi=k*(<|),-<|>,)
(8)
k. where k* is the equivalent spring stiffness defined as: 1
(9)
i=l '^i
The global stiffness matrix can be obtained by adding the values derived for each beam half. Figure 4 exhibits two neighbour elements whose linear system of algebraic equations (v"^q = Q) are given by:
u<"l_jP« i_|k*(-*<'') (2) -]p(2)
(10a)
(10b)
The element stiffness coefficients given in equation (10) (\K^\ {k^^^}, ^^^\ i "=1,2) are the well known values derived for standard beam elements.
219 U2 I U3
(b)
(a)
Us
V2 U4
Vi
V4 V5 V3
L
i7^
J
Figure 4: (a) Degrees of freedom for elements, (b) Assemblage. To derive the stress intensity factor Ki, it is applied the Irwin's relation (Owen & Fawkes 1993): K = VG.E*
(11)
where E* = E / ( l - v ^ ) . G is defined as energy release rate and it is related to the complementary energy of the structure U by: G = cUVaA
(12)
where A is the crack area. The computer code RAST (Rissausbreitung in Stahltrdger), was developed to model the cracked beam element and to compute its stiffness K (see figure 6). The failure due fatigue is usually the result of crack growth fi-om a discontinuity at a stress concentration. The description of the fatigue crack growth phenomenon can be made on the basis of a fi'acture mechanic model. The well-known Paris relationship is given by Paris, 1960: da/dN^QAK)""
(13a)
AK = a.AavTia
(13b)
where a is the crack depth, N is the number of cycles of the applied stress range, AK is the stress intensity range, C and m are constants related with material properties and environments, a is a fiinction depending on the specimen and crack geometry and Aa denotes the stress range.
FREQUENCY ANALYSIS FOR DAMAGE DETECTION The parameters of natural modes, such as mode shape and fi-equency, are products of mass and stiffness of a structure. Damage in a structure can change its stiffness and the modal parameters. It should be noted that each bridge has its own characteristics defined by a set of fi-equency-responses. The crack position and evaluation can be obtained through the closed-form solution of the equation of motion. The natural fi-equency of the cracked beam will be modified resulting to: (Of = (D„ - Ao, where o)„ is the naturalfi-equencyof the uncracked beam, and K = [o^ (Ap/EI)]^^"^. Taking into account the appropriate boundary conditions and the displacements equations resulted from the equation of motion, one can find the characteristic equation of the model, used to obtain the natural frequencies changes: Ao)„=2o)„.n„(P).K-^ where Q„(P) is a fiinction that depends only on the crack position and is given by:
(14)
220
n.(P) =
(15)
5K:
The frequency change for the first mode is given by: ^o),-^2(0^n,{P)
1
(16)
and for the second one, the frequency change is: ACO2 = 2 o 2 QiCP)
(17)
Now, the inverse technique can be applied to recover the equivalent dimensionless stiffness K from the fracture mechanic theory. Bearing in mind that the stiffness K is a static property, from (16) and (17), one obtains:
AcD,VrAQ,^_Q,(P)
(18)
Using equation (18) the crack position can be found. The code EIGFRISS (Eigenfrequenz fur Rissmodelle) has been implemented to identify the crack location and size. Results were compared with numerical solutions computed by using the conmiercial code ANSYS, adopting to represent the 3D structure (Deus, 1998).
NUMERICAL RESULTS The numerical analysis presented in this section consists of studying the Eigenfrequency changes in a simply supported bridge beam due to the presence of a cracked section. The geometric values taken to carry out this analysis, as well as the material properties of that composite-beam, are given bellow (see figure 5): 2400
Figure 5: Beam cross section.
221 Geometric data: Moment of Inertia; I = 0.121 x 10^^ mm"*; Cross section A = 38,332 mm^; span length L = 24,000 mm; Material parameters: Elastic modulus E = 205 kN/mm^; Fy = 0.350 kN/mm^; Z = 13589379.5 mm^ density p = 7.8 xlO"^ N/mm^ Using the code RAST, one obtains the curve displayed in figure 6 exhibiting the stiffness values against the fracture size. r = 0.2a
,, ^gj
IOH
"^ 7200
400
600
a (mm)
800
1000
1200
1400
Figure 6: Crack Length(a) x Stiffness(Log[K]) Obtained by RAST- ^(a) = r = 02a.
The natural frequency can now be achieved using EIGFRISS, as follows: On = 15.7 rad/s (mode I), where f = o/ 27i. The frequency change and the crack position are then obtained from the values of (see figure 7): i=1,11
(o„.p .157D+02
.920D+00 f
6E-05
variation mcde 1
5E-05 D
4E-05
i
3E-05 2E-05 1 E-05 0 1
0,4 0,6 be+a = x / l
Figure 7: Function Q(P) versus P=x/L for simply supported steel beam.
222
CONCLUSIONS A model of structural inspection based on frequency analysis is proposed to simulate crack openings (Mode I) due cyclic loads. The location of a damage in a steel bridge is estimated using the natural frequency variation. The beam stiffness is obtained from fracture mechanics evaluation and it is simulated by an equivalent linear spring. Then, it is applied to solve the inverse problem. The numerical algorithms, RAST and EIGFRISS, were vmtten based on the proposed techniques to verify the accuracy of the simplified models. The results were obtained using the developed codes and compared with other more complete numerical model (FEM simulation). The proposed simplified models were found to be both accurate and very usefiil to be applied to practical problems.
REFERENCES Brinckerhoff, P. (1993). Bridge inspection and rehabilitation, John Wiley & sons, inc. Deus, E.P. (1997). Analise do processo de fraturamento em vigas de pontes de ago sob efeito de fadiga, Tese de doutorado, Escola de Engenharia de Sao Carlos, Brazil. Deus, E.P. , Venturini, W.S and Peil, U. (1998). A Theoretical model to identify fractures due to fatigue in steel bridges using inverse technique. Fourth World Congress on Computational Mechanics, Buenos Aires, Argentina. Deus, E.P. , Venturini, W.S and Peil, U. (1998). Fracturing model to detect and evaluate fatigue damage in steel beam bridges. Fifth Pacific Structural Steel Conference, Seoul, Korea. Ostachowicz, W.M. and Krawczuk, M. (1991). Analysis of the effect of cracks on the natural frequencies of a cantilever beam. Journal of sound and vibration, 150, 191-201. Owen, D.R.J, and Fawkes , A.J. (1993). Engineering fracture mechanics: numerical methods and applications, U.K. Paris, P.C. and Erdogan, F. (1960). A critical analysis of crack propagation laws. Journal of Basic Engineering, 85, 528-534.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
223
FATIGUE DAMAGE RETROFITTING OF RIVETED STEEL BRIDGES USING STOP-HOLES M. Al-Emrani, R. Crocetti, B. Akesson and B. Edlund Department of Structural Engineering Division of Steel and Timber Structures Chalmers University of Technology Goteborg, Sweden
ABSTRACT When a fatigue crack is detected in a steel bridge, it is rather common to temporarily arrest the crack by drilling a stop-hole at the tip of the crack. The aim of this paper is to investigate the efficiency of this method on riveted girders both theoretically and by full-scale fatigue test series. In the experimental part of the investigation, six riveted stringers taken from an old railway bridge were cyclically loaded in four-point bending at a rather high level of the nominal bending stress range, Gj. = 100 MPa, with or without using stop-holes during the fatigue crack propagation. The method of drilling stop-holes was found to be an effective measure for a temporary arrest of the propagation of fatigue cracks in riveted steel girders.
KEYWORDS Steel bridges, fatigue, crack arrest, riveted beams, railway bridges, retrofitting, stop-hole.
INTRODUCTION The propagation rate of an existing crack under cyclic loading is related to the stress intensity at the crack tip, which in turn depends on the stress range under which the structure is loaded and the crack configuration (i.e. the crack length and its location in relation to the main tensile stress). Drilling a stop-hole at the tip of a long crack will significantly reduce the stress intensity by eliminating the crack tip and, as a result, extend the fatigue life of the cracked structure. Based on notched specimen tests, Barsom and Rolf (1987) suggested Eqn. 1 to describe the fatigue-crack initiation threshold, i.e. the stress intensity factor below which no crack re-initiation occurs from a notch.
224
^K (^),,<26.3^/cT,,,,,
(1)
where: AA" is the stress intensity factor at the tip of the crack p is the notch radius o^.^^ij is the yield stress of the material While Eqn. 1 was developed for relatively sharp notches with p « a (a being the notch depth), stopholes with diameters (13-30) mm are usually used to retrofit fatigue cracks in steel bridges. These relatively large hole radii make the term (AK/^p) in Eqn. 1 less significant, if not meaningless. On the other hand, the stress concentration factor at the edge of a drilled hole seems to be more appropriate for evaluating fatigue crack re-initiation since the stress intensity factor for a crack emanating from a hole is proportional to the stress at the hole edge. A conceivable limitation to the stress at the hole edge, in order to obtain a considerable fatigue life extension must then be: ^eclge < ^yield
(2)
where o^j^^ is the stress at the hole edge. Dividing Eqn. 2 by the applied nominal stress o„^^„,, one obtains: ^edge
^ yield
a
G nam
^ .
nom
The stress ratio in Eqn. 3, {o.^^Jo^^^), is a parameter that describes the stress concentration at the hole edge and depends on the dimensions of the cracked component, the crack length and the radius of the hole. The test results presented by Barsom and Rolf indicate that fatigue crack re-initiation from notches is highly dependent on the stress ratio, R. If the value of o„^^^ in the right-hand side of Eqn. 3 is taken as the maximum stress ^max' ^hen Eqn. 3 can be rewritten as: <^^{\-R)
(4)
where: ••o^^{\-R)
and/? =
^
Eqn. 4 suggests that for a cracked structural component, a "permanent" crack arrest can be achieved by drilling a stop-hole if the stress ratio (o^j^^/o„^„,) is less than the right-hand expression of this equation, omitting of course the effect of local stress raisers such as micro cracks at the hole edge due to the drilling process.
225
FATIGUE CRACKS IN RIVETED STRINGERS Fatigue cracks in riveted components usually initiate from the edge of a rivet hole. In five of the tested stringers presented in this paper, fatigue cracks initiated at a "neck-rivet" (i.e. at the flange-to-web connection), and propagated through the vertical leg of the cracked L-profile and then down and out into the horizontal leg. Figure 1(a) illustrates the tension flange in a riveted built-up stringer with a crack of length 2a. If a hole with a diameter 2r is drilled with its centre located at the crack tip, the horizontal part of the tension flange can be approximated as a semi-infinite plate having width b and containing an eccentric elliptic hole with the dimensions shown in Figure 1 (b).
(b)
(a)
Figure 1: An equivalent plate model representing a cracked tension flange of a riveted built-up stringer. An approximation made to calculate the stress at the edge of a stop-hole. The equivalent semi-infinite plate in Figure 1(b) is used to calculate an approximative value for the stress ratio {.o^^^/jO^^^J at the hole edge. The stress at point A can be calculated as: ^ed^e ~ ^net'
(5)
^m
where the expression for the average stress in the net section (o„^,) and the stress concentration factor based on the net section (k^„) are given in Eqn. 6 and Figure 2 respectively, as presented by Peterson.
l/r =1
u
0."1
0.'2
0'.3
0.4
0.5
"o*6 '^^
Figure 2: The stress concentration factors for an eccentric elliptical hole.
226
vl-(-) V
cr.^r =
a-')
C
c e
__
(6)
/ \ 2, \ c
As both the net-section stress (o„^,) and the stress concentration factor (k^,^) are functions of the three parameters a, r and b, the stress ratio (o^^^^ / o„^,„,) is also a function of these three parameters. In analysing the results obtained from the full-scale test series performed at Chalmers and those found in the literature, the equivalent plate model presented above is used to estimate the stress ratio (^ edge/^ no,,,)- ^hls ratlo is then plotted against the number of cycles observed between drilling the holes and crack -reinitiation. In Figure 3 the stress ratio (o^^^^ CJ„om) i^ calculated for the stringers at hand and is plotted as a function of different values of the reduction in section caused by a crack.
0
5
10
15
20
25 " ' ^ ^
Figure 3: The stress ratio {0,jgJo^^,„^)foT different b/2a-ratios for the stringers tested, (b = 238 nmi and 2r = 21 mm).
FULL-SCALE FATIGUE TESTS Description of Original Structure and Test Specimens Two sets of each three riveted stringers were tested at the Department of Structural Engineering at Chalmers University of Technology during the period 1997-98. The stringers were taken from the old riveted railway bridge over the Vindelalven in the north of Sweden. The bridge, built in 1896 and demolished in 1993, had three simply supported arch-type truss spans of 72 m each. The track super-structure consisted of 13 floor-beams supporting 24 (2x12) stringers. Besides the diaphragms connecting the two rows of stringers, lateral wind bracing were attached to the lower flanges of the stringers (referred to as flange-rivet-connections in the following text). Test Set-up and Testings Procedure All six stringers were tested in four-point bending with a distance of 2.0 m between the loading points (i.e. a constant moment region of this length). The distance between the supports was either 4.5 or 5.0 m. The nominal bending stress range chosen for the stringers was 100 MPa with a stress ratio that varied between 0.16 and 0.28. In one case, this stress range could not be achieved due to some limitations in the testing machine, see Table 1.
227 Floor-beam
Stringer .
Figure 4: Dimensions of the stringers tested at the Department of Structural Engineering at Chalmers. While fatigue cracks in the first set were left to propagate, 21 mm stop-holes were drilled to arrest the propagation of fatigue cracks in the second set as soon as these were detected or had propagated to a length that allowed application of the hole-drilling equipment. Before the hole was drilled, the paint layer around the crack was removed. The area was then sprayed with white grease and a magnification lens was used to determine the exact location of the crack-tip. When the crack-tip location had been marked, the tests were temporarily stopped and a 21 mm diameter hole was drilled with its centre located at the crack-tip. No measures were taken to improve the surface of the drilled holes and the loading was then proceeded until total failure. TABLE 1 DATA FROM THE FULL-SCALE FATIGUE TESTING Set no. 1
2
Stringer 6B 7A 12B
0,
100 100 97
R 0.28 0.28 0.17
Notes Previously tested at o^ = 40 MPa for 20 million cycles
1
1 1
12A
100
0.16
Previously tested at o^ = 60 MPa for 10 million cycles
3A lA
100 100
0.16 0.16
The stress range was here reduced to 60 MPa after hole drilling |
1
Except for stringer 3A, the fatigue cracks initiated from a rivet hole at the neck-rivet connection. A typical crack propagation scenario from a test is shown in Figure 5. In stringer 3A, the crack initiated from a flange-rivet-connection where the lateral wind bracing was attached to the bottom (tension) flange of the stringer.
The number of cycles when the cracks were detected, Ndetect» and those corresponding to the total number of cycles to failure, Nfaiiure, are given below in Table 2. Also given is the location where the crack initiated in each stringer and the number of cycles, Nredund, noted from that the first flange angle was completely fractured until a crack was observed in the second flange angle or in the web.
228 TABLE 2 TEST RESULTS FOR THE FATIGUE LIFE OF RIVETED STRINGERS 1 Stringer 6B 7A 12B 12A 3A
1 lA
J^ detect
^failure
^redund
3,627,300 1,428,830 5,799,020
3,712,020 1,587,110 5,959,340
>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. <(^edge I (^ yield)
2.5
Figure 7: Crack arrest length in cycles as a function of the stress ratio cr^^^^ ja^
230
CONCLUSIONS Based on the results obtained from full-scale fatigue tests conducted on six riveted stringers taken from a 100 years old railway bridge, the following findings can be concluded. 1) Although there is a wide scatter in the test results, the fatigue lives of riveted stringers were in accordance with or above the fatigue design curve given in structural steel codes for riveted details (category detail C=71 in Eurocode 3, corresponding to AASHTO category D). This despite the fact that the stringers had been subjected to almost 100 years of loading and environmental effects. 2) Drilling a stop-hole with its centre located at a fatigue crack tip has shown to be an effective measure of retrofitting, when aimed to give a temporary arrest of the propagation of such a crack, even when the crack lies in plane perpendicular to the main tensile stress field. The extension of the fatigue life achieved by this measure is judged to be sufficient, even when the stress range is as high as 100 MPa and with up to 33% of the tension flange cracked. 3) The crack arrest achieved by drilling a stop-hole at the tip of a propagating fatigue crack can be sufficiently large to be considered as "permanent", if the ratio between the theoretical stress calculated at the hole edge and the nominal stress does not exceed the yield stress of the material. Generally, the magnitude of fatigue life extension achieved by this method depends upon the magnitude of the applied stress, the stress ratio, the reduction of the section caused by the crack and the material properties (tensile strength and yield stress). 4) A substantial number of cycles was required to propagate the fatigue cracks from one angle bar of the flange to the other. This inherent redundant structural behaviour of built-up riveted stringers extends their fatigue life and diminishes the probability of brittle fracture.
REFERENCES Al-Emrani M. (1998). Fatigue Damage Retrofitting in Riveted Railway Stringers. Department of Structural Engineering, Chalmers University of Technology. Int.skr. S 98:7. Goteborg, Sweden. Barsom J. and Rolf S. (1987). Fracture and Fatigue Control in Structures, Prentice-Hall, Inc., New Jersey. Fisher J. and Menzemer C. (1987). Bridge Repair Methods. NATO Advanced Research Workshop on Bridge Evaluation, Repair and Rehabilitation, Vol. 187,495-512. Kadir Z. (1997). Riveted Joints / Full-scale Fatigue Tests. Department of Structural Engineering, Chalmers University of Technology. Int.skr. S 97:7. Goteborg, Sweden. Out J., Fisher J and Yen B. (1984). Fatigue Strength of Weathered and Deteriorated Riveted Members. Transportation Research Record 950:2, 10-20. Peterson R. E. (1974). Stress Concentration Factors. John Wiley & Sons, New York. Akesson B. (1994). Fatigue Life of Riveted Railway Bridges. Department of Structural Engineering, Chalmers University of Technology. Publ. S 94:6, Goteborg, Sweden.
Session A4 ANALYSIS OF SHELLS AND FRAMES
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
233
BUCKLING OF AXIALLY COMPRESSED CYLINDRICAL SHELLS MADE OF AUSTENITIC STAINLESS STEEL AT AMBIENT AND ELEVATED TEMPERATURES K. T. Hautala and H. Schmidt Department of Civil Engineering, University of Essen, D-45117 Essen, Germany
ABSTRACT Systematic experimental and numerical investigations on the axial load carrying and buckling behaviour of austenitic stainless steel cylinders both at ambient and at elevated temperatures have been carried out. Based on the evaluation of the results, design recommendations are put forv^ard.
KEYWORDS Cylindrical shells, shell buckling, stability design, austenitic stainless steel, elevated temperatures.
1 INTRODUCTION Austenitic stainless steels are used in tank and process engineering structures to an increasing extent, not only because of the corrosion resistance, but also because of the excellent temperature resistance of the material. One typical application are double-skin chimneys in modem power plants, where the inner liner which carries the flue gas is made of stainless steel. The operating temperature is usually between 400°C and 500°C, and the grades of austenitic stainless steels used in these constructions are up to X6 CrNiMoTil? 12 2. When dealing with some recent damages at flue gas liners, it was found out that there are no suitable design rules available in structural codes or recommendations to be used in stability calculations of cylindrical shell structures made of stainless steels. The supplementary rules of Eurocode 3 for stainless steels [ENV 1993-1-4-1996] are, on the one hand, applicable only in case of ambient temperature, and provide, on the other hand, stability design rules only for beam and column structures. The buckling resistance of stainless steel shell structures at elevated temperatures might be roughly estimated, for example, from ECCS-R (1988) or DIN 18800-4 (1990) by introducing temperature-reduced material properties instead of those specified for structural steels. However, because no verification of such a design procedure exists, high safety margins would have to be used. For the economic and safe use of austenitic stainless steels in shell-like structures it was, therefore, necessary to develop simplified shell buckling design rules at ambient and elevated operating temperatures.
234
The present investigations of which a concise survey is given in this paper have been carried out by the first author during her staying at the University of Essen from 1994 to 1998. They are documented in full detail in [Hautala/Schmidt 1998] and [Hautala 1998]. The latter thesis contains an extensive list of references; only the publications of Robertson (1928), Saal et al. (1979) and - very recently - Salmi/AlaOutinen (1997) on axial buckling tests on cylinders made of austenitic stainless steel are mentioned here.
2 AXIAL LOAD TESTS AT AMBIENT AND ELEVATED TEMPERATURES 2.1 Test program The objective of the experimental study was to examine the three basic parameters "steel grade", "slendemess" and "operating temperature" influencing the buckling behaviour of axially loaded cylindrical shells made of austenitic stainless steel. Three different austenitic stainless steel grades, i.e. AISI 304 (X4CrNil8 10), AISI 316L (X2CrNiMol7 13 2) and AISI 316Ti (X6CrNiMoTil7 12 2), were chosen to the study because of their frequent use in constructions such as silos, chimneys and tanks. Three different geometries with r/t-ratios of 50, 150 and 400 were investigated in order to cover the whole elastic-plastic region. Thus the non-dimensional slendemess X = V(fy/aci) varied approximately between 0.3 and 1.0. The length of the specimens was kept constant 1 = 350 mm. The investigated test temperatures were chosen to be 20°, 100°C, 250°C and 400°C. The first elevated temperature 100°C is the limit value up to which many structural codes assume negligible influence of the steel temperature. The two other temperature values 250°C and 400°C stand for typical applications in processing and power plants. Altogether 37 tests on axially loaded stainless steel cylindrical shells under ambient and elevated temperatures up to 400°C and 6 additional reference tests on mild steel cylinders were carried out.
2.2 Results of material testing Because of its crucial importance for the buckling behaviour, also the material behaviour of the used austenitic stainless steel grades under ambient and elevated temperatures was extensively investigated. Besides providing the factual material properties for non-dimensional evaluations of the buckling test results and for any numerical comparison analyses to the buckling tests, the following direct conclusions may be drawn from the material testing program: • The material properties of all the tested sheets of austenitic stainless steels, both at ambient and at elevated temperatures, were better than specified in the relevant technical delivery codes such as [EN 10088-2-1995] or in relevant design codes such as [ENV 1993-1-4-1995] and [ENV 1993-3-21997]. • The relative temperature dependence of the material properties agreed rather well with the relevant codified informations up to 500°C. • The Ramberg-Osgood material model approximates the real material behaviour of austenitic stainless steels very well up to 2% strain and up to 400°C. • The compressive material properties of austenitic stainless steel sheets may be taken equal to the tensile ones when considering the transverse properties of the sheets. Special care should be exercised when concerning the longitudinal compressive material properties of the sheets, especially at elevated temperatures, because in this case the material behaviour under compression may be somewhat worse than under tension.
235 2.3 Results of shell buckling tests Resulting from the manufacturing procedure, the test cylinders were rather perfect in meridional direction (with maximum deviations from the "best-fit" straight generators of about 1.0/0.5/0.2 times the wall thickness for r/t = 400/150/50), but rather imperfect in circumferential direction (with maximum deviations from the "best-fit" circles of about 4.0/2.0/1.0 times the wall thickness), especially at the longitudinal weld. Because of this complexity of the imperfection situation, conclusions from the buckling tests have to be drawn with special care. Nevertheless some direct conclusions may be drawn from the buckling tests, as follows: • The relative buckling stresses at ambient temperature show no significant differences between austenitic stainless steel and mild steel for r/t = 50 and 150. For r/t = 400, a vague tendency of the austenitic stainless steel specimens to buckle at lower relative stress levels than the mild steel specimens may be recognised. However, the scatter is rather high. • The relative buckling stresses at elevated temperatures (related to the relevant 0.2% proof stress at the same temperature) show no significant dependence on the temperature for AISI 316L and AISI 316 Ti, but a tendency to decrease with increasing temperature for AISI 304. However, the scatter is rather high for r/t = 400. • The temperature reduction of the buckling stresses correlates rather well with the temperature reduction of the 0.2% proof stresses, but does not correlate with the temperature reduction of the elastic modules. However, the scatter is again very high for r/t = 400. • The buckling modes are similar for mild steel and for stainless steel at all temperatures, depending only on the shell slendemess. An observation from the elevated temperature tests that has nothing to do with buckling but could be helpful for practitioners when applying stainless steels at elevated temperatures: the structure looses its bright colour above 150°C and gets brownish-dark at 400°C.
3 NUMERICAL INVESTIGATIONS 3.1 Objective of the numerical investigations Fig. 1 illustrates the material behaviour of mild steel and stainless steel by the aid of a schematic stressstrain diagram. From this figure it becomes clear that the material behaviour of stainless steel strongly differs from that of mild steel when the stress value exceeds the proportional limit. The objective of this numerical study was to define • the slendemess limits within which the lower shape of the non-linear stress-strain-curve causes the axial buckling behaviour of stainless steel cylinders to differ negatively from the axial buckling behaviour of mild steel cylinders, and • how these differences can be taken into account when designing stainless steel shell structures. The partly better material performance of stainless steels in relation to mild steel, resulting from their strainhardening property at higher strains, cannot be utilized in buckling design because of the fact that the stability phenomena occur at rather small strain values. In order to achieve the above-stated objective, a comprehensive numerical parametric study was carried out. It included the same three typical austenitic stainless steels and the same four temperatures that were used for the experimental part of the investigations. Regarding the geometry of the cylinders, r/tratios of 40 to 3000, meaning non-dimensional slendemesses X of ca. 0.2 to 2.5, were included. The main numerical impact from the different temperatures comes from the fact, that with increasing tem-
236 perature the 0.2% proof stress (used as fy, see Fig. 1) drops more rapidly than the initial elastic modulus Eo (used as E). That results in increasing E/fy-ratios when the temperature increases (see Table 1).
0.001
0.002
0.003 strain [-]
0.004
0.005
0.006
Figure 1: Stress-strain-curves of mild steel and stainless steel (schematic) TABLE 1 MATERIAL PROPERTIES OF THE INVESTIGATED STEEL GRADES, AS GIVEN FOR DESIGN PURPOSES IN EUROCODE 3 PART 1-4 (ROW 1) AND AS SPECIFIED FOR DELIVERY IN EN 10088 (ROWS 2 TO 5) AISI316Ti fy E*10'^ E/fy [X] [N/mm^l 20 240 200 833 20 215 200 930 100 185 194 1049 250 157 183 1166 |400 135 172 1274 T
fy
AISI316L EMO"' E/fy
[N/mmT 240 190 166 127 108
200 200 194 183 172
AISI304 fy
E*10*^ E/fy
[N/mmT
833 1053 1169 1441 1593
220 195 157 118 98
200 200 194 183 172
1 1
909 1026 1236 1551 1755|
3,2 Basic philosophy of the numerical analysis It is well-known that, when aiming at realistic calculated buckling strengths of real shells, it is necessary to include the imperfections into the numerical model. It is, on the other hand, just as well-known that, no matter how sophisticated a numerical imitation of an imperfection field may be, it still represents merely a "substitute imperfection" because certain components of the imperfections (residual stresses, inhomogenities, anisotropics, loading and boundary inaccuracies) are not included and must therefore be "substituted" in the imperfection model. Having this in mind, it made sense to use for an extensive parametric study like the present one (ca. 6000 geometrically and materially non-linear shell analyses) an as simple substitute imperfection model as possible. After a series of pilot studies [Hautala 1998] a single axisymmetric inward substitute imperfection with a sinusoidal meridional shape located in the middle of the cylinder length was chosen. It provided the possibility to use a specificly developed shell-of-revolution computer program [Esslinger/Wendt 1984] which consumed about 10% computer time compared to a conventional FE-package. The basic idea was to calibrate the substitute imperfection against codified characterictic buckling curves for structural steel shells and then calculate directly the difference caused by the non-linear stress-strain-relationship of austenitic stainless steels.
237 3,3 Numerical buckling curves for mild steel and stainless steel The calibration yielded a substitute imperfection amplitude WQ = r/500 to be appropriate, independent on the shell slendemess. In Fig. 2 the resulting numerical buckling curve using a bilinear material model with E == 200000 N/mm^ and fy = 240 N/mm^ (upper curve) is shown to agree rather well with the characteristic buckling design curve of ECCS-R for structural steels. A constant ratio wo/r means, in fact, an increasing normalised imperfection amplitude wo/t with increasing X or r/t, respectively. This is a well-known characteristic of real shells. In the parametric study a pair of computer runs were performed for every given set of data r/t and E/fyi firstly using the bilinear elastic-plastic material model and secondly using the strainhardening material model according to Ramberg-Osgood (see Fig. 1). The boundary conditions were taken to be simply supported. The type of the performed shell buckling analysis is in [ENV 1993-1-6-1998] called OMNIA (geometrically and materially non-linear analysis applied to an imperfect structure). Fig. 2 shows, as an example, the pair of numerical buckling curves for E = 200000 N/mm^ and fy = 240 N/mm^. The numerical buckling load of the imperfect shell was defined either as bifurcation load or as limit load or as fully plastic load where yielding has penetrated the whole thickness, whichever is reached first. In the plastic region the latter was relevant, in the elastic-plastic region it was usually the limit load, and in the elastic region always bifurcations were found before getting to the limit load. 1.4-1
•O "3
1
E = 200000 N/mm^ 1.2 -l-fy = 240 N/mm^
0.5
-r
.
\ n = 6.0 -\v^o = r / 500 .
1.0
1.5
2.0
non-dimensional slenderness X Figure 2: Numerical buckling curves for wo = r/500 in comparison with the buckling design curve of ECCS-R. As can be seen from Fig. 2, the numerical Ramberg-Osgood curve is in a certain slendemess region located below the elastic-plastic curve. Within this region, a correction for stainless steel is necessary when using the actual design rules for mild steel cylindrical shells. 3,4 Definition of material buckling correction factors for stainless steels In the next phase the numerical Ramberg-Osgood buckling loads were divided by the elastic-plastic buckling loads. This ratio is now treated as the material buckling correction factor 4^ of a material following the Ramberg-Osgood material model for which Eo and Rpo.2 are taken as being E and fy. These correction factors are plotted in Fig. 3 for the steel grade AISI 304 for all of the four investigated temperatures as function of the non-dimensional slendemess (Fig. 3a) and also of the r/t-ratio (Fig. 3b). The
238 latter is, of course, only valid for the specific E/fy-ratios of AISI 304 (see Table 1). For the other two investigated steel grades the results look rather similar [Hautala 1998]. For T = 400°C, correction factors for a very small imperfection amplitude (wo = r/3000) and for the perfect shell are also included in Fig. 3 - additional to the "standardized" substitute imperfection amplitude Wo = r/500. For these very small or zero imperfections the drop from the elastic-plastic buckling load to the Ramberg-Osgood buckling load is more significant than for the larger imperfection. However, this is merely a scientific aspect because the less reduced buckling load of the more imperfect shell is still smaller than the more reduced buckling load of the less imperfect shell. 1.05 1.00 O)
*HS—Ql
^
= ^0.95o o 3 JS 0.90 4-
T = 20X...100°C
.2 S 0.85 -
T = 100°C...400°C
^ o
« §0.80 + =
o
0.75
1 a) -1
1 0.5
^^^:^
1—
1 1.5
1.0
non-dimensional slenderness A.
D
—0
-O—20X -D-IOO'C —A—250X —O-400°C —X—400°C —1—400°C 1
—0
Wo = r/500 Wo = r/500 Wo = r/500 Wo = r/500 Wo = r/3000 perfect —1
2.0
1.05
3 4S 0.90 +
f= .2 $ 0.85 +
T = 20''C...100''C
-20°C -100X -250X -400X -400°C -400''C 1
T = 100°C...400''C
g I 0.80 +
1500
2000
radius to thickness ratio r/t
Wo = r/500 Wo = r/500 Wo = r/500 Wo = r/500 Wo = r/3000 perfect
3000
3500
Figure 3: Material buckling correction factors for austenitic stainless steels at 20°C...400°C as function of a) the non-dimensional slenderness and b) the r/t-ratio.
3.5 Approximate equations for material buckling correction factors On the basis of the above described observations, in both of the diagrams of Fig. 3 two outer limit polygons are shown (thickened straight lines) which would include the results up to 100°C and above 100°C, respectively, for substitute imperfection amplitudes between wo = r/200 and r/800. This represents about the range of practical imperfection levels as implied by the various shell buckling design codes [Hautala 1998]. The small shortcomings in the regions X = 0.3..0.4 and X = 1.50..1.60 are neglected. The transition between the minimum plateau of 0.80 for T < 100°C and of 0.75 for T > 100°C and the level of 1.0 is set to be linear in order to simplify the equations. Consequently, the approximate equation for determining the material buckling correction factor at temperatures up to lOO^'C may be written as function of the non-dimensional slenderness X as follows:
239 T = 1.00 = 1.00-0.800 (X-0.40) = 0.80 = 0.80+1.000 (X-0.80) = 1.00
0.40 0.65 0.80 1.00
< < < <
X X X X X
<0.40 <0.65 <0.80 < 1.00
(1)
and, introducing E/fy = 909 (see Table 1), as function of the r/t-ratio as follows: 4^ = 1.00 = 1.00-1.389 (O.OOl-r/t-0.088) = 0.80 = 0.80+1.010 (O.OOl-r/t-0.352) = 1.00
88 232 352 550
< < < <
r/t<88 r/t<232 r/t<352 r/t<550 r/t.
(2)
At temperatures above 100°C the material buckling correction factor may be established as follows: 4^ = 1.00 = 1.00-0.714 (A--0.30) = 0.75 = 0.75 + 0.833 (?i - 0.80) = 1.00
0.30 0.65 0.80 1.10
< < < <
X X X X X
<0.30 <0.65 <0.80 < 1.10
(3)
and, introducing E/fy =1755, again as function of the r/t-ratio as follows: ^ = 1.00 = 1.00 - 0.708 (0.001-r/t - 0.096) = 0.75 = 0.75 +0.413 (0.001-r/t-0.680) = 1.00
96 < 449 < 680 < 1285 <
r/t < 96 r/t < 449 r/t < 680 r/t < 1285 r/t.
(4)
It should be kept in mind that these equations have been derived by using the material properties of AISI 304 according to Table 1. However, they may conservatively be used for any other austenitic stainless steel.
4 VERIFICATION OF THE MATERIAL BUCKLING CORRECTION FACTORS The basic idea was to supplement the modem buckling design codes to include stainless steels by adding a material buckling correction factor to the existing design method without further changes of the codes. This factor takes all the relevant influences resulting from the use of the stainless material into account. The actual buckling design stress would be simply multiplied by this factor, and thus, a reduced buckling design stress would be reached. In order to verify the applicability of the proposed equations (1)...(4), the experimental results of the actual study are compared with the design rules of ECCS-R: Fig 4a shows the comparison with the actual characteristic design values and Fig. 4b the comparison with the improved characteristic design values. This kind of presentation shows directly whether the codes are giving safe, exact or unsafe estimations when compared with the experiments. Of course, the measured material properties have been applied when calculating the values of both axes.
240 1.2 -|-relative value of buckling load 1 0 "f-technical tensile material properties
0.4 0.6 0.8 improved ECCS-R
Figure 4: Comparison of the experimental results with a) the actual and b) the improved characteristic design values of ECCS-R. It is obvious from Fig. 4a that ECCS-R gives unsafe results for most of the cylinders with r/t = 400 and 150 and for some of the cylinders with r/t = 50. Thus it is clear that the design procedure of ECCS-R has to be supplemented to include stainless steels. As Fig 4b demonstrates, this can be done by using the material buckling correction factors of equations (1)...(4). The fact that some of the specimens with r/t = 150 and 50 are still slightly on the unconservative side has to do with a known principal shortcoming of the ECCS-curve for medium-thick cylinders. The proposed material buckling correction factors for stainless steels may be used in a similar way in connection with other codes and recommendations. 5 ACKNOWLEDGEMENTS The authors appreciate the financial support of the Academy of Finland and the "Deutsches Institut fur Bautechnik" when carrying out this investigation, as well as the supply of the testing material without charge by the companies OUTOKUMPU POLARIT OY and RAUTARUUKKI OY.
6 REFERENCES DIN 18800-4 (1990). Stahlbauten: Stabilitaetsfaelle, Schalenbeulen. Berlin: Beuth-Verlag. ECCS-R (1988). Buckling of Steel Shells - European Recommendations, 4* ed. Brussels: ECCS. EN 10088-1-1995. Stainless steels - Part 1: List of stainless steels. Brussels: CEN. EN 10088-2- 1995. Stainless steels - Part 2: Technical delivery conditions... Brussels: CEN. ENV 1993-1-4-1996. EC 3: Design of Steel Structures - Part 1.4: Suppl. Rules for Stainless Steels. CEN. ENV 1993-1-6-1998. EC 3: Design of Steel Structures - Part 1.6: Suppl. Rules for the Strength and Stability of Shell Structures. CEN. ENV 1993-3-2-1997. EC 3: Design of Steel Structures - Part 3.2: Towers, Masts and Chimneys. Brussels: CEN. Esslinger, M and Wendt, U. (1984). Eingabebeschreibung fuer das Programm F04B08 "Berechnung der Spannungen und Beullasten unter..." Inst. Strukturmechanik DFVLR Braunschweig, Report No. IB 131-84/29. Hautala, K. T. (1998). Buckling of Axially Compressed Cylindrical Shells Made of Austenitic Stainless Steels at Ambient and Elevated Temperatures. Universitat GH Essen, FB Bauwesen, Dr.-Ing.-Dissertation. Hautala, K. T. and Schmidt, H. (1998). Buckling Tests on Axially Compressed Cylindrical Shells Made of Various Austenitic Stainless Steels at Ambient and Elevated Temperatures. Universitat GH Essen, Research Rep. 76. Robertson, A. (1928). The Strength of Tubular Struts. Roy. Soc. Proc. 121:A13, 558-584. Saal, H, Kahmer, H. and Reif, A. (1979). Beullasten axialgedrueckter Kreiszylinderschalen - Neue Versuche und Vorschriften. Stahlbau 48:9, 262-269. Salmi, X and Ala-Outinen, Y (1997). Cylindrical Shell Structures from Austenitic Stainless Steel under Meridional Compression. VTT Research Note 1879, Espoo.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
241
NONLINEAR ANALYSIS OF GENERAL STEEL SKELETAL STRUCTURES — PART I: THEORETICAL ASPECTS W. Guggenberger Institute of Steel, limber and Shell Stmcturcs, Technical University Graz, A-8010 Graz, Lessingstr. 25, AUSTRL\
ABSTRACT Several versions of 'geometrically exact' beam formulations, based on nonlinear continuum mechanics and including finite rotations, have been presented in recent years. It is the purpose of the present work to compare the underlying theoretical concepts and to established a technical nonlinear rod formulation (characterized by moderate rotations), which is a consistent mathematical approximation of a geometrically exact rod formulation (characterized by finite rotations). By this unique procedure it becomes possible to evaluate the many technical beam formulations of the past and to clarify existing discrepancies and contradictions in an elegant way.
KEYWORDS consistent approximation, finite rotations, geometrically exact, moderate rotations, nonlinear analysis, shear buckling load
INTRODUCTION The three-dimensional static stability and nonlinear load-carrying behaviour of the three-dimensionally loaded rod (acting as beam, column or beam-column) has been a subject of ever increasing research since more than hundred years. Several versions of 'geometrically exact' beam formulations, based on nonlinear continuum mechanics and including finite rotations, have been presented in recent years (e.g. Wunderlich et al. 1980, Reissner 1981, Simo et al. 1991, Wriggers 1988 and 1993, Antman 1995, Jelenic and Crisfield 1997). There remains the task of closing the gap to the great variety of 'technical' nonlinear beam formulations which have been established by many researchers within the past century (e.g. Prandtl 1899, Wagner 1929, Chwalla 1939, Kappus 1939, Schroder 1970, Roik/Carl/Lindner 1972, Pfluger 1975, Vielsack 1975, Gebekken 1988, Ramm and Hoffmann 1995). It is the purpose of the present paper to establish a technical nonlinear beam formulation (characterized by moderate rotations) which repre-
242 sents a consistent mathematical approximation of a geometrically exact beam formulation (characterized by finite rotations) based on the work-conjugate pair of Green-Lagrange strains and Second-PiolaKirchhoff stresses. Transversal shear deformations are naturally included and — in a second step — eliminated by the normality condition. By this unique procedure it becomes possible to re-evaluate the many technical beam formulations of the past and to clarify existing discrepancies and contradictions in an elegant way. The formulation serves as a starting point for subsequent discretization and algorithmic implementation in a Finite Element computer program for stability and nonlinear load carrying analysis of general steel skeletal structures (part II of the present paper by Salzgeber et al. 1999).
GEOMETRICALLY EXACT ROD FORMULATION AT FINITE ROTATIONS
Z=X3; W=U3t
X=Xi=S U=Ui
(a)
R = [ei;e2;e3] finite rotation of the cross section
(b)
super-imposed 'small' cross-sectional warping
(c)
Figure 1: Cross-sectional deformations, a. undeformed position (= Cartesian reference system), b. finitely rotated position and c. 'small' super-imposed cross-sectional warping (schematic view). By a beam a three-dimensional structural member is understood whose state of deformation may be characterized by three displacement and three rotation parameters which permit a mathematically onedimensional description of its mechanical behaviour with respect to a pre-defined reference axis (axial coordinate X=Xi=S in Fig. La, b). There may optionally also exist one or more additional warping degrees of freedom which are super-imposed orthogonally with respect to the finitely rotated cross-sectional plane (Fig. l.c). Therefore, in this classical way of reduced or degenerated structural modelling, any cross-sectional distorsional deformations are excluded from the beginning. Continuum Mechanical Foundations Nonlinear classical continuum mechanics serves as the theoretical framework for the subsequent derivation of the governing equations of the deformation behaviour of the rod. There exist several equivalent possibilities of mechanical description depending on the preferred choice of the work-conjugate pair of stress and strain measures (Ogden 1984). Thereby an invariant representation of the constitutive behaviour is required; i.e. it must be independent of the specific choice of stress and conjugate strain. For 'small strain'-conditions the Saint-Venant-Kirchhoff hyperelastic material modell is sufficient which, in a classical simplified way, expresses the specific strain energy function W - WSVK(C) by a quadratic dependency on the Right-Cauchy-Green tensor C=F^F, thus yielding a linear relationship between Green-Lagrange strain EoLand Second-Piola-Kirchhoff stress S (3rd col. in Tab. 1). There is much evidence that this conjugate pair S <^ C (or equivalently S <-> EQL) is of fundamental importance in the description of constitutive behaviour since both of these quantities are strictly related to the initial undeformed reference state (1st line in Tab. 1; 'basic concept'). Consequently, if alternative strain measures are optionally used, the work-conjugate stresses have to be transformed accordingly in order to
243
preserve a unique and invariant constitutive description which is termed 'materially exact' in the present context (4th col. in Tab. 1). This becomes relevant already for 'small strain and moderate rotation'conditions, as will be shown below, and which has not been taken into account in the existing literature. The work-conjugate pair of Biot stress TBIOT ^^^ Biot strain EBIQT = U-l has been utilized by Wunderlich et al. (1980) and Heins (1991) — U being the right stretch tensor — who explicitely carried out the polar decomposition of the deformation gradient F=R-U but neglected the neccessary stress transformations (2nd line in Tab. 1). The most effective approach to 'small strain' nonlinear beam and shell analysis is obtained, at first glance, by circumventing the geometrically exact polar split of the deformation gradient and replacing it in a straightforward way by a kinematical split F=R U with subsequent symmetrization U=(U+U"^)/2 (e.g. Simo et al. 1991, Wriggers 1988 and 1993), with R denoting the rotation tensor of the cross-section. This yields the concept of so-called Pseudo-Biot strains E BIOT= U -1 and conjugate Pseudo Biot stresses t BIOT which is of approximative nature, in a geometrical as well as in a material meaning (3rd line in Tab. 1). This has consequences is practical applications when shear effects become important. A more elaborate treatment of this topic is given in Guggenberger (1999). TABLE 1 GEOMETRICALLY EXACT VERSUS APPROXIMATIVE CONTINUUM MECHANICAL PROBLEM FORMULATION FOR ROD AND SHELL STRUCTURES AT 'LARGE ROTATIONS BUT SMALL STRAINS'
CONSTITUTION
KINEMATICS
O
strains and virtual strains geometrically approximative exact
material law (linearHsotropic hyperelastic)
PJ
w
basic concept
0 03
5E<5L-F'r§F
E
^BIOT « U - I =
(R from polar split: F«=BU)
EB,OT=U-I
>
Z. CO
I
O
B (0
I
O
B CO
CO O
U=(U + UT)/2 (U =R"^Ffrom kinematical split: F=RU « R U ) pseudo-Biot strain
approximative
>
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
II It
D
ET
I-
3 II
CbzwE^t
o
0} Q.
:
II
P» 4^svit<0)
aF'
»FS»F©:E6t 1-Piola-Klmhlioff
>
73 to Q.'
244 Kinematics and Deformation Gradient of the Rod Starting from the continuum mechanical concept of Green-Lagrange strain and work-conjugate Second-Piola-Kirchhoff stress ('basic concept' No. 1 in Table 1) and implementing the kinematical constraints which are characteristic for the deformation of the rod (i.e. rigid body motion of cross-sections with super-imposed 'small' cross-sectional warping) we arrive at a 'geometrically and materially exact' rod formulation for 'large rotations but small strains'. Due to the imposed restriction of 'small' strains a further approximative step may be carried out at the level of the general strain expressions yielding an extremely compact and accurate overall formulation (Eqns. 6 and 7 below). The structure of these strain equations turns out clear and simple and moreover, it is completely invariant with respect to the magnitude of the rotations.
'^VN
^ *r"*
\ ^ vQr %-
Y=X2; -<
%^^^N
^**^*«**^
/'^lil
t = dx^/dS
p=s
\
63^
I=3y
X=Xi=S^U=Ui
Z=X3 'W=U 3 Figure 2: Kinematical assumption: finite displacement U of the reference point M of the cross-section and finite rotation of the cross-sectional plane about M (rotation tensor R, depending on rotation parameters ct)i, ^2 und (1)3, e.g. Euler parameters as coordinates of a pseudo vector of the axis of rotation). The displacement assumption (Fig. 2) for finite displacements U^ = [u V w] »independent finite crosssectional rotations with rotation parameters (^{sf = U^ (^^ (^^ , as well as superimposed 'small' warping displacements is given in Eqns. l.a, b below. Neglecting warping, we arrive at the geometrically exact expression of the deformation gradient F for any point X of the cross-section (Eqn. 2). It may written in compact form by introducing effective intermediate variables v (Eqns. 3 and 4), which turn out to be the so-called Pseudo-Biot strain quantities. Finally, a kinematic split may be carried out F = RU (Eqn. 5).
AX = R((1)(5))AX = [ei((t)(5));e2(<|)(5));e3((|)(5))] • (X-X^);
with
X'=[S;Y;Z]
(l.a)
X/=[S ;Y^ ;Zj X = XM + AX + X^^J,P = (XM + U(5)) + R((|)(5))(X-XM) + co^(r,Z).v|/(5)-ei_
F = -^x =
ax
(El + U'(5))
+ R'((()(5))(X - XA,) ;e2 ;e3
Xj^\S) = t
= R 1+
(l.b)
...
(Rx^'(5)-E,)
ns)
stretched tangent vector of the rod axis
+ R R'((|)(5)) (X-X5)
^ms))
)Ei
(2)
245
H
T -
e^ t - i
r(5) = ^2 =
"i
e^J
Q{S) = axial(Q.) = " 2
and
["3
hi [«3-M
r," v=
=
= r(5) + a((j)(5))(X-X^) = r + Qx(X-X^)
r2 +
el eg' ejeg' e^ev
62-eg' 63 61'
ejea;
"-(y-yj,)Q3 + (Z-Z„)Q2 -(Z-Z„)Ql
fi
(3)
(4)
(1'-J'M)"I
Ax = RJ|+ ( r(5) + n((^(5))(X-X^)) ®Ei]=
R(l + V(8)Ej) = R
U
(5)
Green-Lagrange Section Strains The kinematical split of the deformation gradient directly yields the components of the Pseudo-Biot strains [EJJ -^y^ ;y^ = [v^ Vg Vg], explaining their preferred usage. In consequence, the Green-Largange strains may be calculated in a geometrically exact way (Eqn. 6). The Green-Lagrange shear strains obviously coincide with the corresponding Pseudo-Biot components. The rigid-body condition of the cross-sectional plane is exactly reflected by the 2x2-zero-entries. However, the normal strain component E11 appears to be enhanced by quadratic terms. At this step the hypothesis of 'small' strains may be invoked and in consequence the underlined term — as the only geometrical approximation — may be neglected in Eqn. 6. The canonical forms of the normal strain are given for the general case (Eqn. 7) and for the special cases of plane deformations (Eqn. 8) as well as shear-rigid spatial deformations (Eqn. 9). EGL = 5 ( F ' ' F - I ) = i ( C - l ) = i((l + Ei (8)V)(l + v<8)E,)-l) =
^11
Y2/2
V l + ^ V l ' + |(v|-^v|);V2/2
Y3/2
Y2/2
0
0
Y3/2
0
0
Vo/2
'
0
I
0
I
V3/2
(6)
v i + i ( v | + v^) =
= r, JiiBlii I p l i i l -(Y-Y^)(n,^ TjOi) + (z - z^)( ^2 - r^Oi) + /^^ • Jin' ||||||||||||||;:|g|||
V
^,,,™.^——'
*
Q. axial strain
^11=
T^ -{Y-YM)^^^{Z-Z^)Q.^ v^
*t
Wagner strain
... 2D-case (Tg = Qg = 0 and ^ 1 = 0; no torsion)
^ ^ \ d
(7)
J
Q9
bi-axiai bending strain
£11 = [ r , + i p j J - y • ^3
.^-.yii iiiiiiiiiii y
... 3D shear-rigid case (Tg = Fg = 0)
Wagner strain term
(8)
(9)
246 CONSISTENT QUADRATIC APPROXIMATION OF THE SECTION STRAINS Consistent approximations are defined as mathematically complete approximations by Taylor-series expansions of the governing equations as well as any of the basic parameters up to a common prescribed polynomial order. Starting from a representation of rotations by Euler parameters (Box 1) the fundamental quantities within a displacement-based approach, i.e. the Green-Lagrange section strains E^* , r2*=r2, Fg* =r3 and section curvatures Q/, Q.2, ^3*; ^w* ^ ^ presented in quadratically approximated form — for the shear-flexible rod (Box 2) and for the shear-rigid rod (Box 3). The strains of the shearrigid rod served as starting point for the development of a nonlinear ¥E program (Salzgeber et al. 1999). By comparison, the consistent approximations of the Pseudo-Biot strain components are also presented (Box 2). There are pronounced differences in the normal strain and bending curvatures due to transverse shear effects. The Wagner-term (Wagner 1929) is completely absent in the Pseudo-Biot approach which means that due to this defect torsional buckling cannot be represented even qualitatively.
,
R = exp(O) = I + 0 + - 0 0 + ...=
2
3
2
2
,
.
_^l2
^^ ~2~
_^(|>1<1>2
^ ^ 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
= r; = r, + i(r^ + r^) = f/' + i(V'' + W^); Q ;
= 02-^2^1
al
= 2' + ^(|'^
= ^ 3 - ^ 3 ^ 1 = 3'-^(4)l'l2')-l'W' *
1
2
1
^
Q^r = :^Oj = ;^<|> • ...Wagner strain term
Box 2: Consistent quadratic approximation of the Pseudo-Biot section strains T^, Pg, Fs, Q.^, Qg* ^3 and the Green-Lagrange section strains (marked by superscript asteriks) of the shear-flexible rod NORMAL HYPOTHESIS: (j)2 = -W{l-ir) r. = r, = 0 =
+ (^^VV2
und
(|)3 = V'(l-f/') + (|)jW/2
MODMED SECTION STRAINS OF THE SHEAR-RIGID ROD:
^ii.o = r; = r, = u^ + i{v\w^)
"1,0 = (t)' + i(WV"-W"V) "2,o = - ( ^ ( l - f ^ ' ) ) ' + ^'
* 1 2 12 V = 2^1,0 = 2^'
"3.o = ( m - ^ ' ) ) ' + (|)W^'
Box 3: Consistent quadratic approximation of the normal hypothesis (F2 = F3 = 0) and the modified Green-Lagrange section strains of the shear-rigid rod (with (^i = (^)
247 EXAMPLE : SHEAR BUCKLING LOAD OF A ROD IN THE Y-Y PLANE The derived strain equations (Box 1-3) may be used to solve practical problems within the framework of a displacement-based approach. This is demonstrated by the classical example of shear buckling of the Euler-rod under constant normal force with hinged supports at both ends. Thereby the effect of (linear) prebuckling deformation at the state of bifurcation is also taken into account (compression Ecr > 0). The shear buckling loads due to various analysis approaches are compared in Box 4 and displayed in Fig. 3. This is a simple yet controversial problem in the history of structural mechanics, the solution of which having been given for the first time by Engesser in 1891, neglecting prebuckling deformations. In the present paper this problem is solved in a unique way by a continuum-mechanically well founded approach. The solution, based on Green-Lagrange strains, has to be viewed as the correct and ultimate solution of the problem (Eqn. 1 in Box 4); it coincides with Engesser's early formula (for shear only). K.
1.GREEN*LA0RANGETf == 2.BI0T: (materially inexact alternative) 3. PSEUDO-BIOT:
4a.ZIEGLER(1982):
1
Ncr ^e
1 i _ p _ p f i , ? £ V i ^ . 3 ^ ^' ""cr]^''^ 4 J 4 S 4S
1
l-2z,, + N,/S ^ ^ = ^, _ , J , , j,^,sy
N^ N.
Jl+4(N/S-2E,)-l 2iN/S-2e,)
4b.ENGESSER(1891): ^ ^ =
^
^
Box 4: Shear buckling loads including the effect of prebuckling deformation (compression Ecr > 0; compression is positive) — comparison of various analysis approaches ABBREVIATIONS: D = EA ...axialStiffness S = GA^.,. shear stiffness K = EJ
...bendingstiffness
Ne = T^ —...Euler buckling load e^ = —^ ...prebuckling strain at ^ Euler buckling load Ncr>0
... critical buckling load
A^ 8^^ = - ^ ...prebuckling Strain at ^ critical buckling load (compressive load/ strain are positive)
1 1.5 2 2.5 3 3.5 dimensionless shear stiffness S/Ng
4
Figure 3: Shear buckling loads for various analysis approaches in dependence of shear buckling stiffness and axial pre-buckling stiffness: 1. Green-Lagrange basic concept (geometrically and materially exact approach), 2. Biot strain (geometrically exact; materially inexact alternative), 3. Pseudo-Biot stress and strain (geometrically and materially inexact) and 4. Engesser (1891), Ziegler (1982).
248 CONCLUSION A technical nonlinear rod formulation has been derived (characterized by moderate rotations), with the cross-sectional strains being consistent quadratic approximations of a continuum-mechanically based geometrically exact rod formulation (characterized by finite rotations). For practical relevance it turns out that the formulation has to be based on the work-conjugate pair of Green-Lagrange strains and Second-Piola-Kirchhoff stresses, assuming the simple hyperelastic Saint-Venant-Kirchhoff material model for 'small strain'-conditions. Consequently, if alternative strain measures are used, the conjugate stress measures have to be transformed accordingly. This was demonstrated by the classical example of shear buckling of a beam, yielding inconsistent buckling loads for these competing analysis approaches.
REFERENCES Antman, S. S. (1995). Nonlinear Problems of Elasticity. Applied Mathematical Sciences 107, Springer. Chwalla, E. (1939). Die Kippstabilitdt gerader Trdger mit doppeltsymmetrischem I-Querschnitt, Forschungshefte aus dem Gebiete des Stahlbaus 2, Springer. Engesser, F. (1891). Die Knickfestigkeit gerader Stabe. Zentralblatt Bauverwaltung 11,483-486. Gebekken, N. (1988). Eine Eliefizonentheorie hoherer OrdnungfUr rdumliche Stabtragwerke. Dissertation, Universitat Hannover. Guggenberger, W. (1999). Nichtlineare Berechnung raumlicher Stabtragwerke — geometrisch exakte Formulierungen und konsistente Approximationen. Baustatik-Baupraxis BB7, March 1999, Aachen. Heins, E. (1991). Eine allgemeine nichtlineare Stabtheorie mit einem dreidimensionalen Rifimodell fur Stahlbeton. Dissertation, Technische Universitat Munchen. Jelenic, G., Crisfield M.A. (1997). Geometrically Exact Strain-Invariant 3D Beam Finite Element. Dept. of Aeronautics, Imperial College, London, 117-124. Kappus, R. (1939). Zur Elastizitatstheorie endlicher Verschiebungen. ZAMM 19:5, 271-285 and 19:6, 344-361. Ogden, R. W. (1984). Nonlinear Elastic Deformations, Ellis Horwood, Chichester. Pfluger, A. (1975). Stabilitdtsprobleme der Elastostatik, 3. Aufl., Springer. Prandtl, L. (1899). Kipperscheinungen, Inaugural Dissertation, Munchen. Ramm, E., Hoffmann, Th. J. (1995). contribution in: Der Ingenieurbau Baustatik/Baudynamik (ed. Mehlhom, G.), Ernst & Sohn. Reissner, E. (1981). On finite deformations of space-curved beams. J. Appl. Math. Phys. 32, 734-744. Roik, K., Carl, J., Lindner, J. (1972). Biegetorsionsprobleme gerader dUnnwandiger Stabe, Ernst & S. Salzgeber, G., Guggenberger, W. (1999). NonUnear Analysis of General Steel Skeletal Structures — Part II: Computer Program and Practical Applications. Int. Conf. ICSAS, Espoo, June 1999, Helsinki University of Technology, Finland. Schroder, F. H. (1970). Allgemeine Stabtheorie des diinnwandigen raumlich vorgekriimmten und vorgewundenen Tragers mit groBen Verformungen. Ing. Arch. 39, 87-103. Simo, J. C , Vu-Quoc, L. (1991). A Geometrically-Exact Rod Model Incorporating Shear and TorsionWarping Deformation. Int. J. Solid. Struct. 27:3, 371-391. Vielsack, P. (1975). Lineare Stabilitatstheorie ealstischer Stabe nach der zweiten Naherung. Ing. Arch. 44, 143-152. Wagner, H. (1929). Verdrehung und Knickung von offenen Profilen. Festschr 25 J. TH Danzig, Danzig. Wriggers, P. (1988). Konsistente Linearisierungen in der Kontinuumsmechanik und Anwendung auf die Finite Elemente Methode. Forsch. Ber Uni. Hannover, F 88/4, Nov. 1988. Wriggers, P. (1993). Kritische Wertung nichtlinearer Stabtheorien bei Anwendung auf baustatische Aufgabenstellungen. Tagung Baustatik-Baupraxis BB5, TU MUnchen, Msirz *93, 18.1-15. Wunderlich, W, Obrecht, H. (1980).contribution in: Nonlinear Finite Element Analysis in Structural Mechanics, Proc. of the Europe-US Workshop, Ruhr-University Bochum, Springer,Berlin, 185-216. Ziegler, H. (1982). Arguments For and Against Engesser's Buckling Formulas. Ing. Arch. 52, 105-113.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
249
NONLINEAR ANALYSIS OF GENERAL STEEL SKELETAL STRUCTURES — PART II: COMPUTER PROGRAM AND PRACTICAL APPLICATIONS G. Salzgeber^ and W. Guggenberger^ ^University Assistant, ^Assistant ssistant Professor, Institute of Steel, Tin Timber and Shell Structures, Technical University Graz, A-8010Graz,Lessingstr. 25, AUSTRIA
ABSTRACT In this paper a new computer program (NLBEAM3D) is presented for nonUnear Finite Element analysis of general skeletal structures. The theoretical background is represented by a consistent quadratic approximation of the geometrically exact finite rotation beam equations (moderate rotation approach). Cross-sectional warping and geometrical imperfections are accounted for in a consistent way. Nonlinear material behaviour including initial residual stress conditions may be prescribed for the individual parts of parametrically defined cross sections. Practical examples demonstrate the basic completeness of the presented element formulation and the modelling capability of the developed computer program. KEYWORDS computer program, nonlinear, beam element, moderate rotation, warping, geometrical imperfection, path following algorithm INTRODUCTION In the following a brief account is given on the theoretical background of the developed Finite Element program NLBEAM3D. The program is aimed at the study of the nonlinear load carrying and stability behaviour of general three-dimensional steel skeletal structures, however, it is designed in such a way that any material or material combination may be analyzed as well. Finally two practical examples are analyzed which serve for verification purposes and demonstrate the robust performance of the program. Theoretical background The theoretical basis of the program NLBEAM3D is given by a moderate rotation formulation as follows. The present displacement-based formulation can be derived from the geometrically exact strain equations, which are valid for finite rotations, as consistent quadratic approximations of these strains.
250 The assumption of 'small strains' is taken into account by a further approximation step which is carried out already at the finite rotation level as shown in the co-paper by Guggenberger (1999). Using GreenLagrange strains and work-conjugate Second-Piola-Kirchhoff stresses the equilibrium conditions are formulated in integral form by the principal of virtual work (Eqn. 1). The effect of transveral shear deformation is suppressed by default by invoking the normal hypothesis which is indicated by the index '0' in Eqns. 1 and 2 (Bemoulli-beam-formulation). -6W.. = j(N5Ej 10 + M26iQ2,o + M36Q3 0 + M(^5Qj^ 0 + K6Q^ 0 + M^aQi o)dL
(1)
The section strain and section force components are defined in Eqn. 2. The according virtual strain components consequently result by variation of the actual section strains. The nonlinear parts of torsional twist take account of the refined effects of the deformation-induced curvature of the initially straight beam and become even more important in the presence of imperfection-induced initial curvature. This essentially means, that also the effects of slightly curved beams may be accurately represented within this theoretical framework. For comparison purposes these nonlinear parts of torsional twist and warping curvature have been optionally neglected (dash-underlined terms in Eqn. 2). Finally, the stiffness matrix results by consistent linearization of the internal virtual work expression for each beam element about the actual deformed state. By simplification, the nonlinear interaction of axial dilatation with bending about both axes has been neglected throughout, which means the term (I-U') is not considered in the bending curvature terms (solid-underlined terms in Eqn. 2). kinematical effect
section forces
section strains
N = jSii(Eo + E)-dA
axial dilatation
E i i , o = U' + ^ - ( V ' ' + W'^)
torsional twist
Q i 0 =
bending about y-axis
^2,0 = - ( W ' - ( i - u ' ) y + v " - ^
A
bending about z-axis warping curvature Wagner effect
2
Mj = G I ^ O i 0
^3,0= (v-(i-u')r + w".$ r. _ ^ ._ v-.w'-v'.w-
^ V o - ^^1,0 - s> • 0
1 A'2
2
M2 =
JSii-ZdA A
M3 =
(2)
-jSijYdA A
Mco=
jSii-codA A
K
=
jSu-R^dA A
Geometrical Imperfections Geometrical imperfections are taken into account at a level of approximation which is consistent with the overall approximation of the section strains and fully integrated into the element formulation. By simplification the section strains of the imperfect structure are calculated as the difference of total strains and imperfect strains, both with respect to the straight reference position: E ~ E^^^^i - Eij^pQrfecv Discretization and Locking Phenomena The bending and torsional deformation quantities are discretized by Hermitian (cubic) shape functions. The axial displacement is discretized by at least quadratic shape functions which is dictated by the basic requirement of accurate strain representation for arbitrary eccentric positions of the reference
251
fibre. The discretized nonlinear normal strain results in polynomial order four, i.e. an axial displacement interpolation of polynomial order five is required to enable a complete interaction of the contributing strain distributions ('enhanced assumed strain' formulation). In order to avoid nonlinear membrane locking reduced integration with two Gauss points only could alternatively be applied. This element is of low cost in comparison to the enhanced formulation. If the enhanced formulation is used four additional internal degrees of freedom are present. These internal unknowns are eliminated by static condensation of the element stiffness matrix and internal force vector. This element formulation is therefore costly and, moreover, five Gauss points are necessary. Both of these methods were incorporated into the developed computer program. The elastic torsional shear must be integrated exactly by three Gauss points to avoid zero-energy modes. Therefore a selective integration scheme becomes necessary. In a general setting all strain components would have to be evaluated at identical integration points in order to model the interaction of these stress components within a general three-dimensional elastic-plastic material law. This is not neccessary in the present Bernoulli-beam formulation since transversal shear is not included and torsional shear is treated as purely elastic — in an inconsistent but well-justified way. COMPUTER PROGRAM (NLBEAM3D) A Finite Element computer code (Salzgeber 1998), written in Fortran 77, was developed which is intended for the nonlinear stability analysis, maximum load and postbuckling analysis of general threedimensional steel skeletal structures. The program was designed with respect to incorporating features which are relevant in real practical applications as described below: A clear and easy-to-use input concept was developed similar to the text-based input concept used in the Finite Element program ABAQUS (Hibbit et al. 1998). The data and variable memory management concepts are originally based on the linear Finite Element code DLEARN (Hughes 1987). Geometrically and materially nonlinear analyses of 3D skeletal structures may be carried out using prismatic straight beam elements which are based on the moderate rotation concept described above including geometrical imperfections. Moreover, a three-dimensional finite displacement truss element and a two-dimensional finite rotation Timoshenko element were implemented. Any of these elements may be used with user defined reference axes which are eccentric with respect to the centre of the cross-section. A robust path following algorithm has been implemented based on the so-called hypercube control with automatic and adaptive selection of the control parameter during the path following analysis (Guggenberger 1992). In the predictor step the new control parameter is automatically determined on the basis of the largest, linearly computed, incremental increase of any of the system parameters (deformation, rotation or load) with respect to pre-defined limits. This control parameter is kept unaltered during equilibrium iterations in the corrector steps. The path following procedure may be initiated by prescribing any of the system parameters to a specified initial value (= initial control parameter). A similar concept is presented by Chroscielewski et al. (1986). Additional built-in stabilization procedures permit the mathematically stable computation of mechanically unstable deformation states such as snap-through or snap-back points and postbuckling behaviour. Dependent on the convergence behaviour of the solution process the size of the controlling hypercube, i.e. the pre-defined limits of the allowable increments of the system parameters, will be enlarged or reduced in order to achieve optimal solution progress during path following. The solution progress may be controlled by simultaneous monitoring of an arbitrary number of solution variables. Geometrical constraints may be prescribed e.g. for the purpose of modelling rigid bodies and so on. Currently, geometrical constraint equations are limited to be linear ones in the program. In order to achieve good performance a symmetric skyline solver was implemented (Bathe 1986). The algorithm has been extended to effectively handle these constraint equations by the Lagrange multiplier method.
252
The ability to deal with geometrical imperfections is a neccessary pre-requisite for performing realistic maximum load analyses. Within the program geometric imperfections may be defined parametrically, as perturbations of the existing displacement and rotation degrees of freedom, in the local element coordinate systems. The imperfections may be generally prescribed with respect to any of the deformation parameters and therefore consist of pre-curvature, pre-twist or transversal discontinuities of the beam axis at the ends of the beam elements. Furtheron a general parametric definition of cross-sectional shapes was implemented. An example is shown for an open thin-walled circular section which is represented by four parametric cubic spline segments (Figure la). Arbitrary combinations of material properties within the cross section as well as arbitrary parametrically definable patterns of residual stress distributions may be prescribed. A cross section may consist of an arbitrary combination of thin-walled and thick-walled cross-sectional parts. A built-in warping processor automatically handles the evaluation of the warping function by a suitable general Finite Element subroutine. The integration over the individual cross-sectional parts is carried out numerically by user-selectable patterns of integration points. A comprehensive library of uni-axial nonlinear material laws of a great variety of constructional materials is implemented. Currently isotropic Mises elastoplasticity including loading and unloading behaviour, different behaviour in tension and compression, as e.g. is neccessary for the modelling of reinforced concrete, or general nonlinear elasticity, again different in tension and compression, may be optionally handled.
a) warping function
b) displacement plot
Figure 1: a) warping function of a thin-walled open circular section, b) displacement plot The loading may be defined either as nodal loads as well as distributed and concentrated element loads. Eccentrically acting element loads are assumed to be conservative, i.e. deformation-independent, however, the nonlinearity due to the rotation of the eccentric points of load application is consistently included. The loading of imperfect beams is defined with respect to the imperfect beam axis but using the length of the perfect beam axis as reference length. In nonlinear analyses it is always neccessary to define a load history. This may be carried out in a sequence of analysis steps, each of them with specific reference loads patterns, within a single analysis job. The program has the possibility to produce 3D graphical output by writing a command file in Mini-Pascal format for the CAD program MINICAD (Diehl 1997). E.g. the warping functions are plotted axonometrically, e.g. with respect to the centre of shear as shown for an open circular section (Figure la). The deformation history for several analysis steps may be plotted automatically by animation-like graphical representations. In order to get a clear impression of the results the cross sectional quantities may be plotted for different levels of abstraction. E.g. a body faced model of the initial configuration and at a postbuckled state of an I-beam under axial and transverse loading is shown in Figure lb. The deformations of the structure as well the imperfections of the initial configuration may be arbitrarily scaled.
253 EXAMPLES Cantilever with an unsymmetric cross-section under concentrated vertical loading The load carrying behaviour of an imperfect cantilever under concentrated vertical tip loading is studied. In order to demonstrate the completeness and accurateness of the mechanical formulation of the beam element implemented in NLBEAM3D numerical comparison analyses were carried out using ABAQUS (Hibbit et al. 1998). The cantilever shown in Figure 2 has a length of 180 cm and a vertical reference load of 1.35 kN. The cantilever has a paraboHc horizontal imperfection with a maximum amplitude at the tip of 0.36 cm (= L/500). The welded cross section is build up by two plates with 8 mm thickness. The web is 140 mm high and the flange is 40 mm wide. There are no restraints along the cantilever out of the plane of loading. At the built-in end the warping degree of freedom has not been fixed. Out
A8 mm
0.9 mm I ^ ^ ^ : ^ - ^ ® c ^ / 6 o
40 E
3§ 3.6 mm fy = 23.5 KN/cm^ E = 21000 KN/cm^ v = 0.3 Figure 2: Cantilever under transversal loading: system, loading and geometrical imperfection It is well known, that unsymmetric cross sections exhibit a different behaviour depending on the direction of the transverse loading. If the web of the T-section is under compression (load acting downwards; positive direction in Figure 3) the load carrying capacity is lower than in the complementary case (load acting upwards; negative direction in Figure 3). In Figure 3, for both of these loading conditions, the load-displacement curves for the out-of-plane tip deflection (V in Figure 2) are shown. The results of the beam analyses carried out by NLBEAM3D are drawn by solid lines. The thin solid lines correspond to the classical thin-walled section approach neglecting thickness effects. The thick solid lines correspond to the more general case when thickness effects are included by treatment as a general thick-walled cross-section. The results of various ABAQUS analyses are drawn as dashed lines. If shell analyses are carried out by ABAQUS using S4R-shell elements limit load factors of A = 3.62 and 5.09 are obtained for the downward case and the upward case respectively. By comparison, the analyses with NLBEAM3D, based on the thin-walled assumption (thin solid lines) results in limit load factors A = 3.41 and 5.03 respectively. The load-deflection curves look very similar to those of the shell analyses but the values of the beam analysis are in both cases on the conservative side. If the effects of plate thickness are included in the beam analysis increased values of the load factors are obtained (A = 3.71 and 5.21 respectively) which are sHghtly on the unconservative side. This may be attributed to the basically reduced kinematic modelling capability of the beam approach, compared to the shell approach, and because the torsional shear is taken into account only elastically. Carrying out ABAQUS analyses using beam element B310S, surprisingly, an invariant limit load factor of A = 4.20 is obtained, independent of the direction of loading. In order to clarify this discrepancy an analysis with NLBEAM3D has been performed neglecting the Wagner part (Eqn. 2). This results in an identical load-deflection curve (thin dashed line) which is the proof that this important term is obviously missing in the ABAQUS beam element formulation. In this example the moderate rotation approximation is sufficient since finite rotations do not actually occur. On the left-hand side of Figure 3 classical buckling eigenvalues are plotted for comparison purposes, as computed by ABAQUS and NLBEAM3D respectively.
254 5.79 5.78 (5.42)
•^6
5.21
]NLB6AM3D ^Wagn^Effect" "^ not iijcluded)
-P
4.33 3.81 3.78 (3.50) Q CO
^ <
LU CO —I
z
CO Ui
+
+P
CO -J
o <
CD
< (/>
NLBEAM3D ABAQUS
© D CO
o c > c CO o D)
•
o ^3
0 0
0
2
4
6
displacement V [cm]
8
10
Figure 3: Load-displacement curves (vertcal tip load P is acting in positive and negative direction) Nonlinear three-dimensional behaviour of a plane steel frame A steel frame which was tested by Friemann et al. (1990) is referenced for further comparison. Friemann et al. measured material data and geometrical imperfections, the latter both within the cross-sections and out-of-plane of the frame. Geometry and material data are given in Figure 4. The nonprismatic I-section of the column is represented by five elements with stepped height of the web. The system is restrained at the upper flange at the points of load application. The base plate of the frame is simply modelled as a hinged support. The specific constructional detail of the joint of the column and the beam strongly affects the geometric coupling of these structural elements (Salzgeber 1999). In the present case the warping amplitudes of both of the joining elements were directly coupled. Otherwise, in a more refined approach, the joint detail could be modelled as an eccentric rigid body by linear constraint equations. The geometrical imperfections of the test frame are introduced as follows: All vertical web plates are positioned 3.5 mm out of the perfectly centered vertical loading plane, which serves as a reference plane, due to a fabricational mismatch. The sign of the horizontal imperfection of the web plates is alternating in subsequent parts of the frame, as indicated by signs within small circles in Figure 4. The out-of-plane imperfections have little effect on the load maxima, however, they have considerable effect on the out-of-plane buckling deformations. The measured imperfections were simplified by cubic splines for each frame element. Residual stresses are not taken into account. The load maximum and the in-plane deformation behaviour primarily depend on the support conditions (warping constraints) and the flexibility of the joint detail. Analysis results of significant deformation components are compared with test results in Figure 5. The maximum load factors differ only by about 2%. Above a load factor of about A = 1 progressive deformations occur in the test probably due to the effect of residual stresses. The history of the vertical deflection W at the ridge is plotted in Figure 5a and corresponds well with the test results up to a load factor of about A = 1. The out-of-plane deflection V2 of the lower flange near the joint is very sensitive to the specific distribution of the geometric imperfections (Figure 5b). The maximum deviation of the out-of-plane displacement of 0.3 cm — with respect to the test result — is very small and amounts to only about 1/3 of the relevant imperfection amplitude.
1.14
3.36
30.8KN/cm^ 19300 KN/cm2 Web fy = 25.5KN/cm2
E = 19500 KN/cm^ Figure 4: Welded steel frame — system definitions and deformed structure at a postbucked state 1.5
1.5
I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' • ' ' I
1.221M-243
1.25
1.25
.
'
'
!
'
'
'
!
'
'
'
!
'
'
'
!
'
'
'
!
'
'
'
.
i o*- n ^ -«ri—^^s— i
- ^ H ;
0.5
•
; \ i i \i i\
i \
:
\ ^\ i\
i1
^ ^ i
i -
I ^ M
0.25 h
ill
1 - o - Experiment 1 Calculation
=,., i . , , i . , , 1 / , , , i , . . 1 , . , ^ 1
2
3
4
5
6 a)
b)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Displacement W [cm] Displacement V2 [cm] Figure 5: Load-Displacement-Curves a) vertical deflection W, b) out of plane deflections V2 In the present analysis the reference axes of the beam elements were artificially located at the outer flanges of the frame members (Figure 4). Alternatively, if the reference axes, in a classical way, are located at the centre lines of the frame members the maximum load factor increases by about 3% (A=1.26). In addition, if the supports of the frame are built-in (including warping constraint) the maximum load factor increases by about 10%. In conclusion, the test results are reasonably well reflected by the numerical analysis. In conclusion we may state that changes in modelling assumptions (support conditions; joint detail) result in minor changes of the according values of the load carrying capacity only.
256 CONCLUSION A recently developed Finite Element computer program is presented (NLBEAM3D) which is aimed at the nonlinear load carrying and stability analysis of three-dimensional skeletal structures made of steel or combined with any other constructional material. The theoretical background of the incorporated beam formulation is based on a consistent quadratic approximation of the general geometrically exact finite rotation strain equations (moderate rotation approach). The normal hypothesis of the cross-sections is implemented by default. Cross-sectional warping, geometrical imperfections, uni-axial nonlinear material behaviour and initial residual stress conditions are consistently taken into account in this beam formulation. Internal properties of the developed computer program like concepts for the management of input, memory and cross-sectional warping (warping processor) are described. The performance of the program is demonstrated by two practical examples. REFERENCES Bathe K.-J. (1986). Finite-Elemente-Methoden, Springer. Chroscielewski J., Schmidt R. (1986). A Solution Control Method for NonHnear Finite Element PostBuckling Analysis of Structures, Euromech Coll. 200, (ed. Szabo J., Caspar ZS., Tamai T.), Elsevier. Friemann H., Schafer P. (1990). Traglastversuch an einem Rahmenbinder (Beitrag zum Biegedrillknikken von Rahmenecken). Festschrift R. Schardt, Darmstadt. Diehl Graphsoft Inc. (1997). MINICAB 6.0, MiniPascal Manual Guggenberger W. (1992). Incremental Iterative Solution of Nonlinear Equation Systems in Structural Mechanics - Application of a New Consistent Automatic Variable-Parameter Control, Inst, of Steel, Timber & Shell Structures, Technical University Graz, Austria. Guggenberger W., Salzgeber G. (1998). Geometrisch und materiell nichtlineare Berechnung von Stabstrukturen mit ABAQUS - Moglichkeiten, Einschrankungen und Verbesserungsvorschlage, 8. Osterr. ABAQUS Anwendertreffen, Wien, Austria. Guggenberger, W. (1999). Nichtlineare Berechnung raumlicher Stabtragwerke — geometrisch exakte Formulierungen und konsistente Approximationen, Baustatik-Baupraxis BB7, March 1999, Aachen. Guggenberger W. (1999). Nonlinear Analysis of General Steel Skeletal Structures — Part I: Theoretical Aspects, Int. Conf. ICSAS 99, Espoo, Finland. Hibitt, Karlson & Sorenson (1998). ABAQUS Users and Theory Manual Version 5.7. Hughes T.J.R. (1987). The Finite Element Method — Linear Static and Dynamic Finite Element Analysis, Prentice-Hall. Salzgeber G. (1998). Nichtlineare Berechnung von raumlichen Stabtragwerken, Dissertation, Technical University Graz, Austria. Salzgeber G. (1999). Modellierung von Konstruktionsdetails in raumlichen Stabberechnungen mittels geometrischer Zwangsgleichungen, Baustatik-Baupraxis 7, Aachen.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
257
PLASTIC DEFORMATION AND LIMIT STATES OF METAL SHELL STRUCTURES WITH INITIAL SHAPE IMPERFECTIONS V. S. Hudramovych^ A. A. Lebedev2, V. L Mossakovsky^ • Institute of Technical Mechanics, National Academy of Sci. of Ukraine, 15 Leshko Popel St., 320600, Dnepropetrovsk, Ukraine 2 Institute for Problems of Strength, National Academy of Sci. of Ukraine, 2 Timirjazevskaja St., 252014, Kiev, Ukraine 3 State University, 72 Gagarin Av., 320625, Dnepropetrovsk, Ukraine
ABSTRACT The behavior of cylindrical shell structures with initial imperfections under plastic deformation is studied. The method of successive loading is used. Some calculated and experimental data for cylindrical shells with shape imperfections are presented. KEYWORDS Shell, plasticity, shape imperfections, deformation, limit state, calculation, experiment. INTRODUCTION The shape imperfections are typical for shell structures of different branches of engineering. Initial imperfections arise from technological processes and from diverse operational conditions. On the other hand, the geometrical imperfections are included in the models whereby the behavior of elastoplastic and other nonlinear systems is investigated (see, for example, Hutchinson (1974)). Allowing for such imperfections is a topical problem in investigations of the strength of structural members. To be specific, we will consider a structure in the form of a cylindrical shell loaded with a compressive force T and external pressure q acting together or separately. We will study the plastic deformation and limit state of such structural elements. The most suitable method for the investigations in question is the step method of successive loading. This method is a version of the methods of parameter continuation of solution in which the acting loads are taken as the parameters (see Lahaye (1948), Oden (1972), Grigoljuk &
258 Shalashilin (1988)). According to the method of successive loading, the structure loading process is divided into loading series and the stressed-and-strained state on each loading step is studied. The increments of the components of this state are added together. For this investigation the theories of plasticity in increments must be used. We will use the deformation theory of plasticity in increments . For this theory the relationships are formulated in terms of the strain Aey and stress Aay increments. The obtained equations make it possible to use other theories of plasticity where relations between strain and stress increments are given( see Hudramovych & Demenkov (1991)). The limit state is defined as the load value at which the shell deflections increase sharply. For one load q or T the limit state is determined by the limiting surface ( see, for example, Bushnell (1981), Hudramovych (1998)).
BASIC EQUATIONS The equations that relate the increments in the forces ATy and in the moment AMy to the increments in the strains Asy and in the curvatures Axy are as follows ATy = Aykl Askl+ Bykl Axkl A My = Cykl A8ki+ Dyki Axkl
(1) (2)
where (A,B,C,D)yki are functionals defined by the level of a stressed-strained state. Determinig Asiki from (1) we, obtain Aey = aykiATki + byki Axki
(3)
aiiii= (A2222Ai2i2-Ai222A22i2)/A,... , ai2i2 = (Aim A2222-Ai 122A2211)/A A= det [Aykl], byki= Biikiayii+ B22ikiay22 + Bi2kiayi2 Substituting (3) in (2) gives AMy = Cykl ATki+ d ykl Axkl
(4)
Cykl = C yi 1 a 1 ikl + C y 22 a22kl + C yi2 ai2kl dykl = C yi 1 b 1 Ikl + C y 22 b22kl + C yi2bi2kl The equations describing the behavior of shells with initial shape im.perfection are obtained using the varitional prinsiple of virtual displacement (see Washizu (1982)). Finally we obtain the nonlinear equilibrium and strain consistency equations in the increments of the stress function AFi and the radial displacement Awi for each loading step in the form (see Hudramovych & Demenkov (1991)) M'i(AFi)+ M'2(Awi)- M' (w+wO+Awi,AFi)-M' (Awi,F)- AFi'7R=Aqi (5) M'3 (AFi)4- M' 4 (Awi)+ M' (w+wH l/2Awi, Awi)+ A wi'VR =0 where w^ is the initial deflection, R is the radius of shell, M'j are the differential operators
259
M r ( v | / ) = Cll22V|/""+ C22l\\\f""+
(Cllll-2C12I2+ C2222)V|/" "
+(Cl 112+2 C1222)V|/"' • + ( 2 C1211- C2212)vi/'-+2(c'1122+ C ' 2212)V|/"' +2(c'l222+ C* 2 1 I ) H / * " + 2 ( C ' I I 1 I + C * 1211- C'l212- C' 2212)V|/'** + 2(C'1112- C'l222+ C- 1212- C- 2222)V1/" ' + ( c " l 122+2 C' ' 1222+ C"2222)v|/" +(C"1111+2C'- 1211+C •• 221 l)V}/ - - ( C " 1112+2 C'- 1212+ C"2212)V1/'-,
(6)
M y (V|/)=a2222V}/"" + ail22Vl/""+ (ail22- ai212+ a221l)V|/""
-(a22i2+ ai222) v|/"' • -(an 12+ ai2ii)vi/' —+ (2a'2222- a* 1222)11/'" +(2a- iiii-a'i2ii)v|/--(2a- iii2-2a'22i2-a* 1211+a'i2i2)M/'" + (2a- ii22-2a'22i2-a'i222 + a- i2i2)v|/"- +(a-iiii-a'-1211+a"22ll)v|/•• +(a•• 1122-a'' 1222+ a" • 2222)M/ " -(a** 1112-a' • 1212+ a"22i2)M/' • v|/""=a4 v|//ax4, v}/ " •• =54 y / ^ 2 ^2^ v|/' - =54 vj// ^x ^y^, a' • ijki= 52 aijki/5x Sy, a • ijki= d ayki/ 5 y , . . . The operator M2'(9) will be obtained from Mi'(cp) if dyki is substituted for Cijki and M'4((p) will be obtained from M'sCcp) if byki substituted for ayki. For a plane stressed state that is realized in shells and plates we have Asii= CiiAaii + Ci2Aa22+ Ci3Aai2,.. . , A8i2= CsiAan + C32Aa22+ C33Aai2
(7)
from which we find Aan = DiiAeii + . . . + D13AS12,. . . , Aai2= D31A811 + . . . + D33A812 (8) Dii=(C22 C33- C23 C32)/Ai,..., D32=(C3i Ci2- Cn C32)/Ai , Ai=det[Csi] For the deformation theory in incremenrs we have the following simplified relationships in (8) Dii = D22=4/3fi, D12 =D2i = 2/3f2,D33=Ek/3,Di3=D23 =D3i = D32=0 (9) f 1 = f3 (1 + ©)/( 1 + 4co), f2= f3 (1 -2 ©)/( 1 + 4o)), f3= Aai/A8i, co = f3/9K, K=E/3( 1 -2v), Ek= aai/a8i where E is the elasticity modulus, v is the Poisson coefficient,CTI, 8i are the stress and strain intensities. The equations (5-8) form the basis for study of the deformation and limit state of cylindrical shell structures according to the accepted methodology. RESULTS OF INVESTIGATIONS Numerical analysis Let us present some results of the numerical analysis of the behavior of shells with initial shape imperfections. Calculations were performed using the above relationships.
260
w"
^A/KA/N AW*- lO'^
Figure 1: Development of the deflections Figure la shows the development of the deflections of the cylindrical shell with the imperfections shown at the top of the figure (w^*= w^/h). The results were obtained for the shell parameters: R/h=62, L/R=l, where h, L are the thinkness and length of the shell. The material of shell is the AMG-6 aluminium alloy. Shown in the figure are the deflection increments on the loading stages preceding the carrying capacity reaching its limit. On the y axis the dimensionless deflection increment Aw*= Aw/h is plotted. The shape imperfection shown at the top of the figure is obtained as a result of the preliminary dynamic loading with a pulse of external
261 pressure. Figures lb, c depict the schemes that characterize the development of the deflection increments of the shells with the shape imperfection shown in at the tops of these figures on the loading stages just before the carrying capacity reaching its limit. The initial shape imperfections are given analytically. For figures lb,c we have respectively wO= 0.5 f^(l+5cos9) COS9 sin TCX/R w^= f^ C0S9 sin Tix/R for -nil < 9 < nil
(10)
and wO=0 on the other side of the circular cross-section. Experimental data Let us present some experimental results on the carrying capacity of a shell from the AMG-6 aluminium alloy with initial shape imperfection obtained as a result of the preliminary pulse dynamic loading in a hermetically sealed chamber (see Mossakovsky et al. (1974)). This loading was realized with the help of electric discharge in a liquid (water). The shape imperfection is shown at the top offigure2.
:tf^~*nr""^zrinnz:iz:z*iz'a:=. •/
X X
Vl.
— y ^
0
2
X
4
6
"^
Figure 2: Experimental and calculated data The solid curves show the calculated results and the dashed curves approximate the experimental data. On the y-axis the value q*=q/qo where q,qo -critical loads for shell with and without imperfection and on the x-axis ^0= f^/h where f^ - maximum deflection are plotted. Circles, triangles and crosses designate the experimental data for shells with L/R =1.5 and R/h equal to 75, 100, 150, respecdvely. The shell with R/h=150 exhibited no plastic deformation. But in this case the above relationships are suitable and we must put Ek=Es=E in them. There is a large difference between the calculation and experiment (~ 18%) for the shell with R/h=150. There is better agreement between the calculation and experiment in the other cases. Let us present some results of investigation of the mechanical properties of the AMG-6 and D16H aluminium alloys (see Lebedev et al. (1984)).
262
8 a 0
8
0
8
b 16
0
8
16
16
0
8
d 16
c
24
24 a-10~^kg/cm^
Figure 3: Yield diagrams for the D16H alloy
Figure 3 shows the limit yield diagrams for the AMG-6 aluminium alloy. When testing tubular specimens for axial load, internal pressure and torque at different stress ratios k=CTii/a22; a - k = o o , b - k = -l, c - k =0.5, d-k = 2, a=-^afi +022 .
S/10"^kg/cm2
1
0
-1
-2
-3 Syl0"^kg/cm2
Figure 4: Yield curves for the D16H alloy Figure 4 shows the yield curves in the plane Si - S2 (Si= J ? ^ (an- 0.56a22), S2 = a22/V2) for the D16H aluminium alloy. Curves 1 and 2 correspond to the well-known Mises and Tresca yield conditions. Curve 3 corresponds to the empirical yield condition — V^2 + xH^l + CTs) - G^G2 - aja3 - (2x - l)a2CJ3 + (1 - r|)—
^=1
263
k = l , j = 3 for k = l , j = 3 for
X = ^\shi (a,-a3)/(a2-a3)>x
(11)
(a,-a3)/(a2-a3)<x<(cJ,-a2)/(a2-a3)
k = l , j=2 for ( a i - a 2 ) / ( a 2 - c y 3 ) < x Tl = (cT„-XCT2,^'^)/[(A/2x'-X-a2.^'^]
aic (k= 1-3) are the principal stresses, ais is the yield point for ai, ci^^ is the stress corresponding to the yield point for biaxial tension with oixlai = %. The plastic deformation increment vector is normal with respect to the yield curve determined by condition (11). This follows from the well-known Drucker postulate. The test data shown in figures 2-4 are obtained at a temperature of 20^0. References Bushnell D. (1981). Buckling of shell-oitfall for designers. AIAA Journ. 19 : 9 , 1183-1226. Grigoljuk E.I. and Shalashilin V.I. (1988). Nonlinear problems of deformation (in Russian), Publ. Nauka, Moscow. Hudramovych V.S. and Demenkov A.F. (1991). Elastoplastic srtuctures with imperfections and residual stresses (in Russian), Acad. Publ. Naukova Dumka, Kiev. Hudramovych V.S. (1998). Plastic and creep instability of shells with initial imperfections. Solid mechanics and its applications 64, lUTAM Symposium on rheology of bodies with defects (ed. Ren Wang), 277-289, Kluwer Acad. Publ., Dodrecht/ Boston/ London. Hutchinson, J.W. (1974). Plastic buckling. Adv. Mech. 14, 67-114. Lahaye M.E. (1948). Solution of system of transcendental equations. Acad. Roy. Belg. Bull/Cl Sci. 5, 805-822. Lebedev A. A et al. (1984). Mechanical properties of the structural materials under complex stressed state (in Russian), Acad. Publ. Naukova Dumka, Kiev. Mossakovsky V.I. et al. (1974). Experimental investigations of the preliminary dinamic loading influence on carrying capacity of the cylindrical shells under lateral external pressure (in Russian). Izv. AN SSSR. Mech. Tverdogo Tela 4, 170-175. Oden I.T. (1972). Finite elements of nonlinear continua, Mc Graw-Hill Book Co. Washizu K. (1982). Variational methods in elasticity and plasticity, Pergamon Press Publ., Oxford.
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
265
BUCKLING ANALYSIS OF SHELL TYPE STRUCTURES UNDER LATERAL LOADS M.Ohga\ Y.Miyake^ and T.Shigematsu^ ^Department of Civil and Environmental Engineering, Ehime University Matsuyama 790, Japan ^Mitsui Construction Ltd., Tokyo 101, Japan ^Tokuyama Technical College, Tokuyama 745, Japan
ABSTRACT In this paper, an analytical procedure to estimate not only the buckling loads but also buckling mode shapes of the thin-walled members composed of cylindrical shell panels, v^hich are connected each other, by the transfer matrix method is presented. The transfer matrix is derived from the differential equations for the cylindrical shell panels, and the point matrix relating the State vectors between consecutive shell panels are presented. Therefore, the interactions between the shell panels of the cross sections of the members can be considered exactly. Although the transfer matrix method is naturally a solution procedure for one-dimensional problems, this method is applied to thin-walled members by introducing the trigonometric series into the governing equations of the problems. The buckling loads and mode shapes of the thin-walled members with shell type cross sections are presented for various number of shell panels, and the effects of the number and curvature of shell panels on the buckling loads and mode shapes are examined.
KEYWORDS Transfer matrix method, thin-walled member, shell structure, local buckling, global buckling, buckling mode, instability phenomenon
INTRODUCTION Thin-walled members with shell type cross sections are widely used in a broad range of structural applications to reduce the material cost as well as the dead weight of a structure. A thin-walled member, whose cross section is made up of thin plates or curved panels, may be subjected to local and overall types of instability. Therefore, the investigation of the buckling loads and mode shapes of the members is very important to clarify the instability phenomena of these members. In this paper, an analytical procedure to estimate not only the buckling loads but also buckling mode shapes of the thinwalled members composed of cylindrical shell panels, which are connected each other, by the transfer matrix method is presented. The transfer matrix is derived from the differential equations for the
266
^x
^9
•*
\/ (a)forces
V
(b)displacements and lateral load Figure 1 Cylindrical shell panel
cylindrical shell panels subjected to the lateral loads, and the point matrix relating the state vectors between consecutive shell panels are presented. Therefore, the interactions betw^een the shell panels of the cross sections of the members can be considered exactly. Although the transfer matrix method is naturally a solution procedure for one-dimensional problems, this method is applied to thin-walled members by introducing the trigonometric series into the governing equations of the problems [Ohga (1995a,b,c);Tesar(1988);Uhrig(1973)]. The buckling loads and mode shapes of the thin-walled members with shell type cross sections are presented for various number of shell panels, and the effects of the number and curvature of shell panels on the buckling load and mode shape are examined.
ANALYTICAL PROCEDURE Equilibrium Equations for Shell Panels under Lateral Loads The equilibrium equations of forces for the shell panel subjected to the lateral load are described as follows (Figure 1):
K+N'^+co'ptu = o, M;+M;-a=o,
.. . Q. N',^+N;+^
= O,
N^
—^+Q'^+Q;-cT^(^)tw**=o
M',^+M;-Q^=O,
(ld,e,f)
N,^-N^-^-^=O R
where N^,N^,N^^,A^^ :in-plane forces, M^,M^,M^^,M^:bending
(la,b.c)
and twisting moments, Q^,Q^:
shear forces, a^{= pR/t):in-iplsine force in cp direction,/? ilateral load, t ishell thickness, u, v, w \ displacements in x, (p, z directions, R '.radius of shell panel, ' = — , • = . dK Rdcp Relation between Strains and Displacements The relations between the strains and displacements for shell panel used in this paper are given as follows (Figure 1(b)): w
£x=u'^ ^'p^^^'^J^ ^ ^ = ^ ' + «*' /xz=^' + ^ . = 0 , ,
V
r^=>^'+^^-- = 0
(2a'-e)
267
Figure 2 Relation between consecutive panels
Figure 3 Cross section with shell panels
v'
\
where e^, e^ inormal strains, y^^, y^, y^^ :shear strains, K^,K^^^^^^
w
(2f-i)
xurvatures of displacements.
Transfer Matrix for Shell Panel From Eqns. 1 and 2, the partial differential equations for the state vector are obtained as follows (Figure 1): Z = Z- = A(^). Z , dcp
Z=
{w,cp^M,y,,v,u,N^,Nj
(3)
When the sides x=0 and x=a of the shell panel are simply supported, the state variables can be assumed in the form: (4)
w{x, (p) = w {(p) sin ax where a =
, a: shell panel length, m: buckling mode in x direction. a
Substituting Eqn. 4 into 3, the following ordinary differential equation referred to the variable (p only are obtained: d Z = Z*=A Z (5) Rd(p Integrating Eqn. 5, the transfer matrix, F, is obtained as follows:
Z = exp(A^)-Zo=F-Zo
(6)
where exp( A^) = I + (A^) +—(A^) +—(A^) +
,
I :unit matrix
(7)
Point Matrix As the state vectors for each panel are referred to the local coordinate system, the relations between the
268 kgf/cm
T
'
kgf/cm'
1 — ' — r -
TMM VM Pignataro
2
a^=1.0
4 6 8 (a)Buckling loads
a^=2.0 a/b=4.0 (b)Mode shapes
10 a/b
2
a^=1.0
a^=10.0
4 6 8 (a)Buckling loads
a^=3.0
10 a/b
a^=5.0 a^=10.0
(b)Mode shapes Figure 5 Buckling loads and mode shapes (R/t=100)
Figure 4 Buckling loads and mode shapes (R/t=1000)
state vectors of consecutive two panels are required, in order to allow the transfer procedures of the state vectors over the cross section of the member. Considering the relation between the state vectors at the left and right hand sides of section / (Figure 2), the point matrix, P, is obtained as follows: Z[^=Pj-Zf
(8)
where superscripts L and R indicate left and right hand sides at section /. Stability Equation and Buckling Mode Shape Applying the transfer and point matrices described above, the relation between the state vectors at both ends of the cross section of the member can be obtained as follows (Figure 3): Z3=F3 P2 F, Pi Fi Z o = U Zo
(9)
Considering the boundary condition of the member, the stability equation for the member is obtained as follows:
u' z;=o In buckling problems, it is required that the determinant of the matrix U' of Eqn. 10 to be zero.
(10)
269
(a) N=3
(b) N=5
(c) N=10
Figure 6 Analytical models
1 k .
•
1 '
1 '
. \
1J J
ZIZ^N=4 :
V \^ N
N=5
:
x / \ ^ \*-
:I
\X
; 0.01
1 ' " rvFi
:• ^\'S\ H
0.1
1 '
1
xVu^I^ "" i \ XN^^^^"^ J
• b/t=1000
• e=2.o 1
i
2
1
1
4
"-^. 1
'^C'^-"^"-^ i1
1
6
^^ 1
j i
1 1
8 ,.10 a/b
Figure 7 Buckling coefficients Substituting the buckling load obtained above, and setting an unknown variable of the initial state vector, for example WQ = 1, the relative values of the remaining unknown initial state variables can be obtained. The state vector at any point of the cross section is obtained by further transfer procedure and then the mode shape corresponding to the buckling load can be obtained simply.
NUMERICAL EXAMPLES In Figure 4(a), the buckling loads of the cylindrical shell panels under lateral load (Figure 1(b): <^=1.0, R/t=\000, ^^=1.0'^ 10. 0), obtained by the proposed method (TMM) are compared with the results obtained by the variational method (VM) and by Piganatoro (1991). The boundary conditions at (p =0, ^ are assumed to be simply supported (y=w=0). Although in the transfer matrix method the state variables are assumed by the trigonometric series in one direction only, in the variational method and Piganatoro those are assumed in both directions. The numbers indicated in the figures are the mode numbers in the ^-direction obtained by VM and Piganatoro. As shown this figure, good agreements exist between these results. Figure 4(b) shows the buckling mode shapes corresponding to the buckling loads in Figure 4(a) for some aspect ratios a/b, and the mode numbers in the ^-direction determined from Figure 4(b) are agree with those by other two methods, shown in Figure 4(a). In Figure 5, the
270 a/b=1.0
a/b=2. 0
a/b=3.0
a/b=4. 0
a/b=5. 0
a/b=10.0
N=1
N=2
N=3
N=4
N=5
N=10
Figure 8 Buckling mode shapes
results for R/t=lOO are also shown and good agreements exist between the results by TMM and other methods. Figure 7 shows the buckling coefficients A: (=^^/^/(;r^Z)),D = £rV{12(l-v^)}) of the thin-walled members composed of cylindrical shell panels under lateral loads (Figure 6: Z?=2000cm, ^=1.0, b/t=\000, a/b=l.0^lO. 0) for the shell panel numbers A^=1'^10. The analytical models for the shell panel numbers A^=3, 5, 10 are shown in Figure 6. The boundary conditions at both ends (Z>=0, 2000cm) are assumed to be ti=v=w=0. As shown in Figure 7, the buckling coefficients for each shell panel number intersect each other, and decrease as the aspect ratios a/b increase. In Figure 8, the buckling mode shapes corresponding to the buckling coefficients shown in Figure 7 are shown for A^=l, 2, 3, 4, 5, 10, and for ^^=1.0, 2.0, 3.0, 4.0, 5.0, 10.0. In the case of iV=l, the mode number in circumferential direction decreases as the aspect ratio increases from n=7 to «=3, and the deformation of middle part of the cross section is grater than that of end part. In the case ofN=2, the mode shapes for a/b=l.O, 2.0 and 3.0 show the local deformation of the shell panels composing the
271
0.011-
Figure 9 Relation between buckling coefficients and numbers of shell panels members, and those for a/b=4.0, 5.0 and 10.0 show the distortional deformations (combined deformations of local and global ones). In the case of A^=3, adding the local (a/b=l.O, 2.0) and distortional deformations (a/b=3.0,4.0, 5.0), global deformation, where the members deform as the flat plates, can be seen (a/b=lO.O). Various mode shapes are shown for N=4, 5 as the case ofN=3. On the case of A^=10, every mode shape shows the global deformation. From mode shapes in Figure 8, although in the case of small aspect ratios and number of shell panels the mode shapes show the local deformations, the global deformations become to govern the mode shapes of the members as the aspect ratio and number of shell panels increase. In Figure 9, the relations between the buckling coefficients and the numbers of shell panels for the aspect ratios a/b=l.O, 2.0, 3.0, 4.0, 5.0 and 10.0 (Z>=2000cm, /?=1.0, b/t=\000). In every aspect ratio, the buckling coefficient indicates the peak value at certain number of shell panel N„^, and the value of the number N„^ decreases as the number of shell panels increases, N^^=l, 4, 3, 3, 2, 2 for a/b=l.O, 2.0, 3.0, 4.0, 5.0, 10.0, respectively. The mode shapes corresponding the peak values are indicated in Figure 8 except a/b=l.O (the mode shape for a/b=l.O is shown in Figure 9). As shown these figures the mode shapes of the peak values indicate the distortional deformations. Figure 10 shows the relations between the buckling coefficients and the numbers of shell panels for b/t=2000, 6^=-2.0 (Figure 10(a)) and for b/t=\000, ^=1.0 (Figure 10(b)), and similar results to Figure 9 are obtained. From Figures 9 and 10, the buckling coefficient and the shell panel number of peak value increase as the angle ^ and thickness t of shell panel increase.
CONCLUSIONS In this paper, an analytical procedure to estimate aot only the buckling loads but also buckling mode shapes of the thin-walled members composed of c /lindrical shell panels is presented, and the effects of the number and curvature of the shell panel on the buckling loads and mode shapes are examined. From the numerical examples presented in this pa per, the following conclusions are obtained. 1. The exact buckling coefficients and mode s lapes of the members are obtained with very small computational efforts. 2. In the case of small aspect ratios and numl ers of shell panels, the mode shapes show the local
272 1
1
1
1
1
1
1
r b/t=2000 k : 0=2.0
•
1
•
J
j
^ ^
0.1
V ^SD^ 0.01 r o
]
ci
..^^^
0.01
r ^ 10^^^^ 4
0.001 •
i
l
l
2
1
1-
1
1
1
1
1
3
4
6 8 ^ 10 2 4 6 8 NlO (a) (b) Figure 10 Relation between buckling coefficients and numbers of shell panels
deformations, on the other hand in the case of large aspect ratios and numbers of shell panels the global deformations govern the mode shapes of the members. 3. The mode shape corresponding to the peak buckling load shows the distortional deformation, i.e., the combined deformation of local and global ones. 4. The mode shapes obtained by the transfer matrix method are very effective to clarify the complicated buckling phenomenon of the members composed of cylindrical shell panels.
References Ohga M., Hara T. and Kawaguchi K.(1995a). Buckling Mode Shapes of Thin-Walled Members. Computers & Structures 54: 4,767-773. Ohga M., Kawaguchi K. and Shigematsu T.(1995b). Buckling Analysis of Thin-Walled Members with Closed Cross Sections. Thin-Walled Structures 22:1, 51-70. Ohga M,, Shigematsu T. and Kawaguchi K.(1995c). Buckling Analysis of Thin-Walled Members with Variable Thickness. J. Struct. Engrg., ASCE 121:6,919-924. Pignataro M., Pizzi N. and Luongo A.(1991). Stability, Bifurcation and Postcritical Behavior of Elastic Structures, Elsevier, London. Tesar A. and Fillo L.(1988). Transfer Matrix Method. Dordrecht: BQuwer Academic. Uhrig R.(1973). Elastostatik und Elastokinetik in Matrizenschreibweise. Berlin: Springer-Verlag.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
273
STRENGTH OF THIN RECTANGULAR BOX-COLUMNS SUBJECTED TO UNIFORMLY VARYING EDGE DISPLACEMENTS N. E. Shanmugam' and R. Narayanan^ ' National University of Singapore, Singapore ^ Duke University, Durham, North CaroUna, USA
ABSTRACT This paper deals with the ultimate strength of rectangular box-columns subjected to uniformly varying load across the cross-section. Short columns with fixed ends are considered in the analysis. An approximate method to predict the load carrying capacity of such box-columns is presented. The method is based on effective width concept for plates subjected to uniformly varying load. Equations are derived for effective width of plates clamped at the loaded ends and simply supported along the unloaded edges. The edges are assumed to be free to pull-in. The effective width equations thus derived are applied to the component plates in a rectangular box-columns to obtain their ultimate load. Perforated plates are also considered in the analysis and results for box columns with openings are presented. Predicted results are compared with experimental failure loads.
KEYWORDS Plates with loaded edges clamped, uniformly varying load, effective width, energy method, ultimate strength, rectangular box-columns.
INTRODUCTION A rectangular box-column may be considered as an assemblage of plates supported along the edges. When such column is subjected to an eccentrically applied load, the component plates undergo uniform or uniformly varying edge-displacements depending upon the nature of eccentricity, uni-axial or bi-axial. Such plates with simply supported edges, were investigated for the buckling behavior by Timoshenko (1961). The uniformly varying axial force, acting in the middle plane of the plate, was assumed as Nx. ==N„ 1-af N„(l-«^]
(1)
274
In the above equation, setting a = 0 and 2, will correspond to uniform compression and pure bending load, respectively, a > 2 corresponds to a combination of bending and tension and a < 2 to a similar combination of bending and compression. Difficulties faced by the designers to evaluate post buckled behavior of plates opened the way for the development of simplified approximate methods. Plates under linearly varying loads have been investigated by Walker (1967), Rhodes and Harvey (1971), Rhodes et al.(1975), Kalyanaraman and Ramakrishna(1984) and Narayanan and Chan (1985). The method proposed by Narayanan and Chan considered the plates simply supported on all four sides. Shanmugam and Narayanan (1998) extended the method to include clamped condition along the loaded edges. They assumed, in order to simplify the solution, that all edges are free from applied or restraining moments and that the finite buckled shape in the post-buckling shape is identical with the shape of infinitely small buckles during incipient buckling. These assumptions are in addition to the usual ones and for the purpose of analysis, only a single wave of buckling was considered (i.e., the plate is an approximately square one). Brief description of the method is presented in the following sections. . The elastic buckling load for an initially imperfect plate which has its unloaded edges simply supported and the loaded edges clamped as shown in Fig. 1 can be obtained by using the energy method. As no law governs the general pattern of plate imperfections, it is usual to assume a suitable shape for the sake of simplicity of analysis, which takes the same form as the final buckled shape. When a^ reaches the elastic
Figure 1: Plate Subjected to Uniformly Varying Edge Displacement critical stress a^^ the plate will be assumed to buckle in a series of half wavelengths the deflected form being given by: ,, 27uxY • "^ . • 27iy w = 1-cos aiSin—=^ + a2Sin
'
a A
b
(2)
b
in which w is the out-of-plane deflection of the plate and a,,a2 refer to amplitudes of the buckling wave. The initial surface of the plate before any loads are applied can be expressed by an expression similar to Eqn. (2), i.e.,: w„=[l-cos2^|Afsin^ + A5sin^
(3)
in which A,", Aj" = amplitudes of the wave of initial imperfections. The final buckled shape will be given by Eqn. (2).
275 ht Using the energy method the expression for buckHng coefficient K (= C7^^, —z—) is given as K'D
4K,K2b^a'
K=
(4)
((|) + 1) (K, + K2) + J ( K , + K2) - 4 K I K 2 1-0.13
(t) + l
in which a„i is smaller edge stress at buckling, 't' thickness and 'D' the flexural rigidity per unit length of the plate, 4 4
2 2, 2
3 >i, 4
a a b ^ b
4 ' ^2
4
8
12
2, 2
1 4
^ '
a a b b
^
82 -,
s,
ULTIMATE STRENGTH OF A PLATE FIXED ALONG LOADED EDGES WITH SIMPLY SUPPORTED EDGES FREE TO PULL-IN The analytical treatment described below pertains to an initially imperfect plate clamped along two loaded edges whilst the two unloaded edges remain simply supported. A theoretical analysis based on an ideally flat plate is useful only to determine upper bound solutions. All plates, in practice, have some irregularity in their surfaces. While these irregularities follow no general pattern, it is usual to assume for the sake of simplicity, that initial imperfection is similar in form to the buckled shape. Thus the initial surface of the plate before any loads are applied will be assumed to be in a form given by Eqn. (3) and the final buckled shape is given by Eqn (2). Considering a half wave length of the plate subjected in the post buckling stage to stresses which vary along the width, the total strain energy (Ui„t) is the summation of energy due to bending (Ub) and the energy due to strain in the middle plane of the plate (UJ. The strain energy due to bending stored in a half wavelength is given by:
Ub
dxdy
(5)
In the expression for strain energy the actual deflection (i.e., the final minus the initial deflection) may be replaced by: 7iy
. 27cy
w = ( a , - A ; ) [ l - c o s ^ ] [ psm-^ + sm—b b
(6)
where P = : ^ = A!L a2
A^
It is assumed that the buckled shape remains the same as at incipient buckling. Therefore, the ratio P remains constant although the actual values increase with the increase of loading. The expression for strain energy due to bending is therefore obtained by the following expression
276
"•-^ri-vO'"-'''''* Tty .
27ty
(7)
. 2 27ty
»2 • 2 ' t y R = P' sin' -r + 2Psin - f sin — i + sin' - r ^ b b b b
The strain energy due to mid-plane strain is given by. Us = ^
(8)
U(ax + aj - 2v axo,)t dxdy
Since the edges are free to pull-in, a = 0 along AD and BC and we have, therefore, . Us = ^ U V , d x d y The total strain energy is given by: Ui„, = Ub + Us
U.. = ^
£
(
(9)
l + ^ y ] ( a . - A ° . y R d y + ^a'a^
The incremental change of the total energy (d Uint) due to incremental change of amplitude (from a2 to (a2 + da2)\ is equated to the corresponding incremental change of external work done (i.e., stress multiplied by the change in volume) as was done by Narayanan and Chan. We obtain the relation between the applied edge strain and the corresponding stress at any section. ^e _
3yA°/(m^-l)(P' + 4 p V l ) ^
^
acri(m-l)
(10)
Em
Plil.(^.l)fP!.M^l a x - E Sx ^ ( a ^ A ° / ) R a
or
,=E
37t'A°2'
(m'-l)(p' + 4p' + l)
,o„,(m-l) Em
2
^^ ''1^4 9-^
y-^MM-.-;y
A}\ (H)
277 The mean stress G^ is given by:
3u'A''2'(m'-l)(P' + 4 p ' + l) . ^ a c ( m - 1 ) Em
am'
P!ll + (^.,)i!.M,l 4
z
Za
9, (12)
Ritchie and Rhodes (1975) have demonstrated that the effect of the presence of hole in a buckled plate may be approximated by assuming the stresses released by the introduction of the hole in an unperforated plate to be redistributed over the remaining width of the plate which carry the stresses. For a given load, this produces an increase in the maximum stress across a section passing through the hole. The above concept is extended to compute the elastic stress distribution across the post-buckled stage. Earlier studies by Narayanan and Chow (1984) have validated this approximation. When an edge strip reaches yield, after the above redistribution, the applied load will be increased no further and the plate will be deemed to have reached the ultimate capacity. ( in other words, the criterion of first yield is employed for estimating the ultimate capacity). Application to Design Problems Based on the theory proposed above, an approximate method of estimating the ultimate capacity of plates subjected to uniformly varying edge displacement is developed. It is postulated that when the longitudinal stress at any section has just reached the yield stress (oyy), the plate is no longer able to sustain any load increment. This would be particularly true of wide plates, where the loss of stiffness beyond the onset of first yield is rapid. The effective width of the plate will therefore be computed as the ratio of the mean stress at first yield a„ to the yield stress cTyj. The ultimate strength of a plate can, therefore, be determined as the product of effective width, thickness and the yield stress. By using the above criterion, the effective width of plates with unloaded longitudinal edges free to pull-in is obtained from Eqn. (12) as
Kbs
371^ A f (m^ - 1) (P' + 4p^ + 1)
~ CTys
2a^
P' + 1
(*-l)
^P^ ^ P -
At U - 1) (P' + 1)
^ a,,^ (m - 1) Em
(tf (13)
278 For any chosen value of initial imperfection, effective width curves can be obtained simply from Eqn (13) for a number of parameters such as plate slendemess, edge strain ratio etc. Variations of K^s for plates in which the unloaded edges are free to pull-in are given in Fig 2 for a number of varying edge displacements using v --= 0.3, E = 200, 000 N/mm^ and a^, = 250 N/mml Application to Rectangular Box-Columns Short rectangular box-columns of 720mm long and 240mm x 60mm cross-sectional dimensions (a = 240mm and b = 60mm) were tested to failure under eccentrically applied loading. The thickness of the plate was so adjusted that plate slendemess ratios ranging from 20 to 150 were obtained. The load was applied with varying degree of eccentricity so that the longer plates in the column cross-section were subjected to trapezoidal in-plane loading of different magnitudes. The type of loading on the plates is characterised by the eccentricity of its line of action, 'e' defined in the non-dimensional form (e / b) or
c|^!lTnw
J 1.00 0.50 033 0.01
20
to
60
100
120
UO
160
T p. '-
TTTTTT^
Figure 2 Variation of Effective Width Factors Plate with unloaded edges free to pull-in
T
/
180 200 90
b/l
-*
'
^
Figure 3 Loading on Component Plates in a boxColumn subjected to eccentric Loading
by the non-dimensional ratio of edge strains (([) = 82 / ^i) of the two large plates. Initial imperfections in component plates were measured before testing. Some of the box-columns were tested to determine the effect of perforations in component plates. Openings of 80mm diameter were thus made in longer plates of some of the box-columns. The complete details of the test specimens are summarized in Table L As mentioned previously, a box-column may be considered as an assemblage of plates, as shown in Fig. 3, simply supported along the edges. It can be assumed that the unloaded edges are free to pull-in and Eqn. 13 for effective width developed in the previous section may, therefore, be used to compute the effective widths of component plates in the box column. The ends of the plates may be assumed to be clamped since the top and bottom were welded to rigid plates. Longer side-plates in a rectangular boxcolumns subjected to eccentrically applied load are under uniformly varying compression and tension as shown in Fig.3 whilst the shorter plates are under uniform tension or compression. The box-column specimens listed in Table 1 were analyzed by using the effective v^dth values for long plates as per the Eqn. 13 with the measured imperfection values. The shorter plates in tension were assumed frilly effective and those in compression were analyzed usmg the effective width equations given by Narayanan and Shanmugam (1979). The ultimate load that can be carried by the columns were computed by multiplying
279 the yield strength by the effective area of cross-section obtained based on the effective width equations. The values of ultimate loads (Pth) thus obtained are summarized along with the corresponding experimental values (Pexpt) and the comparison between the two values (P^^p, / Pt^) in Table 1. Table 1 SUMMARY OF THE BOX-COLUMN TEST SPECIMENS AND EXPERIMENTAL AND THEORETICAL RESULTS Specime n
B2 B3
B4 B5 B6 B7 Bg
B,
B,o B„ 1 B,2
t
a/t
b/t
1.59 1.59 1.59 2.97 2.97 2.97 2.97 2.97 2.97 1.69 1.69 1.69
d
e/b
mm
mm 151 151 151
38 38 38
80 80 80
20 20 20
80 80
20 20
0.0 0.0 0.0 80.0 80.0 80.0 0.0 0.0
0.0 0.1 0.2
0.2 0.1 0.0 0.2 0.0
80
20
0.0
0.1
142 142
36 36
0.0 0.1
142
36
80.0 80.0 80.0
0.2
E kN/mm 2 202 1.000 202 0.822 202 0.673 0.817 210 210 0.904 210 1.000 210 0.817 210 1.000 0.904 210 208 1.000 0.835 208 208 0.696
<^ys kN/mm 2 226 226 226
p '^ expt
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<24 PTY=0.8fyAi PTU=fyAi acr=7i2E(t/b)2/3(l-v2) be=b(acr/fy)l/2
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
Where, E is elastic modulus of steel; v is Poisson's ratio; fy and fu are yield and tensile strength respectively; Li is the length of the inelastic element; Ai is the sub-area that concentrate on the spring; Aie is reduced area that is the product of Ai and a discount factor of effective width, be, over the width of straight line, b. These formulae are based on the stub tests. For example the effect of residual stress is evaluated by factor 0.8 in Equ.(6) and (8). Discount factors of stiffness in inelastic and degrading stages as Equ. (2) through (5) are also calibrated by the tests. Comparison with Test Results Two sets of numerical computation results, stub columns and eccentrically compressive loaded columns, are shown in Table 4. Here, Nn is the maximum loading value by numerical analysis. When computing the model parameters, the nominal gross area of test specimen is used for simplicity. Except for the specimen of LEQ3-2 whose initial imperfection resulted in lower tested ultimate, the analysis results have satisfied agreement with the tests. Loading-deformation curves by analysis dotted in Figure 2 show that not only the resistance but also the deformability are well simulated. TABLE 4 MAXIMUM LOAD CAPACITY BY ANALYSIS AND THE COMPARISON WITH TESTS
Specimen Code Nn(kN) Relative error(%) Specimen Code Nn(kN) Relative error(%) -7.43, -6.94 SCR3-l,-2 885 1050 13.75, 10.41 SC03-l,-2 SCR5-l,-2 SC05-l,-2 1420 -19.50, -19.63 1930 -2.03, -0.10 SCR7-l,-2 2310 -0.56, -0.77 2740 SC07-l,-2 -5.84, -5.39 4.14, 34.02 LEQ3-l,-2 579 LER3-1 705 6.98 908 -14.7, -16.3 LER5-l,-2 LE05-l,-2 1230 -5.38, -3.91 LEQ7-l,-2 1420 -5.33, -7.97 LER7-l,-2 1690 -14.20, -10.43
298 HYSTEREIS CHARACTER OF THIN-WALLED RHS MEMBERS The inelastic multi-spring element model is then used for study of the hystereis character of thin-walled rectangular section. Using the set skeleton curves above and adopting Ramberg-Osgood function as unloading rule, a series of numerical simulation of columns are performed. The section and length of the analyzed column are taken the same as loading test specimen, while the yield stress of steel is specified as 235 MPa or 345 MPa. As for boundary constraint, simulated column is fixed to rotation in ends but can be shifted in lateral direction in one end. A constant axial load is given on the column top. Figure 4 illustrates the hystereis behaviors of 3 mm thickness columns. Figure 5 illustrates those of 7 mm thickness columns. The lateral shift route is imposed to the column as undergoing story relative deformation in the sequence, 1/400, 1/200, 1/100, 1/70, or larger. We can observe that with the increase of the axial load ratio the cyclic loop loses its stability early, that is to say, it deteriorates in the first two or three loops. Under the same axial load ratio, the thicker column keeps its stable capacity better owing to its less width-to-thickness ratio than thinner column. We can deduce that even the relative drift reaches to 1/70, the second order moment is not large enough, and deterioration is almost caused by local buckling. Qc (kN)
100 .fy=235MPa Nc/Ny=0.1
fr
Nc/Ny=0.3
Li'J
-100
;
Qc (kN)
Qc (kN)
Qc (kN)
100 .fy=345MPa
100 fy=235MPa
Nc/Ny=0.3
Nc/Ny=0.1/
ttl
^ •100
•100 clis.(cm)
dis.(cnn)
100 .fy=345MPa
jrZHi
/Vj
•100 clis.(cm)
dis.(cm)
-5.0 -2.5 0.0 2.5 5.0 -5.0 -2.5 0.0 2.5 5.0 -5.0 -2.5 0.0 2.5 5.0 -5.0 -2.5 0.0 2.5 5.0
Figure 4 Hystereis loops of 3 mm thickness column '^nn
0
-300
Qc (kN) fy=235MPa Nc/Ny=0.1 /
/
III
1
Qc (kN)
Qc (kN)
300 .fy=235MPa Nc/Ny=0.3
11 1
/
/
-300
r/A
•4i v.-
-2.5
0.0
2.5
Nc/Ny=0.3
i^3l7
/ / /
/ /
/7
-300
-300
dis.(cm)
dis.(cm) -5.0
300 .fy=345MPa
300
5.0
-5.0
-2.5
0.0
2.5
5.0
-5.0
-2.5
0.0
2.5
5.0
-5.0
-2.5
0.0
dis.(cm) 2.5 5.0
Figure 5: Hystereis loops of 7 mm thickness column Because of the importance of the stability of cyclic loop, it should be discussed that in what lateral load level the column can keep its stable loading capacity. To a ten-story building built on soft ground, the first structural period can be reasonably estimated about 1.0 second. If input ground acceleration is 100 to 120 gal, the base shear response factor is about 0.2 by spectrum analysis. In an average view, we can set the lateral load acting on a column is near to 0.2 times of the vertical load that the column borne. To 7 mm thickness column, the numerical results are shown in Figure 6. We can see that when axial load ratio is pressed below 0.3, the cyclic loop can run stably even the lateral load goes over 0.2 times of axial load. But if axial load ratio increases to 0.5 the lateral resistance becomes down and the plastic deformation develops.
299 25oLQc(kN)
250 Qc (kN) j
/
0
I ^^
-250 -3.0
^
-1.5
/
fy=235MPa Nc/Ny=0.3
0.0
dis. (cmp -250 1.5
3.0
-3.0
-1.5
0.0
1.5
3.0
-3.0
-1.5
0.0
1.5
Figure 6: Hystereis loops under constant repeated displacement routes
TESTS ON JOINTS Six joint specimens were experimentally studied by the authors. The specimen is shown in Figure 7. Rectangular column section is the same as those in member test. Two beams connected with the column are composed of double channels back to back, with section size 400x100x35x5. A pair of outer ring plates are welded to column at the places where beam's flanges are connected. The panel zone, however, is not strengthened by any other reinforced ribs. A constant vertical load, Nc, is exerted on the column. A pair of reversed lateral loads, Qb, are put on the beams as repeated loads. The shear in column, Qc, can be computed by balance condition. In the test, beam end displacement, panel zone shear deformation were measured. The tested specimens are listed in Table 5. The ratios of constant vertical load Nc to column axial yield force Ny defined as in Table 3 are also shown. Qcmax refers to the maximum value of column shear during the loading process.
. positive load 3 negative load
^*
^U
W^ tq.
B-
Figure 7: Specimens for beam-to-column joint tests
Figure 8: Story shift and beam displacement
Two examples of hystereis loops of column shear with computed relative 'story drift' are shown in Figure 9. Story drift is approximately calculated by beam end displacement, referring to Figure 8. By Figure 9, we can read that the joint did not appear to degradation until the final loading cycle in the test. But it can be found that the stiffness becomes softer in lower level of lateral loads after relative drift exceeds 1/100. It can be explained by the large panel zone deformation which were measured to be near to 1/150 after the first two cycles. So, in the frame using these cold-formed members, the joints should be strengthened including the panel zone and connecting plates of outer ring.
300 TABLE 5 JOINT TEST SPECIMENS AND THE LOADING CONDITION
Specimen Code Nc(kN) CCJ03-1 300 CCJ05-1 800 CCJ07-1 800 CCJR3-1 300 CCJR5-1 800 CCJR7-1 800 150 r
Nc/Ny 0.23 0.49 0.40 0.20 0.35 0.30
Qcmax (kN) 73.3 98.7 112.8 81.8 70.5 134.0
Notice
Beam lateral buckling occurred in test
Qc(KN)
150 - Qc(KN)
1 i 100
40
CaQ7-l
-20^
//
-150
Rt(%o)
20
40
CCJR7-1
Figure 9 : Column shear and relative story drift
SUMMARY Thin-walled cold-formed RHS columns and beam-to-column joints are tested, in monotonically static loads and repeated static loads respectively. An inelastic multi-spring element model is used to research the hystereis behaviors of these members imder cyclic loads. The analysis model parameters are careftilly calibrated by test results of RHS stub columns, to reflect the characters of residual stresses and local buckling. Numerical simulation reveals that if the axial load on columns can be limited in not very higher level, the column can keep efficient loading capacities and deformability to resist the cyclic horizontal loads. The results of joint tests show that strengthened panel zone is necessary for the frame system. It is understood that to middle-high rise building having ten stories or more the thin-walled cold-formed steel members can be used as main frame members under some necessary limitation..
References Ohi K, Takanashi K (1992). Multi-spring joint model for inelastic behaviors of steel members with local buckling. Stability and ductility of steel structures under cyclic loading, CRC Press Takanashi K, Ohi K, Chen Y, Meng L (1992). Collapse simulation of steel frames with local buckling, Proceedings of 10th WCEE, 4481-4484 Chen Y (1996). Numerical simulation for strength damage of steel structures subjected to disastrous loadings (in Chinese), Journal ofTongji University 24:5,487-491
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
301
FLEXURAL WRINKLING FAILURE OF SANDWICH PANELS WITH FOAM JOINTS D. McAndrew and M. Mahendran Physical Lifrastmcture Centre, School of Civil Engineering, Queensland University of Technology, Brisbane QLD 4000, Australia
ABSTRACT In Australia, sandwich panels are commonly made of steel faces and expanded polystyrene foam cores. The presence of joints between the polystyrene foam slabs in the transverse direction of sandwich panels introduces two potential sources of weakening in the panels. First, they introduce a possible local reduction in the shear strength of the panel. The second potential source of weakening is associated with their effect on the flexural wrinkling failure stress of the panel. This paper presents the details of an investigation using full scale experiments and finite element analyses into the flexural wrinkling phenomenon of sandwich panels with flat steel faces, the results and comparison with available theoretical and design solutions.
KEYWORDS Flexural wrinkling, sandwich panels, foam joints, finite element analysis, experiments
INTRODUCTION hi Australia, sandwich panels are commonly made of steel faces and expanded polystyrene foam cores (Fig.l) and their use as roof and wall claddings has increased in recent times. Flexural wrinkling is a unique failure associated with flat and lightly profiled sandwich panels when subjected to bending or compressive loads. This wrinkling failure occurs well below the yield stress of the compressive face, and thus is the governing design criterion for most sandwich panels. The method of using polystyrene foam slabs leads to the presence of transverse joints in sandwich panels. The presence of these transverse joints implies an imperfection in the continuously supported plane of the faces (i.e., a 'gap' between the foam slabs and a *step' due to the difference in the heights of foam slabs) as shown in Fig. L The joint introduces two potential sources of weakening into the panel. The 'gap' between the foam slabs in the transverse direction introduces a possible reduction in shear strength of the panel. Davies and Heselius (1993) indicate that if such a joint were to extend completely across the panel, the shear strength at that point would be close to zero. The second potential source of weakening is associated with the effect the 'gap' and 'step' has on the flexural wrinkling failure stress of the panel.
302 Steel Face
J
i
i
I ^^^P
mmm I Foam Slab
11 GaD Gap
Figure 1. Transverse Joints in Sandwich Panels Although extensive sandwich panel research has been carried out in Europe and America (Davies, 1987,1993, Hassinen, 1995, ECCS, 1991), the above mentioned weakening effects have not been investigated to date. Therefore, a series of full scale experiments and finite element analyses were conducted to evaluate the effects of these joints on the flexural wrinkling stress of sandwich panels with flat faces. This paper presents the details of this investigation, the results and comparison with available theoretical and design solutions.
Figure 2. Flexural Wrinkling failure WRINKLING THEORY Wrinkling is a form of local instability of the flat and lightly profiled steel faces of a sandwich panel associated with short waves of buckling of the faces. It can occur when a sandwich panel as a whole is subjected to a compressive load or when one of the steel faces is in membrane compression under bending of the panel (Fig. 2). The compressive face of a sandwich panel in bending develops a series of buckling waves at a relatively low stress level. The amplitude of these waves gradually increase until one of them fails and forms a wrinkle at the location of greatest bending moment and/or imperfections in the panel. The flexural wrinkling failure occurs rather suddenly, and this characteristic must be considered in design. The plane faces of sandwich panels carry axial forces. They do not carry any bending moment. Their failure mode, therefore, is considered like that of a thin compressed beam on an elastic foundation (i.e., the foam core). Equation (1) gives the theoretical expression for the wrinkling stress Gwr of sandwich panels with flat faces that was based on an energy method (Davies, et al., 1991). It is assumed that the steel face is supported on an infinitely deep elastic support (core). It is important to realise that Eqn.(l) has been derived assuming a continuous elastic medium. Currently, no formulae or design methods account for joints in the foam core of sandwich panels. ]_
o^ = 1.89{
MzZcl!
rl'(E,E,G,)i
(1)
where Ef and Ec are the modulus of elasticity of the face and core; Vf andvc are the Poisson's ratio of
303 the face and core, and Gc is the shear modulus of the core. When Vc = 0 and Vf = 0.3, Eqn.(l) becomes a „ = 0.819(EfE,GJ^
(2)
A general form of the wrinkling stress equation is given by (3)
a^=K(E,Efi,y
where K is a numerical constant, Ec and Gc are characteristic values when used for design purposes and are the 5% fractile values of the population (ECCS, 1991) For practical design, K is given a value of 0.65 (ECCS, 1991). This is due to practical considerations such as finite core depth, lack of flatness of the face, and non-linearity of the core material (Davies et al. 1991), but not intended for the presence of transverse joints in the foam.
FULL SCALE EXPERIMENTS Details of Experiments Nineteen sandwich panels with flat faces were considered in this study. The foam thicknesses considered were 75 and 150 mm, and the spans varied from 2700 to 4700 mm. The width of all panels was 1215 mm. Steel thickness was constant at 0.6 mm. The joint location was varied, however, it was kept at midspan for the majority of test panels. Panels with transverse joints extended full-width with others extending only half-width to study the observations of Davies and Heselius (1993). A vacuum chamber was used to produce a uniformly distributed transverse loading of the panels (see Fig. 3), enabling flexural wrinkling failures (see Fig. 2) to occur in bending. Test panels were simply supported over 70-mm wide rectangular hollow sections and were not restrained by the timber casing. Li each test, the steel face to be tested was placed on the top so that it was subjected to a compression force. Once the panel was positioned in the chamber, a polyethylene sheet was placed loosely over the panel. The sheet was sealed to the sides of the timber casing and a vacuum pump used to decrease the air pressure in the chamber.
LVDTs Polyethylene sheet
TT
D
I
Vacuum Chamto
r
Timber Casing
Figure 3. Experimental Set-up
Sandwich Panel
Supports
304
For all panels without joints, flexural wrinkling occurred within the middle third of the panel. It is interesting to note that no panel without a joint actually failed at midspan at the location of greatest bending moment. Most of the panels with joints failed at the joint. The others failed within 250 mm of the joint. In all the tests, the panels collapsed suddenly, indicating the nature of these flexural wrinkling failures. Failure involved a distinct inward fold of the steel with tensile failure of the foam occurring directly beside the fold. This failure type occurred in every test panel under bending. No glue failure was apparent in the tests. Table 1 shows the results for five panels tested in this investigation. Other experimental results will be presented at the conference. TABLE 1. EXPERIMENTAL RESULTS
Panel Characteristics l/P/2700/75 2/P/2900/75/FW 3/P/2700/75/FW 4/P/3700/75/FW 5/P/3700/150
Distance X
(mm) 300 670 0 80 500
Failure Pressure (kPa) 4.77 3.33 3.45 1.38 4.75
Wrinkling Stress Theory (MPa)
Wrinkling Stress Expts. (MPa) ^wr-wr
86.6 86.6 86.6 86.6 86.6
91.1 61.3 69.9 54.7 82.8
0"xvr-midspan
95.8 77.9 69.9 54.8 89.4
Notes: 1) x is the distance between the wrinkle and midspan, FW indicates full-width joint 2) awr-wr and awr-midspan are the calculated failure stresses at the wrinkle and midspan based on measured failure pressures 3) Theoretical wrinkling stress is based on Eqn. (2) using material properties of Foam: Ec = 3.44 MPa, Gc = 1.72 MPa, Vc = 0 (measured); Steel: Ef = 200000 MPa, and Vf = 0.3 (assumed).
Experimental Results and Discussion Results fi-om fiill scale experiments clearly indicated that sandwich panels, which contained transverse joints, had significantly reduced flexural wrinkling capacity than panels, which contained no joints. This is shown in Table 1 by comparing Panels 1 and 5 (i.e., Gwr = 95.8 and 89.4 MPa) which contained no joints with Panels 2,3 and 4 (i.e., G^ = 77.9, 69.9, 54.8 MPa) with transverse joints. The failure stress at midspan awr-midspan is used in the comparison of results as this is the maximum stress in the panel that initiates wrinkling. The mechanism for the observed reduction in flexural wrinkling capacity due to the presence of joints is not obvious. As seen in Table 1, the nominal calculated stresses at the wrinkle based on failure pressures are considerably less than the corresponding midspan stresses when the joint is located awayfi*ommidspan. However, larger locaUsed stresses could have been present at the joints, which caused these premature failures. The failure mechanism for panels with joints must be considered to be different to the theoretical mechanism whereby buckling waves are formed with one becoming unstable at the critical stress to form a fold. Further research is being conducted to study the failure mechanism in detail. A comparison of design wrinkling stresses with experimental wrinkling stresses for flat faced panels containing joints has shown that in some cases the experimental stress to be less than the design value. Therefore, for panels containing joints, designers should be aware that the design equation may not be conservative and that a more generous empirical reduction factor may need to be applied. Another approach is to modify the transverse joint so that an improved form of connection between the slabs of foam is achieved.
305
A comparison between experimental results for flat faced panels (awr = 95.8 and 89.4 MPa) without joints with the theoretical result (awr = 86.6 MPa) based on the energy method (ECCS, 1991) provides reasonable agreement. Experiments confirmed that the effects of foam thickness and span on the flexural wrinkling failure stress of a sandwich panel are negligible. It was found that increasing the foam thickness beyond 75 mm lead to only a marginal increase in wrinkling strength for sandwich panels considered in this investigation.
FINITE ELEMENT ANALYSES Flexural wrinkling behaviour of sandwich panels with transverse joints was also investigated using finite element analyses (FEA). It was considered that FEA could be used to isolate the effects of joints that was not possible in the experimental investigation. A finite element program Abaqus (HKS, 1998) was used for this study. S4R5 shell elements with four nodes and five degrees of fi-eedom per node were chosen for the steel faces whereas C3D8 3D brick elements with 8 nodes and 3 degrees of freedom per node were used to model the foam. The material properties of foam as measured in the experimental study were used in the analysis, ie., for SL Grade: Ec = 3.44 MPa, Gc = 1.72 MPa and Vc = 0 whereas for steel they were assumed to be Ef = 200,000 MPa and Vf = 0.3. Both materials were considered to be isotropic. Since wrinkling occurs in the elastic region (well below the yield stress of steel faces) and is a bifurcation phenomenon, an elastic buckling analysis was used. As predicted by the theoretical approach (Davies et al. 1991), the FEA also showed that the first buckling mode to be the most critical one, however, other modes were also viewed. The initial finite element model simulated a steel face with all four sides simply supported subject to a compression load as for the theoretical approach using energy method so that it can be calibrated against the available theoretical predictions (Eqn.2). The foam core was considered to be sufficiently wide and deep to simulate the conditions assumed in the theoretical method. A single half-wave buckle was modelled with appropriate boundary conditions including that of symmetry. Figure 4a shows the model geometry (half the panel width b/2 x half-wave buckling length a/2 x model depth = foam depth tc + steel face tf) and mesh used. This enabled a finer mesh to be used while enabling both the wrinkling stress and the associated half-wave length to be compared with theoretical predictions. A convergence study was undertaken to justify the adequacy of the mesh used. The minimum buckling stress was obtained by varying the model length a/2. Figure 4b shows the buckling mode of the halfwave buckle model. Edge of panel face (tf)
Foam core (tc) Model width b/2 (a) Model
(b) Buckle Shape Figure 4. Half-wave Buckle Model
306 The results showed that a model using b/2 = 300 or more and tc = 75 mm or more gave a wrinkling stress of 87 MPa that Hed within 1% of the theoretical prediction of 86.6 MPa. The half-wave length a/2 of 24 mm also agreed well with the theoretical prediction of 23.8 mm. As seen from Eqn.2, steel face thickness has no influence on the wrinkling stress, which was also verified by FEA of models with different thicknesses of 0.4, 0.5 and 0.6 mm (87.4, 87.2,. 87.1 MPa). All these results confirm that a half-wave buckle model can be successfully used to model sandwich panels without transverse joints. Panel edge
Steel face
Foam core
Model width b/2 (a) Model
(b) Buckle Shape
Figure 5. Finite Element Model including the Gap Imperfection hi order to model the gap and/or step imperfections due to the presence of transverse joints (see Fig.l), a larger model was chosen as the half-wave buckle model could not be used, hi this case, a longer length (, was chosen to include the joint at mid-length while being able to accommodate sufficient multiple buckles. Figure 5a shows the model used with appropriate boundary conditions. The panel length i of 300 mm was found to be adequate to obtain accurate results. Even though the model was large, resulting in a slightly coarser mesh density, results were still found to be adequate, hi order to validate the model used, it was initially used without a transverse joint. Panel lengths (J) of 300, 600, 900 and 1000 mm gave wrinkling stresses of 89.1, 89.7, 90.1 and 90.2 MPa, which compared well with the corresponding result of 88.8 MPa obtained from the half-wave buckle model with the same mesh density. Figure 5b shows the buckling mode of the panel. The gap imperfection was studied first using the model shown in Figure 5 a with the gap placed at midlength. The model dimensions used were: b/2 = 300 mm, I = 300 mm and foam depth tc = 75 mm. The gap was modelled by leaving a physical gap between the solid elements representing the foam slabs on either side of the gap. It was also modelled with solid elements of very low stif&iess and the results were identical. Effects of the gap size on the wrinkling stress was studied by analysing the model with varying physical gaps. It was found that a significant reduction in wrinkling stress occurred even when the gap was very small. With increasing gap size from 0.1 to 10 mm, the reduction was very gradual after the initial significant reduction. This observation was the same and magnitudes were similar for steel faces of varying thicknesses (0.4 to 0.6 mm). Examination of the buckling mode of panels containing a gap shows that buckling becomes concentrated at the gap location. In order to verify whether the panel length chosen had any effect on the wrinkling stress results, it was also varied from 300 to 600 and 1000 mm, but the resuhs confirmed the observations made using the 300 mm length model. The foam depth was also varied from 75 mm to 50, 100, 150 and 200 mm, but the results for different foam depths compare very closely with each other.
307
Models containing step imperfections were then created. They had the same geometry, mesh and boundary conditions as for the models with gap imperfections. Since the step size was not expected to be greater than 1 mm in practice, it was used in the model with varying gap widths of 1, 5 and 10 mm. Liitial task was to study only the effect of step imperfection, and hence the foam was modelled as continuous, ie. the gap also had solid elements within the gap width. Results showed that there was little change to the wrinkling stress and the mode of buckling due to the presence of step imperfections. This study was continued with gap and step imperfections, which showed that the wrinkling stresses were reduced by similar magnitudes observed with the models with only the gap imperfections. Further analytical modelling is continuing and the results will be presented at the conference. The model used in the FEA and the theoretical methods of Davies et al. (1991) assumed infinitely wide and deep panels. The sandwich panels used as claddings in buildings are considerably wide and have foam depths greater than or equal to 50 mm with another steel face in tension. Therefore it is expected that the theoretical predictions from Eqn.2 and FEA models used so far could adequately predict the wrinkling stress of these sandwich panels. However, these panels are in fact subjected to lateral uniform wind pressures rather than end compression loads. Therefore it may be necessary to investigate the wrinkling behaviour of sandwich panels subjected to a uniform pressure loading. Mid^an
Free edge
Foam core
Steel faces Supported
Model length L/2
Model width b/2 Figure 6. Model of a Sandwich Panel without Joints For this purpose, a finite element model shown in Figure 6 was used with a finer mesh at midspan region. Both steel faces and the foam core were modelled with appropriate boundary conditions. The model length L/2 and width b/2 were chosen as 1400 mm and 600 mm, respectively whereas the foam depth tc and steel face thickness tf were taken as 200 mm and 0.6 mm, respectively. The wrinkling stress determined from this model was 90.3 MPa. The slight difference in wrinkling stress compared with earlier results from FEA and theoretical prediction of 87.1 and 86.6 MPa might have been due to the difference in loading (lateral pressure loading versus end compression loads) and the coarser mesh used. However, the difference is small and thus continued use of theoretical formulae (Eqn.2) and FEA using simplified models (Fig.4a) is satisfactory in determining the wrinkling stress of sandwich panels as used in practice. Th model under a lateral pressure loading was also used in a series of parametric studies to investigate the effects of foam depth, steel thickness and panel length. Again, the results showed that difference in wrinkling stresses is small for each of these parameters. However, this model showed that the reduction of wrinkling stress observed using half-wave buckle model with one steel face and a foam depth less than 50 mm was not present due to the presence of both steel faces. Li order to study the effects of gap and step imperfections, a fiiU-length model was used, but with other details remaining the same as for panels without joints (see Fig.6). The fiiU-length model was first
308 validated by comparing the wrinkling stress from this model without a gap or step. The result of 92.4 MPa compared well with that of the half-length model of 90.3 MPa. Following this, effect of gap and/or step imperfections was investigated using the full-length model with the gap at mid-length. Results from this model also confirmed that gap imperfections cause reductions to wrinkling stresses. The magnitude of reductions was very similar to what was obtained with the half-wave buckle models subjected to end compression loads.
CONCLUSIONS An investigation into the flexural wrinkling behaviour of sandwich panels with transverse foam joints has been described in this paper. Experimental results indicate that significant reduction in flexural wrinkling capacity is caused by the joints. Finite element analyses also confirmed this finding. However, strength reductions observed in some experiments are much greater than the predictions from the finite element analyses. Further experimental research is needed to improve the design of sandwich panels with joints.
REFERENCES Davies, J.M. (1987) Design Criteria for Structural Sandwich Panels, Journal of Structural Engineering, ASCE, Vol. 65A, No.l2, pp.435-441 Davies, J.M. (1993) Sandwich Panels, Thin-walled Structures, Vol. 16, pp.179-198 Davies, J.M. and Heselius, L. (1993), "Design Recommendations for Sandwich Panels", Building Research and Information, 21 (3) 157-161 Davies, J.M., Hakmi, M.R. and Hassinen, P. (1991), "Face Buckling Stress in Sandwich Panels", Nordic Conference Steel Colloquium, pp. 99-110 ECCS, (1991), "Preliminary European Recommendations for Sandwich Panels", Part 1, Design, European Convention for Constructional Steelwork (ECCS), No. 66 Hassinen, P. (1995) Compression Failure Modes of Thin Profiled Metal Sheet Faces of Sandwich Panels, Sandwich Construction 3 - Proceedings of the Third International Conference, Southampton, pp.205-214. Hibbitt, Karlsson & Sorensen, Inc. (HKS) (1998), Abaqus User Manual, USA
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
309
DESIGN AND DEVELOPMENT OFA COLD-FORMED LIGHTWEIGHT STEEL BEAM R.F Pedreschi Department of Architecture, University of Edinburgh 20 Chambers Street Edmburgh,EHllJZ
ABSTRACT Cold-formed steel structural elements such as purlins and sheeting rails can be manufactured and produced very efficiently. There is growing interest in the use of cold-formed steel sections in fabricated structures such as frames and trusses. A considerable proportion of the cost of fabricated structures lies in the cost of fabrication itself and thus the potential savings and benefits of efficient material usage may be lost The effective manufacture of such systems lie in designs that both optimise the material usage and the fabrication process. This paper describes the development of a fabricated cold formed steel beam that considers manufacture as a principle design criterion. The beam consists of cold-formed steel flanges attached to a profiled steel web. The flanges and web are attached using a mechanical clinching technique that facilitates rapid fabrication and produces a reliable and easily checked structural connection. A series of full scale prototype beams were manufactured and subjected to structural tests to destruction. The deformation and failure mode is discussed. KEYWORDS steel, cold-formed, beams, connections, mechanical clinching, prototype, tests INTRODUCTION Fabricated cold-formed steel structures are an increasing part of the modem construction. Steel stud framing and cold-formed steel trusses are now often used in residential and industrial buildings. Roll forming and folding can produce lightweight structural elements efficiently. When using these components to produce fabricated structures many of the potential benefits can be lost by inefficient assembly techniques. By considering the manufacturing process as a principal design criteria it is possible to produce structures that are efficient in both fabrication and material usage. A key aspect of efficient fabrication is the nature of the connections used. Mechanical clinching techniques can provide a simple, low cost form of connection that is suited to the fabrication of cold-formed steel structures. In order to study the potential application of mechanical clinching in fabricated cold-formed steel structures a system for lightweight cold-formed steel beams was developed. A series of full scale prototypes were produced and tested and a cost study was undertaken to compare with existing alternative systems. DESIGN OF THE SYSTEM For lightly loaded beams in short spans, for example, floor joists or purlins, roll-formed sections such as Zed sections are used extensively. As the span increases then these sections may no longer be appropriate, given the size limits of current roll formers. Hot rolled steel section or fabricated steel
310 lattice joists are then often used. In addition there is a number of proprietary fabricated timber beams that also operate in this short to medium span range. The present design utilises cold-formed steel components specifically designed to suit mechanical clinching. The main design criteria adopted were: • • • • •
efficient use of steel minimise post fabrication corrosion protection use of mechanical clinching range of depths and spans without changing production methods span range 6 - 1 5 metres be adaptable to incorporate further changes and modifications.
Figure 1 shows a typical section of the beam. flange
Figure 1 Isometric of beam A key aspect of the beam is the ability to utilise different material thicknesses in different sections. The flanges of the beam are typically produced from 2.0 mm steel whilst the web is made from 1.0 mm steel. The elements of the system consists of flange and web components. The flange component is manufactured from two stiffened angle sections connected back to back. A symmetrical and torsionally stiff flange is formed. The web is produced from a profiled section, figure 2. The return angles provide buckling resistance to the flat portions of the web. 60
n-^
42
100
100
30
length varies to suit depth of beam
-4
12.5
flange section
web section
Figure 2 beam components In production both elements could be produced by roll forming, however for the prototypes the sections were folded. Connection Techniques
311 Mechanical clinching was used to connect the components together. This technique was originally developed for use in the automotive industries but in recent years research has been carried out into its application in cold-formed steel structures, Pedreschi et al (1996), (1998), Davies et al (1996) and Lu et al. The detailed structural behaviour of mechanical clinching is explained in these references. The key benefits of mechanical clinching over other methods such as welding andrivetingare: • • • • •
galvanised and painted coatings are left intact low energy - approximately 10 percent of an equivalent spot weld process can be readily automated it requires only semi-skilled operatives the connection is formed using the parent metal of the elements being connected - the use of consumables such as screws andrivetsis therefore eliminated • readily checked for quality by non destructive techniques. Figure 3 shows the process of forming a mechanical clinch.
die
shearing of metal phase 1
latetal defonnation of steel as die spreads phase 2
finished press join
Figure 1. Process of forming a mechanical clinch The mechanical clinch is formed using hydraulically operated tools to operate a punch and die. There are two phases to the process. During phase one the punch closes on the die causing shearing along. In phase two the punch pushes the sheared material on base of the die. Under compression the steel deforms laterally and the dies spreads to accommodate this movement. Thus the finished connection is formed as the material on the die side is wider than the gap it passed through. There is a variety of differeht forms of clinch. In the beam a standard Eckold type S mechanical clinch was used. The flange of the beam is assembled by clinching two of the angle sections. The flanges have projecting tabs which are connected, using clinching, to the flat portions of the web, figure 4.
mechanical clinches
Figure 4 Detail of flange to web connection PROTOTYPES A series of twelve full scale prototype beams were manufactured and tested. All the beams were 6.0 metres long and the overall depths were 300 mm, 450 mm and 600 mm. Producing a range of
312 prototypes with span to depth ratios from 10 to 20. The flange and web components were manufactured using standard brake presses. The maximum length of the available brake press was 2.0.. Three lengths of flange section were butt welded to produce a six metre length. The web sections were manufactured in various depths to suit the depths of the beams. The material used for the each of the beam components is given in table 1 TABLE 1 MATERIAL PROPERTIES FOR PROTOTYPE BEAMS 1 Beam element flange flange web ^web
steel thickness mm
Modulus of elasticity kN/mm^
Yield stress N/mm^
1.5 2.0 1.0 0.7
il8
iss 388
226 237 258
352 417
Ultimate Tensile 1 strength N/mm^ |
460 391 380 466
1 1 1 1
The web element was manufactured in lengths of 600 mm for ease of fabrication. The beam was assembled by hand and the elements connected together using a large floor mounted hydraulic pressjoining tool. Three clinches were formed in each of the tabs connecting the flange to web. Each tab is 100 mm long therefore there are three clinches per 100 mm. Shear tests were carried out on the clinches applied to the various steel thicknesses used and are summarised in table 2. The values are the average of a number of tests. TABLE 2 SHEAR STRENGTH OF MECHANICAL CLINCHES USED die side thickness mm 1
punch side thickness mm
1 1 1 1
0.75 1.0 1.5 2.0
0.75 1.06 kN 0.96 kN L15kN
1.0 0.88 kN 1.28 kN 2.26 kN L95kN
1.5 1.32 kN -
] 2.0 1.41 kN :
The strength of mechanical clinching is dependent on the UTS , thickness of steel angle of applied shear and orientation of the layers to the punch and die Pedreschi et al (1996, 1998). The values in bold type in table 2 are appropriate to the prototype beams.
STRUCTURAL TESTS ON PROTOTYPES The aims of the test programme were: • to study the general structural behaviour of the beams • to study the influence of depth on the strength of the beam • to study the effect of varying flange and web thickness • to study the influence of holes in 3ie web on the strength of the beams All the beams were tested in a specially designed rig. For beam series land 2 load was at four points, using hydraulic rams to simulate uniformly distributed loading. Load was measured using electronic load ceils. Deflections were measured using mechanical dial gauges reading to 0.01 mm. The beams were restrained laterally at three positions along their spans, midway between each of the load points. For the beams in series 3 the load was applied as shown infigure5
I
im^ 4800
^1200 I "1
Figure 5 Test arrangement for series 3 beams A typical test arrangement is shown infigure6.
313
Figure 6 Typical test arrangement In this short paper it is not possible to cover all thefindingsof the research programme and only some particular points are presented. The test results are summarised in table 3. TABLES SUMMARY OF TEST RESULTS ON PROTOTYPE BEAMS beam number 1.1
1 1 1 1 1 1 1 1 1
1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2
bottom top web flange thickness flange mm thickness thickness mm mm 2.0 2.0 1.0 2.0 1.0 2.0 2.0 2.0 1.0 2.0 2.0 1.0 1.5/1.5 1.5/2.0 0.7 1.5/2.0 1.5/2.0 1.0 2.0/2.0 1.5/2.0 0.7 1.5/2.0 1.5/2.0 1.0 2.0 2.0 1.0 2.0 2.0 1.0 2.0 2.0 1.0 2.0
2.0
1.0
failure mode of failure load kN
depth of beam mm 300 300 600 450 300 300 300 300 300 300 300
span m
6.0 6.0 6.0 4.8
32.0 34.4 61.0 47.5 12.4 27.0 15.5 19.0 35.9 30.0 20.6
450
4.8
42.6
6.0 6.0 6.0 6.0 6.0 6.0
6.6
flange buckle flange buckle shear of clinch shear of clinch shear of clinch shear of clinch shear of clinch shear of clinch flange buckle shear of clinch shear of clinch
remarks
1
span/depth 20 span/depth 20 span/depth 10 span/depth 15
175x120 hole each end shear of clinch 100x100 hole at one end
The influence of span depth ratio Beams 1.1 - 1.4 had span/depth ratios varying from 10 - 20. As the span depth ratio of the beams decreased there was an increase in failure load. However the mode of failure also chimged. Beams 1.1 and 1.2 failed by buckling of the compression flange at mid-span of the beam. Beams 1.3 and 1.4 span/depth ratios 10 and 15 respectively, failed by shear of the mechanical clinching at the support points. Where failure occurred byflangebuckling it was possible to re-straighten theflange,invert the
314 beam and re-test. In both cases where this was done the failure load was again by flange buckling and the failure loads were almost identical. The effect of variation in steel thickness The beams of series two had flange thicknesses varying between 1.5 mm and 2.0 mm and webs varying between 0.7 mm and 1.0 mm. All the beams failed by shear of the mechanical clinching. Reductions in either the web of flange thickness result in a reduction in strength. The greatest reduction in strength occurred with 0.7 mm web thickness. In these beams, 2.1 and 2.3, buckling of the webs in the area around the junction of the flange to web at the supports occurred followed by shearing of the connection Influence of weh penetrations Holes in the webs of beams are known to be points of significant weakness. Two initial tests were carried out on beams with holes in the web, primarily to determine their influence on the performance of the beams. In both cases holes were positioned near the points of maximum shear. Beam 3.1, 300 mm deep, had holes 175mm long by 120 mm high . These holes were the largest that were practical, effectively removing two sections of web. Failure occurred due to web buckling around the area of the penetration. There was a considerable reduction in load carrying capacity of the beam. Compared with previous tests on beams without penetrations there was a reduction in strength of approximately 35%. Beam 3.2, 450 mm deep, had a smaller hole, 100 by 100 mm, through one section of the web only. Failure occurred by shear of the clinches and again there was a reduction in shear strength compared with previous results although in this case it was much smaller, approximately 9%. beam 1.2
Applied load kN
ou50-
"-""'^""- beam 1.3
0-^
6
40-
beam 1.1
^^^^
,o
beam 1.4
A—
^r
302010-
P '\K^
#v
o^
— 1
10
1
20
1
1
30 40 mid span deflection mm
Figure 7 Load-deflection results for series 1 beams Flexural behaviour of the beams Load was applied to the beams in a series of cycles. In the initial cycles the beams were loaded until they deflected 17 mm at mid-span, just over the normally allowable span-deflection limit of span/360 under service load. This cycle was applied at least three time and all beams showed linear, repeatable behaviour and short term recovery of deflection in excess of 95%. In the final cycle the beams were loaded to failure. The load-deflection responses of the beams in series 1 are shown in figure 7. Comparing the test results with calculated behaviour based on gross second moment of area some deflection due to shear also was evident. For the shallower beams this was estimated at 25 -30%. Increasing to 48 % for beam 1.3, having a span /depth ratio of 10. Comparison between shear force in clinch and shear force obtained from lap shear tests The shear force at failure in the mechanical was calculated using expression 1 and presented in table 4.
315 F = S.A.Y/Ixx.(100/3)
(1)
A = cross sectional area of flange Ixx = second moment of area F = force in clinches at failure S = shear force at failure There are three mechanical clinches for every 100 mm offlange.The shear forces in the press joins at failure for each beam and were 1.57 for beam 1.3 and 1.73 for beam 1.4. The shear strength values calculated from the tests on the beams are generally less than corresponding results from the shear tests. The effective stiffness of the beams was less due to shear deformations. TABLE 4 SHEAR FORCE AT FAILURE IN MECHANICAL CLINCHES 1
beam 1.1
1 1 1 1 1 1 1 1 1 1 1
1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2
Y mm 135 135 285
iio
135 135 135 135 135 135 135 210
S shear at failure kN 17.1 16.0 30.2 23.7 6.2 13.5 7.8 9.5 17.9 15.0 10.3 21.6
A area of the flange mm2 482 482 482 482 361 421 421 421 482 482 482 482
Ixx mm^.lO^ 18.40 18.40 89.91 46.60 14.96 16.28 17.20 16.28 18.40 18.40 18.40 46.60
F shear force/per clinch kN 2.00 1.90 1.57 1.73 0.67 1.57 0.85 1.11 2.11 1.81 1.24 1.56
shear force from test kN
1.95 1.95 1.95 1.95 0.96 2.26 0.96 1.95 1.95 1.95 1.95 1.95
1 1 1 1 1 1 1 1 1 1 1 1
COST COMPARISON WITH ALTERNATIVE BEAMS One of the driving forces of the project has been to consider the effect that efficient manufacturing methods may have on the economic viability of fabricated cold-formed steel beams. A cost study was undertaken to compare the likely costs of the proposed design with alternative beams currentiy available in the UK. It is not possible to go into much detail in this short paper and only the findings are summarised below. Costs were sought from suppliers of the beams on the basis of an actual contract on which steel lattice joists had previously been used. The loads, span and spacing of the beams were all specified. The costs for the proposed beams were established in conjunction with a steel fabricator and an equipment manufacturer to obtain realistic but conservative labour rates and material prices. The specification for the proposed beam was based on an extrapolation of the actual test results ignoring any further refinements and improvements possible by increasing the number of clinches for example. The results are summarised in table 5. The proposed beam appears cheaper than the alternative systems available. Two sets of prices were in fact obtained from different suppliers of steel lattice beams and the lowest price is presented in the table. The costs of the laminated veneer beam in the roof were actually based on beams at 600 mm centres rather than 1200mm as specified. Although it is difficult to obtain accurate production costs based on prototype design this study does never the less indicate that an economic design was obtained. SUMMARY AND CONCLUSIONS A system for the manufacture and design of fabricated cold-formed steel beams has been developed, using fabrication processes as a key design parameter. A series of full-scale tests on prototypes was carried out and the following conclusions can be drawn. • Mechanical clinching can be used in the production of efficient cold-formed steel beams • The calculated shear force in the clinches correlates reasonably well with the values obtained from small scale tests.
316 For deeper beams, span depth ratios less than 20, shear failure of the clinches dictated the load carrying capacity. TTie present design allows the number of clinches to increase by more than 100%. Penetrations in the web of the beam near the support can lead to signiticant reductions in strength of the beams and further work is needed to understand the behaviour and develop methods of providing effective local stiffening. The cost study indicates that the present design is economically viable. TABLE 5 SUMMARY OF COST STUDY type of beam
1
plywood I
beam
plywood I beam steel lattice beam glue laminated timber glue laminated timber laminated timber veneer laminated timber veneer proposed design proposed 1 design
floor beam cost/m dimensions and £ span 600x100 28.47 8.0m 800x100 i^.6S 9.8m 400 mm ii.6$ 8.0 and 9.8 m 115x540 32.52 9.8 m
cost/m roof beam dimensions and span 500x100 24.71 11.2 600x100 28.47 9.8 m 400 mm 23.09 11.2 and 9.8 m 115x585 35.35 11.2m
115x405 8.0m
24.61
115x495 9.8m
30.11
75x600 9.8 75x500 8.0m 600 mm 9.8 450 mm 8.0m
25.16
75x450 11.2m 75x400 9.8m 600 mm 11.8 600 mm 9.8m
18.86
20.96 17.72 16.55
16.65
notes
1
The same size was specified throughout
16.65 is based on beams at 600 mm centres. |
17.72 17.71
REFERENCES Davies, R. Pedreschi, R. and Sinha B.P. (1996) The Shear Behaviour of Press-joining in Cold-formed Steel Structures. Thin-Walled Structures 23:3, 153 -170. Lu W, Kesti J and Makelainen Shear and cross Tension Tests for Press -joins (1998) Helsinki University of Technology Report 7 56 pages. Pedreschi R, Sinha B.P. and Davies R. (1996) Advanced Connection Techniques for Cold-formed Steel Structures Journal of The Structural Division, ASCE 123:2138 -144. Pedreschi R. Sinha B.P. and Lennon R, (1998) The Shear Strength of Mechanical Clinching in Coldformed Steel Structures (in press), Pedreschi R., Sinha B.P., Davies R and Lennon R (1998) Factors Influencing the Strength of Mechanical Clinching Recent Research and Developments in Cold-formed Steel 14th Int. Speciality Conference St Louis 549 - 562.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
317
STEEL-BOARD COMPOSITE FLOORS Dr G H Couchman^ Ir A W Toma^ Ing J W P M Brekelmans^ and Ir E L M G Van den Brande^ ^ Steel Construction Institute, Ascot, SL5 7QN, UK 2 TNO Bouw, Delft, NL
ABSTRACT Work currently underway in Holland and the UK aims to increase the already considerable potential for light steel framing in buildings by developing rules for so-called "dry composites". This paper discusses both theoretical and experimental work to develop validated design rules for floor systems comprising cold formed joists and timber boarding. Preliminary results show that for practical boardto-joist connection details, a bending stiffness enhancement of around 30% can be expected. KEYWORDS Light steel framing, cold-formed, composite, shear connection, partial interaction, experimental, finite element modelling INTRODUCTION Composite construction is often adopted in hot-rolled steel framed buildings, where the provision of mechanical shear connectors, typically studs welded to the top flange of the beams, allows composite interaction with the concrete floor slabs. The connectors transfer longitudinal shear between the steel and concrete, and by so-doing typically the bare steel beam can see its strength and stiffiiess increased by a factor of two or three. There is an increasing use of light gauge steel members in the floors and walls of buildings, particularly in the residential sector, in many European countries. The benefits of light steel framing include its potential for off-site prefabrication of panels, a high strength-to-weight ratio, the ability to accurately control member dimensions (which leads to a quality product), and an ability to achieve high standards of thermal and acoustic insulation. These make light gauge steel an attractive alternative to the traditional timber and blockwork solutions adopted in the residential sector. A further advantage of light steel framing is its ability to achieve long-span solutions, due to the inherent strength and stiffness of steel itself. Moreover, progress could be made towards achieving even greater spans were appropriate guidance available, and appropriate systems validated, to consider composite action between the steel joists and adjacent flooring.
318 This paper will report progress already made in an on-going research project, one of the aims of which is to produce design rules, supported by testing, for so-called dry composite systems. To-date a basic analysis method has been identified, and its validity has been supported by a series of tests. Both analytical and experimental aspects are discussed below, as are plans for further testing. PARTIAL INTERACTION THEORY General The structural behaviour of an element comprising two primary components depends, amongst other things, on the properties of the shear connection between them; its strength, stifftiess and deformation capacity. Because real shear connectors are not infinitely stiff, slip takes place at the interface between the components, leading to what is called "partial interaction". This should not be confused with the terms "partial or full shear connection", which are associated with the strength, rather than stifftiess, of the connection. In the following discussion the influence of partial interaction on the bending stiffness of a composite member will be considered; the connection is assumed to provide full shear connection. Besides the stifftiess of each connector, the degree of partial interaction depends on the number of connectors. Both effects are included in the "slip modulus" k^, which is equal io k la, where k is the stifftiess of a single connector, and a is the spacing of the connectors. Consider a beam comprising two components of the same material, each with the same rectangular cross-section, and no interaction between the two (connection stifftiess is zero). The total bending stifftiess is simply the sum of the individual bending stiffnesses. When there is full composite interaction between the two components (connection stifftiess is infinite), the bending stiffness of the beam is four times the sum of the stiffnesses, or in other words eight times the stifftiess of each individual component about its own axis (see Figure 1). i Io,Wo
^
,
^
. ^ _ _ _ ^ .
t
No composite interaction (W = 2 Wo), U = 2 Io
r~~_
i
/-^____^
Io,Wo
T^
I0.W0
•
^
\
Full composite interaction (W = 4 Wo), U = 8 Io
Figure 1: Two extremes of composite interaction. Linear elastic shear connection The behaviour of real composite sections lies in between the two extremes represented in Figure 1. The curves in Figure 2 show composite stifftiess enhancement as a function of interface shear connection stifftiess for a cross-section comprising a cold-formed C section with timber board, assuming linear elastic material and connection behaviour. Three distinct regions can be identified: /.•
^^ < 1.0:
//;
1.0 < ^^ < 200:
///;
k^ > 200:
changes in connection stiffness have little effect on the composite bending stiffness (partial interaction is negligible). changes in connection stifftiess have a more significant effect on the composite stifftiess. changes in connection stifftiess have little effect on the composite stifftiess
319
load'
p
/ A t.—1
slip %
I — ,
' 77.:
L— '
. ..'. 11
""_:. 1 .
..IJTl'l
'; ~' ' 1 "
"
"
:
"
*-~-II-
\V '
.
Mr^fi:
:
:
:
1
',-.:* wr'~"
WF:; r" _: r: _ ~
r*
L
r
:. J.K/:
-r
f
"^A
^^ "
"
J i d
--'M
.- -7.:.i
^l,5kN/m^
3 ^
•
:-f:*
:r^
1 _~7.
- ::f .17
:!":-::
• * "
-.i-r.r::.: -
. -; ).TTp
T*
^
:-- : .-:r •
"T 1
w^ff^
^^^
J J
Ek - JOOO H/bm>
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
iJim^
|
min(li2 + li4,li3 + li5)
ll3 + ll5
1
338 T
ex
H N. • N
1
^
est
m2(3)
1^2(1) ^^tf^^m2(2)
0-
- ^
e
^w
- ^
T
Figure 3: Dimensions needed to determine an equivalent T-stub 2.2 Compression Part of the Base The bearing strength of the joint fj is be determined as
fj = /J • kj • f^^
(10)
where: f^^ - design value of the concrete cylinder compressive strength y0j - the joint coefficient, which may be taken as 2/3 provided that the characteristic strength of the grout is not less than 0,2 times the characteristic strength of the concrete and the thickness of the grout is not greater than 0,2 times the smallest width of the base plate kj - the stress concentration factor may be taken as kj = ^aj h^liab) « 1.5, provided that the conservative values of the active dimensions of the concrete block ai,bi available for Astron foundations are taken into account (a,b - base plate dimensions) It is assumed that the stress in compression zone is uniformly distributed over the effective area, which is composed from the column and stiffeners cross-sectional part and a strip of width x around the column and stiffeners outlines. x = tp-7fy/(3-rMo-fj) (11) When the value of the effective area is determined from the equation (1), computer routine starts with it's distribution from the compression edge of the base plate towards the tension edge (see Figure 2). Consequently the center of gravity of the obtained surface is established.
3. INTERACTION CHARTS Equations (1) and (2) assume the tension part of the base reaching it's full resistance and the bolts in the compression side in no action. Then their domain is bound by the normal force equal to this resistance and the one equal to the difference of the full compression resistance of the base and the a.m. tension resistance: Interaction of the normal force and the bending moment for the symmetrical base is shovm in Figure 4a and for the asymmetrical base in Figure 4b. In both some significant points can be recognized. It is evident, that the equilibrium of the internal forces can be reached outside the domain (curves between the points TI and CE, respectively TE and CI) of the equations (1) and (2) too. Point T represents the full tension in both base parts (concrete in no action) and point C the full compression in both parts (anchor bolts in no action). Thus the closed interaction charts in Figure 4 were obtained using linear interpolation in between points T—TI, T-*TE respectively C-*CE, C—CI.
339 ^ M^d [kNm]
C NRd[kN] compression (+)
web: 600.5 flanges: 180.8
baseplate: 180.20 stiffeners: 200.90
Figure 4a: Interaction M and N for the symmetrical base MRd[kNm]
NRdlkN] 1500 compression (+)
* I * * I • I • • I
* •
Figure 4b: Interaction M and N for the asymmetrical base Significant charts points: - concrete in fiiU action ( Ag^ = Afuu ): C - pure compression ( no tension in anchor bolts) CE - full compression + tension resistance is reached in the internal base part CI - full compression + tension resistance is reached in the extemal base part - concrete in no action (A^^ = 0): T - pure tension (tension resistance is reached in intemal and extemal base part) TE - tension resistance is reached in the extemal base part TI - tension resistance is reached in the intemal base part - concrete in action on compression side + tension resistance is reached in the tension side: Mmax - tension resistance is reached in the intemal base part + Ag^is distributed between the column neutral axis and the extemal edge of the base plate Mmin - tension resistance is reached in the extemal base part + Ag^ is distributed between the column neutral axis and the intemal edge of the base plate
340
The actual combinations of the normal force and the bending moment coming from the frame calculation has to sit inside the particular interaction chart to be covered by the base resistance. As some parts of the interaction chart are only a fiction and no really possible base stage, these parts were excluded from the practical design procedure as follows: - symmetrical base resistance was limited by the region: TI—Minax~'CE—C-*CI—Mmin-*TE-^TI - asymmetrical base resistance was limited by the region: TI—Mmax-^CE-^CI—Mniin-"TE—TI. Figure 5 shows the interaction charts for the base with the column cross section used in Figure 4 and several anchor bolt configurations. Two symmetrical (case 1/1 and 5/5) and two asymmetrical (case 1/5 and 5/1) cases were considered. Charts are limited by a.m. assumptions.
-1000
2(1)0
I'i T L1L1 • • • •
I
I
• • • •
*I * * I •I • • I
* •
1• 1*
Figure 5: Fixed base with four different anchor bolt configurations
1* * 1* *
• 1
•1
1
341 4. PRACTICAL APPLICATION Lets consider the fixed base with symmetrical bolts distribution and geometry indicated in Figure 6. Figure 6 shows the different evolution of the interaction charts for the default bolt diameter 18 mm and varied base plate thickness t = 16, 20, 24, 30 mm and contrary for the default base plate thickness 16 mm and varied bolts diameter d = 18, 24, 30 mm. It is evident that the raising bolt diameter considerably increases the moment resistance by decreasing normal force resistance. Such chart evolution fits for the "wind column" used for the building stabilization in its longitudinal direction. Wind column is apart from its own weight loaded by the horizontal force on its peak only. Chart evolution for the varied base plate thickness is wanted for the portal frame, for which both the increase of the moment and the normal force resistance is needed.
• I • • I •
web: flanges: base plate: stiffeners:
* I* • I •
-1000
600.5 180.8 180.20 200.90.9
Figure 6: Sensitivity study for the base with varied bolts diameter and base plate thickness Fixed bases are used for Astron buildings with eaves height to span ratio h^ > 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: M
5. CONCLUSION The computer program for the fixed base resistance determination which comes out from the a.m. principles has been developed. Shear force between the column and the foundation is supposed to be transmitted by the group of anchor bolts in the compression side of the base. In the frame calculation the fixed base is considered as fully rigid and its initial stiffness should be greater then the limit given for unbraced frames in Annex J [1]. Up to now is the estimation of the sufficient initial stiffness based on the engineering practice. Its more precise determination will be the next step of the presented analysis. Acknowledgment The author would like to express thanks to Jean-Pierre Jaspart, Dr.Ir. for his advice and valuable discussions concerning Annex J [1] during the meetings held in Liege and Diekirch. References [1] EUROCODE 3, ENV 1993-1-1, Part 1.1, Revised Annex J: Joints in Building Frames [2] EUROCODE 3, ENV 1993-1-1, Part 1.1, Annex L: Column Bases [3] Wald F. (1995) Column Bases (Patky Sloupu), Edicni stfedisko, CVUT Praha [4] Wald F. and Sokol Z. (1997) Tuhost kotveni sloupu, 18* Czecho-Slovak International Conference on Steel Structures and Bridges, Proceedings p. 1/137 -1/142 [5] Sawczuk A. and Sokol-Supel J. (1993) Limit Analysis of Plates, Polish Scientific Publishers PWN Warszawa
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
343
NEW STRUCTURAL SOLUTION OF LIGHT-WEIGHT STEEL FRAME SYSTEM, BASED ON THE SIGMA PROFILES Sc, Eng. Zdzislaw Kurzawa^ M. Sc, Eng. Katarzyna Rzeszut^ D. Sc, Eng. Andrzej Boruszak^ D. Sc, Eng. Wojciech Murkowski^ ^ Department of Civil Engineering, Poznan University of Technology Poznan, Poland ^ Designing Office „ Metalplast- Inwest", Poznan, Poland
ABSTRACT Tliis paper presents a light-weight steel frame system with a span ranging from 6,0 m to 18,0 m and height ranging from 3,5 m to 6,5 m, which is fully designed and produced in Poland. All structural elements of the frame are made from thin wall cross section „Sigma". Sandwich plates are used as light sheeting of the frame. The main structural elements are made from combined profiles and have a very effective cross section. The system frames with rigid joints are available in two versions, i.e. with or without bowstring. All joints of the structure are universal solutions which fit easily for every kind of system frame. The paper provides examples of a^ new joint solution and methods of calculation. It also presents a comparison of our own structural solutions with the well-known system „PRACTA".
KEYWORDS Thin walled cross section, joint rotation, frame, joint flexibility.
INTRODUCTION The main aim of a designer of an industrial building is to ensure the optimal steel consumption for cross section frame structure and sheeting. Economical designing is often limited by the range of light steel profiles available on the market. Frame steel structures without cranes and a span up to 20 m can be constructed from light-weight thin wall profiles produced from 1,5 to 3,0 mm thick cold formed steel sheets. This type of frame steel structures is not very common in Poland due to the lack of well designed profiles with height over 250 mm which would allow to construct bars (cross section columns and beams) longer than 6,0 m.
344
Limited use of light-weiglit steel profiles in industiial buildings is also due to their high cost. Their higli price results from the use of expensive techniques of anticorrosion protection. The most popular method is hot galvanising. PR ACT A, SADEF and a Czech HARD are the examples of systems based on light-weight profiles. They are constructed from bars with a „sigma" cross section (PRACTA, SADEF) or „sigma"-like cross section (HARD). Figure 1 shows the cross sections of the profiles as well as systems used to construct bars of the frame. CROSS SECTIONS OF PRACTA OR SADEF SYSTEMS y
^\
b)
CROSS SECTIONS OF HARD SYSTEM
' A y
2 •fH—^
%
QJ
\^
i\iacing Figure 1. PRACTA and SADEF are veiy similar systems and the vertical axial symmetry (Y-Y) of profile systems in SADEF makes the basic difference. HARD, unlike the other systems, makes use of profiles produced from steel sheets of thickness over 3 mm which allow to construct a closed system of profiles joined by welding. This makes HARD system rigid enough in the perpendicular plane to the loading plane and, therefore, resistant to lateral buckling (Fig. Id). METALPLAST-PRACTA system is a system based on „sigma" profiles manufactured by Rautaruukki from Finland. Therefore, we can compare two frame steel systems: PRACTA and METALPLASTPRACTA. In both systems in most cases complex bars (Fig. lb) were used to achieve better stability. Collaboration of the two parts of the cross section in tiansmitting load was ensured by the use of steel lacing spaced at approximate^ 1000 mm intervals along the whole height of the cross section.
A DETAILED DESCRIPTION OF THE CONSTRUCTION PRINCIPLES OF METALPLAST-PRACTA SYSTEM The basic construction and installation assumptions are as foUows: - all components are made from „sigma" profiles as single or combined elements, - all construction joints are designed for high tensile bearing-type bolts, - use of hot-dip zinc coating of all elements for anticorrosion protection, - all elements used in cross and vertical section as well as sheeting are light enough to enable installation with liglit-weight constmction equipment or without it. - the system can be applied for one bay industrial buildings with a span ranging from 6 m to 18 m and a fLxed spacing of frames at 4 m interx-^als in vertical section. A round bar can be used as an alternative bowstring for bays with a 16 m span. Bays with a 18 m span have all a bowstring at the comer joint. It is also possible to construct an extended frame structure with a 20 m span. The useable heiglit of bays ranges from 3,5 m to 6,5 m depending on the span of the ft-ame structure. For detailed information see the catalogue. Although there are two alternative versions of the system, they are both based on one structural solution in which a cross section with three welded joints of the following type: an alloy joint (WS), a comer joint
345
(WN) and a roof ridge joint (WK) is combined with a system of columns and beams designed with combined „sigma" profiles and where bolts are used to fasten all the structural parts and to anchor them to joints and to the bearing elements of the sheeting. Figure 2 shows the version of the system with 2 m long units. Figure 3 shows a system frame where a beam and column are an integral whole. Figure 4 presents a structural solution used in PRACTA system in Finland.
Figure 2.
Figure 3.
346
Figure 4. The bearing structure of the sheeting in METALPLAST-PRACTA system was different than in the Fmmsh system. In the PoUsh system spandrel beams, purlins as well as-columns and spandrel^eamst gable walls were ludden m the cross section structure (frame) and not left outside. Adoptmg this t>pe 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
i_c>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
(2)
R - appropriate resistance in a simple case; let it be bending resistance MR for cross-section design and compression resistance NR for of member design, Seq - equivalent bending moment Meq or equivalent compression force Neq, respectively. They are linear flinctions of loads Fi. Neither the reduced resistance moment MRR nor the MV.R are necessary any more. Product of the normed interaction matrix crs and combination matrix vi/yF gives equivalent load effect Seq=v|/YFci- crsiv
(3)
The three-dimensional matrix crs is reduced to the appropriate two-dimensional matrix after selection of the most unfavourable variant v for every load i = 0, 1^ 2, 3, 4 one after another and the matrix crsie is reduced to a vector after selection of the most unfavourable load effect interaction range e . The matrix of combination values of loads vj/yF is reduced to a vector after selection of the most unfavourable load combination c = 0 , l , 2 , 3 , 4 . I n such a way the scalar product xi/yFi- crsi can give the value Sea.
362 Cross-section Design Interaction curves m-n , m-v and n-v are shown on Fig.2 for standard rolled I or H sections according to the points EC3/5.4.8 and EC3/5.4.9 of the Eurocode 3. The curves are represented in non-dimensional coordinate system
where
m = MS/MR ,
n = NS/NR ,
v = VS/VR
(4)
MR = Wc-fy/l,l ,
NR = A„-fy/l,l ,
VR = Av-fy/(V3-l,l);
(5)
Wc = Wpi for Class 1 , 2 or Wc = Wd for Class 3 of the cross-section; An = A if Ns> 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<^,wF.
,,^.
'
^''^
where the matrix ij/yF is reduced to an appropriate vector for the selected load effect combination c and both the 1-st order influence matrix a(|) and the 2-nd order influence matrix b<|) are reduced to respective vectors for the selected load variants v. Finally, the trial value (j) is corrected in the line of variables as well as other preliminary parameters of #1 , #2 and #3 lines of variables.
366 CONCLUSIONS Safety of the whole structure is all right if inequality (2) is true for every structural element at the ultimate limit states and the increment sway Ac]) = (|) - (t)o does not exceed the limit value for the frame in serviceability limit state. The advantage of computer templates will be evident if many trials are necessary to select economical and safe structural elements. Pilot design templates have helped to discover deficiencies of the European prestandards, e.g. • superfluous requirements like the EC/5.4.6 bending and shear interaction condition (Fig.2), • imaginary results in some cases of the EC/5.5.4 bending and axial compression equation (11), • risky recommendations like the ECl/9.9.4 combination rule v|/i (Table 1). Many other improvements of the Eurocodes are possible taking into consideration design assisted by computer templates, e.g. • deleting classification of sway and non-sway frames and ehmination of other exemptions which complicate the computer-aided matrix calculations.
REFERENCES Books and Papers Ferry-Borges J., Castanheta M. (1971). Structural Safety (2-nd ed.), LNEC, Lisbon 1971. Murzewski, J. (1989). Niezawodnosc konstrukcji inzynierskich., Arkady, Warszawa Murzewski, J. (1992). Imperfections and P-Delta effects in multistorey steel frames, Archives of Civil Engineering 38:3, 191-203. Murzewski, J. (1996). Upper bound for combination of action effects. Basis of Design and Actions on Structures lABSE Colloquium Delft 1996. Reports 74, 279-290. Murzewski J. (1997). Computer-aided design of framed steel structures. Probability-based design of structures in matrix formulation. Janowice' 97, Politechnika Krakowska & KJLiW PAN. Turkstra C.J. (1972). Theory and structural design decision. Solid Mechanics Study 2, University of Waterloo, Ontario. Standard Specifications Eurocode 1. (1993). Basis of Design and Actions on Structures. Part J : Bases of design. ENV 1991-L Eurocode 3. (1992). Design of Steel Structures. Part J.J: General rules and rules for buildings. ENV 1993-1-L
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
367
CONSTRUCTION OF A 60.000 m^ STEEL STORAGE TANK FOR GASOLINE
J. Vojvodic Tuma Department for the Revitalization of Industrial Facilities and Equipment, Institute of metals and technology, Lepi pot 11, 1000 Ljubljana, Slovenia
ABSTRACT The paper deals with the use of the new micro-alloyed steel niomol 490 K for the construction of above-ground tank for the storing of gasoline.The volume of the storage tank is 60.000 m^, its inner diameter is 60,1 m, and the height of its curved surface 22 m. Fine grained micro-alloyed steel niomol 490 K has a ferrite-bainit microstructure and exhibits the following mechanical properties : tensile strength 560-700 MPa, impact toughness more than 200 J and brittle transition temperature at -133 °C. The storage tank has a fixed domed roof, floating dock and double bottom. The elements of the storage tank are made of niomol 490 K, whereas the domed roof and the floating dock are made of aluminium.The storage tank has double curved surface. In case of damage of the tank this prevents the spilling of the fluids and their trickling into the earth. The reservoir is the largest steel structure of this kind in Slovenia, and the same time also the most modem one.
KEYWORDS cylindrical storage tank, construction, welding, double bottom, double shell, fine grained microalloyed steel
INTRODUCTION Above-ground reservoirs are designed in the first place for the storage of liquids and liquefied gases. Most often they are vertical stand-alone cylindrical vessels with inclined bottom and fixed or selfcarrying domed roof with or without floating dock. In the last few years the difficulties due to the lack of space for the construction of reservoirs with classical earth or concrete capturing pools have been solved by the construction "reservoir in reservoir", i.e. with a cylindrical capturing pool where the distance between the reservoir shell and the shell of the capturing pool is between 1,5 to 3,0 m. The most appropriate material for reservoirs is steel, while for the roof structure and the floating dock recently aluminium has gained importance due to its low weight. All this was considered at the
368 construction of the reservoir with a volume of 60.000 m^ for the storage of petrol at Sermin near consl Koper.
DESCRIPTION OF THE STRUCTURE The reservoir has a volume of 60.000 m^, internal diameter 60.100 mm and shell height of 22.000 mm, and is designed for the storage of unleaded petrol. The shell of the reservoir, the capturing pool, the heel of the reservoir and the reinforcing rims are made of the steel Niomol 490 K^ , the domed roof and the floating dock are made of aluminium. The reservoir has a double bottom v^ith a thickness of 7 mm, v^hereas the basic ring is 13 mm thick. The internal diameter of the capturing pool is 67.000 mm, the shell height is 17.500 mm, and the volume of the capturing pool corresponds to the reservoir volume. The shell of the reservoir is consisted of totally eleven rings with a height of 21.975 mm, where the height of the first 10 rings is 2000 mm and the height of the last one is 1975 mm. The thickness of the reservoir shell is between 19 and 10 mm. On the top the reservoir shell is reinforced by a square with a height of 100 mm and along the height with two more reinforcing rims made of Niomol 490 K. The shell of the capturing pool has a height of 17.478 mm and consists of 8 rings with a height of 2000 mm and the last ring with a height of 1478 mm. The shell of the capturing pool has a thickness of 16 to 10 mm and is reinforced by five reinforcing rims at a distance of 3300 mm, 2900 mm, 1900 mm and 2550 mm, measured from the upper primary rim, and on the top the shell is reinforced by a 100 mm high square. The reservoir has a self-carrying fixed aluminium domed roof of the type Vacondome and internal floating aluminium roof Vaconodeck. During the construction of the reservoir also the adequate technological pipelines and fire-resistant system were constructed.
RESERVOIR CALCULATIONS When calculating the elements of the reservoir the following norms were considered : • DIN 4119, part 2 - for loadings, proof of stresses and calculus of strains • DIN 18800, part 4 - for the stability control and combination of stresses • BS 2654/1989 - for the distribution and determining of strengthening rims and control of the earthquake load • ANSI/API 650 for the calculus of strengthening rims In the structural calculations the following loadings were taken into account: For the reservoir: • dead load of the shell . weight of the domed roof (assumed) 500 kN . weight of the equipment (assumed) 200 kN • weight of snow 0,35 kN/m^ • hydrostatic pressure of the medium in exploitation with g = 8,44 kN/m^ • hydrostatic pressure of water at hydrotest with g = 9,81 kN/m^ • wind load at a speed of wind 150 km/h • earthquake load for the 8* earthquake zone according to MCCS and poor ground For the capturing pool of the reservoir: . dead load of the shell and the equipment • hydrostatic pressure of the medium with g = 8,44 kN/m^ • hydrostatic pressure of water at hydrotest with g = 9,81 kN/m"^ • wind load at a speed of wind 150 km/h
369 The thicknesses of the reservoir shell (from above downwards) are: 3 rings with a thickness of 10 mm - total height 6000 mm 2 rings with a thickness of 11 mm - total height 4000 mm 1 ring with a thickness of 12 mm - total height 2000 mm 1 ring with a thickness of 14 mm - total height 2000 mm 1 ring with a thickness of 15 mm - total height 2000 mm 1 ring v^th a thickness of 17 mm - total height 2000 mm 1 ring with a thickness of 18 mm - total height 2000 mm 1 ring with a thickness of 19 mm - total height 2000 mm The thicknesses of the capturing pool shell (from above downwards) are: 5 rings with a thickness of 10 mm - total height 11.500 mm 1 ring with a thickness of 12 mm - total height 2000 mm 1 ring with a thickness of 13 mm - total height 2000 mm 1 ring with a thickness of 15 mm - total height 2000 mm 1 ring with a thickness of 16 mm - total height 2000 mm These calculations do not take into account the additions for possible negative tolerances of the plates (0,3 mm) and for the shell corrosion (1 mm). This means that only positive tolerances are to be allowed when ordering plates, and the best anti-corrosion protection of the shell is to be carried out. Further more, a special attention has to be paid to the maintenance and the protection control in the life-time of the capturing pool. The secondary reinforcement rims are foreseen at a distance of 3300 mm, 2900 mm, 1900 mm and 2550 mm, measured from the upper primary rim. All the reinforcement rims are to be assembled successively with the shell. The designed dimensions of the reservoir and the foreseen reinforcement rims provide safety and stability of the finished reservoir, provided that the following conditions are met: • •
no negative pressure will occur in the reservoir the plate thicknesses will not be smaller than the designed ones the tolerances of measures and forms will be within the foreseen allowable limits
THE BASE MATERIAL The use of high strength structural and fme-grained micro-alloyed steels for pressure vessels in Slovenia has increased significantly since microalloyed carbon-manganese steels were introduced in the 1969's. The first fine-grained micro-alloyed steel made in Steelwork Jesenice, Slovenia, was nioval 47, the last one is niomol 490 K, which is comparable to ESTE 500 and will be presented in this article. Recognizing that designs should be based on yield stress resulted in significant changes in our and foreign national design and application codes. The need to develop steels of higher yield strengths without impairing weldability has resulted in alternatives which use microalloys such as niobium, titanium and vanadium. Since the occurence of fast fi-acture in ferritic steels depends on the total strain operating in regions of flaws and discontinuities, steels of improved fi-acture toughness are required as thicknesses and design stresses increase. As the performance requirements of highstrength steels become more severe, the need to establish rational levels of toughness in the base metal, heat-affected zones, and weld metals is vital. Fracture-mechanics concepts that develop realistic fitness-for-purpose criteria offer the best approach. Company ACRONI - Jesenice, Slovenia, has developed special steel named niomol K which is especially made for the petrochemical industry service. Its chemical composition is given in Table 1. Fine grained micro-alloyed steel niomol 490 K of ferrite - bainit microstructure is very suitable for
370
fabrication of high risk welded structures, for example, pressure vessels, storage tanks, spherical tanks, car and waggon mounted tanks and piping systems in petrochemical plants. This steel is resistant to the hydrogen and induced stress corrosion, and exhibits the following mechanical properties : tensile strength 560 - 700 Mpa, impact toughness more than 200 J and brittle fracture transition temperature at -133°C. Niomol 490 K is resistant to the hydrogen embrittlement. This steel is very useful for other constructions because of the excellent notch and crack toughness, and high weldability without preheating, especially for the dynamic loaded ones.
TABLE 1 CHEMICAL COMPOSITION (WEIGHT %) C
Si
Mn
P
S
Cr
0.08 0.34 0.36 0.01 0.004 0.54
Ni
Cu
Al
Sn
As
0.17 0.36 0.05 0.015 0.017
Mo
N
0.27
0.007
carbon equivalent CE=0,25-0,28 The mechanical properties of Niomol 490 K at room and low temperatures are shown in Table 2. TABLE 2 MECHANICAL PROPERTIES AT ROOM TEMPERATURES Yield stress- Rp
MPa
Tensile strength - R^^
MPa
604 522 Notch tough ness (ISO-V): T=+20°C - 260 J, 232 J, 249 J;
e^xlOO
%
Z
%
22 79,2 T=0°C-224 J, 231 J,244 J
MECHANICAL PROPERTIES AT LOW TEMPERATURES Yield stress- Rp MPa
Tensile strength - Rjj^ MPa
euX 100 %
Z
Kic
%
MPaVrr
T=-133°C 660 713 17 73 Notch toughness (ISO-V): T=-60°C - 209 J, 165 J, 225 J; T=-80°C - 54 J, 62 J, 66 J
102
Due to its low contents of carbon and manganese the steel niomol 490 K is distinguished for good and reliable welding characteristics. Niomol 490 K can be classified according to the DIN norms to the quality ES tE 500 (WNR 1.8919), and according to the EURO norms to the quality S 500 WLl. The main properties of this steel are: • • • • •
the steel has very good mechanical properties, especially toughness at low temperatures, the steel allows to be reshaped when cold, due to low C equivalent it can be cut by flame without pre-heating, the steel has good welding properties; when using the correctly chosen welding material the weld has a good toughness up to -40°C, the edges allow to be treated by flame.
371 . • •
welding demands no pre-heating, when using the correct welding technology, the strengths in the transitional zones do not exceed 220 HB, the steel is not sensitive to cracking in cold, since in no case martenzit occurs.
Niomol 490 K steel is quenched (950°C) and tempered (630°C). Experiments confirm that niomol 490 K steel exhibits excelent resistant against all kinds of corrosion atach including hydrogen embritelment. Microalloyed steels have given excellent results at normal- and low-temperature conditions. Research work showed that niomol 490 K has good mechanical properties, weldability, fracture toughness, hydrogen embrittlement and the ability of crack arresting with which the conditions for the use in the petrochemical industry are fulfilled.
ASSEMBLY OF THE RESERVOIR Figure 1 presents the preparation of the concrete foundation.
Figure 1: Preparation of the concrete foundation For the electric arch-welding the electrodes EVB Ni Mo ^ 3.25 mm and for the semi-automatic welding in protection of 80% argon and 20% CO2 the welding wire FILTUB 28 B (j) 1.2 mm were used. The welding of the reservoir bottom is presented in Figure 2. The welding was carried out without pre-heating. The lower bottom plate of the reservoir was inspected visually and by using a vacuum device. After welding the upper bottom plate of the reservoir and after the hydrostatic test of the reservoir and the capturing pool tightness a vacuum of 400 mbar has to be ensured between both bottom plates. If during its operation an increase of the sub-pressure occurred, this is a sign that the reservoir is leaking. The welds in the shell were radiographically checked, and the welds at the joints were tested by ultrasound^ and by penetrants.
372
Figure 2: Welding of the reservoir bottom The stability of the reservoir shell and the capturing pool shell is achieved besides by using the adequate shell thickness also by providing adequate distribution and adequate dimensions of the reinforcing rims. The process of assembling the capturing pools and the reservoir is presented in Figure 3.
Figure 3: The process of assembling the capturing pool and the reservoir
373 The reservoir has a self-carrying fixed aluminium domed roof of the type Vaconodome and internal floating roof Vaconodeck. The composition of the roof is presented in Figure 4.
Figure 4: Composition of the reservoir roof The constructed reservoir and capturing pool before the assembly of the roof are presented in Figure
Figure 5: Reservoir before the assembly of the roof
374 Figure 6 presents the finished reservoir with anti-corrosion protection at Sermin near Koper.
Figure 6: Reservoir with a volume of 60.000 m^ at Sermin near Koper
CONCLUSION The paper presents the construction of an above-ground cyhndrical reservoir with a volume of 60.000 m designed for the storage of petrol derivatives. Due to increasingly rigid ecological demands and for economic reasons the structure of a reservoir with a double shell, double bottom and self-carrying aluminium roof was chosen. References 1. J. Vojvodia Gvardjan5i5,1993, Conditions for a transition to brittle fracture of fine-grained microalloyed steels. Ph. D. dissertation, Ljubljana, Slovenia. 2. J.Vojvodic-Gvardjancic, B. Ule, S. Azman, 1993, Fracture toughness of fine grained microalloyed steels at nil ductility and transition range. Journal de Physique IV, 3, pp.105-108. 3. J.Vojvodic-Gvardjaneid, B. Ule, S. Azman, 1994, The influence of the ageing of the fine grain low alloy structural steels on the nil ductility temperature, Proc. Conf. on Material Testing, Balaton pp 1251-1259. 4. B.Ule, J. Zvokelj, J.Vojvodi5 Gvardjanaic, A. Zajec, F. Krzic, D. Beg, F. Hladnik, J. Banovec, V. Nardin, R. Turk, S. Azman, A. Lagoja, 1993, Development and Introduction of High Strength Microalloyed Steels for Applications in Process Industry and Civil Engineering, Research work 420399-91/93, pp.1-21 and 1-69. 5. J.Vojvodic Gvardjancic, S. Azman, 1987, The Influence of Technology Treatment on the Properties of Niomol 490, Proc. Conf Weldability and Material Testing, Beograd, pp.53-61. 6. J.Vojvodic Gvardjancic, 1995, Application of ultrasonic measurements for safety evaluation of cylindrical tank for oil derivates, Proc. Conf. Inservice Inspection, Pula, Croatia, pp.245-255.
Session A6 DEVELOPMENTS IN DESIGN
This Page Intentionally Left Blank
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
377
MULTIMEDIA DATABASE USING JAVA ON INTERNET FOR STEEL STRUCTURES
Y. Itoh' and H. Wazaki' ' Center for Integrated Research in Science and Engineering, Nagoya University, Nagoya 464-8603, Japan ^ Department of Civil Engineering, Nagoya University, Nagoya 464-8603, Japan
ABSTRACT Recently, the Japanese seismic deign guidelines have greatly been improved, and the seismic tests such as the cyclic loading test and the pseudo-dynamic test have been conducted in a lot of research organizations. This paper presents the preparation and development of a multimedia database for the seismic test results about the steel piers. The database is made available on the Internet, and Java language enables the interactive retrieval efficiently. The system takes in an existing database for steel structures and includes the ultimate strength test results of structural steel members such as steel columns, steel beams, and steel plates.
KEYWORDS Databases, Earthquake, Internet, Java, Multimedia, Structural experiment
INTRODUCTION Since the disaster of the great earthquake at Hanshin district in 1995, the Japanese seismic deign guidelines have greatly been improved. The seismic tests such as the cyclic loading test and the pseudo-dynamic test have been conducted in a lot of research organizations. With the increase of the test data gathered and managed in the form of databases, some statistical approaches have been applied for recognizing the characters of materials and structures such as the ultimate strength and
378 ductility analyses in by Suzuki and Usami (1997). However, only a part of data on these tests has been published as reports or papers, and most parts of those data such as dimensions and failure types of specimens have not been published. In order to preserve and share experimental data, a Numerical Database of Steel Structures (NDSS) has been developed at Nagoya University since the 1980s (Itoh 1984, Itoh et al. 1996). This database has been referred as the basic data for the revision of various specifications on steel structures all over the world. However, taking into account of the access from outside, this database is not efficient from the viewpoints of both hardware and software. On the other hand, with the recent rapid extension of information networks, Internet has become one of the most effective media. The recent progresses in the performances of hardware and software like browser enables easy handling of multimedia like image, video and sound. In this research, a multimedia database that includes the valuable data of seismic test results relating to strength and ductility has been developed to collect and combine various types of test data. The data for the ultimate strength tests in the above-mentioned existing database have also been made available on Internet using Java. This new database is named as Multimedia Database on Internet for Steel Structures (MDISS), which is useful for the study and education in civil engineering as well as research. This database is still under development to contain more structural experimental data.
INFORMATION ACCESS ON WWW In 1989, the Conceil Europeenee Pour la Recherche Nucleaire (CERN) in Europe published the first World-Wide Web (WWW) system which made multimedia data including images, sounds, and literature document as well, easily treatable on Internet (Masuoka and Kibakura 1995). However, in the WWW system based on the HyperText Markup Language (HTML) the server can deliver only created documents consisting of the text, images, and so on. When the interactive operations such as searching data are carried on, special programs are needed. One of such special programs is Common Gateway Interface (CGI) that is usually developed with C or Perl language as shown in Fig. 1(a). In this interface program system, when the request from the client is made, the server can carry out the required operation and send the information interactively. Another program is Java that is an objectoriented program language available for the Internet use. Original programs called "applet" can be made using Java, which are sent to the client. Those original programs can also be exerted at the client and shown on the browser as if they are a part of the document. The architecture of applet is shown in Fig. 1(b). Java enables to deal with more issues than CGI does. In addition, since applet is exerted at the client's CPU, the operation at the network level CGI can not function, and the socalled "Distribution Computing" can be performed realized (Laura 1995). As far as the development of the user-interface and the distribution computing system of this database concerns, using Java is considered to be a suitable adoption. Therefore Java is used in this research for developing the database. This database is not developed only to display the experimental numerical data, but also to make the
379 experiment procedure understood through various types of information including the image data, the experimental process, and the purpose of each experiment by taking the advantage of WWW. In this way, the integrated information related to each experiment can be provided. In addition, the system has been developed aiming flexibility and generalizing in order to be able to respond the updated progress in the multimedia environment. The development platform use for the preparation of this system is Sun workstation.
Server •
Client Browser
® Request
Receive request, Execute CGI program
® Document
(b) Java Figure 1: Differences between CGI and Java
IDENTIFICATION OF EXPERIMENTAL DATA The actual system includes seismic experiments such as the cyclic loading experiment and the pseudo-dynamic experiment of the steel pier for the single-column type that includes the effect of local buckling. It also includes the data of the ultimate strength tests which are handled in Numerical Database for Steel Structures (NDSS) developed at Nagoya University. The results of 431 new individual seismic tests are categorized into 5 types according to their specimen type shown in Table 1. Some of seismic test results include the displacement history in the dynamic loading. The number of those results is shown within parenthesis in Table 1. In addition, the number of data of the ultimate strength tests imported from NDSS is shown in Table 2. These data from 5653 individual tests are categorized into 5 types according to their specimen types. Information of structural tests consists of four types of data (Itoh et al. 1995): (1) numerical raw data including the cross-sectional properties, the model material data, and the load data; (2) text data such as the purpose of the experiment, how to set up the specimen, how to load, and bibliographies that
380 includes list of papers; (3) images of arrangement test equipment and failure specimen, and diagrams to determine the position of the measurement point such as strain gages and displacement transducers; and (4) videos of the behavior of the specimen collapsing in the pseudo-dynamic experiment. All these data are useful for users to know the details about structural tests. The database developed in this research consists of only three types including data of the numerical data, the textual data, and the image data. All above data are related one another to understand the entire test from various sides. Figs. 2(a) and 2(b) show examples of graphics, and image data treated in the system, respectively. These figures show the history of displacement and restored force of a specimen of the cyclic test and the picture of the test specimen after failure, respectively. TABLE 1
TABLE 2
SPECIMENS OF CYCLIC LOADING AND PSEUDO-DYNAMIC TEST
LIST OF ULTIMATE STRENGTH TESTS
Unstiffened column
Cyclic loading test 54(21)
Stiffened column Unstiffened column with concrete Stiffened column with concrete
142 (37)
Shape of specimen Steel column
Pseudo dynamic test 28(3)
Number of specimens 1665
25(2)
Steel beam
554
41(5)
22(3)
Steel plate
793
70(1)
Pipe
13(7)
0(0)
Plate girder Material experiment
333
36(2)
Total
286(71)
j
145 (9)
Total
2308 5653
••••••••••••lEiklll^ C-C-3
Load-Displaconoit
2.0
H/HyO
1
10.
•jso
-^^^^^L^:^/ "'i^ li
1T
H i S/ j^^j/^^L--^^
^
:
^'
d/dyO
-2 9
JiSfiPMnf; Apptel Wfidoi*
>•
(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 <»»»««*? tBwjt
CotMftft£(a£tilit h
Choose parameter Show the retrieved data
W
^^'*^t~Show the reference number of the data cHcked
Figure 7: Ultimate strength test results of steel columns
384 CONCLUSION In this paper, a multimedia database was developed on Internet for the steel structures. This research can be concluded as follows: 1) Seismic experimental data have been gathered and arranged, and a prototype system of multimedia database for seismic experiment was developed on WWW. An existing ultimate strength test database of ultimate strength of structural steel members was shifted to this system. 2) It was proved that Java is an effective programing language to develop such an interactive system for retrieving test data interactively and making the graphical user interface. 3) The system was developed to be able to generate some valuable coefficient and formulation from the statistical viewpoint. With the increase of available data in this database, the system will be able to supply more information to the users.
REFERENCES Itoh Y, (1984). Ultimate Strength Variations of Structural Steel Members. Doctor Dissertation, Department of Civil Engineering, Nagoya University, Nagoya, Japan. Itoh Y, Usami T. and Fukumoto. Y (1996). Experimental and Numerical Analysis Database on Structural Stability. Engineering Structures 18:10, 812-820. Itoh Y, Hammad A. and Liu C. (1995). Development of Information Environment for Structural Tests Data and Numerical Analyses Results. Proceeding of the Fifth Asia-Pacific Conference on Structural Engineering and Construction, Queensland, Australia, 1839-1844. Masuoka R. and Kibakura K. (1995). World-Wide Web (WWW). Journal of Information Processing Society of Japan 36:12,1155-1165 (in Japanese). Nakai H., Kawai A., Yoshikawa T, Kitada T., and Miki T. (1982). Investigation on Steel Rigid Frame Piers (Part 1). Bridge and Foundation Engeneering 82:6. 35-40 (in Japanese). Laura L. (1995). Teach Yourself More Web Publishing with HTML in a Week. Prentice Hall Japan, Tokyo (in Japanese). Suzuki M. and Usami T. (1997). Study on the Behavior of the Steel Pier Subject to the Severe Earthquakes. NUCE Research Report 9701, Nagoya University, Nagoya (in Japanese).
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
385
DESIGN OPTIMIZATION OF NONUNIFORM STIFFENED STEEL PLATE GIRDERS LRFD vs. ASD PROCEDURES S. A. Alghamdi, M.ASCE and M. H. El-Boghdadi Department of Civil Engineering, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
ABSTRACT The paper presents the results of a design procedure (nongradient-based) that has been automated to obtain a minimum weight design of nonuniform stiffened steel plate girders. The procedure allows the use of either the ASD guidelines or the LRFD guidelines to obtain the required design. The results of numerical studies performed to obtain a minimum weight design of a three-span nonuniform stiffened plate girder are summarized to: (1) highlight differences between values obtained from these two design procedures; (2) emphasize effectiveness of applying the procedure to a design problem as compared to the use of a generalized gradient-based design procedure.
KEYWORDS Plate girders. Stiffened steel plate. Minimum weight design. Nonuniform girders, LRFD vs. ASD.
INTRODUCTION Recent decades have witnessed a fast growth in the number of structures built of structural steel. Reasons for this trend are many and vary from country to country, but it seems that key issues leading to this trend include: (1) predisposition of structural designers to use a structural material with wellknown characteristics; (2) needs for more durable material; (3) relative ease to satisfy constraints often imposed to provide optimal designs (e.g.: minimum weight design). And since structural steel is a construction material that does meet the requirements of these issues, it is always possible (even required) that designers utilize the well-known attributes of this materiall to specify optimal engineering designs that would minimize costs without compromising on structural safety and design requirements (Gallagher & Zienkewicz, 1973; Cornell et al., 1966). This process of structural design requires the satisfaction of optimal design requirements at the component level (Anderson & Chong, 1986), as well as at the structural level (Haflka et al., 1980). It is, therefore, necessary that designers have efficient tools to obtain optimal designs of key structural components (e.g.: beams/girders;
386 columns; plates). For this purpose, the psLperfirstpresents a general review of previous studies that have addressed the issue of optimum design of structures in general, and the minimum weight design of plate girders in particular. Then an alternative design procedure (which does not require evaluation of gradients of constraints or weight function) is presented. This procedure has been automated in a computer code that allows for selecting either the method of Allowable Stress Design (ASD) (AISC, 1989), or the method of Load and Resistance Factor Design (LRFD) (AISC, 1986) to obtain a minimum weight design of a three-span nonuniform stiffened steel plate girder. A summary of several numerical studies using this procedure is used to: (a) compare results of the two design methods (i.e.: ASD and LRFD); (b) indicate the effectiveness of this ad hoc procedure as compared to the use of a gradient-based design procedure.
OPTIMAL DESIGN OF PLATE GIRDERS In recent years, the search for more economical and rational structural designs (optimal designs) has grown to unprecedented levels. It is believed that this trend is attributed to two major reasons: (1) stringent financial constraints are often imposed on designers to provide the most economical designs without compromising on design objectives and structural safety; and (2) better understanding of material and structural behavior in addition to availability of computational and search tools have made designers more predisposed to always search for optimal designs. One should, however, note that most of available methods for optimal structural design [e.g.: Gallagher & Zienkiewicz (1973); Haftka et al. (1980] are iterative in nature and require a great deal of care in the formulation of design problem. It has also been noted that, without proper human interaction, these tools usually suffer from numerical difficulties. Such difficulties may make convergence of a search procedure to an optimal solution impossible and may even lead to convergence to nonoptimal designs (Vanderplaats, 1984). Some of these difficulties have been reported in a previous comparative study on the optimal plastic design of nonuniform plate girders using the reduced gradient procedure (Alghamdi, 1996). Based on the above remarks, and due to known high structural efficiency of built-up steel plate girders in carrying loads, key design considerations that play a major role in optimal proportioning of these structures have been addressed in previous studies [e.g.: Easier (1961, 1963); Holt & Heithecker (1969); Schilling (1974); and Smith (1979)]. It is noted in these studies that an optimal design may be obtained from certain proportioning of cross section components (such as: web depth d; plate thickness /; ratio of web area to total area AJA), steel strength, and/or support locations. But since web stiffness is a key factor in determining optimality of a design, it is necessary to use adequate number of web stiffeners in order to mobilize tension-field action against web and compression flange buckling. This design consideration and the need for more efficient (automated) design procedures require the development of more robust algorithms that account for all design constraints specified by design codes [e.g.: ASD (1989); LRFD (1986)] in a unified form using the tools of mathematical programming (Khot, 1981). A recent review of literature on the subject indicates that several studies have been performed to automate the design process of plate girders. Notable among these studies is the use of General Geometric Programming (GGM) technique by Abuyounes and Adeli (1986) to obtain a minimum weight design of unstiffened and stiffened welded steel plate girders with uniform cross-section area. This procedure was first extended by Adeli and Chompooming (1989) to obtain a minimum weight design of continuous multispan nonprismatic steel plate girders, using the AISC-ASD specifications. In a later study, Adeli and Mak (1990) extended their work to obtain a minimum weight design of multispan steel plate girders - often used in highway bridges - using the AASHTO specifications for moving loads and design constraints (AASHTO, 1983).
387 In a recent study, Alghamdi (1996) formulated the design problem of an unstiffened plate girder in a nonlinear mathematical programming format and used the generalized reduced gradient method to obtain a minimum weight plastic design. The results obtained from several parametric studies were noted to be always higher than those of an ad hoc direct line search procedure. It was then concluded that the ad hoc procedure would yield designs that are consistently more economical in comparison to designs obtained by a generalized search algorithm. This study presents an extension of the direct line search algorithm and its application to the design of nonuniform stiffened steel plate girder according to the two versions of AlSC-design guidelines (i.e.: ASD and LRFD). Application of this algorithm is illustrated via a set of comparative studies of these two design specifications.
FORMULATION AND DESIGN ALGORITHM In order to obtain a minimum weight plastic design of a three-span built-up nonuniform stiffened steel plate girder, the structure geometry, loading and an assumed moment profile are shown in Figure 1. It is noted that the objective function (steel volume) for a typical span may be written in the following generic form
v = \ tKd,^Y.'^A,J,^Y.n'AJ„
(1)
in which: /„ is part of a span length having web depth hx and plate thickness t, Af. is a flange area extending over a length /,; and Ast is the area of a typical stiffener having a length 1st used n' times in a group defined by «,. The solution procedure is based on an assumed moment profile defined in terms of a moment coefficient a. The moment expression used is
M^=awl,^
(2)
where a can assume only positive values not exceeding 0.25p^ where P is the ratio of span length /2//1. Design constraints include flexural and shear strength provisions stipulated by AlSC-specifications including provisions to prevent web and/or compression flange buckling that may arise due to a slender web behaviour. These constraints include provisions to: (a) utilize post-buckling strength of web plates (mobilization of tension-field action); and (b) to prevent web yielding or crippling, and/or web buckling in regions carrying concentrated loads, by the design and proper placement of adequate number of transverse stiffeners. This design algorithm has been implemented in a computer code the main features of which are summarized in the flow chart shown in Figure 2. It is noted that the designer may select either the ASD specifications or the LRFD specifications to complete a design cycle. Key steps of the two design procedures are summarized in Figure 3 (a and b). Within this form, the code can be utilized to: (a) specify the geometry of an optimal design; and/or (b) perform comparative studies on the results obtained by the ASD and LRFD specifications.
388
Figure 1: Geometry and loading of the beam
i s
389
Input: A,aQ,a„,L^,L^,Fy,w,t
(^StopJ)^I
No.
Construct the shear force and bending moment diagrams
Determine: M\;M.j,\Mi (magnitudes and locations)
Design the components of the cross-section: Af{, Afj, Af^ and design cutoff points
Yes Design stiffeners (bearing and shear types; welding size)
M Evaluate objective function
Yes_
k = k+\
Print: all optimum design variables
Figure 2: Flow chart of design procedure
390
1 Limit State Lateral Tortional Buckling
Limit State Flange Local Buckling
X = Z>,/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
<
Y e s _ _ / — ^234
J v„=Q.6A^F^\
JFTFI
1
44000*
^' Tension field action not permitted K„=0.6^^Fr
Figure 3: continued - (b) Shear design requirements
K=0.6A^F^-
392 COMPARATIVE DESIGN STUDIES Several comparative design studies have been executed to investigate how the minimum weight design problem is influenced by key design variables (e.g.: plate thickness /; span ratio p; moment profile coefficient a; and steel yield strength F^. Then the design results of the two design procedures (namely: ASD and LRFD) are compared with each other in terms of steel volume. The results are summarized in Figures 4-8, and the following discussions highlight key features of these studies. Design Study 1 This design study includes several design examples to investigate the infiuence of span ratio p and plate thickness t on the final design. The results obtained are summarized in Figure 4 (a-c) for plate thickness of 0.313 in., 0.472 in., and 0.630 in., respectively. It is noted that the use of LRFD specifications results in a more economical design. It is also noted that the degree of material saving by using the LRFD method is consistently reduced (relative to the ASD method) when higher values of plate thickness are used. The results indicate that a plate thickness of 0.630 in the two methods yield identical minimum weight designs for various values of p. On the other hand, the results indicate that: (a) for low to medium values of plate thickness, the optimum range of values of p is 1.0-L2 and 1.1L40 for the LRFD method and the ASD method, respectively; and (b) for high values of plate thickness, the optimal value of p is unity (i.e.: equal span lengths) as shown in Figure 4-c. Design Study 2 This design study includes several design examples to investigate the influence of moment coefficient a on the minimum weight design using plate thicknesses of 0.313 in., 0.472 in., and 0.630 in. as shown in Figure 5 (a-c), respectively. The results confirm that economical benefits gained by using the LRFD specifications range fi'om 4% for medium size of plate thickness to 20% for small size of plate thickness. For higher values of plate thickness, the two design specifications yield identical results at a value of a equal to 0.13. The results also indicate that the LRFD method yields a minimum weight design within a short range of a values for all sizes of plate thickness. It is noted that, while the optimal values of a for the ASD method range fi'om 0.13 to 0.21, the values of a for the LRFD method range only fi-om 0.13 to 0.15. These numerical values indicate that results obtained by the ASD method are more sensitive to size of plate thickness t as compared to results obtained by the LRFD method. For practical purposes, therefore, the optimal value of a may be taken as 0.14. Design Study 3 This design study includes several design examples to investigate the dual influence of plate thickness / and span ratio P on the ratio of minimum weight designs provided by the ASD method and the LRFD method. The results are summarized in Figure 6. It is noted that more economical designs are obtained by using the LRFD method with low values of/, and for values of p between 1.8 and 2.2. Design Study 4 This design study includes several design examples to study the influence of flange plate thickness //on the minimum weight designs obtained by the ASD and LRFD specifications. The results summarized in Figure 7 indicate that, while the ASD method yields an optimal design at a value of 0.473 in., the LRFD would yield more economical designs at lower values of //. It is noted that the economical advantages offered by the LRFD method range fi-om 22% for small sizes of flange plate thickness to less than 1% for higher sizes of plate thickness.
393
10.00
t= 0.313 in.
9.00
- X - LRFD
0
•X
^>C^
8.00
=5f
^«-
-X--
ASD
(a)
7.00 0.80
1.20
1.60
2.40
2.00
Span Ratio p 12.00
^ "
—
11.00
•
(U 3
>
a3 e c
t= 0.472 in.
10.00
o
^
-
- 4V ' ^ ,
9.00
0
ASD
- X - LRFD
-
\^ — "
8.00
5
.X 7.00
1
0.80
^ ^x
(b) 1
1 1.20
1.60
1
2.00
2.40
Span Ratio p 12.00
*^ ^4>—'
11.00
b
10.00
t= 0.630 in.
9.00
- X - LRFD
S3
o
>
B
0
=3
a
^c
5
8.00
ASD
(c)
7.00 0.80
1.20
1.60
2.00
2.40
Span Ratio p Figure 4: Effects of span ratio and web plate thickness
394 7.00
6.50
•7^
0
6.00
ASD
- X - LRFD
CO
B S
5.50
t
(a)
x~x
5.00 0.08
x-ix-x-x^k-xy-x-^x-x-x 0.12
0.16
Moment Coefficient
0.20
0.24
a
7.00
6.50
-T;
6.00
% 5.50
0 0 j> ^ ^ X x.< fx^xx->k><-x-x>^-x-xx
(b)
5.00 0.08
0.12
0.16
Moment Coefficient
0.20
0.24
a
7.00
6.50 o
> T.
6.00
3
5.50
0
ASD
• X - LRFD
(c)
5.00 0.08
0.12
0.16
Moment Coefficient
0.20
a
Figure 5: Effects of moment coefficient
0.24
395
- X
0.80
1.20
1.60
2.00
-
t = 0.630 in.
—Q—
t= 0.472 in.
0
t = 0.313 in.
2.40
Span Ratio p Figure 6: Variation of ratio of steel volume with span ratio 7.00
0.315
0.394
0.473
0.551
0.630
Flange Thickness (in.) Figure 7: Effect of flange plate thickness Design Study 5 This design study includes several design examples to study the influence of steel yield strength Fy on the optimal results obtained from the two design specifications. The results, summarized in Figure 8, indicate that the LRFD procedures lead to more optimal design values. It is noted that use of LRFD procedures would result in at least 22% saving in steel volume required by the ASD procedures.
CONCLUSIONS & CLOSURE In recognition of proven high structural efficiency of built-up steel plate girders, this work summarizes the results of wn ad hoc (line search) design procedure to optimize the design of nonuniform stiffened steel plate girders using either the ASD procedures or the LRFD procedures. Through the automation of the procedure, a set of several combinations of systematic parametric studies were executed to: (1) study the sensitivity of design results to key design variables (namely: moment coefficient; web depth;
396
Yield strength Fy(ksO
Figure 8: Effect of steel yield strength Fy plate thickness; ratio of areas A J A', steel strength; support conditions; web stiffeners); (2) compare design results provided by LRFD procedures with those provided by ASD procedures; and (3) provide some reconunendations for designs of stiffened plate nonuniform steel girders. Based on the results reported herein, the following remarks are drawn: (1) The design code does allow the designer to select either the LRFD specifications or the ASD specifications and provides a complete proportioning of the structure including locations of cut-off points of flange plate, and design of web stiffeners. And in comparison to a general purpose mathematical program code, this code has been proven to yield more economical designs at less computing costs. (2) The LRFD method provides more economical designs but the economical benefits decrease when higher plate thicknesses are used. The two methods give similar results for plate thickness of 0.63 in. (3) Optimal value of span ratio p for medium values of plate thickness are within L0-L40 for both methods. And for high values of plate thickness the optimal value of P of unity (equal spans) is indicated by both methods of design. (4) Investigations of effect of moment coefficient a indicate that economical gains by using LRFD can be as high as 20% for low values of plate thickness. The optimal value of moment coefficient a for both methods is within the range of 0.13-0.2L (5) The LRFD procedures allow for a wide range of span ratio p when a low value of plate thickness t is used. The optimal value of p vary from L8 to 2.2 depending on size of plate thickness. Moreover, as the LRFD procedures are based on more realistic strength factors (to account for actual type of material loading which, in turn, depends on type of structural component under load), the resuhs they provide are more optimal. The design studies performed indicate a material saving of at least 22% in comparison to the results of ASD procedures. (6) Analysis of design sensitivity to size of flange plate thickness //indicates that in comparison to the use of ASD method, the use of LRFD specifications offers economical advantages that may be as high as 22%.
397 Moreover, the totality of the above remarks indicates high effectiveness of the ad hod design code to perform the design of nonuniform stiffened steel plate girders. This effectiveness is of prime importance for the design process since: (a) the code gives the designer the choice to use either the ASD or LRFD specifications; (b) the code does not require the evaluation of computationally expensive design gradients; and (c) the code design output includes complete proportioning of beam cross sections, web stiffeners and cut-off points of flange plates.
ACKNOWLEDGEMENT The authors would like to record their appreciations to King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, for providing the support to complete this research.
REFERENCES AASHTO (1983). Standard Specifications for Highway Bridges, 13 th edn., American Association of State Highway and Transportation Officials, USA. Abuyounes S. and Adeli H. (1986). Optimization of Steel Plate Girders via General Geometric Programming. Journal of Structural Mechanics 14:4, 501-524. Adeli H. and Chompooming K. (1989). Interactive Optimization of Nonprismatic Girders. Computers & Structures 31:4, 505-522. Adeli H. and Mak K.Y. (1990). Interactive Optimization of Plate Girder Bridges Subjected to Moving Loads. Journal of Computer-Aided Design 22:6, July/August, 368-376. AISC (1986). Manual of Steel Construction: Load & Resistance Factor Design, Chicago, IL. AISC (1989). Manual of Steel Construction: Allowable Stress Design, Chicago, IL. Alghamdi S.A. and Alghamdi AS. (1996). Optimal Plastic Design of Nonuniform Beams via the Reduced Gradient Method. Proceedings, 3rd Asian-Pacific Conference on Computational Mechanics, Seoul, Korea, 16-18 September, 591-597. Anderson K.E. and Chong K.P. (1986). Least Cost Computer-Aided Design of Steel Girders. Engineering Journal, AISC, 23:4, Fourth Quarter, 151-156. Basler K. (1961). Strength of Plate Girders in Bending. Journal of Structural Division, ASCE, Paper No. 2913, August. Basler K. (1963). Strength of Plate Girders in Shear. Transactions, ASCE, 128:Part H, 683-719. Cornell C.A., Reinmschmidt K.F. and Brotchie J.F. (1966). Iterative Design and Structural Optimization. Journal of Structural Division, ASCE, 92:ST6, 281-318. Gallagher R.H. and Zienkewicz O.C. (1973). Optimum Structural Design - Theory and Applications, John Wiley & Sons Ltd. Haftka R.T., Gurdal Z. and Kamat M.P. (1980). Elements of Structural Optimization, 2nd edn., Kluwer Pub., Dordrecht. Holt E.C. and Heithecker G.L. (1969). Minimum Weight Proportions for Steel Girders. Journal of Structural Division, ASCE, 95:ST10,2205-2217. Khot N.S. (1981). Algorithms Based on Optimality Criteria to Design Minimum Weight Structures. Engineering Optimization 5, 73-90. Schilling C.G. (1974). Optimum Properties for I-Shaped Beams. Journal of Structural Division, ASCE, 100:ST12, 2385-2401. Smith E.A. (1979). Minimum-Weight Design of Nonuniform Beams. Journal of Structural Division, ASCE, 105:ST7,1559-1562. Vanderplaats G.N. (1984). Numerical Optimization Techniques for Engineering Design, McGrawHill, 250-282.
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
399
DESIGN RULES FOR TANK STRUCTURES - DIFFERENT APPROACHES H. Saal ^ and U. Hornung ^ ^^^ Lehrstuhl fiir Stahl- und Leichtmetallbau, Universitat (TH) Karlsruhe, D-76128 Karlsruhe, Germany
ABSTRACT Within the scope of European standardization both CEN/TC 265 and CEN/TC 250/SC 3/PT 4 claim to be responsible for the development of design rules for site built storage tanks. However, they use two completely different approaches for this same subject. The CEN/TC 265 rules use the allowable stress concept whereas the CEN/TC 250 rules are based on the partial safety factor concept. For tank structures the relation between these two concepts is different from that in other structural buildings because of the distinct upper limits to the dominating actions (liquid contents). The very different danger potentials, e.g. water tank or LNG tank, are reflected by the reliability classes of the CEN/TC 250 rules. The challenge for both of these standards is that in contrast to most of the existing standards they not only have to apply for the petrochemical industry but for all types of liquid products. The approach to solve this problem is completely different in both of these standards again. CEN/TC 265 uses simple rules based on experience with built tanks of a special type whereas the CEN/TC 250 rules are based on laws of physics and laboratory tests with small samples which are rather general, but no real structures. The comparison of the intention, the background and the rules of these two standards also refers to other national and international standards for this subject. It shows the different approaches to thin shell design in application standards and compares it to recent large scale tests with tank structures of 10 m and 70 m diameter respectively which were tested to failure.
KEYWORDS tank design, shell buckling, internal negative pressure, allowable stress, partial safety factor, large scale tests, CEN/TC 265, CEN/TC 250/SC 3/PT 4
INTRODUCTION In the past there has been a strong demand for metallic, site built, flat bottomed vertical cylindrical above ground tanks for the storage of liquids. This is especially true for the petrochemical industry where the largest of these buildings have been built. For these buildings the API-standards have been used world-wide. There were also other national standards which more or less corresponded to the
400
API-standards (Figure 1). All of these standards were based on the concept of allowable stresses. Some of these standards were more similar to each other whereas BS 2654 is more strongly related to API. Early in 1991 at the first meeting of CEN/TC 265 the resolution was taken that the scope of its work for an European standard should be „Standardization of material, design, fabrication, erection and testing requirements of site built, flat bottomed, vertical cylindrical above ground welded metallic tanks for the storage of liquids, in various sizes and capacities for internal gas pressure approximating atmospheric pressure". The work started with tanks at ambient temperature and was strongly related to BS 2654 due to the persons involved. Thus the allowable stress concept was used regardless of the statements that indicated the possibility of overlap and conflict with the work of CEN/TC 250/SC 3/PT 4. The overlap was to be anticipated because CEN/TC 250 which has the overall responsibility for structural design rules in the civil engineering field was expected to cover the design of steel tanks in SC 3/PT 4. The conflict was to be anticipated because throughout CEN/TC 250 the concept of partial safety factors is used. However, these objections were brushed away with the comment that compatiblity would be established by later modifications if necessary. Three years later in spring 1995 CEN/TC 250/SC 3/PT 4 started its work on structural design rules for shells and plates with special application to silos, tanks and pipelines (Figure 1). Figure 1 shows the parts of ENV 1993 which will result from this work including masts and chimneys which are covered by project team 3. At that time with the Anpassungsrichtlinie to DIN 18 800 the first design rule for tanks based on partial safety factors was in force for half a year already.
1.6 Shells
1.7 Plates
ENV 1993
Figure 1: European standards for tank design TC 250 and TC 265 and related ones
MATERIAL SELECTION As a consequence of its origin in BS 2654 the work of CEN/TC 265 concentrates on tanks of the type used in petrochemical plants. This means that most of all it deals with C-Mn-steels. Therefore the design against brittle fracture coveres a lot of this work (Figure 2). This applies to the choice of the design temperature, Saal &
401 Wacker (1993), as well as to the material selection. The toughness requirements for C-Mn-steels are much more restrictive than those in Annex C of ENV 1993-1-1, in CEN/TC 54 or in DIN 4119, Saal & Hornung (1996). On the other hand stainless steels and aluminium alloys are only incidentally dealt with. With respect to tanks in the food industry this will be a major obstacle. Furthermore steels with characteristic values of yield strength higher than 390 N/mm^ will not be used with the application of CEN/TC 265 because of the limitation of allowable stresses in this standard. This restriction reflects a limitation of the radial displacements of the tank shell. For cryogenic tanks the material selection of CEN/TC 265 corresponds to BS 7777. There is hardly any reference to material standards for the five groups of ferritic steels which are provided depending upon the different purposes of use. Austenitic stainless steels and aluminium alloys are provided for very low operating temperatures. The toughness requirements are different from those for ambient temperature tanks with more concern of the toughness of the weld. Eurocode 3 Part 4.2 is restricted to steel because of Eurocode 9 dealing with structures from aluminium alloys. It includes structural steels, pressure vessel steeels and stainless steels. The requirements are in any case in accordance with the material standards. For ambient temperatures the toughness requirements of Annex C of ENV 1993-1-1 for structures have to be satisfied whereas for lower temperatures those given in CEN/TC 54 for pressure vessels have to be satisfied. The characteristic value of the yield strength to be used is limited to 460N/mm^ which is 18% beyond the value in CEN/TC 265.
TC265
TC 250 MATERIALS
fyk^390N/mm2
'y,k - 460 N/mm^ ambient and cryogenic
ambient ferritic steel:
ferritic steel:
very detailed toughness requirements partially outside material standards
according to ENV 1993-1 -1 and material standards
stainless steel:
stainless steel:
only incidentally dealt with
according to ENV 1993-1 -4 and material standards
aluminium: only incidentally dealt with cryogenic 5 groups of ferritic steels partially outside material standards stainless steel aluminium alloys
Figure 2: Material requirements
402 LOAD CASES AND DESIGN With the scope of CEN/TC 265 it became evident that it is based on the experience of preexisting standards. The pressures above the Hquid level according to these standards were in the range between 8.5 mbar negative (vacuum) and 60 mbar positive (Figure 3). However, for ecological reasons there is a strong demand for much higher values of both extremes of the pressure. Thus it was decided that there should be no gap between tanks and pressure vessels. This gave an upper limit of the positive pressure of 500 mbar. The maximum negative value was assumed to 100 mbar according to the operating conditions which are known so far. Besides the increase of danger potential with the pressure above the liquid level the increased maximum pressure would change the situation at the anchorage and at the top edge of the shell as well. This situation was outside the existing experience and therefore not covered by the design rules of CEN/TC 265. It perhaps would require a toroidal transition zone at the top edge of the shell. With the increase of vacuum pressure the buckling design of the tank shell and the roof becomes more important. In this context it has to be kept in mind that the allowable negative pressure including wind and internal vacuum according to CEN/TC 265 is very close to the theoretical buckling pressure without reduction due to imperfections and application of a safety factor, Saal (1996). This may be acceptable on the basis of good experience for tanks with dominating wind load if the loading situation is practically less severe or a postcritical load carrying capacity may be expected. However, with dominating vacuum pressure and missing experience it would be risky to apply the buckling design formulae of CEN/TC 265. This risk is increased by the fact that for tanks at ambient temperatures - in contrast to cryogenic tanks - CEN/TC 265 does not take into account the destabilizing effects of negative axial compression in the tank shell.
TC 250
TC265
1
ACTIONS internal pressure: -8,5 < p < 60 mblar
thermally induced loads: only > 100°C distributed live load: snow: wind:
internal pressure: -100 < p < 500 mbar simplified design: -8,5 < p < 60 mbar ik
EC 1-2-1
EC1
EC 1-2-3 ^5D=4-«2HS^ > 45 m/s
...
•
by agreement between purchaser and nrianufacturer: concentrated life load, wind, settlement load, emergency loadings by specification of the purchaser: seismic
wind pressure coefficients for specific tank situations 1
emergency loadings specified by the relevant authority
seismic:
EC 8
Figure 3: Loading conditions and design
'
403
Therefore the buckling design rules of CEN/TC 265 only apply for vacuum pressures up to 8.5 mbar. This range of applicability of the design rules is given in Figure 3 for CEN/TC 265 despite the larger range which is defined in its scope. Eurocode 3 Part 4.2 gives very general design rules which are not subject to the same restrictions like those of CEN/TC 265. Its rules for the design of shell stability, however, are based on the results of laboratory tests to account for the detrimental effect of imperfections. This may be questioned because of the obvious difference in size and fabrication between test samples and actual structures. For the range where CEN/TC 265 permits the application of its design rules these are modified in Eurocode 3 Part 4.2 to comply with physical laws, e.g. interaction of circumferential and axial compression in buckling design, and partial safety factor concept to be used in a simplified design, which will be applicable for a large number of tanks. This simplified design also includes rules from other standards, e.g. for the roof design. Eurocode 3 Part 4.2 refers to Eurocode 1 and Eurocode 8 (seismic loading) except for the emergency loads which have to be specified by the relevant authority. It gives some additional information for the wind pressure coefficients for the specific tank situation, e.g. tank with catch basin, which is hardly considered in CEN/TC 265. Because of the rather scanty information in Eurocode 1 Part 4 it gives in its Annex A some additional information on loading conditions. CEN/TC265 only refers to Eurocode 1 for snow and distributed live load. It argues that wind loads on a tank are different from those on a building and gives its own definition for the wind speed to be used in tank design. Perhaps this has something to do with the buckling design in so far that the wind loads are that large that they include the safety factor, i.e. they are not characteristic values but design values. - This would complicate the analysis because the vacuum pressure used in the buckling analysis is a characteristic value. Finally it has to be mentioned that tanks outside the petrochemical industry, e.g. food industry, would also require design rules for structural details different from those in BS 2654. This still seams to be an open question to CEN/TC 265. This is no problem, however, with the very general rules of Eurocode 3 Part 4.2.
LARGE SCALE BUCKLING TESTS Because of the detrimental effect of imperfections the actual buckling loads of shell structures are smaller than the theoretical buckling loads. This has been verified by thousands of laboratory tests with small samples. Wherever theoretical buckling loads are used as a design basis they are adjusted by a knockdown factor derived from the results of such tests. However, because of the differences in material, fabrication and size the applicability to large shell structures may be questioned. Recent tests with two tank structures may a little bit reduce this lack of experience, Hornung & Saal (1998). Both of the tanks had a fixed roof of spherical shape. The dimensions and materials were D = 10 m, H = 13.3 m, t = 10.0 mm, boiler plate HE for the smaller tank, which had a constant wall thickness, and D = 70 m, H = 17.1 m, t = 10.6 mm and St 37, StE 29 and StE 36 for the larger tank with seven stepped shell courses of equal height (D = diameter of the tank shell, H = height of the tank shell, t = thickness of the top course of the tank shell). Thus they were representative for tanks at both extremes of usual dimensions. The tests were performed with vacuum pressure in both cases. With the small tank (tank 1) buckling occured suddenly at a negative pressure of 116 mbar and after some unloading and reloading it failed completely at a negative pressure of 118 mbar. Because of the narrow time schedule it was not possible to measure imperfections of this tank. However, from visual inspection it seemed rather perfect. With the large tank (tank 2) the first buckle appeared at a negative pressure of 7 mbar. More buckles occured and increased with increasing vacuum pressure. Before failure the wall of the buckled shell formed a kind of columns between the buckles. The failure of the shell occured at a negative pressure of 17 mbar when one of these „columns" buckled, see left side in Figure 4.
404
Table 1 compares these test results to the analytical results according to various standards. Obviously there are only minor differences between the values according to DIN 18 800-4, Eurocode 3 Part 4.2 and the ECCS rules. The first buckles in tank 2 occured below the design value according to these standards although the imperfections at this location were within the tolerances, e.g. of DIN 18 800-4. For the comparison with the results according to CEN/TC 265 it must be kept in mind that these are not design values but allowable values. The comparison with the experimental results shows that the safety margin with the formulae of CEN/TC 265 is close to or even less than unity. - However, the pressures are without the scope of CEN/TC 265. - The design loads according to the other standards are only about half the failure loads. This means that the safety margin for complete failure is nearly twice the required value. TABLE 1 COMPARISON OF ULTIMATE LOADS
tank 1 2
pressures in mbar buckling failure buckling failure
test 116 118 7.0 17.0
DIN 18800 76.6
ECCS 63.1
CEN/TC 265 123
EC 3-4.2 65.0
9.7
-
14.0
7.4
Figure 4: Failure of tank 2
CONCLUSION The comparison of the two European standards for design of site built, flat bottomed, vertical cylindrical metallic tanks shows big differences (Figure 5): The main purpose of CEN/TC 265 is to supply a basis for tender and competition which is confined to the field of experience. With this it is necessary to have design rules which are easy to handle - even for unqualified personal - and may be based on experience with built tanks. These rules of good practice for the sake of simplicity may be dimensionally inconsistent and do not require any physical
405 background. A Finite-Element-analysis or any other computer analysis was never considered as a possible means for this design. As a consequence this standard adheres to the traditional allowable stress design. The main purpose of Eurocode 3 Part 4.2 is to supply a general basis for design for resistance and stability. This requires dimensionally consistent design rules which are based on the laws of mechanics. For stability design due to the detrimental effect of imperfections the design rules will need modifications which will be based on experimental results mostly from laboratory tests. Different approaches will be used with the type of analysis depending upon the reliability class of the tank which reflects the possible social and economical consequences of a failure. Consequently the partial safety factor concept will be used with this standard. For loading due to the liquid content this allows a 14% increase compared to the allowable stress concept.
TC265
TC250 MAIN PURPOSE
basis for tender and competiti on confined to the field of experie nee
design for resistance and stability
1 DESIGN RULES dimensionally inconsistent
dimensionally consistent
based on experience with built tanks
based on laws of mechanics and experimental results
easy to handle even for unqualified personal
three alternative approaches on different levels of analysis
allowable stresses
partial safety factors
Figure 5: Overall comparison of CEN/TC 265 and Eurocode 3-4.2 (TC 250)
REFERENCES Anpassungsrichtlinie Stahlbau. Anpassungsrichtlinie zu DIN 18 800 - Stahlbauten - Teil 1 bis 4. Mitteilungen Deutsches Institutfur Bautechnik. 1995:11. API Standard 650. Welded steel tanks for oil storage BS 2654 (1992). Manufacture of vertical steel welded storage tanks with butt-welded shells for the petroleum industry CEN/TC 54AVG B+C/SG-LT N..., Requirements for prevention of brittle fracture, prEN UFPV-2, Unfired Pressure Vessels. Annex D.
406 CODRES (1991). Code francais de construction des reservoirs cylindriques verticaux en acier. DIN 4119-1 (1979). Oberirdische zylindrische Flachboden-Tankbauwerke aus metallischen Werkstojfen; Grundlagen, Ausfiihrung, Priifungen DIN 4119-2 (1980). Oberirdische zylindrische Flachboden-Tankbauwerke aus metallischen Werkstojfen; Berechnung. DIN 18800-1 (1990). Stahlbauten, Bemessung und Konstruktion DIN 18800-4 (1990). Stahlbauten, Stabilitdtsfdlle, Schalenbeulen ECCS (1988): Buckling of Steel Shells; European Recommendations; ECCS - Technical Commitee 8 Structural Stability; Technical Working Group 8.4 - Stability of Shells; Fourth Edition ENV 1993-1-1: (1992). Eurocode 3 : Design of steel structures, Part 1.1 : General rules for buildings ENV 1991 (1993). Eurocode 1: Basis of design and actions on structures Eurocode 3 Part 1.6: (Draft 10/98): Design of steel structures, General rules: Supplementary rules for the Strength and Stability of Shell Structures Eurocode 3 Part 4.2: (Draft 10/98): Design of steel structures, Tanks Eurocode 8 (1998): Design provisions for earthquake resistance of structures Eurocode 9 (1998): Design of aluminium structures Hornung, U., und Saal, H. (1998). Ergebnisse von Beulversuchen mit zwei GroBtanks. Stahlbau 67:6, 408-413. ONORM C 2125 (1982). Oberirdische zylindrische Flachboden-Tankbauwerke aus metallischen Werkstoffen Saal, H., und Wacker, M. (1993). Karte der tiefsten Tagesmitteltemperaturen LODMAT fiir Deutschland. Universitat Stuttgart. Saal, H. (1994). Erlauterungen zur Anpassungsrichtlinie zu DIN 4119. Mitteilungen Deutsches Institut fur Bautechnik 1994:4, 121-125. Saal, H., und Hornung, U. (1996). Vermeidung von Sprodbruch im Tankanlagenbau. In: Moderne Konzepte zur Vermeidung von Sprodbruch in harmonisierten europaischen Regelwerken. VDEh, Diisseldorf. Saal, H. (1996). Aktuelle Forschungsvorhaben zur Auslegung von Tankbauwerken DGKMFachtagung „Sicherer Bau und Betrieb von Flachbodentanks"; 09.05.1996, Hannover,; DGKMTagungsbericht 9604.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
407
FAILURE MODES OF SLENDER WIND-LOADED CYLINDRICAL SHELLS W. Schneider, S. Bohm and R. Thiele Institute for Statics and Dynamics of Structures, Leipzig University 04109 Leipzig, Germany
ABSTRACT The paper presents the failure patterns of wind-loaded, slender, thin walled, circular cylindrical steel shells. These modes are distinctly different from those of shells with slendemess ratios h/r < 15. The structural analyses regarding geometrical and physical non-linearities reveal three different forms of instability in the range of h/r>15 and r/t<400. The main failure pattern of the slender wind-loaded cylindrical steel shell in the transition range from beam to shell is characterized by lee-ward buckling in the base area. Very slender and thin walled shells however dent at the lee-ward side up to 60% of the height. On relatively squat and thin walled shells luff-ward buckling processes occur in the upper half of the shell before further loading leads to system collapse in one of the two modes previously mentioned.
KEYWORDS Steel Shells, Chimneys, Wind Load, Structural Stability, Local and Global Buckling, FEM
INTRODUCTION Slender, thin walled, wind-loaded circular cylindrical shells are often used in chemical industry and power supply. The widely spread applications contrast with the fact that load bearing capacity is still not clearly defined. As an example the ECCS-Recommendations "Buckling of Steel Shells", ECCS 4.6 (1988), do not contain statements of this stability case. A systematic investigation in the wind tunnel is almost impracticable because of the different scale relations of statical and dynamical properties. Thus parametrical studies of the bearing and collapse behaviour have to be carried out nearly exclusively numerically. The program system FEM AS has been used for these numerical studies. High interpolation order elements as well as finite rotation kinematics and non-linear constitutive relations are implemented. For the material of the selected mild steel an elasto-plastic material-law including von Mises-yield-criterion and associated yield rule with slight linear kinematic hardening is assumed. Circular cylindrical shells with slendemess ratio h/r > 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<400 (Fig. 3). They correspond with the critical areas of local compression force maxima mentioned above. 350 The main collapse mode of the slender ^ wind-loaded circular cylindrical shell in i_ -^ the transition range beam-shell is the leeluff-ward buckling k— lee-ward denting in the300 in the upper half h] V- lower half of the shell ward buckling close to the clamped supof the shell . y^ port, mostly in the shape of so called -y^ NHKN; „elephant foot"-pattem. In contrast very N/ r/t ZTt slender, thin walled shells dent at the 1/ rear lower half of the cylinder. Relatively 1/ squat and thin walled shells start with a lee-ward buckling in the base area 150 luff-ward buckling process in the upper > half of the cylinder that succeeds with 100 one of the previously described failure y modes while the load is increased. The A^ transition of the geometrical ranges of 50 25 35 40 45 55 luff-ward buckling to the other modes of h/r system collapse is out of focus. The amFig. 3 Limits of the main failure patterns of the wind-loaded pHtude of the buckles of the luff-ward
:z:
K'
wards the range limit. But also the limit between the system collapse modes „lee-ward denting in the lower half of the shell" and „lee-ward buckling in the base area" is not drawn sharply. Transition ranges of the modes may be found at both sides of the border. This is caused by the relatively low difference of the buckling loads between the both failure
410 modes. The range limits are therefore sensitive to disturbance by geometrical and structural imperfections as well as load variations. Dynamic analyses show that minor load variations are sufficient to change the failure mode of a real shell from lee-ward denting to lee-ward buckling in the base area, Schneider & Thiele (1998a). Real structures of the investigated geometrical range will never fail in a separate mode of luff-ward buckling or lee-ward denting, it is always a combination of lee-ward buckling in the bottom area and one of the patterns mentioned before. Therefore the lee-ward buckling in the base area is the main collapse mode of the investigated slender wind-loaded shell. If an additional ring stiffener is arranged the collapse is characterized only by the lee-ward buckling in the base area in nearly the whole geometrical range, Schneider & Thiele (1998b). Lee-ward buckling in the base area of the shell Designing the cylindrical shell using the beam theory with the yield conditions as ultimate limit state, the exceeding stresses at the luff meridian calculated by a linear shell theory lead to material plastification in the base area. Fig. 4 shows the non-linear global behaviour of the structures presented in Fig. 2, plotting the displacement of the ring stiffened top edge. The larger a^, the wider extended are plastic zones and the more the paths deviate from the linear path. Although the post-buckling paths of the thin walled structures remind on a secondary path of axial compressed cylindrical shells it is not a case of load bifurcation. For all geometries the collapse is characterized by snap-through.
///
V/ \
y^
//A-J // /^
1.0
o
\ /
0.6
/A
//
#
X
at=1.4r/t= 61
0.4i
1 / ]/
0.2 0.0
a, = 1.4;r/t=61 at=1.8; rA=123
y
0.0
•
—
a, = 2.2;r/t=188
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Displacement [m]
Fig. 4 Load-displacement-paths of the ring stiffened top of the shell for varying Ot; non-linear shell theory; h=50m, h/r=30 apl.Sr /t= 123
a,=2.2r/t=188
Fig. 5 Post-buckling deflections (non scaled) of the whole cylindrical shell at a displacement of the shell top of 2.5m, view perpendicular to wind direction; h=50m, h/r=30, ring stiffened top
Fig. 6 Post-buckling deflections (non scaled) of the 1.5m high base area at a displacement of the shell top of 2.5m for varying o^, view perpendicular to wind direction; h=50m, h/r=30;ringstiffened top
411 The load decrease in the post-buckling area is determined by the ratio of wall-thinness. This is obvious by observing the collapse mode. The global failure pattern represents tipping around the base area (Fig. 5). The modifications causing the system collapse happen in the lower 3% of the shell height. As soon as the yield condition at the lee-ward base area is satisfied a plastic bulge above the clamped support cross section emerges, a so called elephant foot pattern (Fig. 6). The normalized post-buckling equilibrium load is dropping with the decrease of wall bending stiffness, i.e. with an increasing wall-thinness ratio. The location of the plastic bulge is determined by the pre-buckling deformation caused by edge moments. The decay length related to the shell height decreases with rising h/r and rising / r / t causing the ring dent to move closer to the clamped support cross section for constant slenderness ratio and rising wall-thinness ratio. Lee-ward denting in the lower half of the shell If the shells are of higher slenderness ratio than the shells described before and the wall-thinness ratio is large enough a failure pattern occurs, which has not been investigated up to now: lee-ward denting between 0.14h and 0.60h, Schneider & Thiele (1998b). For the example of Fig. 7 the dent is situated at 0.25h. The term „denting" of this phenomenon does not mean a beam instability. The classical beam instability happens while the cross section shape is kept. In this case a shell instability is initiated by the lee-ward denting, which is visible in the top view of the lower first quarter of the shell. The location of the dent depends mainly on the position of the lee-ward meridional compression maximum, i. e. on a^. It is slightly moved up relative to this lee-ward comFig. 7 Post-buckling deflections (non pression maximum. The thinner the wall of the shell the scaled) of the whole cylindrical larger the offset is for an equal a^. This is caused by the shell (view perpendicular to pre-buckling deformation. The wind pressure distribuwind direction) and the lower tion on the cross section with pressure on the luff meriquarter of the shell (top view) at dian and suction on the flanks effects the ovalisation of a displacement of the stiffened the cross section. The maximum amplitude is reached upper edge of 2.5m ; h=50m, between 0.5 and 0.6h. For squatter shells the ring stiffeh/r=30,r/t=332, 0^=1.80 ner at the top constrains the ovalisation in a considerable manner, therefore the lee-ward denting does not occur as failure pattern. Both conditions for the creation of the described failure pattern - relatively high compression forces above the base area and sufficient ovalising of the cross section - are not satisfied in the case of a lateral loaded cantilever shell with sinus shaped meridional forces as they occur if the load is transmitted by bracing members and not around the circumference. Thus this collapse pattern can not be observed and investigated in this load case. A separate loading „wind" should be part of the shell stability standards. The failure pattern lee-ward denting is initiated only if the associated buckling load is lower than the critical load of buckling in the base area. Below the range limit indicated in Fig. 3 this condition is not satisfied. The transition between the failure pattern lee-ward denting and lee-ward buckling in the base area is explained by several examples with the same slenderness ratio h/r=50 but different wall-thickness ratios. Fig. 8 shows the load-displacement-paths, Fig. 9 presents the associated post-buckling deformations of the 1.5m high base area (view opposite to the wind direction for better illustration).
412
1.0
1.5
2.0
Displacement [m] Fig. 8 Load-displacement-paths of the ring stiffened top of the shell for varying c^; non-linear shell theory; h=50m, h/r=50
MityBfflli tniiiLiiiM
at=l. 2
r/t=91
a,=1.5 r/t=210
a,=1.8 r/t=332
Fig. 9 Post-buckling deflections (non scaled) of the 1.5m high base area at a displacement of the shell top of 2.5m for varying at*, view opposite to the wind direction; h=50m, h/r=50;ringstiffened top The shell with the thickest wall (r/t=91) fails with the elephant foot pattern. The border between lee-ward buckling in the base area and lee-ward denting is announced by the transition from elephant foot buckling to lee-ward diamond like buckling near the clamped support while the wall-thinness ratio is rising. If the decay length of the edge moments is to small the bracing influence of the clamped support prevents the raising of a ring bulge at the location of the first maximum of the radial deformation. For growing wall-thinness ratio, i.e. increasing ovalisation, the critical buckling load of this diamond like buckling is finally higher than the buckling load of lee-ward denting. Thus lee-ward denting causes the system collapse. The shell with medium wall-thinness ratio (r/t=210) is situated in this transition zone and presents successively lee-ward denting and diamond like buckling in the base area. For further wallthinness decrease only lee-ward denting occurs (r/t=332).
413 Luff-ward buckling in the upper half of the shell If the slendemess ratio of the shell decreases and the wall-thinness ratio increases high luff-ward meridional compression forces occur in the upper half of the shell (see Fig. 2). These stresses and the additional stresses in ring direction may cause luff-ward buckling of the shell. This process marks the transition to wind-loaded circular cylindrical shells with h/r<15 investigated by Greiner & Derler (1995). Though this buckling phenomenon has influence on an extended area of the shell it does not lead to the collapse of the system. It has only a local effect, further loading is possible. The global collapse occurs by increasing load in the mode of lee-ward buckling in the base area.
0.8 1
x=h; luff and lee x=0,7h:luff first local snap-through point (LSP) X global snap-through point (GSP) 4- point in the post-buckling area (PP) 0.0
1 — I — I — I — I — I — I — I — I — I — I — I — I — I
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Displacement [m]
Fig. 10 Load-displacement-paths of the displacements in wind direction at x=h and x=0.7h; h=50m, h/r=20, r/t=200, a^S.Q, ring stiffened top
GSP Fig. 11 Deflections (non scaled) of the whole shell at global snap-through point (GSP) and at point PP of the post-buckling path ace. to Fig. 10, view perpendicular to wind direction; h=50m, h/r=20, r/t=200, a^S.Q, ring stiffened top
Fig. 10 presents the load-displacement-paths of the ring stiffened shell top and a point at the luff meridian at 0.7h for an example shell of this geometrical range. Because of the different influences of local and global buckling processes on the displacements of these points their paths are distinctly different. Once the first local snap-through point at normalized load factor NLF=0.60 is reached the shell starts buckling at the luff meridian (Fig. 11) without appreciable loss of overall stiffness of the structure. Until the global snapthrough point at NLF=0.72 is reached no buckling processes in the base area occur (comp. GSP in Fig. 12). Exceeding the global limit point lee-ward buckling starts. No stable ranges exist on the postbuckling path, thus the system collapses very quickly.
414
GSP Fig. 12 Deflections (non scaled) at the 1.5m high base area at global snap-through point (GSP) and at point PP of the post-buckling path ace. to Fig. 10, view perpendicular to wind direction; h=50m, h/r=20, r/t=200, at=3.9, ring stiffened top
References ECCS 4.6, (1988). Buckling of Steel Shells. ECCS-TWG 8.4, Brussels, Belgium. Eurocode 1 Part 2.4 (1994). Actions on Structures - Wind Actions. CEN, Brussels, Belgium. Eurocode 3 Part 3.2 (1997). Chimneys. CEN (unpublished draft), Brussels, Belgium. Greiner, R. and Derler, P. (1995). Effect of Imperfections on Wind-Loaded Cylindrical Shells. ThinWalled Structures 23, 271-281. Peil, U. and Nolle, H. (1988). Stress Distribution in Steel Chimneys. 6th Int. Chimney Conf, Brighton, UK. Schneider, W. and Thiele, R. (1998a). Kollapsanalyse quasistatisch windbelasteter schlanker stahlemer Kreiszylinderschalen (Collapse Analysis of Quasi-Static Wind-Lx)aded Slender Cylindrical Steel Shells in German). Finite Elemente in der Baupraxis - Modellierung, Berechnung und Konstruktion - FEM'98, 501-510, Emst&Sohn, Berlin, Germany. Schneider, W. and Thiele, R. (1998b). Eine unerwartete Versagensform bei schlanken windbelasteten Kreiszylinderschalen (An Unexpected Failure Mode of Slender Wind-Loaded Cylindrical Shells - in German). Stahlbau 67:11, 870-875.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
415
Laminated Constructions on the Basis of Thin Metal Sheets in Building Chistyakov A.M^, Rass F.V'., Konovalov P.N J, Chernoivan N.V.^ •Central Research, Design and Technological Institute for Light-Weight Metal Structures, Moscow, Russia. ^Brest Politechnical Institute, Brest, Belarus.
ABSTRACT In the report the space-planning and design solutions of the worked out system of Monopanel buildings and the final results of the tests of load-bearing wall panels are presented. A special emphasis has been made upon the theoretical and experimental analysis of a supporting layer loosing its local stability in the whole construction. This aspect of a critical limit characterizes compressed-bent laminated structures within a definite parameters range. Its value was underestimated up till now. KEYWORDS Monopanel, laminated structures, compressed bent, critical force, displacement, discrete joints, supporting sheet. INTRODUCTION Central Research, Design and Technological Institute for Light-Weight Metal Structures, Moscow, Russia developed a frameless Monopanel building which is erected within a short period of time. The basic bearing and protecting element of the system is a double-layered panel consisting of a corrugated galvanized steel sheet - 0.55...0.8 mm and fire-proof foam plastic of Penorezol type up to 100 kg per cu.m. [1] The constructions are up to 12.4 m long, 107 to 197 mm thick, 750 mm wide (in axes) with 50 to 140 mm heat blanket insulation. The panel weighs 17 kg per sq. m. They are manufactured on production line with a speed of 6 m per min. An annual production makes 1.2 million sq. m. of the panels. A specialized panel producing factory is situated in Taldoma, Moscow region. The panels are designed for buildings operated in areas in which there are wind loads of less than 0.6 kPa and snow loads of less than 1.5 kPa, temperatures of the outside air being -55 to
416 +45°C. The buildings are of II-V class of fire-resistance according to Building Codes 2.01.02-85. The panels are issued a sanitary certificate JvrQ3152-16 from 21.04.97. Unlike the laminated panels with foam polyurethane used at the present time, fire-proof foam plastic is used in the worked out construction. Its fire spreading index is zero. The metal consumption and the weight of the protecting structures are reduced in comparison with frame panels and sheet assembly protections used at the present time. Assembage labour input is considerably reduced too. DESIGN OF BUILDINGS The basic modification of Monopanel buildings have a bearing corrugated metal sheet facing the interior. (Fig. 1 a, c)
Figure 1: Schemes of protecting constructions design: a - polimeric roofing; b - metal roofing; c load-bearing wall; 1 - corrugated galvanized painted metal sheet; 2 - fire-proof foam plastic; 3 - polimeric waterproofing; 4 - polimeric fish (for sealing joints); 5 - loadbearing constructions; 6 - cement-chip plate or fibre-gypsum plate, water-proof plywood, phosphate fibre-glass plastic; 7 - stucco; 8 - brick; 9 - self-tapping screw. The exterior side of the wall is treated in several modifications: by putting special liquid substances of natural stones (marble, granite, sand, ect.) and acrilic resins (Fig. Ic); by coating with ceramics, corrugated metal, siding, plank, slab, ect. The interior side of the panel is supported with flat cement-chip plates or fibre-gypsum plates, water-proof plywood, phosphate fibre-glass plastic and other constructional sheet materials jointed together by discrete ties. This inner layer is a supporting and finishing element which can be painted or papered when needed. The double-layered panels can be used for making covering of various geometric shapes like tucks, shells and traditional flat design solutions. The roof can be designed with both traditional rolled roofing and polymeric materials or metall sheets which are more effective. (Fig. 1 a, b) A polymeric roof can be designed in two modifications: by polimeric fishing after the installation of the panels which are provided with polimeric pre-coating or by polimeric rolled
417 roofing after the installation of the panels which are covered with ruberoid, kraftpaper, or film materials. A metal roof can be designed in three modifications: by metal tiling, by using corrugated metal sheets with their faces up or by using fluted metal sheets which are set on a slope of the roof and fastened to the mortage elements of the panels with self-tapping screws. Then they are rolled up by a set of machinery. It is advised to use the latter, especially when a slope is extended ( more than 12 m). The monopanel allows to design the following modifications of warm floors: by placing floor boards on joists, by placing cement-chip plates, modified veneer or other construction plastics on a corrugated metal sheet or joists, by placing concrete on a corrugated metal sheet with reinforcement when needed. At the customer's request the floor can be covered with parquet, linoleum, synthetic carpets, tile or bulk syntetic mixes and suspended ceilings are possible. The installation of partitions is not connected with the load-bearing constructions. It allows a flexible design and meets all customer's requirements. Ordinary partitions consist of bent channel made by rolling 0.5...0.8 mm galvanized steel sheet and card-board-gypsum sheets or fibre-gypsum sheets and other materials. The basic accepted modification is a rigid metal frame out of channel or a Z-shaped steel profile, that serves as a foundation of the building. The design of a foundation depends on a type of the building, a construction site and soils. The easiest and cheapest way of preparing the construction site is to remove the surface soil, to level the site, to fill it with sand or crush gravel, to install cushions or brick pillars on which the metal frame is to be mounted. The bent steel profiled frame is mounted on the wall panels. It is provided with surfacing panels. The monpanels are jonted to each other or to other load-bearing constructions with self-taping screws or special rivets. Hermetic sealing compounds and single-component foam sealing compounds are used for sealing and compacting the joints and construction conjugations. Thus, the erection of the basic building constructions is reduced to assembling prefabricatied and marked light weight laminated structures. It doesn't requre the use of cranes and skilled labour. A cottage with a floor space of 103 sq. m. can be assembled by 4...5 men within a week. CALCULATIONS The development o^ Monopanel constructions requires necessary calculations and tests. An important feature of Monopanel constructions for calculations and eleboration is the fact that a monopanel is the main bearing element of a building (primarily of its wall). From a static point of view the wall panel reinforced with an additional sheet on the interior side is a compressed-bent composite plate with two bearing layers - the main one is a corrugated steel sheet, and the additional one is out of various sheets. When calculationg the total stress deformation of the panel, you should not take into account a light weight flller - foam plastic, because of its low stiffness. But you should take into account its influence when calculating the local stability of the profile sides. The basic load-bearing layers are connected with ties which are elastic at deformation and absolutely rigid in the direction from the normal to the middle plane of the plate. At the first stage of calculation of the total stresses, the discrete ties can be replaced by equivalent continually distributed ones on the total area of the plate. The materials of the bearing layers of the plate are ortotropic. The hypothesis of straight normals is correct for each of them. In connections with the fact, that the panel is supported only by two short edges and works in conditions of uniaxial compression and cylindrical bending (Fig. 2a), the panel deformation can be calculated for a unit band width.
418
9
M Figure 2: Structural model of the load bearing wall panel - a; diagram of panel deformation - b. In the case of cylindrical bending a set of equations for the double-layered laminated plate can be written as follows [2]: CM, 5
E,F,
DoW"
(1)
-M. + CT
5: C^ 1 where 5 = — + —. Do E,F, E1.2
EB
E,F,
1 ;— E,F,
C- a distance between the middle planes of the layers,
• a reduced modulus of layer elasticity, F^ 2 " ^ cross-section area of layer 1 and
^ - l^xil^yi
2 per unit of plate width at i=l,2; |ixi.l^yi - Poisson's coefficients for the layers of the plate, i=n
A
DQ = 2]—EJJF^^ - total bending stiffness of the layers in the main system without displacement
:ri2
ties, ^ = -i=i a coefficient of stiffness of displacement ties on 1 sq. cm., ^j - a coefficient of LB stiffness of i-discrete tie, B - panel width, L - a plate span, N- a longitudinal force applied to the whole plate, N^
—^—^-^—N , N, = —.—^"-^—N- a longitudinal force in the layers, M.- a E^Fi-hE^Fj
E^F.+E^F^
bending moment occuring due to eccentric longitudinal load, T - the total longitudinal shear force. The influence of the longitudinal force on plate bending can be calculated accurately by means N of the following function VNL = WK, where K = '^^—. In the paper we present only an approximate solution for (1) in series, which is compact and practically accurate. The tests of strength of the layers being satisfied, the tests of the local stability of the panels as a component part [2] and the local stability of the corrugated steel sheet sides [3] must be performed.
419 A significant feature of the panel is discrete ties along its length between the layers. The ties are distributeted along its width so freqently that they can be considered equally distributeted on the line. As a consequence of the discrete ties there is the difference of the layer displacement between the junction lines. This is quite possible if the layers are bent before assemblage (i.e. initial deflection). In this case the critical limit of supporting sheet 1 is characterized not only by its strength but also by the local loss of stability between the junction lines of the layers. While analysing the behaviour of the supporting sheet of the compressed-bent panel, the location of the ties should be taken into account. Depending on the eccentrical application of the longtudinal loads and the initial deflection of sheet 1, its deformation can have a form of Euler buckling, a fast increase of deflection due to the prevaling influence of the initial deflection, transient buckling, bending stiffness of the layer being sufficient. We'll consider the latter case (Fig. 2b), as it can be significantly characterized. As the plate being hinged along two edges and the layers being joined together along the lines, the construction works in cylindrical bending. Thus, the calculations are made for unit band width as it has been made earlier. As it has been said, the ties from the normal to the middle plane are considered to be absolutely rigid. It is convenient to calculate by a force method. To define the deformation of layer 1, we use the derived solution [4]. Having calculate the stress deformation of layer 1 in the system of the composite plate, we test its stability in the whole construction. We consider it as an isolated plate where the influence of the rejected ties upon it is substituted by forces occuring in them. The calculation is made by a strain-energy method as it has been made earlier for the given unit band width. We write the functional of energy for the layer under consideration:
3 = J^[D,(W^ - w;'+ Wof - N,(w^ - w; + Wofldx + (2) 2
^(DT^D;)!^^-^^'^^^^^
where Wyy, W,, WQ - the total deflection of a the middle plane of the plate, the deflection and initial deflection of layer 1, Tj - the total shear force occuring in i-tie, ^j - a distance from the support to i-tie at symmetrical location of the latter. We use Riesz method to calculate the critical force of the local loss of layer 1 stability in the whole construction. As it has been made ealier, we approximate the elastic deflection of the layer and the initial deflection in series of sine flexings. Having substituted the series into (2) and integrated along the length of the plate across trigonometrical function, we derive a strain-energy equation in the following final form: 4
n't' L
„^iV
>.
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
-<^-0=0.31
70
P 2 -1 1.5 4 «
0=0.01
1 \
-•-0=0.1
'
0.5 H
-^0=02 -*-0=0.31
, 0
10
.'^r-r^^^ 20
30
40
50
60
70
P
(a) Simply supported junction (b) Fixed junction Figure 5: Non-dimensional budding stress and strain versus slendemess limit for fabricated section Box Sections The analysis for the plates of a box section where it 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 6. It is illustrated that the effect of residual stresses is influential in causing a reduction in the local buckUng stress in the elastic region. This also affects the
428 yield slendemess limit for the cross-section as evidenced in Figure 6 (a) and (b). The resultant yield slenderness limits are 45 and 65 respectively.
-0=0.01 -Qt=0.1 -0=0.21 -0=0.3
20
40
60
80
100
120 140
-(X=0.0| -0=0.11 -0=0.2 -0=0.3!
20
40
60
80
100
120
140
(a) Simply supported junctions (b) Fixed junctions Figure 6: Non-dimensional buckling stress and strain versus slendemess limit The analysis for the plates of a box section where it 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 7. As for the hot-rolled section, the effects of the residual stresses on the elastic local buckling stress is to cause a reduction in this value. Furthermore, a lower value of yield slendemess limit is caused through the inclusion of residual compressive stresses. The resultant yield slendemess limits are unchanged from the hot-rolled case and are determined as 45 and 65 respectively. T
1.25 1
y
0.75 0.5
V^
t ,, , ^ ^ « T
0.25 0
0
20
40
60
80
100 120
0=0.0
-0=0.01
-ii-0=0.1
-0=0.1
-^0=0.2
-0=0.2
- 1 ^ 0=0.31
-0=0.3
20
140
40
60
80
100
120 140
P 2 -r
k. ,
1.5 ]
o
1 J 0.5 \
0
1
1
1
20
40
60
i 80
^
^
^
^
0=0.01
-0=0.0
-*-0=0.1
-0=0.1
-•-0=0.2
-0=0.2
-*-0=0.31
-0=0.3
^
100 120 140
20
40
60
80
100
120
140
P
(a) Simply supported junctions (b) Fixed junctions Figure 7: Non-dimensional buckling stress and strain versus slendemess Umit
CODE COMPARISONS Table 1 illustrates the various slendemess limits obtained from the analyses of flange outstands, as well as those outlined in various intemational steel codes. It is shown that the use of a rational local
429 buckling analysis has many advantages if one is to include the effects of local budding to determine plate dimensions or to incorporate local budding in an advanced frame analysis. Table 2 illustrates the various slendemess limits obtained from the analyses of box sections, as well as those outlined in various international steel codes. This table also highlights the benefits of the use of a rational local budding analysis. It should be noted that the plastic slendemess limits derived in this paper have used the limiting strain criterion of normal strength concrete. For higher concrete strengths, the crushing strain is reduced and thus it would be expected that the plastic slendemess limit may be able to be shghdy increased. However, the limits in Table 1 and 2 will be able to be applied to any composite stmcture, as they are conservatively developed regardless of the concrete strength. TABLE 1 SLENDERNESS LIMITS FOR FLANGE OUTSTANDS
-1^
METHOD
t V 300
YIELD CLAMPED FSM (SS) CLAMPED FSM (FIXED) WRIGHT ( 1 9 9 3 ) (SS) WRIGHT ( 1 9 9 3 ) (FIXED)
AS4100(1990) BS5950: PARTI (1990) EUROCODE3(1992) EUROCODE4(1992)
PLASTIC
HOT-ROLLED
WELDED
HOT-ROLLED
WELDED
15 25 20 42 18 14 13 19
13 22 18 39 15 12 12 18
15 25 11 26 11 8 9 9
13 22 11 26 9 7 8 8
TABLE 2 SLENDERNESS LIMITS FOR BOX SECTIONS
1
b 1A.
METHOD
300 YIELD CLAMPED FSM (SS) CLAMPED FSM (FIXED) WRIGHT ( 1 9 9 3 ) (SS) WRIGHT ( 1 9 9 3 ) (FIXED)
AS4100 (1990) BS5950: PARTI (1990) EUROCODE3(1992) EUROCODE4(1992)
PLASTIC
1
HOT-ROLLED
WELDED
HOT-ROLLED
WELDED
45 65 50 69 49 37 37 NA
45 65 46 64 49 27 37 NA
45 65 37 50 33 25 29 46
45 65 37 50 33 22 29 46
CONCLUSIONS This paper has highUghted the benefits of the use of concrete in restraining local budding of steel sections. A finite strip method has been used which incorporates the effects of residual stresses associated with both hot-rolling or welding. A parametric study was undertaken to consider the effects
430
of both residual stresses and boundary conditions. The results were compared with those of existing international codes and were shown to be advantageous as far as structural behaviour is concerned. The analysis undertaken herein has assumed both simply supported and fixed edges for the junctions of plate elements in the various assemblages. This was undertaken as one can not be entirely confident as to the exact nature of the boundary conditions which are existent. To determine, the effects of boundary conditions with confidence, it is suggested that further experimental tests be undertaken and that these are calibrated with the model presented herein.
ACKNOWLEDGEMENTS The authors would like to thank the Australian Research Council, International Project Grants Scheme which has funded the exchange visits of the two authors to Scotland and AustraUa to prepare this paper.
REFERENCES British Standards Institution, (1990) British Standard, Structural use of steelwork in building, BS5950: Part 1, Code of practice for design in simple and continuous construction: hot rolled sections. Cheung, Y.K. (1976) Finite strip method in structural analysis, Pergamon Press. European Committee for Standardisation, (1993) Eurocode 3: Design of Steel Structures, Part 1.1 : General rules and rules for buildings, ENV1993-1-1. European Committee for Standardisation, (1994) Eurocode 4: Design of Composite Steel and Concrete Structures, Part 1.1: General rules and rules for buildings, ENV 1994-1-1. Matsui, C.K. (1985) Local buckling of concrete filled steel square tubular columns. International Association for Bridge and Structural Engineering, ECCS Symposium, Luxembourg, 269-276. Standards Australia, {1990) Australian Standard-Steel Structures, AS4100. Trahair, N.S. and Bradford, M.A. (1998) The behaviour and design of steel structures, 3rd Edition, Chapman and Hall. Uy, B. (1998) Local and post-local buckling of concrete fiUed steel welded box columns. Journal of Constructional Steel Research, 47:1-2,47-72. Uy, B. and Bradford, M.A. (1994) Inelastic local budding behaviour of thin steel plates in profiled composite beams. The Structural Engineer, 72:16, 259-267. Uy, B., Wright, H.D. and Diedricks, A.A. (1998) Local buckhng of cold formed steel sections filled with concrete. Proceedings of the 2nd International Conference on Thin-Walled Structures, Singapore, Thin-Walled Structures, Research and Development, 361-314. Wright, H.D. (1993) Buckling of plates in contact with a rigid medium. The Structural Engineer, 71:12, 209-215.
Session Bl ALUMINIUM STRUCTURES
This Page Intentionally Left Blank
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
433
ALUMINIUM MULLION-TRANSOM CURTAIN WALL SYSTEMS: 3-D F.E.M. MODELLING OF THEIR STRUCTURAL BEHAVIOR Baniotopoulos C.C., Koltsakis E., Preftitsi F. and Panagiotopoulos P.D.^ Institute of Steel Structures, Department of Civil Engineering, Aristotle University, 54006 Thessaloniki, Greece ^(Deceased 12.8.1998)
ABSTRACT A parametric study of aluminium mullion-transom curtain wall systems by means of a 3-D finite element model under static loading is presented herein. Concerning the material properties, a Ramberg-Osgood isotropic stress-strain lav^ and the nonlinear evolution law proposed by F. M. Mazzolani have been applied to account for work-hardening. A numerical application illustrates the proposed analysis scheme for aluminium mullion-transom curtain wall systems. A detailed description of the analysis model and the definition of the loads are also herein presented. The respective finite element model was implemented with the CASTEM 2000 code. The computed values of displacements have also been compared with the restrictions imposed by the Eurocode 9.
KEYWORDS Aluminium, curtain walls, Eurocode 9, folded shell model, work-hardening, Ramperg-Osgood law
INTRODUCTION The use of aluminium alloys in Civil Engineering projects represents a modem trend having a lot of advantages in structural applications. In many applications aluminium alloys could compete with steel which is now the most widely used material in metal fabrication. Aluminium's corrosion resistance is a great advantage in comparison with steel and that's the reason why aluminium structures usually don't need any protection against atmospheric or chemical corrosive agents even in a marine environment. Aluminium is a very light material, its specific weight is equal to one-third of that of the steel, a fact that makes this material more attractive as far as it concerns problems of transport. Although pure aluminium's strength is very low for structural applications, a sufficient way to increase this strength is to alloy it with other elements ( Mg, Mn, Zn ); its behavior can be fijrther improved if heat treatment is applied.
434
On the other hand, the limited knowledge about the structural response of aluminium structural members that exhibit a nonlinear character, has induced hesitation for the use of aluminium alloys as a structural material. Nowadays, thanks to the development and the spread of advanced computational methods any detailed, accurate inelastic analysis for aluminium structural problems can be carried out with the F.E.M. Curtain wall systems which are herein presented are made-up of a grid of aluminium muUions and transoms that act as the glazing support. The design of such structural systems is usually performed in several countries according to a variety of different rules and recommendations, whereas the framework of the European Prestandard "Eurocode 9: Design of aluminium structures. Part 1.1, 1.2 and 2" only very recently entered its three years ENV period. Within such a landscape, the target of the present research is to contribute to the investigations aiming to support the serviceability and ultimate limit state rules and principles dictated by EC9 for the design of such aluminium curtain wall structures.
ON THE INCREMENTAL THEORY OF WORK-HARDENING PLASTICITY As said before, the material nonlinearity is a fine point in any numerical simulation attempt. As it is well known, mild steel exhibits plastic flow under constant stress. Aluminium alloys exhibit a workhardening behavior as well. A number of techniques that addresess work-hardening plasticity have been developed in order to determine: i) an initial yield surface that defines the elastic limit of the material. In the herein proposed model, the value usually used by structural engineers is fo.2'=175N/mm^, where the 0.2 index denotes a permanent deformation of 0.2 percent. ii) a work-hardening rule that describes the evolution of the yield surface. The determination of the subsequent yield surface is probably one of the major problems in the aforementioned approach. For this reason, several hardening rules have been proposed in the past for use in any type of isotropic, kinematic and mixed hardening plastic analysis approaches. The model herein proposed works with isotropic hardening, i.e. the initial yield surface, as plastic flow occurs, remains self-similar with increasing plastic deformation. This type of hardening cannot account for the Bauschinger effect, which is a particular type of directional anisotropy induced by plastic deformation. This phenomenon can be neglected with structural applications at the absence of predominant cyclic loading (e.g. machine bearing structures). iii) a flow rule that is related to a plastic potential fiinction defining the direction of the incremental plastic strain vector in strain space.
DEFINITION OF THE ALUMINIUM ALLOYS CONSTITUTIVE LAW The definition of a unique o-8 law for every type of aluminium alloy is not an easy task as their mechanical properties differ drastically between them. In our model a continuous law of the form 8 = 8(o) proposed by F. M. Mazzolani is also mentioned because, due its simpler formulation, it is used in many structural analysis. The stress-strain evolution curve proposed by Ramberg and Osgood (see Figure 1) is mathematically described by the exponential formula:
435 a
fa
(1)
where B and n are material dependent constants determined by experimental testing, and E is the initial Modulus of Elasticity ( E = 70000 N/mm^ ) i.e. the E measured around the c=0 area.
Figure 1: Stress-strain curve (Ramberg-Osgood model) After some algebraic manipulation, Eqn. 1 takes the form: In 2 In
(2)
fo.2 fo.i.
expressing the fact that the exponent n is a different constant for each alloy. The parameter B shows the extent of the strain area where the first term (the linear one) is more significant than the second The value — = 1 can be deemed to correspond to the "boundary" between the elastic and the plastic B fo.2
region. It can easily be proved that B= ,
- and then, the following relation is obtained: f G
a
(3)
where fo.2 is the conventional yield limit stress, used in aluminium alloys, corresponding, as said before, to a stress that leaves a permanent deformation of 0.2 percent. The model proposed by Mazzolani is based upon the idea that the o-8 expression can be derived separately for different regions.
£p
Ee
Figure 2: Stress-strain curve ( Mazzolani model) The first term refers to the elastic behavior of the material and the upper limit of this first linear portion is the proportionality stress limit fp; the second region describes the inelastic behavior, a characteristic
436 "knee" behavior which is valid between the values of fp and the convential elastic limit of 0.2 percent fe, and the third region describes the strain-hardening behavior of the material till the upper ultimate strength fu limit, see Figure 2.. The expressions for these three different laws are affected by the initial E value, and by the
/ f o i ^^^^^ which is known as strain-hardening parameter, different for each
aluminium alloy and only determined by experiment
THE NUMERICAL MODEL The following paragraphs deal with the analysis of curtain walls having a grid of axb, as shown in Figure 3, where a and b take the values 1.00, 1.25, 1.50, 1.75, 2.00 and 1.00, 1.50 respectively. The vertical members, termed "muUions" are interconnected by discontinuous horizontal members, termed "transoms" which act as the glazing support. The cross section for these elements are shown in Figures 4a and 4b along with a simple load-transfer model for the application of the glazing-bom wind load onto the mullion-transom grid. mullions
-•X
H L:_
transoms
Figure 3 Curtain wall's grid The columns are supported on the floor slab and therefore translations along x, y, z axes are restrained. 120.0 50.0
18.0
1 30.0 34.0
t= 1.2mm
t=2.0mm
44.0
58.0 12.0
Figure 4a Profile of aluminium mullions
Figure 4b Profile of aluminium transoms
Transoms are jointed to the mullions, so they were examined seperately and the respective forces were transfered to the mullion-transom connection. There, also the translations along the x-axis were restrained because of the presence of the traverses. A folded shell structural model was used so as to give us the possibility to describe initialisation of plastic behaviour with accuracy; in fact a simpler, bending beam model, would have to incorporate effective values for either the Young Modulus or the moment of inertia; as such an association is delicate we chose the accuracy offered by the folded shell approach. Production of effective c/s tables requires extensive numerical work and will, hopefully, be done by our Institut, in the future.
437
For the purpose of the present analysis, the mullion and transoms were divided into 4582 and 1995 4-node shell elements of proper size, shape, i.e it has taken care so that the elements side's length would have a ratio near to 1.2. From the standard library of the Castem software a four-noded thin shell element with global displacements and rotations as degrees of freedom has been selected. In the present analysis the weight of aluminium members has not been taken into account because of its small influence in the static behavior of this system and the influence of the glazing has been considered as an edge load at the horizontal members. The wind loading acting as pressure at the facade of the whole structure is assumed to be 0.0012N/mm^.
REMARKS ON THE RESULTS OF THE NONLINEAR ANALYSIS The nonlinear analysis carried out, shows that the computed values of stresses are significantly lower than the aluminium yield stress (see Table 1, 2 and Figures 5, 6) and therefore, no significant plastification phenomena appear neither on the aluminium muUions, nor on the aluminium transoms. On the contrary, displacements value in some cases are not negligible being much higher at the traverses, as expected, and even exceed Eurocode's 9 maximum values. The latter proves that the serviceability limit state is more critical than the ultimate limit state and that the appearance of large deformations even in the elastic range is one of the major problems of aluminium structures. TABLE 1 TRAVERSES GRID 1000mm
Von Mises stress Eurocode's 9 valued Displacements z 1 traverse Displacements y (mm) (N/mm2) (mm) 1 1 length (mm) 1 (nmi) columns traverses columns traverses columns traverses columns traverses 1 26.33 1.02 0.13 0.46 28.22 4.00 4.00 1000 1.78 43.93 0.16 1.04 32.91 4.00 1250 2.20 2.42 5.00 2.10 39.49 63.26 4.00 1500 4.97 0.19 6.00 2.67 3.83 46.07 86.16 1750 3.11 9.17 0.23 4.00 7.00 2000 15.59 0.26 6.48 52.65 112.60 3.55 4.00 8.00 1 TABLE 2 TRAVERSES GRID 1500mm
1 traverse Displacements y Displacements z Von Mises stress Eurocode's 9 valued 1 length (mm) (N/mm2) (mm) (mm) traverses columns traverses columns traverses columns columns traverses 1 1 (mm) 1000 1.52 0.46 23.56 40.68 1.49 0.11 6.00 4.00 1250 1.86 3.63 29.45 63.42 0.15 1.04 6.00 5.00 1500 2.23 7.45 0.17 2.10 35.34 91.33 6.00 6.00 1750 2.60 13.74 41.23 124.38 0.20 3.84 6.00 7.00 2000 2.98 23.37 0.22 6.49 47.12 165.52 6.00 8.00 1
438
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1000
1250
1500
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Traverse length (mm)
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O ¥ 1500
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Traverse length (mm)
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1500
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Traverse length (mm)
Figure 5: Traverse grid 1000mm
2000
439
25-
B 20 ^
15 -
columns
s
—X—traverses
i 10-
o a & Q
5- < >00 10
—
__—xr^^^^ 1250
1500
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1 -\ S
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o
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Traverse length (mm)
-O— columns -X—traverses
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Traverse length (mm)
Figure 6: Traverse grid 1500mm
2000
440 REFERENCES Chen W.F, Han D.J. (1988). Plasticity for structural engineers, Springer-Verlag, New York, USA Kammer C. (ed) (1996). Aluminium Taschenbuch I, II, Aluminium Zentrale, Dusseldorf, FRG Mazzolani F.M. (1985). Aluminium Alloy Structures, Pitman Publishing Ltd., London, UK Preftitsi F., Baniotopoulos C.C, Kohsakis E.and Panagiotopoulos P.D. (1997). A Nonlinear Numerical Analysis of the Structural Response of Aluminium Curtain-Wall Systems. Proc. S'^ National Congress on Steel Structures (K. Thomopoulos, C.C. Baniotopoulos, A. Avdelas (eds)), Thessaloniki, 63-71.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
441
COLUMN CURVE FORMULATION FOR ALUMINIUM ALLOYS K.J.R. Rasmussen^ and J. Rondal^ ^ Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia ' MSM Institute of Civil Engineering, University of Liege, Quai Banning 6, 4000 Liege, Belgium
ABSTRACT The paper describes a column curve formulation capable of producing accurate strengths for extruded aluminium alloys failing by flexural buckling. The formulation uses a simple extension of the Perrycurve and is valid for the full range of alloys used in practice. The material properties are assumed to be expressed in terms of the Ramberg-Osgood parameters (£"0, ao.2, «), typically obtained from a stub column test of the finished product. It is shown that firstly, the formulation is capable of reproducing closely the ECCS a-, b- and c-column curves for aluminium alloys. Secondly, by adopting the ECCS Recommendations of basing the column curve selection on the type of alloy (heat-treated or non-heat-treated), it is shown that a better agreement with tests can be obtained by using the column curve formulation proposed in the paper than the column curves of the ISO Recommendations and the current pre-standard Eurocode9 for aluminium alloy structures. A new improved a-curve is also proposed. KEYWORDS Aluminium, Columns, Flexural buckling. Design, Eurocode 9, ECCS column curves.
INTRODUCTION Extensive experimental and numerical research was conducted during the 1960s and 1970s under the auspices of the European Convention for Constructional Steelwork (ECCS) to determine the flexural strength of aluminium alloy columns. The research was undertaken in response to the lack of test data for aluminium alloy columns and the realisation that the strength curves of aluminium columns were different from those of conventional carbon steel columns. The fact that different column curves applied to carbon steel and aluminium alloy columns was primarily a consequence of the different stress-strain curves of the two materials: The stress-strain curve of carbon steel was essentially bilinear whereas the stress-strain curves of aluminium alloys were nonlinear. The latter type of stressstrain curve can be modelled closely using the Ramberg-Osgood expression. On the basis of tests performed during the 1960s and 1970s, several column curves were proposed, referred to as the ECCS a-, b- and c-curves for aluminium alloys (ECCS, 1978). The choice of using
442
multiple column curves reflected the necessity made apparent by the tests to distinguish between the strengths of heat-treated and non-heat-treated alloys, open and closed cross-sections, and symmetric and asymmetric cross-sections. The combination of these material and geometric factors led to a complicated classification system for the choice of column curve. Consequently, in the final recommendations, only the a- and b-curves were adopted by the ECCS, applying to heat-treated and non-heat-treated alloys respectively. The recommendations took into account the effect of asynmietry by multiplying the column strength by a factor which depended on the degree of asymmetry. In the final recommendations, no distinction was made between open and closed cross-sections. The ECCS column curves were presented in tabular form and thus were not suitable for design. An analytic expression (Frey & Rondal 1978, Rondal & Maquoi 1979b) was presented and adopted by the ECCS (1978), reproducing the a- and b-curves to an accuracy of 2%. Subsequently, a simpler but slightly less accurate expression was suggested by Rondal (1980). The latter was based on a Perry-type column curve using an imperfection parameter in the form of,
Tl = a{p-X){X'-KT
(1)
While this expression could approximate closely the ECCS a- and b-curves, it suffered from the shortcoming of producing quasi-vertical tangents of the column curves at X=Xo. ]n developing the Eurocodes during the 1980s and 1990s, an effort was made to unify the rules of the various codes, including EurocodeS (1992a) for carbon steel structures and the pre-standard Eurocode9 (1995) for aluminium alloy structures. In regard to columns failing by flexural buckling, the prestandard Eurocode9 uses the same Perry-type curve as that specified in EurocodeS and the same linear form of the imperfection parameter, r]:=a{X-\)
^=0.2
(2)
except that different values are chosen for the a-constant. The a-, b- and c-curves of Eurocode3 are defined by a=0.21, 0.34 and 0.49 respectively. For unwelded sections, the pre-standard Eurocode9 specifies a=0.20 and 0.45 depending on whether the cross-section is symmetric (or mildly asymmetric) or asymmetric. The column curve classification adopted in the pre-standard Eurocode9 is quite different from that of the ECCS Recommendations in that, firstly, there is no distinction between heattreated and non-heat-treated alloys and, secondly, the classification of cross-sections differs in so far that the ECCS Recommendations distinguish between symmetric and asymmetric sections by applying a factor to the column strength while the pre-standard Eurocode9 uses different column curves. The reason for distinguishing between heat-treated and non-heat-treated alloys in the ECCS Recommendations is mainly the fact that non-heat-treated alloys exhibit greater softening, or more gradual yielding, than heat-treated alloys. This is reflected by the exponent (n) of the Ramberg-Osgood expression (DeMartino et al., 1990) which is usually in the ranges [8-15] and [20-40] for non-heattreated and heat-treated alloys respectively. Because of this difference and because heat-treated alloys have distinctly higher proof stresses than non-heat-treated alloys, the strength curve of heat-treated alloys is higher than that of non-heat-treated alloys. As far as symmetric and asymmetric cross-sections is concerned, the factor used in the ECCS Recommendations is a continuos function of the degree of asymmetry. In contrast, by using different column curves for mildly and strongly asymmetric crosssections in the pre-standard Eurocode9, a discontinuity in strength occurs at the limit between these two classes of cross-section. From a physical and design viewpoint, such a discontinuity is undesirable. The linear form of the imperfection parameter given by Eqn. 2 was also adopted in the ISO Recommendations (ISO 1992), although the classification was still based on the type of alloy (heattreated or non-heat treated) as in the ECCS Recommendations. The ISO Recommendations suggested that the values of a=0.2, Xo=0.3 and a=0.4, A,o=0.3 be used for heat-treated and non-heat-treated alloys respectively.
443
While the rules of the pre-standard Eurocode9 and the ISO Recommendations for columns failing by flexure are attractive from the viewpoint that they resemble those of EurocodeS, they do not describe accurately the column strength of aluminium alloys. This is particularly the case for non-heat-treated alloys. The main problem is that the linear form of the imperfection parameter defined by Eqn. 2 is not generally applicable to nonlinear materials, albeit proven accurate for carbon steel (Rondal & Maquoi 1979a, 1979b). Rasmussen & Rondal (1997a) showed that the appropriate form for nonlinear materials is,
n=a({;i-\y-\)
(3)
where the constants a, p, XQ and X\ can be expressed in terms of material parameters. In the limit, « -> oo, leading to a bi-linear material, the definition of a, p, XQ and Xi is such that Eqn. 3 simplifies to Eqn. 2 (Rasmussen & Rondal 1997a). The imperfection parameter defined by Eqn. 3 has been shown to be accurate for a wide range of material parameters which includes all structural aluminium alloys. The range of application of Eqn. 3 is wider than that of Eqn. 1 and the expression offers the advantage of simplifying to Eqn. 2 for bi-linear materials. The main purpose of this paper is to present a rational design procedure for extruded aluminium columns failing by flexural buckling. The procedure uses the ECCS column curve classification and strength curves that are close approximations to the ECCS column curves. The column curve formulation is the same as that used in Eurocode3 and the pre-standard Eurocode9, except that the imperfection parameter is defined by Eqn. 3 rather than Eqn. 2. The paper includes a comparison with tests which shows that the proposed procedure is generally more accurate than those of the prestandard Eurocode9 and the ISO Recommendations. The same procedure has been shown also to be accurate for the design of stainless steel columns (Rasmussen & Rondal, 1997b, 1999). Thus, the proposed procedure is capable of unifying the rules of the Eurocodes for different metals while maintaining accuracy, involving only a simple generalisation of the linear imperfection parameter expression used in Eurocode3.
COLUMN CURVE FORMULATION The design procedure described in Rasmussen & Rondal (1997a) is applicable to metal columns in general, and is applied to aluminium alloy columns in the present paper. In using the procedure, the mechanical properties are firstly assumed to be defined in terms of the Ramberg-Osgood parameters, comprising the initial Young's modulus (EQ), the 0.2% proof stress (ao.2) and the parameter (n) which controls the sharpness of the knee of the stress-strain curve. The Ramberg-Osgood parameters are assumed to have been obtained from curve fits of measured stress-strain curves obtained from stub column tests of the finished product. Secondly, a Perry curve is adopted as strength curve by modifying the imperfection parameter to be expressed by Eqn. 3 where the constants a, p, Xo and ^1 are expressed in terms of the Ramberg-Osgood parameters (EQ, ao.2, n). Thus, the nondimensional column strength is calculated using,
.p = ^ ( l + »7 + A 0
(5)
where the imperfection parameter {r\) is given by Eqn. 3 and the constants (a, p, XQ, Xi) are expressed in terms of the parameters e=Go2/Eo and n as detailed in Rasmussen and Rondal (1997a). In Eqns 4-5, % and X are defined as
444 (6) (7) (8)
{LIrr
where QU, L and r are the ultimate stress, effective length and radius of gyration respectively. The expressions for a, p, XQ and X\ given in Rasmussen and Rondal (1997a) were derived so as to produce close fits to column strength curves obtained using advanced finite element analyses of square hollow sections. The expressions covered the range of structural aluminium alloys used in practice. In Rasmussen and Rondal (1998), they were used to generate strength curves for CHS and I-section columns of 7020, 5454 and 5083 alloys. The strength curves were shown to be in excellent agreement with established solutions and tests (Bernard et al., 1973). 1.2
-I
1
r
n
\ 1 Proposed curves ECCS curves
1.0 Euler (E Q)
0.8 0.6 0.4
0.2 Ol
\
\
0.5
\
\
1.0
L
1.5
2.0
2.5
Fig. 1: Proposed curves and ECCS a-, b- and c-curves
ECCS COLUMN CURVES Aiming for close fits to the a- b- and c-curves of the ECCS Recommendations, an approximation to the a-curve was obtained usingCTo.2=200MPaand w=20. The value of n was consistent with the ECCS Recommendations. The b-curve was obtained using ao.2=100MPa and n=10. In the ECCS Recommendations, the b-curve was associated with a value of n of 15. However, the b-curve was derived by also considering variations of wall thickness, which are common for extruded profiles and which lead to eccentricities of the applied load with an associated decrease in strength. Consequently, a smaller value (n=10) was chosen in this study as a means of accounting for loading eccentricities induced by variations of wall thickness. The c-curve was obtained using ao.2=50MPa and n=8. Both of these values were lower than those on which the ECCS c-curve was based. For all three curves, the value of EQ of 70000 MPa was assumed.
445 Using the stated values of £"0, cTo.2 and n in conjunction with the expressions for a, p, >.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,<0.3 is minor for the a- and b-curves. TABLE 1 a, P, Xo AND Xi VALUES FOR ANALYTICAL APPROXIMATIONS TO THE ECCS COLUMN CURVES Column curve a-curve b-curve c-curve
a
P
0.4 0.7 0.95
0.2 0.15 0.25
XQ
Xi
0.55 0.55 0.35
0.2 0.2 0.2
PROPOSED DESIGN CURVES Classification and parameters The classification of aluminium alloy extruded columns contained in the ECCS Recommendations and the ISO Recommendations is used in the present proposal. The classification, which distinguishes between heat-treated and non-heat-treated alloys, is considered appropriate because heat-treated alloys have distinctly higher values of e=Go2/Eo and n than non-heat-treated alloys and consequently, distinctly different strength curves apply to the two types of alloys (Rasmussen & Rondal 1997a). It is considered more rational to base the column curve selection on the type of alloy than the degree of asymmetry, as is the case in the pre-standard Eurocode9. The degree of asymmetry does not lend itself to distinct classes and leads to an undesirable discontinuity in strength when used for column curve selection. Thus, it is proposed to use the a- and b-curves for heat-treated and non-heat-treated extruded alloy columns respectively and to compute these curves using Eqns 3-5 in conjunction with the values of a, P, A-o and Xi shown in Table 1. Furthermore, it is proposed to take the detrimental effect of asymmetry into account by applying a factor to the column strength as proposed in the ECCS Recommendations and described in detail in Mazzolani (1995). Comparison with tests, pre-standard Eurocode9 and ISO Recommendations In Figure 2, the proposed curves are compared with tests and the column curves of the pre-standard Eurocode9 and the ISO Recommendations. All tests were performed on symmetric sections (I-sections and circular tubes) and consequently, only the curve of Eurocode9 defined by Eqns 2,4-5 and a=0.2 is included in the figure. The column curves of the ISO Recommendations were defined by Eqns 2,4-5 using a=0.2, A-o=0.3 and a=0.4, A,o=0.3 for heat-treated and non-heat-treated alloys respectively. The tests (Bernard et al. 1973, Djalaly & Sfintesco 1972, Kloppel & Barsch 1973, Arnault 1967) are divided into groups containing heat-treated (2xxx, 6xxx and 7xxx series) and non-heat-treated alloys (5xxx series). Many of the test points are mean values of several tests conducted at particular slendemess values (Djalaly & Sfintesco 1972, Arnault 1967). The scatter of test strengths for each slendemess value is not shown in the figure but is taken into account in the statistical evaluation described below. As shown in the figure, the test strengths of non-heat-treated alloys are clearly lower than those of heat-treated alloys and generally, the a- and b-curves follow closely the distributions of the test strengths. The a=0.2 curve of the pre-standard Eurocode9 and the (a=0.2,Xo=0.3)-curve of the
446 ISO recommendations follow fairly accurately the distribution of test strengths for heat-treated alloys. However, the a=0.2 curve of the pre-standard Eurocode9 and the (a=0.4,A.o=0.3)-curve of the ISO Recommendations are not close representations of the test strengths for non-heat-treated alloys. 1.2 Tests, h-t 0 Tests, n-h-t +
1.0 0.8
0.6
Proposed a-curve, h-t Proposed b-curve, n-h-t
0.4 0.2 h h-t, heat-treated n-h-t, non-heat-treated
0.5
1.0
1.5
2.0
2.5
X Fig. 2: Comparison of column curves with tests.
TABLE 2 MEAN AND COEFFICIENT OF VARIATION OF TEST STRENGTH TO DESIGN STRENGTH Design procedure Proposal (a=0.4 for h-t alloys) Pre-standard Eurocode9 ISO Recommendations New a-curve (a=0.3)
Heat-treated alloys S 1.11 1.12 1.12 1.07
Non-heat-treated alloys
Vs
s
0.077 0.082 0.073 0.071
1.11 1.00 1.12
0.066 0.101 0.109
STATISTICAL EVALUATION OF PROPOSED STRENGTH CURVES Annex Z of Eurocode3 (1992b) describes a procedure for calculating the partial coefficient factor (YM) to be applied to the nominal strength in order to take into account random variations of material and geometric properties as well as the design model. The statistical assessment of the design model requires evaluation of the ratio (5) of test strength to design strength where the design strength is calculated using measured values of material and geometric properties. This ratio has been calculated for the design strengths determined according the pre-standard Eurocode9, the ISO Recommendations, and the design procedure proposed in the present paper by using the test strengths reported in Bernard et al. (1973), Djalaly & Sfintesco (1972), Kloppel & Barsch (1973) and Arnault (1967). Full details of the calculation are given in Appendix I of Rasmussen & Rondal (1998). The mean (5) and coefficient of variation (V^) of the ratio of test strength to design strength are shown in Table 2.
447
It follows from Table 2 that the coefficient of variation (V5) obtained using the design strengths of the proposed column curve is smaller than those obtained using the pre-standard Eurocode9 and the ISO Recommendations for non-heat-treated alloys (b-curve) and is comparable with these specifications for heat-treated alloys (a-curve). Thus, the proposed design curves are generally more accurate than those specified in the pre-standard Eurocode9 and the ISO Recommendations.
IMPROVED a-CURVE In selecting a column curve for heat-treated aluminium alloy columns, a better agreement with tests was obtained by reducing the a-value for the a-curve from the value of 0.4 shown in Table 1 to 0.3. The effect of reducing a was to slightly raise the strength curve. The coefficient of variation (V5) of test strength to design strength obtained using the higher curve (a=0.3) is shown in Table 2. It appears that by using a=0.3, the coefficient of variation is reduced from 0.077 to 0.071 compared with using a=0.4. Such reduction should lead to a smaller value of YM and thus more economical designs.
CONCLUSIONS A column curve formulation has been described which involves a simple extension of the linear imperfection parameter expression currently used in Eurocode3 for carbon steel structures. The formulation is shown to be capable of producing accurate strength curves for extruded aluminium alloy columns failing by flexural buckling and close approximations to the ECCS column curves for aluminium alloys. It is proposed that the formulation be adopted in Eurocode9 for aluminium alloy structures. This involves a column curve classification that distinguishes between heat-treated and non-heat-treated alloys rather than symmetric and asymmetric cross-sections as is the case in the current pre-standard. The proposal is shown to produce lower coefficients of variation of the ratio of test strength to design strength than the column strength rules of the pre-standard Eurocode9. Thus, it offers a more efficient design procedure. In Rasmussen & Rondal (1997b, 1999), the proposed formulation has been shown to be accurate for stainless steel alloys, which commonly have lower values of the exponent (n) of the Ramberg-Osgood expression than aluminium alloys. Furthermore, the formulation reproduces the linear form of the imperfection parameter for bi-linear materials and so is in accord with the rules of Eurocode3 for carbon steel columns. It can be concluded that the formulation offers a generalisation of the imperfection parameter expression currently used in the Eurocodes which is capable of unifying the rules for carbon steel, stainless steel and aluminium columns while maintaining high accuracy.
REFERENCES Arnault, P., (1967), "Recherche sur le Flambement des Profiles en Alliages Legers", Centre technique Industriel de la Construction Metallique (CTICM), Paris. Bernard, A., Prey, P., Janss, J. & Massonnet, C , (1973), "Recherches sur le Comportement au Flambement de Barres en Aluminium", lABSE Memoires, Vol. 33-1, Zurich, pp 1-32. DeMartino, A., Landolfo, R. & Mazzolani, P.M., (1990), "The use of the Ramberg-Osgood law for materials of round-house type". Materials and Structures, Vol. 23, pp 59-67.
448
Djalaly, H. & Sfintesco, D., (1972), "Recherches sur Flambement de Barres en Aluminium", Reports of the working commissions, lABSE, Vol. 23, International Colloqium on Column Strength, Paris. ECCS, (1978), European recommendations for aluminium alloy structures, European Convention for Constructional Steelwork, 1st ed., Brussels. EurocodeS (1992a), Design of Steel Structures, Part 1.1: General Rules and Rules for Buildings, ENV 1993-1.1, European Committee for Standardisation (CEN), Brussels. Eurocode3 (1992b), Design of Steel Structures, Part 1.1, Annex Z: Determination of design resistance from tests, ENV 1993-1.1, European Committee for Standardisation (CEN), Brussels. Eurocode9 (1995), Design of Aluminium Alloy Structures: General Rules, ENV 1999-1, European Committee for Standardisation (CEN), Brussels. Prey, F. & Rondal, J., (1978), "Aluminium alloy buckling curves a,b,c: table and equations", ECCS Committee 16, Doc. 16-78-1. ISO (1992), Aluminium Structures: Material and Design, Part 1: Ultimate Limit State - Static Loading, Technical Report, Doc. No. 188, International Standards Organisation Committee TC 167/SC3. Kloppel, K. & Barsch, W., (1973), "Versuche zum Kapitel "Stabilitatsfalle" der Neufassung von DIN 4113", Aluminium, Vol. 10, pp 690-699. Mazzolani, P.M., (1995), Aluminium Alloy Structures, 2nd ed., E & FN Spon, London, Rasmussen, K.J.R. & Rondal, J., (1997a), "Strength curves for metal columns". Journal of Structural Engineering, American Society of Civil Engineers, Vol. 123, No. 6, pp 721-728. Rasmussen, K.J.R. & Rondal, J., (1997b), "Explicit approach to design of stainless steel columns", Journal of Structural Engineering, American Society of Civil Engineers, Vol. 123, No. 7, pp 857-863. Rasmussen, K.J.R. & Rondal, J., (1998), Strength curves for aluminium alloy columns. Research Report No. R771, Department of Civil Engineering, University of Sydney. Rasmussen, K.J.R. & Rondal, J., (1999), Column curves for stainless steel alloys, Proceedings, 4* International Conference on Steel and Aluminium Structures, ICSAS'99, Helsinki. Rondal, J., (1980), "Formulation Simplifiee des Courbes Europeennes de Flambement des Barres en Alliage d'Aluminium", Report 95, MSM Institute of Civil Engineering, University of Liege. Rondal, J. & Maquoi, R., (1979a), "Single equation for SSRC column-strength curves. Journal of the Structural Division, American Society of Civil Engineers, Vol. 105, STl, pp 247-250. Rondal, J. & Maquoi, R., (1979b) "Formulation d'Ayrton-Perry pour le Flambement des Barres Metalliques, Construction Metallique, No. 4, pp 41-53.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
449
AN EXPERIMENTAL STUDY OF STRENGTH AND DUCTILITY OF WELDED ALUMINIUM BEAMS M. Matusiak^ and P. K. Larsen^ * Kvaemer Oil & Gas a.s, N-4003 Stavanger, Norway ^ Department of Structural Engineering, Norwegian University of Science and Technology, N-7034 Trondheim, Norway
ABSTRACT Four-point bending tests with simply supported beams with a single butt weld in the tension or compression flange, a bracket welded to the tension or compression flange and welded stiffeners were carried out for an I-section in aluminium alloy 6082-T6. It was shown that both the strength and ductility was significantly reduced by the influence of welding. The ultimate moment was independent on whether the welds were located in the tension or compression flange. Tension flange rupture at low values of the vertical displacement was the failure mode for all specimens with weld on the tensile flange. Local instability followed by subsequent large plastic deformations occurred for beams with welded details located on the compression flange. It can be concluded that plastic design may require more attention for aluminium than for mild structural steel structures. Cross-sectional classification is traditionally based on stability only, and expressed in terms of the with/thickness ratio and the yield strength of the parent material may not be sufficient for aluminium sections. Local material softening in heat-affected zone may drastically limit the member ductility.
KEYWORDS Aluminium, weld, HAZ, softening, strength, ductility, beam, connection.
INTRODUCTION AND BACKGROUND Inelastic design of statically indeterminate structures requires that plastic hinges possess sufficient deformation capability without losing the resistance. In case of steel beams this capability is expressed by the rotation capacity, and represents the ability of a member to withstand large plastic rotations before its bending resistance drops below the fully plastic moment of the section. Lukey & Adams (1969) used the term *hinge capacity' for the inelastic hinge rotation of the full span beam under moment gradient.
450
Mazzolani & Piluso (1995) proposed a definition for the rotation capacity in aluminium beams based on the yield moment M0.2 and the corresponding rotation 60.2, Figure 1. It was postulated that due to the hardening behaviour of aluminium a fully plastic moment could not be defined because the 'fibres' never reach a yielding plateau. Consequently, in case of aluminium alloy beams the rotation capacity should be related to the inelastic rotation accumulated before the bending resistance falls below the yield moment MQ.IvM/Mo.2
1
Qu /80.2
Qm /^.2
Figure 1: Definition of rotation capacity As seen in the figure, the rotation capacity R is defined by the rotation 0m at which the descending branch of the response curve equals the elastic limit M0.2 (1) ^02
The stable part RQ of the rotation capacity gives an additional characteristic of the inelastic behaviour and is related to the ultimate rotation 9u
/^=A__i
(2)
9o.2
For beams made of material with very low strain hardening, containing softened zones in tension, the stable part of the rotation capacity may be a useful ductility measure. This is the case for welded artificially aged heat-treatable aluminium alloy members. It will be shown in the following that tensile fracture may occur before the descending branch of the moment-rotation curve attains the value M0.2. Very little experimental data is available on the structural response of aluminium members containing local transverse welds. Lai & Nethercot (1992) performed a series of four-point bending tests on a rectangular box cross-section of aluminium alloy 7019. In order to introduce heat-affected zones two plates of varying lengths were fillet-welded to both flanges of the section using the MIG process. The plates were either welded along the entire span, or 25-200 mm long plates were welded at mid-span, symmetrically at quarter-span or close to the ends. Beams with heat-affected material along entire length showed a reduction in strength but not in ultimate deformation. Several beams with transverse welds at mid-span fractured at relatively small deflections. Referring to a limited four-point bending test series by Edward on a 6082-T6 I-section containing a weld bead around its perimeter or welded stiffeners, Robertson (1985) pointed out that welding primarily caused a change of failure mode from 'flange/web buckling' to tensile failure of the flange. As instability limited the resistance of the unwelded specimens the welded and unwelded beams had similar ultimate capacities. Moen, Langseth and Hopperstad (1998) also found tensile failure for 6082T6 I-beams with welded stiffeners tested under moment gradient. Compared with unwelded specimens both the ultimate capacity and especially the ductility were reduced as a consequence of material softening.
451 The present study constitutes a part of a larger investigation on strength and ductility of welded structures in aluminium alloy 6082-T6. The whole project comprises testing of: • Single butt and fillet welds having various orientations relative to the load direction • Variations of the mechanical properties of the material in the vicinity of a weld-bead • U-shaped fillet weld groups subjected to centric and eccentric loading. For comparison reason a parallel test series was carried out also for similar connections in mild structural steel • Joints with a transverse tensile force on an unstiffened column • And finally, members and joints exposed to bending moment and shear force. It was shown in a previous study, Matusiak & Larsen (1998) that welding of aluminium alloy 6082-T6 reduces the yield strength fo.2, the ultimate strength fu and the corresponding strain £„ to approximately 50%, 30%, and 10% respectively of the nominal value. The extent of softening depends on the temperature distribution during welding, and varies with the distance from the weld bead. The main objective of this study was to provide information on the influence of welds on the load bearing capability of aluminium alloy members when subjected to bending moment. A number of relatively simple details that are commonly found in more complex configurations were studied. Besides the softening extent also the possible influence of the position of the HAZ was addressed.
EXPERIMENTAL TEST PROGRAMME Four-point bending tests on simply supported beams with a single butt weld in the tension or compression flange, a bracket welded to the tension or compression flange and welded stiffeners were carried out for an I-section. A span width of 1000 mm was used for all specimens, and the distance between the two point loads was 400 mm, as depicted in Figure 2. Four parallel tests were performed for all welded configurations and three parallel tests were carried out for the unwelded members.
tp/2
^ ^ P/2I
^7V 200
i2&l
4pb-2
4pb-4
4pb-3
4pb-5
3^
^::^ 4pb-6
V
4|V_
Figure 2: Specimens with welds The welds were made using MIG pulsed arc with filler alloy 5183 by a commercial fabricator. Marine Aluminium AS. Butt joints with groove angle of 60° were made with one pass on each side, and fillet joints were made in a single pass. Uniaxial tensile test was carried out for the parent material, and the mean values of the mechanical properties are given in Table 1. As expected, the strain hardening of the material was low. The test rig consisted of a standard vertical loading frame with two supporting columns, a transverse beam and a hydraulic actuator. At the end points rolling bearings prevented the vertical upward displacement and lateral movement of the beam, i.e. out-of-plane displacement and rotation about longitudinal axis. Symmetric loading was obtained by using two long chains connected to the actuator through a transverse pin-fastened beam. The loads from the chains were transferred to the beam by a
452
specially designed device, Figure 3. A number of ball- and rolling bearings were used in order to prevent lateral movement, leaving all other degrees of freedom unconstrained. TABLE 1 MEAN VALUES OF PARENT MATERIAL PARAMETERS
Web Flange
/o.2
/u
/u //0.2
Eu
(N/mm^) 279.2 277.6
(N/mm^) 289.3 290.4
1.04 1.05
0.066 0.065
Figure 3: General view of the test rig In order to prevent local deformations at the loading points, stiffeners were bonded to the beams using a West System epoxy. Bonding was used instead of welding in order to avoid softening of the material. In order to prevent lateral torsional buckling of the beams additional lateral supports were provided using slip bearings at distances of 50 mm from the beam centre. In order to minimise friction the contact surfaces between the slip bearings and the specimen were polished and lubricated by Teflon. Both the global response of the specimen, measured as force versus displacement, and the local behaviour close to the welded detail were of interest. The applied load P was measured using a load cell. Rotations at the loading points were measured using two Lucas AccuStar electronic clinometers, see Figure 5. An ME-46 Full Image Video Extensometer was used to measure both the vertical displacement of specimen mid-point and the deformations in the region containing a welded detail. This is a non-contact measuring system comprising a monochrome video camera fitted with a high precision Charge Coupled Device chip with photosensitive cells arranged in an accurate grid. The camera is attached to a 'frame-grabber' interface card fitted into the video extensometer controlling PC. The card converts the PAL video signal into an 8 Bit digital format, and a 640x480 pixels image is simultaneously generated on the monitor and software. The interface card is capable of resolving the grey scale level of each pixel in 256 shades, which results in a theoretical displacement resolution better than 10"^ of the camera field of view, Messphysik Laborgerate GES.m.b.H. (1996).
453
In the present investigation white dots with a diameter of 10 mm were attached to the specimen. Since the measuring accuracy is strongly dependent on contrast difference between the targets and background, all other elements in the camera field of view were kept black; the specimen was sprayed with a high-adhesive and high-elastic paint. In all tests a target placed in the 'centre' of the specimen was used for measuring the vertical displacement. Eight additional dots were used to determine local deformations in the welded region, and were located on both the tension and compression flange at distances of 25 and 75 mm from the symmetry axis. In order to verify the reliability of data generated by the video extensometer two non-welded specimens were additionally instrumented with strain gauges and tested in pure bending. The gauges were mounted on both flanges; at the symmetry axis and at a distance of 50 mm from the axis. Good agreement between both data sets has been achieved. A distance of about 5 m between the camera and specimen was used for all tests. The PC controlling the video extensometer was connected to the testing machine in order to save the loading data simultaneously with the recording of the dot co-ordinates. Another PC based logging system was used for data from the clinometers and was synchronised with the loading. Monotonic loading was applied with displacement control and a cross head speed of about 0.03 mm/sec, ensuring a quasi-static loading condition. In order to measure friction between the slip bearing and the specimen a pair of strain gauges was mounted on the bolt that fastened the supporting console to the girder. The friction force was than calculated from the variation of strain in the bolt. The maximum values of the friction force between the specimen and the slip supporting fixtures were about 1% of the ultimate load Pu. This was considered as insignificant and the influence of friction is neglected in subsequent presentation of test results.
RESULTS AND DISCUSSION No lateral torsional buckling occurred for the welded beams. The entire HAZ was kept between the slip bearings. Welding reduced the ultimate moment and outside the HAZ the beam was stiff enough not to buckle laterally. Higher moment capacity was reached for the unwelded beams reducing the lateral stiffness of the beam. Lateral buckling occurred between the slip bearing and the end point. Consequently, insufficient lateral supporting of the unwelded beams caused a reduced rotation capacity. According to Eurocode 9 (1997) both the web and flange of the cross-section is classified as compact. Thus, a higher ability to inelastic deformation should be expected provided no lateral movement condition maintained. Tension flange rupture at low load values was the failure mode for all specimens with weld on the tensile flange of the specimen. The beams with welded details located on the compression flange showed no fracture even at large deformations. For all these specimens local buckling of the compression flange occurred first, followed by subsequent web buckling. Further loading caused bending about section weak axis of a narrow region of the compression flange containing the weld zone. Typical moment-rotation curves for all specimens are shown in Figure 4. The data were normalised using calculated values of the elastic moment M0.2 and the associated rotation ©0.2. Assuming a constant curvature over the mid-span / the rotation was calculated as
Eh
454 where E is the Young's modulus and h is the height of the beam. Measured values of both the cross sectional dimensions and yield strength /0.2 of the parent material in the flange were used in the calculations. The 'plastic' moment resistance Mp was computed as a product of the elastic moment M0.2 and the shape factor of the cross section r| = 1.15.
Mp/Mo.2
1.0 Four-point bending test Virgin material
M/M0.2
Butt weld in tension flange Butt weld in compression flange Brackett welded to tension flange Brackett welded to compression flange Welded stiffeners
0.0 4 e/0b2
Figure 4: Moment versus rotation curves for representative specimens As seen in the figure both the strength and ductility were significantly reduced by the influence of welding. The rotation capacity was primarily dependent on whether the welded detail was located in the tensile or compression flange. As expected, the extent of welding in the section was the governing parameter of the ultimate capacity. The greatest reduction in capacity was found for specimens that contained welded stiffeners, where the ultimate moment was lower than the elastic value Mo.iHowever, the moment capacity did not significantly depend on whether the welds were located in the tension or compression flange. Mean values of the measured vertical mid-span displacement Au at ultimate load Pu and the ratios Au/Ao.2 and Pu/Po.i are given for all tests in Table 2. Both bending and shear deformations were taken into account in the calculation of the vertical displacement A0.2 at load P0.2 giving An. =
J0.2
{L-iy
{2L-l)l , fo.2W
Eh
(4)
where L is the distance between the span width, W is the elastic section modulus and G is the shear modulus. The shear area Ay is given as
A^=(h-2t,X
(5)
where tf and t^ denote the thickness of the flange and web respectively. In addition, mean values of the rotation capacity R and its stable part RQ, as defined in Eqns. 1 & 2, are given in the table. The results show a significant difference between the behaviour of the unwelded and welded specimens. No rotation capacity is computed for the members with welded stiffeners, as their ultimate resistance was lower than the elastic limit M0.2. For all response parameters, except Au and Au/Ao.2, beams with welded stiffeners show the most detrimental effect of welding. Such a detail, which is commonly used in steel structures, shows a very poor structural behaviour in aluminium.
455 TABLE 2 MEAN VALUES OF TEST RESULTS FOR UNWELDED AND WELDED BEAMS UNDER PURE BENDING
Specimen series 4pb-l 4pb-2 4pb-3 4pb-4 4pb-5 4pb-6
P. (kN) 60.8 50.3 53.9 53.5 56.5 47.8
Au
Pu/Po.2
Au/Ao.2
Ro
R
(mm) 45.4 15.5 24.3 20.8 27.6 20.3
1.22 1.09 1.09 1.08 1.13 0.96
4.69 1.94 2.52 2.16 2.87 2.09
4.24 0.53 0.87 0.92 1.58
6.54 ^0.69 2.25 ^1.17 4.22
^ The rotation at rupture 9t limited the rotation capacity
In addition to the rotation over mid-span / = 400 mm, the rotation was also studied over a shorter part of the specimen that contained a welded detail. Using data generated by the video extensometer, rotations over a length of both 50 and 150 mm were calculated. Typical moment versus rotation diagrams as functions of beam segment length are given for all specimen types in Figure 5. As seen, when using the 50-mm length the onset of non-linear behaviour was registered at lower moments than for the longer gauge lengths. Similarly, for specimens with welds on the tension flange the shortest gauge length shows significantly larger rotations at M0.2 than the larger ones. Generally, higher rotations at M0.2 and lower rotations at Mu were found for specimens with the welds located in the tension than in the compression flange. n—I—I—1—I—I—I—I—I—I—I—r-
n—I—I—I—I—I—I—I—I—I—I—T"
-1—I—I—I—I—I—I—I—I—I—I—T"
1.0 (Welded stiffene
^
Clinometers; I = 400mm
Butt weld in tension flange
0.0
_l
0
I
I
I
t
2
4
I
I
6
I
I
8
I
I
10
-J
L_
12 0
2
4
1
1
1
1
1
1
I
6
e/Gbz 1
Video extensom.; I = 150mm
j Bracket welded I to tension flange I
I
8
I
Video extensom.; I = 50mm
i_
10
_J
12 0
2
©/0b2 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
I
I
I
I
6
8
I
I
\
10
L
12
150 50
MA
1
\ C
1
i 1
0
I
1
f 0.0
I
0/0b2 1
~j
1.0
\
1
2
1'
Butt weld ^ f un compression flange^ 1
1
4
1
1
1
6
1
1
8
0/eb2
1
10
1
1 Bracket welded | Uo compression flange J
1
12 0
(^ Clinometer Target for video extensometer
2
4
6
8
10
12
0/0b2
Figure 5: Gauge length dependence of moment versus rotation curves for welded specimens.
CONCLUSIVE REMARKS From both the results from the present investigation and data from literature it can be concluded that plastic design may require more attention for aluminium than for mild structural steel structures. Cross-sectional classification traditionally based on stability only, and expressed in terms of the
456 with/thickness ratio and the yield strength of the parent material may not be sufficient for aluminium sections. Local material softening in HAZ may drastically limit the member ductility. The extent and location of the HAZ in either the tensile or compression component are of primary importance for the strength, and particularly for the ductility of welded members. Existence of a HAZ subjected to tension is the most critical. Even a non-structural element, e.g. a bracket welded to the member may significantly lower the load bearing capacity. On the other hand, when the material softening occurs in the compression part of the critical section the behaviour of the member is relatively unchanged.
ACKNOWLEDGEMENTS The authors would like to thank Hydro Aluminium Metal Products and Hydro Aluminium Maritime for the financial support that made this work possible and Marine Aluminium for preparing the welded specimens.
REFERENCES Eurocode 9 (1997). Design of aluminium structures - Part 1.1: General rules, PrENV 1999-1-1, European Committee for Standardisation (CEN). Lai Y.F.W. and Nethercot D.A. (1992). Strength of aluminium members containing local transverse welds. Engineering Structures 14:4, 241-254. Lukey A.F. and Adams P.P. (1969). Rotation capacity of beams under moment gradient. Journal of the Structural Division, Proceedings of the American Society of Civil Engineers June, 1173-1188. Matusiak M. and Larsen P.K. (1998). Strength and ductility of welded connections in aluminium alloys. Proceedings of the 7th International Conference on Joints in Aluminium INALCO '98, Cambridge, 6^/^3,291-302. Mazzolani P.M. and Piluso V. (1995). Prediction of the Rotation Capacity of Aluminium Alloy Beams. Proceedings of the Third International Conference on Steel and Aluminium Structures, Istanbul, 179-186. Messphysik Laborgerate GES.m.b.H. (1996). ME-46 Video Extensometer Dot Measuring Software, User Manual, RDP-Howden Ltd. Althorpe Street, Leamington Spa, Warks, CV31 2BA, UK. Moen L.A., Langseth M. and Hopperstad O.S. (1998). Cross-secfional Classification of Aluminium Beams - An Experimental Investigation. Proceedings of the Nordic Steel Construction Conference 98, Bergen, 643-654. Robertson I. (1985). Strength Loss in Welded Aluminium Structures, Ph.D. Thesis, Girton College, University of Cambridge.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
457
PLASTIC DESIGN OF ALUMINIUM MEMBERS ACCORDING TO EC9 ^ Mazzolani P.M., ^ Mandara A., ^ Langseth M. ^ Department of Structural Analysis and Design University of Naples Federico H, Naples, ITALY ^ Department of Civil Engineering Second University of Naples, Aversa (CE), ITALY ^ Department of Structural Engineering Norwegian University of Science and Technology, Trondheim, NORWAY
ABSTRACT The innovative aspects of EC9 are discussed with regard to methods and procedures provided by the code on the plastic design of aluminium alloy structures. These include the classification of cross sections, revised on the basis of experimental data, as well as methods for the calculation of intemal actions and for the evaluation of member rotational capacity. Special emphasis is given to the determination of the ductility demand and to the use of the plastic hinge method, suitably modified in order to account for strain hardening and ductility features of material. KEYWORDS Aluminium alloys, Design codes. Plastic design. Plastic hinges, Ehictility demand. Rotation capacity INTRODUCTION The calculation of structures in inelastic range up to failure is one of the most important aspects of the new Eurocode 9 "Design of Aluminium Alloy Structures". This is mainly a consequence of the necessity to take into account the actual properties of aluminium alloys, namely the strain hardening feature and the available ductility, which involve a rather different behaviour as observed for steel structures (Mazzolani, 1995). This results in the usual concentrated plasticity idealisation being not fully reliable owing to the risk of premature failure due to excess of deformation. At the same time, the possible structural overstrength due to the hardening feature of the material has to be taken into account, since neglecting this aspect could yield bad information on the collapse mechanism (Mandara & Mazzolani, 1995). For this reason, some special provisions for the calculation of both members and cross-sections in inelastic range have been adopted (Mazzolani, 1998a,b). The main innovative aspect of Eurocode 9 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
458 structure as a whole. The cross-sections have been classified according to the possibility to rely upon their post-elastic resistance without local bucking phenomena. In addition, the Eurocode 9 represents the first code on aluminium alloys dealing extensively with the problem of the evaluation of the rotation capacity, which is also strictly connected to the inelastic behaviour of structures (Mazzolani & Piluso, 1995). As far as this problem is concemed, the majority of the aluminium codes hardly introduces the behaviour of cross sections beyond the elastic limit, by using a classical approach very similar to that adopted for steel. This way is very limiting because both the actual behaviour of aluminium structures is not clearly interpreted and also nothing is given for analysing the global behaviour of members and structures in inelastic range. The answer to these needs has been, in first place, 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 a different extension of behavioural ranges defined on the basis of experimental evidence. Furthermore, the problem of the evaluation of intemal actions has been faced by considering several models for the material constitutive law, depending on the degree of approximation required. For the global analysis of structural systems in the inelastic range a method similar to the well known method of plastic hinge is provided. It considers the typical parameters of aluminium alloys, like absence of yielding plateau, continuous strain-hardening behaviour and limited ductility of some alloys. By using this method the designer has the possibility to evaluate the ductility demand of the structural scheme and, hence, the requested rotation capacity. This can be calculated via a simplified method based on a closed form equation.
Figure 1: Classification of cross sections in terms of generalised force F and displacement D CLASSIFICATION OF CROSS SECTIONS The classification has been defined on the basis of experimental results, coming from an ad hoc research project supported by the main representatives of the European Aluminium Industry, which provided the material for specimens (Mazzolani et al, 1996). The following classification has been proposed in EC9 Annex G (Figure 1): -
Class 1 (Ductile): sections failing in fully plastic range reaching the required rotational capacity before local instability occurs; Class 2 (Compact): sections which buckle in inelastic range reaching the proof strength fo.2 of the material with good plastic deformation capacity; Class 3 (Semi-compact): sections which buckle in inelastic range reaching the proof strength y^j of the material with limited plastic deformation capacity; Class 4 (Slender): sections not able to develop the proof resistance^^j and failing prematurely due to the occurrence of local buckling in elastic range.
459 The limits between classes are defined as a function of the parameters P = b/t and £ = -^250/f0.2 • It has been shown (Mazzolani & Piluso, 1995) that the classification of EC9 is more appropriate than that of BS 8118 code, whose intermediate class covers a too small range. A similar criticism has previously been done also for steel sections. From the point of view of global structural analysis, the use of plastic methods can be allowed provided that cross-sections belong to class 1 or class 2. On the contrary, some minor plastic redistribution can be allowed in the case of class 3 sections. Finally, the use of elastic methods of structural analysis is strictly necessary in the case of class 4 sections. The load-bearing capacity of the cross-sections under axial load and bending moment is referred to the limit states defined for the assessment of the section behavioural classes (Mazzolani, 1998a,b). These limit states and the corresponding behavioural classes are listed in Table 1. According to the rules supplied in EC9, the axial load and the bending moment for a given limit state can be expressed through the generalised formulas: M=aMjfdW
N=aNjfdA;
(1)
where a^ andttMjare correction factors (see Figure 1), depending on the assumed limit state,^ is the design stress and A and W are the cross sectional area and the resistance modulus of the section, respectively. The expressions for ttNj andttMjare supplied in Table 1 where, in addition to already defined symbols,/ is the material ultimate tensile stress, Z is the section plastic modulus and Agg and Wef are the cross sectional area and the section resistance modulus, respectively, evaluated accounting for local buckling phenomena. TABLE 1 EVALUATION OF ULTIMATE AXL\L LOAD AND BENDING MOMENT ACCORDE^G TO EC9
Limit state
Section Class
Axial load
Correction factor
Bending moment
Correction factor
Collapse
Class 1
Nu
aN,i = f/fd
Mu
OCM.! = a()Clim)
Plastic
Class 2
Npi
aN.2=l
Mp,
Elastic
Class 3
Nel
aN.3 = 1
Me,
Elastic Buckling
Class 4
N^
aN,4 = AefP^A
Mred
OCM,2 = 0 0 =
ZAV
aM,3 = 1 aM.4 =
Wef^
CALCULATION OF INTERNAL ACTIONS AND STRESSES Both elastic and plastic global analysis are considered by EC9 for the structural analysis of redundant structures. When elastic global analysis is used, in order to allow for moment redistribution in inelastic range, the peak elastic moment can be increased or decreased by up to 15%, provided the new internal forces and moments remain in equilibrium and the cross sections have sufficient ductility to allow for the plastic redistribution. For this reason all members where the moments are reduced must have Class 1 or at least Class 2 cross-section. For more refined calculations, non-linear global analysis may be also applied. In these cases the member cross sections have to satisfy the requirements specified for Class 1. in addition, the adopted alloy should have sufficient ductility. Plastic analysis may be carried out by assuming for the material the models shown in Figure 2. The models of Figure 2 differ from each other by the assumption made on the material behaviour in the elastic range, which can berigid,elastic or inelastic. A wider generality can be achieved if the elastic branch of the material law is assumed to be non-linear (Figure If). In this case, called "Generically inelastic", both material and cross sections are idealised according to their actual stress-strain and generalised force-displacement relationship, respectively. The ultimate limit state is defined by the attainment of a given limit value of strength or deformation. The assumptions made on the materisd behaviour have a direct influence on the idealisation of cross-section response in terms of generalised force-displacement relationship. In order to
460 supply provisions consistent with the adopted material law, EC9 suggests the most suitable section behavioural model for each of the material assumptions outlined in Figure 2. In order to assess the possibility to apply calculation methods based on a concentrated plasticity approach, the above material models involve several methods of global analysis, which are also classified in EC9 into two groups, depending on whether it is possible to idealise the structure as made by elements whose behaviour is known or not (Table 2). In the first group there are the methods that operate on the structure considered as an assemblage of simple structural members (beams, columns, plates, etc.) whose individual structural behaviour in terms of nodal stiffness or defomability is fully known and can be expressed through closedform relationships. In the second group there are the methods requiring the structure to be discretised into finite elements, whose response is defined by means of a suitable numerical idealisation.
a) Rigid-Plastic
d) Rigid-Hardening
b) Elastic-Plastic
e) Elastic-Hardening
c) Inelastic-Plastic
f) Generically Inelastic
Figure 2: Material behavioural models for plastic analysis according to EC9 TABLE2 METHODS OF PLASTIC GLOBAL ANALYSIS CONSIDERED IN EC9
Concentrated plasticity methods Rigid-perfectly plastic analysis; Elastic-perfectly plastic analysis; Rigid-hardening analysis; Elastic hardening analysis.
Spread plasticity methods Non linear elastic analysis; Inelastic-perfectly plastic analysis; Generically inelastic analysis.
In the first group the plastic strains are considered as concentrated in single sections (end sections, loaded sections, changes of cross-section, etc.) in the form of plastic hinge. Within two of these sections, the behaviour remains perfectly elastic or rigid, depending on the material behaviour assumptions. The methods of the second group allow for the actual inelastic behaviour of the structure to be taken into account, with a degree of accuracy that increases as far as the degree of discretisation increases. In particular, the generically inelastic method is considered the only actually rigorous approach for the description of aluminium alloy structures in postelastic range and, for this reason, it is considered as a reference method with regard to the evaluation of both load bearing capacity and ductility demand.
THE METHOD OF PLASTIC HINGES The problem of how to apply a concentrated plasticity approach based on the plastic hinge method is faced in Annex D of EC9. This part of EC9 is concemed with Class 1 cross-sections, only. The procedure is mainly
461 based on the result of a wide research work started in the mid Seventies at the University of Naples and recently developed with new advances aiming to put the existing knowledge into a form suitable for codification (Mandara & Mazzolani, 1995, Mandara, 1995). The suggested procedure still relies upon the concept of concentrated plasticity but is based on the use of a correction factor for the yield stress assumed in the plastic hinge method. The amount of this correction is fitted in such a way to take into account both ductility and hardening features of the material at the same time. According to EC9, the plastic hinge method can be applied provided that the structural ductility is sufficient to enable the development of the full plastic mechanism. For generical truss- or beam-made structures, EC9 indicates the following conventional ways to evaluate the ductility demand (Figure 3): 1) The ductility demand is defined as the required rotation in the most developed plastic hinge when the plastic mechanism is attained (see line a) of Figure 3). The structure is solved by means of a concentrated plasticity approach based on the concept of plastic hinge. The maximum required strain can be evaluated if a convenient length for the plastic hinge is assumed. 2) The ductility demand is defined as the required rotation in the most developed plastic hinge evaluated when the plastic hinge idealisation provides the same load bearing capacity as predicted by a more accurate, inelastic method of analysis based on a discretised model (see line b) of Figure 3). 3) The ductility demand is not evaluated on the basis of the structural response, but it is defined a priori as a function of the elastic limit strain of the alloy (see lines c) of Figure 3). The corresponding load bearing capacity can be evaluated in a simpler way by applying the plastic hinge method in which a modified value of the conventional yield stress is adopted, in order to take into account the actual behaviour of the alloy in terms of both available ductility and strain hardening. Discretised metliod l\/letliod of plastic hinges
Figure 3: Evaluation of ductility demand Of the three methods suggested by EC9, the first one is purely conventional, since is based on a concentrated plasticity idealisation, which hardly corresponds to the actual structural behaviour at collapse. The second one requires for the structure to be calculated twice, with a concentrated plasticity approach, as well as with a F.E.M. numerical simulation. For this reason, it would result in a hi^er computation cost, which is not suitable for practical applications. Eventually, the third one could represent a good method for an accurate inelastic analysis of aluminium alloy structures, without disregarding the actual mechanical features of the material. Furthermore, because of its inherent simplicity, it can be profitably used as a design method for structures at the ultimate limit state. In fact, from the practical point of view, it is quite identical with the classical plastic hinge method applied for steel structures. Furthermore, allowance for the strain hardening effect of tile material is also made.
462 For the application of the plastic hinge method, the conventional yield stress^ to be used in the analysis is corrected and is expressed as follows: fy = Tlfo.2
if
Tlfo.2
fy = ft/'>k
if
r|fo.2>ft/7M
(2)
where/ is the material ultimate tensile stress and r| is a correction factor depending on the geometrical shape factor oco, as well as on the conventional available ductility of material; JM is the partial safety factor. The plastic hinge method may be applied by assuming either the elastic (rigid)-perfectly plastic or the elastic (rigid)-hardening behaviour for the material. In the first case, the ultimate section moment is assumed given by: (3)
Mu=CXoTlfo.2W
W being the cross-section resistance modulus. In the second case, a conventional yielding moment, corresponding to the starting of the strain hardening behaviour, is defined as: (4)
My=aoTifo.2W The section ultimate moment can be calculated through the equation: Mu=a^r|fo.2W
(5)
^ being equal to 5 or 10 depending on the available ductility of the alloy, as and aio are the so-called generalised shape factors, depending on the assumed limit curvature Xu-
1-5 T
h
1.4 \ ^ ' rXu=5%e %u=10Xe %ii=5Xe %u= lOXel 1.3 h \ X Q = 1 . 4 - 1 . 5 ao = 1.4-1.5 a o = l . 1 - 1 . 2 a o = l . 1 - 1 . 2 [ N \
L
1.2 •\ ^\ \ \ ks. 1.1 \ ** •••->-..
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
bu<150:0,9 bi,>150:0,7
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
1-160x80-01/Mi
1-160x80-02/Mi
1-160x80-01/ Mi
618
413
208
614
422
204
1.007
1.022
0.981
I-I60x80-C1/Minor (measurement point 2) 700 1
y^p*'*''''''^^
600
Ir
500 400 O O U.
300 1 f 200
-*- Numerical Analysis, Residual Stresses Neglected
[
-
100
-5
{)
5 Displacement [mm]
Figure 7:
Deformed shape and horizontal deflection of specimen I-160x80-Cl/Minor at mid-span
10
15
509 PARAMETRER STUDIES AND COMPARISON WITH ENV 1993-1.4 After the Finite Element model had been approved, parameter studies were carried out to investigate the influence of the variation of parameters and thus generate "electronic tests results". Bending Moment Resistance Table 4 gives the ultimate bending moment resistance Mu from these calculations versus various b/t-ratios. Figure 8 gives a comparison with the limits specified in ENV 1993-1.4. TABLE 4 CALCULATIVE RESISTANCES FOR VARIOUS PARAMETERS cross section
Mp, (ENV 1993-1.4)
(Numerical)
[kNm]
[kNm]
M,
d/tw
c/t,
1-160x80
22.3
3.4
46.2
57.1
1.23
1-160x120
22.3
5.4
64.7
74.7
1.15
Mp,
1-160x140
22.3
6.4
81.1
74.0
1.09
1-160x160
22.3
7.4
83.2
88.2
1.06
1-320x160
49.0
7.4
195.0
215.0
1.10
1-320x185
49.0
8.65
218.9
232.9
1.06
1-320x200
49.0
9.4
233.3
238.4
1.02
Mu/Mp,(EC3) l.*f •
1
1
A d/tw=22.3; 1.4301 A
Od/tw=49;
A O A
A
<:
1.4301 H
O
Cla: sification line a :cording to ENV 1993-1.4
M., Mpi
CI.2 1
Class 1
0.2-
3
Figure 8:
4
5
6
7 C/l
1 8
c ass 4
Class 3
9
^*>>s.
10
Comparison of b/t-ratios for outstanding flanges in compression with £A^V 799J-7.4
11
510 Lateral-torsional Buckling of Beams In table 5 results are given for the ultimate resistance of beams loaded in lateral-torsional buckling with the relative slendemess being the parameter. Figure 9 shows the results together with buckling curve (d) as specified in ENV 1993-1.4. TABLE 5 CALCULATIVE RESISTANCES FOR VARIOUS PARAMETERS total length [mm]
cross section
(ENV 1993-1.4)
rel.'^LT
1024
1-160x80
[kNm]
(Numerical) [kNm]
Mu(FE) Mu(EC 3)
0.30
42.5
46.6
1.097
1500
1-160x80
0.50
36.0
40.0
1.111
2000
1-160x80
0.69
30.0
37.1
1.237
2522
1-160x80
0.86
24.9
31.7
1.273
2520
1-160x160
0.44
66.5
77.4
1.164
3000
1-160x160
0.52
62.0
74.7
1.205
Lateral torsional buckling
^ A
1-
^
0.5-
- , ' • - . -
\ ; - ^-.:;./ • ^ ^ v
Figure 9:
J
n 1-160x160
A^
\ " /^ \"^
0,2 '
1 1
A 1-80x160
L°°
^<^V
X
()
1
1
0.5
1
y _-:><..
/ b
c
/^
"-
1.5 ret.ALT
- c i ^ iiii:^
2
-^'^g.---.
^ i-^-*--.
2.5
Comparison of calculative results with the relevant buckling curve (d) of ^A^V 7995-7.^
:J
511 Buckling of columns about the weak axis In table 6 the numerical results for columns in buckling about the weak axis are given. Figure 10 shows in what way these results fit to the buckling curve specified in ENV 1993-1.4 for column buckling about the weak axis. TABLE 6 CALCULATIVE RESISTANCES FOR VARIOUS PARAMETERS total length [mm]
Nu (ENV 1993-1.4)
Nu (Numerical)
[kNm]
[kNm]
Nu(FE)
cross section
rel.'^B
1-160x80
0.42
609.2
614.0
1.008
1000
1-160x80
0.65
492.5
530.9
1.078
1248
1-160x80
0.82
415.8
421.8
1.014
1700
1-160x80
1.11
302.8
267.1
0.882
2046
1-160x80
1.34
235.6
204.3
0.867
650
Nu(EC 3)
1=lexural buckling of welded l-sectlons (weak axis)
1 11 A 1-80x160
1 •
Ni"< •^'N
X
K~
\"--.
/
V"N 0.5"
S^^^N
A'^
X ~ -. """'
\ : ; .--:.x^ A
(^)
0.2
• 0.5
1
1.5
reJAe
a
y-^
^^
y c
d
""""^i - i : ^ ii:£i£; ^•..-.. 2
2.5
:3
Figure 10: Comparison of calculative results with the relevant buckling curve (d) of ENV 1993-1.4
CONCLUSION It could be demonstrated that testing for code-purpose can be reduced to pilote-testing only as the influence of parameter variations can be easily checked by reliable Finite Element simulations that result in "electronic test results". From parameter studies carried out with the "electronic test generator" comparisons with the design rules in ENV 1993-1.4 could be performed that will allow in a next step to modify the design rules in ENV 1993-1.4 where appropriate.
512 REFERENCES /I/
Eurocode 3: Design of steel structures - Part 1-4: General Rules - Supplementary rules for stainless steel
111
VTT Building Technology (1997): Test report on welded I and CHS beams, columns and beam-columns, report to the ECSC
/3/
MARC Analysis Research Corporation, Sheridan Avenue, Suite 309, Palo Alto ,CA 94306 USA: MARC Version K7 (1998)
Poster Session P4 STRUCTURES AT AMBIENT AND ELEVATED TEMPERATURES
This Page Intentionally Left Blank
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
515
EC9 PROVISIONS FOR FLAT INTERNAL ELEMENTS: COMPARISON WITH EXPERIMENTAL RESULTS R. Landolfol, V. Piluso^, M. Langseth^, O.S. Hopperstad^ 1 DSSAR, Faculty of Architecture, University of Chieti, Italy 2 Department of Civil Engineering, University of Salerno, Italy 3 Department of Structural Engineering, Norwegian University of Science and Technology, Trondheim, Norway
ABSTRACT An experimental investigation was undertaken to study the ultimate strength of aluminium alloy flat internal elements. Thin walled square hollow section and rectangular hollow section profiles were stressed into the post-buckling range under axial compression. The local buckling of the plate elements constituting the member section was investigated considering the restraining action deriving from their interaction. Different alloys and different profiles covering the practical range of variability of the width-to-thickness ratios were considered. The experimental ultimate strengths were compared with the effective thickness approach, recently introduced in Eurocode 9. The results presented in this paper and their comparison with the experimental evidence show the degree of accuracy of the codified approach.
KEYWORDS Local Buckling, Stub Column Tests, Aluminium Box Sections, Eurocode 9, Effective Thickness
1. INTRODUCTION The plastic deformation capacity and the ultimate resistance of aluminium alloy members are strictly dependent on the width-to-thickness ratios of the plate elements constituting the member section and the strain hardening of the material. Depending on the magnitude of these local slenderness parameters, local buckling can develop in elastic or in plastic range. As a consequence, the ultimate resistance has to be evaluated accordingly. Eurocode 9 on aluminium structures divides cross-sections into four behavioural classes. Sections belonging to the first class, also namely ductile sections, are able to develop their whole plastic resistance with high plastic deformation capacity. The whole plastic resistance can be also attained in the case of second class sections, also namely compact sections, but with a limited plastic deformation capacity. In the case of third class or semi-compact sections, local buckling occurs in
516 plastic range, but before the complete development of their plastic reserves. These sections are able to completely exploit their resistance reserves in elastic range. Finally, fourth class sections are characterized by high values of the width-to-thickness ratios so that local buckling occurs in elastic range. They do not exhibit any plastic deformation capacity and, in addition, are not able to completely exploit their elastic reserves. For this reason, these sections are often referred to as slender sections. An experimental research dealing with the influence of the width-to-thickness ratios on the local buckling of aluminium alloy RHS (rectangular hollow sections) and SHS (square hollow sections) members has been recently carried out both at the University of Salerno (Italy) and at the Norwegian University of Science and Technology in Trondheim (Norways). Thanks to this cooperation, a great number of tests on aluminium alloy members subjected to local buckling under uniform compression has been performed. This has allowed to cover all the range of variation of the widthto-thickness ratios of flat internal elements constituting box sections. This activity has been developed under the auspicious support of CEN-TC250/SC9 (chaired by Prof. Mazzolani of Naples University), i.e. the european committee charged of the preparation of Eurocode 9 on "Aluminium Structures" (ENV 1999-1.1, 1998), with the aim to provide the code with the background data necessary for setting up a classification criterion specific of aluminium members and for calibrating the effective thickness approach which has been adopted in Eurocode 9 for computing the resistance of aluminium sections. In this paper, the main results of the stub column tests carried out in Salerno and Trondheim are presented. In addition, the effective thickness approach is briefly summarized and is successively applied for computing the resistance of box sections subjected to local buckling under uniform compression. The comparison between the numerical results and the experimental evidence allows the evaluation of the accuracy of the codified approach.
2. CROSS SECTIONAL AND MATERIAL PROPERTIES OF TESTED PROFILES The whole experimental program deals with SHS and RHS members made of 6000 and 7000 series alloys. In addition, different tempers have been considered. With reference to the profiles used for specimen preparation, the nominal geometrical properties and the measured ones are given in Table 1, where SHS members and two kinds of RHS members can be recognized (RHS with sharp corners and RHS with rounded corners). In this table, Onom and hu om are the nominal dimensions of the large and small side, respectively, of the hollow section. The corresponding thicknesses are denoted with ta and th, respectively. In addition, re and n are, respectively, the external and the internal radius of the corners when rounded. An om is the nominal area of the member section. RHS27 profile has an intermediate plate element stiffening the largest side of the cross-section. In the same table, the measured geometrical properties are given. These are the actual dimensions a and b of the sides and the corresponding thicknesses r^, r^ and r^, r^. As result of the measured geometrical properties, the actual area A of the cross-section is obtained. Tensile tests have been carried out to evaluate the material properties. The measured values of the elastic modulus E, the conventional elastic limits /02 and /Q 1 the ultimate strength /^ and the exponent n of the Ramberg-Osgood law are presented in Table 2. As usual, the exponent n has been calibrated according to the values of /QI and /02 (Mazzolani, 1995). The evaluation of the ultimate strain e^ has not been carried out, but it has been roughly observed that the nominal value has been always exceeded.
517 3. STUB COLUMN TESTS The compression tests have been performed in Salerno (Mazzolani et al., 1998) with a Schenck RBS4000-E2 testing machine (maximum load 4000 kN, piston stroke ± 100 mm) and in Trondheim (Hopperstad et al., 1998) with a Dartec 500 kN testing machine (accuracy ± 1% of 500 kN). For each profile, a minimum of two stub column tests have been carried out under displacement control. The axial displacements have been measured by means of three inductive displacement trasducers (stroke ± 10 mm, sensitivity 80 mV/W). The mean value of the three measures has been considered. Neither top nor base rotation of specimens has been observed, because, as expected, all specimens have failed due to pure local buckling without any coupling phenomenon. The test results are summarized in Table 3, where the specimen height and the experimental value of the ultimate resistance A^^^p are given. 4. THE EC9 APPROACH 4.1 Definition of slender sections The classification of cross-sections has been done on the basis of experimental results, which belong to 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 slenderness 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. Accounting for the examined behaviour of tested specimens, the following classification is therefore proposed (Mazzolani et al., 1996): Class l : p / e < 11 Class 2: l l < p / e < 1 6 Class 3: 1 6 < p / e < 2 1 Class4:p/e>21 being P = b/t and e = /250 7 " ^ . The above classes are characterised by the following behaviour. Class 1 sections buckle in plastic range developing at least the ultimate tensile resistance and the ductility of material. Class 2 sections buckle in plastic range developing at least the proof strength/0.2 of the material. In addition, they are able to withstand quite large plastic deformations. Class 3 sections, as well as class 2 sections, are able to reach the proof strength/a2, but they have limited plastic deformation capacity. Finally, class 4 sections are not able to develop the proof resistance/a2 and prematurely fail due to the occurrence of local buckling in elastic range. 4.2 The design curves In order to assess design curves for slender sections, distinction is made between non heat-treated (NHT) and heat-treated (HT) alloys as well as between welded and unwelded profiles. According to this distinction, three curves (A, B and C) have been assumed as design curves in EC9 for welded or unwelded internal and outstand elements (Fig. 1). First of all, it can be observed that the buckling curves are provided as function of the factor p/e according to BS8118 method. Besides, the ones related to outstand non-symmetrical elements are limited by the branch pc=(p/£)~^, which corresponds to their critical stress. In every case, the effect of local buckling in slender members is allowed for by replacing the true section by an effective one, which is obtained by using a local buckling coefficient pc to factor down
518 the thickness of any slender element that is wholly or partly in compression:
where j3 is a slenderness parameter that depends both on the b/t ratio and the stress gradient for the element concerned and £ = (250/fo2f'^\ ^1 and 82 are two numerical coefficients whose values are reported in Table 4 together with the limit value of the p/e corresponding to Pc=l-
[ r
® unwelded heat-treated (HT) (D welded HT and unwelded non HT Ns^ © welded non heat-treated \\V
r ® - \ ^ \ ^ ^ \ ^ ^ ^ INTERNAL ELEMENTS ^ V . , ^ ^ ^ v ^ | \ ^ N D ROUND TUBES
§ 600
P 400
UN-SYMMETRICAL^ ^^^-^ OUTSTANDS
SYMMETRICAL OUTSTANDS
200
400
600
800
EXPERIMENTAL VALUE (kN)
Fig. 1
Fig. 2 TABLE 4
NUMERICAL COEFHCIENTS FOR EC9 DESIGN CURVES
FCURVE A B
1 .c
61 32 29 25
Internal elements 61 (P/e)o 22 220 198 18 150 15
61 10 9 8
Outstand elements 62 (P/e)o 24 6 5 20 4 16
|
1
1
The influence of welding is estimated by accounting globally all the effects in a higher value of the generalised imperfection factor, which reduces the buckling curves respect to non welded profiles. Such a reduction is practically the same for both NHT and HT alloys and is not substantially affected by the plate slenderness parameter.
5. COMPARISON WITH EXPERIMENTAL EVIDENCE The ultimate axial resistance of the tested specimens has been evaluated by means of the effective thickness approach as codified in Eurocode 9. The computations have been carried out by considering the measured geometrical properties given in Table 1 and the measured proof stress fo.2 of the material, reported in Table 2. Regarding the influence of the strain hardening, T4 temper has been considered as a non heat-treated alloy (curve B), while all other specimens are heat-treated (Curve A). The comparison between experimental and theoretical results is depicted in Fig.2. In addition, in Table 3 the corresponding predicted values for each specimen are reported. Such a comparison emphasises the reliability of EC9 approach in predicting the axial resistance of aluminium box sections, leading to an average value of the ratio NECQ/ Nu,exp, between the numerical prediction and the corresponding experimental value of the ultimate axial resistance, equal to 0.96
519 with a standard deviation equal to 0.10. This means that the design bukling curves suggested by EC9 are able to reflect the actual behaviour of flat internal elements, provided that the temper is properly accounted for. Under this point of view, the experimental evidence shows that alloys in T4 temper behaves as a non-heat-treated materials. Therefore, the type of temper should be explicitly considered in assessing the design buckling curves in EC9.
6. CONCLUSIONS The results of a wide experimental progi:am dealing with box sections subjected to local buckling under uniform compression have been presented and compared with the numerical prediction provided by the effective thickness approach codified in EC9. This comparison has evidenced a satisfactory degree of accurancy of the design buckling curves provided that the strain hardening properties of the alloy are properly accounted for.
7. REFERENCES prENV 1999-1.1, (1998), Eurocode 9: Design of Aluminium Alloy Structures - part 1.1, European Committee for Standardisation Ghersi, A. and Landolfo, R., (1996), Thin-walled sections in round-house type material: a simulation model. Costruzioni Metalliche n.6, pp. 17-27. Hopperstad, O.S., Langseth, M. and Moen, L., (1998), Strength of Aluminium Extrusions under Compression and Bending, in Thin-Walled Structures. Research and Development, N.E. Shanmugam et al (eds.), Elsevier, pp. 85-92. Landolfo, R. and Mazzolani, F.M., (1997), Different approaches in the design of slender alluminium alloy sections, Thin-Walled Structures, Elsevier Science Limited, Vol.17, No. 1, pp.85-102. Landolfo, R. and Mazzolani, P.M., (1998) The background of EC9 design curves for slender sections (pubblication in honour of Prof. J. Lindner, Berlin. Langseth, M. and Hopperstad, O.S., (1995), Local Buckling of Square Thin-Walled Aluminium Extrusions, ICSAS '95, Proceedings of the Third International Conference on Steel and Aluminium Structures, Istambul, May Mazzolani, P.M., (1995), Aluminium Alloy Structures, E&PN Spon, an Imprint of Chapman & Hall Mazzolani, P.M., Paella, C , Piluso, V. and Rizzano, G., (1996), Experimental Analysis of Aluminium Alloy SHS-Members Subjected to Local Buckling Under Uniform Compression, 5th International Colloquium on Structural Stability, SSRC, Brazilian Session, Rio de Janeiro, August 5-7 Mazzolani, P.M., Paella, C , Piluso, V. and Rizzano, G., (1998), Local Buckling of Aluminium Members:Experimental Analysis and Cross-Section Classification, Department of Civil Engineering, University of Salerno.
520 TABLE 1 GEOMETRICAL PROPERTIES OF TESTED SPECIMENS A a b tl (mm) (mm) (mm) (mm) (mm) (mm) (mm^) (mm) (mm) (mm) (mm) (mm) (mm) SHSl 15 15 2 2 1.9 104 15.0 1.9 1.9 1.9 15.0 SHS2 40 40 4 4 576 40.1 40.05 4.05 4.15 4.10 4.05 SHS3 50 50 3 3 564 3.1 3.1 3.05 3.15 51.15 50.0 SHS4 4 4 50 50 50.4 50.35 4.45 4.1 4.2 4.3 736 SHS5 4 4 4.2 70 70 4.1 4.2 4.0 1056 70.15 70.1 4.2 80 80 4 4 4.2 SHS6 4.3 4.35 1216 79.9 79.85 SHS7 4 4 100 100 4.0 3.95 3.85 100.0 99.8 3.9 1536 2 2 SHS8 60 60 464 60.4 60.35 2.3 2.25 2.25 2.25 SHS9 2 2 2.1 2.1 2.1 2.0 80 80 624 80.4 80.2 100 6 6 6.0 6.1 SHSIO 100 2256 100.3 99.9 6.0 5.95 4.7 SHSll 150 150 5 5 150.2 150.1 5.2 4.75 5.0 2900 I SHS12 150 150 5 5 5.2 5.0 5.25 2900 149.9 149.9 5.0 RHSl 34 3 3.0 20 3 3.0 3.0 3.0 288 34.0 20.0 RHS2 30 4 4 40 4.1 4.0 496 3.9 4.0 39.9 29.9 4 4 RHS3 50 20 4.2 4.4 3.9 3.9 496 50.1 20.0 -. RHS4 50 30 3 3 444 2.9 50.0 30.25 3.0 3.05 3.25 3 2.8 RHS5 50 40 3 2.6 2.6 425 50.25 40.3 2.8 34 60 3 34.1 3.0 3.0 RHS6 3 60.2 3.0 3.0 528 RHS7 60 40 3 2.5 3 564 60.2 2.6 2.5 2.6 40.1 40 4 4 RHS8 80 3.9 896 4.0 4.0 3.9 80.25 40.1 4 RHS9 40 4 4.0 3.9 100 4.0 4.0 1056 99.80 40.1 4 4 RHSIO 120 50 4.3 4.25 120.3 50.6 4.15 4.15 1296 4 4.1 RHSll 150 40 4 4.1 4.1 4.0 1456 150.5 40.8 4.3 4 4 4.2 4.2 3.9 RHSI2 180 40 1696 181.1 40.8 4 4 4.0 RHS13 100 50 4.0 3.9 1136 100.1 50.25 3.9 RHS14 2 2.1 60 40 2 384 2.3 2.0 60.1 40.1 2.0 4 4 3.9 RHS15 80 40 4.0 3.9 3.9 896 79.9 40.0 2 2.0 RHS16 80 40 2 464 1.85 2.15 80.2 40.25 2.3 RHS17 60 40 2 2 2.0 2.0 384 40.0 2.0 1.90 59.9 2.3 2.3 100 25 2 2 484 2.3 2.3 RHS18 100.3 25.7 2.6 RHS19 120 60 2.5 2.5 2.7 2.6 875 119.9 61.0 2.7 RHS20 200 100 5 5 5.0 4.7 4.8 5.0 2900 200.0 99.9 RHS21 47 40 2.5 2.5 2.95 2.95 2.8 410 47.0 40.0 2.8 RHS22 180 70 4.5 4.5 4.5 4.65 4.65 4.65 2169 179.5 70.0 RHS23 153 70 4.5 6.5 6.0 3.0 2146.82 153.0 71.6 4.35 5.35 6.85 6.85 RHS24 200 9.3 180 15.1 9 17.0 6.0 8519.22 200.5 179.2 15.3 15.3 9.15 RHS25 120 100 4.5 6.5 15.0 10,0 2155.70 120.5 100.35 4.6 6.7 6.85 4.9 RHS26 200 180 6 6.2 6.5 5.7 6 17.0 11.0 4271.79 201.0 181.5 6.0 RHS27 219 68 4.5 6 3.0 4.4 6.0 6.0 1.5 2940.64 219.8 68.0 5.0 SHSOIC 100 100 6 6 4.0 1.5 100.4 99.3 5.99 5.90 5.95 5.91 SHS02C 100 100 6 6 4.0 1.5 100.4 99.1 5.98 5.86 5.93 5.90 SHS03C 100 100 6 6 4.0 1.5 100.1 99.9 5.91 6.04 5.89 5.94 4 100 4 SHS04C 100 5.0 1.5 102.6 102.3 3.76 3.91 3.63 3.62 3 SHS05C 100 100 3 5.0 2.0 99.5 99.5 2.87 2.86 2.84 2.80 SHS06C 100 100 3 3 5.0 2.0 99.9 98.9 2.78 2.82 2.85 2.84 100 2 2 SHS07C 100 99.5 99.5 1.95 1.96 1.91 1.95 100 2 2 SHS08C 100 99.6 99.6 1.90 1.96 1.98 1.98 100 SHS09C 100 1.5 1.5 100.4 100.4 1.48 1.42 1.45 1.44 SHSIOC 80 80 2.5 2.5 80.1 80.1 2.46 2.50 2.46 2.48 2 2 80.2 2.02 2.05 2.05 2.06 S H S l l C 80 80 80.3 RHSOIC 120 60 2.5 2.5 2.60 2.30 2.11 2.89 119.5 60.8 3 |RHS02C 100 60 3 3.0 1.0 2.99 2.89 2.93 2.89 99.1 61.0 -
A 1 (mm^) 99.56 588.4 582.49 786.22 1089.0 1289.19 1506.82 525.92 649.28 2262.78 2853.95 2960.9 288.0 494.4 504.6 451.3 459.8 529.8 485.52 892.38 1051.99 1360.16 1498.16 1787.76 1125.36 405.21 881.59 482.65 378.01 558.44 936.58 2823.96 467.19 2208.33 2308.95 8659.79 2268.47 4373.45 3070.18 2218 2214 2222 1457 1073 1074 761 761 570 766 641 866 908
521
ALLOY SHSl SHS2 SHS3 SHS4 SHS5 SHS6 SHS7 SHS8 SHS9 SHSIO SHSl I SHSl 2 RHSl RHS2 RHS3 RHS4 RHS5 RHS6 RHS7 RHS8 RHS9 RHSIO RHSll RHS12 RHSl 3 RHSl 4 RHSl 5 RHSl 6 RHSl 7 RHSl 8 RHS19 RHS20 RHS21 RHS22 RHS23 RHS24 RHS25 RHS26 RHS27 SHSOIC SHS02C SHS03C SHS04C SHS05C SHS06C SHS07C SHS08C SHS09C SHSIOC SHSllC RHSOIC
1 RHS02C
6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 6060 6060 6060 T6 6082 6060 6060 6082 6082 T6 6082 TF 6082 T6 6082 TF 6082 6061 T6 6082 T6 6082 T4 7108 T7 7108 T7 6082 T6 6082 T4 6082 T6 6082 T4 6082 T4 6082 T4 6082 T4 6082 T6 6082 T6
TABLE 2 MATERIAL PROPERTIES OF TESTED SPECIMENS COUNTRY E /o.i fo.2 D D D D D D D I I D D N D D D D D D D D D D D D D F F F I N N N N N GB GB GB GB GB N N N N N N N N N N N N N
(N/mm2) 67520 72265 64863 64090 70211 71733 70757 71963 65125 65321 75250 68368 62814 69750 68439 70873 69695 77760 62761 63508 70203 68945 68796 74543 68504 62446 69329 60000 69263 68037 69318 65234 67488 72038 71850 71360 68841 71601 69049 70129 72944 68155 70534 69013 71323 68325 72484 70609 74765 69083 68296 65648
(N/mm2) 214.4 223.6 222.5 202.6 175.7 194.2 209.8 158.2 186.7 293.5 208.9 258.4 218.7 202.0 210.7 217.4 221.6 212.5 234.6 222.0 216.6 215.8 224.6 212.3 216.0 219.6 188.9 225.4 234.3 264.8 209.7 235.2 251.3 320.0 309.2 340.0 323.0 132.7 297.9 312.1 178.7 311.2 306.6 197.9 168.3 252.3 135^7 1^39.3 111.4 138.5 245.2 283.2
n
(N/mm2) 207.8 215.6 217.2 198.1 169.9 189.2 204.8 149.3 182.1 286.0 186.5 245.5 212.4 197.5 205.3 209.8 218.4 204.8 229.4 216.3 213.3 209.8 213.4 204.6 211.5 215.2 184.0 222.4 230.0 258.3 205.5 224.2 246.0 317.3 306.7 336.9 315.7 131.6 294.1
22.4 19.1 28.9 30.6 20.6 26.8 28.4 12.0 27.5 26.9 11.3 13.4 23.6 31.1 26.5 19.5 48.4 18.6 31.3 26.6 45.2 24.7 13.5 18.7 33.3 34.7 26.8 53.0 37.5 27.8 33.7 14.6 32.9 83.7 90.7 77.4 30.3 84.4 53.2
— — — — — — — — — — — — —
— — — — — — — — — — — — —
(N/mm2) 241.3 244.3 244.8 225.2 202.9 220.3 228.3 186.6 203.9 323.7 252.1 300.1 250.9 214.3 233.3 242.5 244.5 235.0 258.9 258.6 242.2 227.3 255.5 246.8 236.6 242.8 212.4 260.5 253.3 285.0 229.4 282.8 276.9 353.4 329.9 362.1 342.8 184.8 325.0 325.2 292.7 352.1 349.4 240.1 282.5 281.4 242.6 263.3 226.2 259.1 282.8
290.2
1
522 TABLE 3: EXPERIMENTAL RESULTS AND COMPARISON WITH EC9 h SHSOl SHS02 SHS03 SHS04 SHS05 SHS06 SHS07 SHS08 SHS09 SHSIO SHSll SHS12 RHSOI RHS02 RHS03 RHS04 RHS05 RHS06 RHS07 RHS08 RHS09 RHSIO RHSll RHS12 RHS13 RHS14 RHS15 RHS16 RHS17 RHS18 RHS19 RHS20 RHS21 RHS22 RHS23 RHS24 RHS25 RHS26 RHS27 SHSOl C* SHS02C* SHS03C* SHS04C* SHS05C* SHS06C* SHS07C* SHS08C* SHS09C* SHSIOC* SHSllC* RHSOIC*
(mm) 44.50 120.00 149.40 149.40 209.50 239.00 296.00 179.00 239.50 303.00 437.00 451.50 46.80 73.60 52.30 80.70 104.5 88.80 179.50 234.50 236.00 361.00 225.00 242.00 299.00 191.00 234.50 239.00 180.00 125.00 359.00 601.00 140.00 540.00 411.00 531.20 361.00 601.00 508.00 376.30 376.90 376.00 383.80 387.70 388.00 392.00 392.20 393.80 310.00 310.10 470.50 387.90
SPECIMEN A NEC9(kN) Nexp (kN) 21.35 30.6 131.57 160.8 129.60 132.4 159.29 186.6 191.34 213.8 250.36 264.4 316.13 300.2 82.7 83.20 96.49 84.7 664.13 728.5 540.47 605.5 666.99 626.5 62.99 78.7 99.87 124.3 106.32 134.8 98.12 109.8 101.89 108.5 112.58 122.4 113.90 120.6 198.11 212.0 227.86 222.6 279.03 271.2 275.92 290.8 284.01 313.2 248.1 243.08 85.95 85.1 166.53 185.7 90.45 92.5 81.81 89.4 104.43 92.7 154.47 137.7 529.50 513.5 117.40 115.3 525.37 493.2 625.85 621.5 2944.33 2939.0 704.72 669.0 576.22 865.0 894.43 831.0' 696.12 726.2 396.90 470.1 696.04 733.6 390.97 408.2 182.58 190.4 148.52 147.3 108.25 102.8 68.28 67.6 40.37 38.2 81.37 81.9 70.35 70.4 152.83 146.8 220.16 236.3
NECQ/NCXP
0.6976 0.8182 0.9789 0.8536 0.8949 0.9469 1.0531 1.0061 1.1392 0.9116 0.8926 1.0646 0.8003 0.8034 0.7887 0.8936 0.9391 0.9198 0.9445 0.9345 1.0236 1.0289 0.9488 0.9068 0.9798 1.0100 0.8968 0.9779 0.9151 1.1265 1.1218 1.0312 1.0182 1.0652 1.0070 1.0018 1.0534 0.6662 1.0763 0.9586 0.8443 0.9488 0.9556 0.9590 1.0083 1.0530 1.0101 1.0568 0.9935 0.9993 1.0411 0.9317
1 RHS02C* Experimental results are mean values of three parallel tests.
h (mm) 46.00 115.70 149.40 149.30 209.50 239.00 299.00 179.00 239.0 303.00 451.00 452.00 46.80 120.40 52.30 149.00 211.00 179.00 176.00 233.50 236.00 361.00 224.50 237.00 298.00 176.00 233.90 238.00 178.00 127.00 355.00 601.00 141.00 540.00 411.50 535.00 361.00 601.00 508.00
— — — — — — — — — — — — —
SPECIMEN B NEC9(kN) Nexp (kN) 29.7 21.35 158.4 131.57 129.60 131.3 159.29 180.9 191.34 208.7 250.36 263.8 304.8 316.13 83.3 83.20 84.7 96.49 664.13 731.5 540.47 592.5 666.99 643.5 62.99 77.5 122.4 99.87 106.32 136.8 109.2 98.12 109.1 101.89 112.58 122.9 113.90 118.7 198.11 212.0 227.86 224.9 279.03 255.6 275.92 261.2 284.01 315.6 248.2 243.08 85.95 79.7 185.2 166.53 90.45 92.8 88.6 81.81 89.4 104.43 139.6 154.47 529.50 506.5 117.40 116.5 525.37 497.0 612.0 625.85 2944.33 2934.0 704.72 670.5 576.22 852.0 808.5 894.43
— — — — — — — — — — — — —
— — — — — — — — — — — — —
1 NEC9/Nexp 1 0.7187 0.8306 0.9871 0.8805 0.9168 0.9491 1.0372 0.9988 1.1392 0.9079 0.9122 1.0365 0.8127 0.8159 0.7772 0.8985 0.9339 0.9160 0.9596 0.9345 1.0236 1.0917 1.0563 0.8999 0.9794 1.0785 0.8992 0.9747 0.9233 1.1681 1.1065 1.0454 1.0078 1.0571 1.0226 1.0035 1.0510 0.6763 1.1063
— — —
j
— — — 1 — — — — — — —
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
523
STAINLESS STEEL COMPRESSION MEMBERS EXPOSED TO FIRE Tiina Ala-Outinen VTT Building Technology, P.O. Box 1807, FIN-02044 VTT, Finland
ABSTRACT There is increasing interest in using stainless steel in buildings as load-bearing structures because of their corrosion resistance, ease of maintenance, attractive appearance and low life-cycle costs, hi addition, the fire resistance of austenitic stainless steels is known to be better than that of conventional carbon structural steels. The aim of this study was to find out whether any austenitic stainless steels are available for use in buildings as load-bearing structures without fire protection. Fire resistance tests were performed to develop and verify the calculation method for determining the strength of the structures exposed to fire. Rectangular and circular hollow sections cold-formed from austenitic stainless steel of type EN L4301 and EN 1.4571 were tested. Both concentric and eccentric compression tests were carried out. The use of austenitic stainless steels in load-bearing structures without fire protection is possible when the parametric fire exposure 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 in the case of standard fire exposure might cause overestimation in normal temperature design. KEYWORDS Stainless steel, fire resistance, load-bearing structures, design method, fire tests, unprotected structures INTRODUCTION The purpose of studies performed at VTT Building Structures was to determine whether any austenitic stainless steels are available for use in buildings as load-bearing structures without fire protection. This is a critical issue, as aesthetic considerations often actuate the use of stainless steels in buildings. Eliminating the fire protection of structures would result in lower construction costs, a shorter construction period, more effective interior space utilization, a better working environment and more aesthetic building design. Furthermore, the life-cycle costs of unprotected stainless steel structures are lower than protected conventional carbon steel structures.
524 MECHANICAL MATERIAL PROPERTIES OF STAINLESS STEEL The stress-strain relationships of stainless steel at elevated temperatures are necessary for determining the load-bearing capacity of structures under fire conditions. On the basis of transient-state tests carried out at Helsinki University of Technology at the Laboratory of Steel Structures (Outinen & Makelainen 1997) the strength values of material EN 1.4301 were determined. The transient-state tests were carried out for both virgin sheet and strongly strain-hardened material. Figure 1 shows the reduction factor of 0.2%-proof stress. The reduction factor of the modulus of elasticity at elevated temperatures was determined from stressstrain curves obtained from transient-state (Outinen & Makelainen 1997) and steady-state tensile tests (Ala-Outinen 1996). The exact determination of the modulus of elasticity at elevated temperatures is very difficult, since the proportion limit of austenitic stainless steel is very low. Even the smallest inaccuracy in the measured curves has a very significant influence on the modulus of elasticity and therefore the dispersion in values of the modulus of elasticity determined from measured stress-strain curves is quite remarkable. The reduction factor of elastic modulus in Figure 1 is determined from linear regression analysis of test results. The numerical values of the curves in Figure 1 are shown in Table 1.
0.90.8 o
1 0.6 1 0-5
o •g 0 . 4 a>
^^0.30.2 0.1 0
0
100
200
300
400
500
600
700
800
900
1000
Temperature [0] -Stainless steel E - 0.2%-proof stress, virgin sheet 0.2%-proof stress, cold-formed material
Figure 1: Reduction factor of 0.2%-proof stress (virgin sheet and cold-formed material) and elastic modulus
525 TABLE 1 THE NUMERICAL VALUES OF THE REDUCTION FACTORS OF THE0.2%-PROOF STRESS AND THE MODULUS OF ELASTICITY Temperature [°C]
Reduction factor of the modulus of elasticity
Reduction factor of virgin sheet
Reduction factor of cold-formed material
20
To
To
To
100
0.981
0.739
0.829
200
0.963
0.643
0.777
300
0.944
0.622
0.752
400
0.915
0.553
0.728
500
0.868
0.529
0.701
600
0.795
0.474
0.610
700
0.688
0.399
0.377
800
0.539
0.220
0.171
900
0.339
0.107
0.061
DESIGN RESISTANCE The same formulae are 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 reduction factor % for column buckling is calculated using the buckling curve c (a = 0.49) and the limit slendemess XQ = 0.40. The value of the elastic modulus according to Eurocode 3, Part 1.4 (ENV-1-4 1996) at normal temperature is 200 OOON/mm^. It has been recommended to use a reduced modulus of elasticity of 0.85 E when considering structures in compression (Talja & Salmi 1995). The yield strength/y may be based on the actual value (stub column test or tensile test) or on the nominal value. The reduction factor for the yield strength is determined from transient-state tests. If the strain-hardening is utilized in strength values, the reduction factor for the yield strength is taken as that for cold-formed material (Figure 1). The modified (non-dimensional) slendemess X is determined at the temperature in question. The simple calculation method is valid when the modified slendemess X is below 1.5 at normal temperature. TEST RESULTS OF COMPRESSION TESTS The concentric and eccentric compression tests were performed for hollow sections cold-formed from austenitic stainless steel of types EN 1.4301 and EN 1.4571 (Table 2). Fire resistance tests on steel columns were performed at VTT Building Technology at the Laboratory of Fire Technology. In the tests freedom
526 TABLE 2 TEST SPECIMENS Section
Section number
Material
Modified slendemess
RHS 40x40x4
RHSl
EN 1.4301 (AISI 304) and
A;2O=1.17
RHS 40x40x4 Ti
RHSl
EN1.4571(AISI316Ti)
X20=l.ll
RHS 30x30x3
RHS2
EN 1.4301 (AISI 304)
^20=1.52
CHS 33.7x2.0
CHSl
EN 1.4301 (AISI 304)
X;2o=1.37
The axial load was concentric or eccentric and during the test kept constant. Various levels of load ranging from 8 kN and 125 kN were applied to the columns. The test loads and temperatures at the moment of buckling are summarized in Table 3. The temperatures in Table 3 are mean values of four temperature measurements in mid-column. The columns were protected by rock-wool sheets to prevent the effects of sudden variations in temperature at the start of the test and thus to ensure a uniform temperature rise in the columns. The furnace temperature rose from 20 °C to 300 °C in 3 min and subsequently by 5 °C/min. All the columns were tested in the vertical position. The columns were heated in a model furnace specially built for testing loaded columns and beams. The test furnace is designed to simulate conditions to which a member might be exposed during a fire, i.e. temperatures, structural loads, and heat transfer. It comprises a furnace chamber located within a steel framework. leading was applied by a hydraulic jack of 2 MN capacity located outside and above the furnace chamber (see Figure 2). A steel unit with circulating water was placed between the column and the loading jack. Sideways support of the steel unit was achieved by the furnace roof element.
Water-cooled steel load equipment Support sideways
Figure 2: Test set-up in furnace tests The axial deformation of the test specimen was determined by measuring the displacement of the top of the water-cooled steel unit using transducers. The axial deformations thus included the deformation of the steel unit. The load was controlled and measured using pressure transducers. The temperature of each column was measured at three cross-sections with four thermocouples at each level, one on each
527 side. In Table 3, the reference temperature of the column was measured in the middle of the column because this is the one most critical for buckling. TABLES SUMMARY OF TESTS FOR HOLLOW SECTIONS AT NORMAL TEMPERATURE AND AT ELEVATED TEMPERATURES
Cross-section
Test specimen
Material
Load [kN] CC = concentric EC = eccentric
The end temperature in midcolumn [°C]
RHS 30x30x3
RHS2-N1-CC
EN 1.4301
66/C
20
RHS 30x30x3
RHS2-T1-CC
EN 1.4301
41/C
610
RHS 30x30x3
RHS2-T2-CC
ENL4301
33/C
693
RHS 30x30x3
RHS2-T3-CC
ENL4301
21/C
810
RHS 40x40x4
RHSl-Nl-CC
ENL4301
184/C
20
RHS 40x40x4
RHS1-T2-CC
EN 1.4301
129/C
579
RHS 40x40x4
RHS1-T3-CC
EN 1.4301
114/C
649
RHS 40x40x4
RHS1-T4-CC
EN 1.4301
95/C
710 766
RHS 40x40x4
RHS1-T7-CC
EN 1.4301
75/C
RHS 40x40x4
RHS1-T5-CC
EN 1.4301
55/C
832
RHS 40x40x4
RHSl-Tl-CC
EN 1.4301
45/C
873
RHS 40x40x4
RHSlTi-Nl-CC
EN 1.4301
167/C
20
RHS 40x40x4
RHSlTi-Tl-CC
EN 1.4571
102/C
720
RHS 40x40x4
RHSlTi-T2-CC
EN 1.4571
73/C
834
RHS 40x40x4
RHSlTi-T3-CC
EN 1.4571
63/C
873
CHS 033.7x2
CHSl-Nl-CC
EN 1.4301
43/C
20
CHS 033.7x2
CHSl-Tl-CC
EN 1.4301
26/C
668
CHS 033.7x2
CHS1-T2-CC
EN 1.4301
12/C
850 716
CHS 033.7x2
CHS1-T3-CC
EN 1.4301
20/C
RHS 40x40x4
RHSl-Tl-EC
EN 1.4301
lOOEC
20
RHS 40x40x4
RHSl-Tl-EC
EN 1.4301
55/EC
672
RHS 40x40x4
RHS1-T2-EC
EN 1.4301
31/EC
795
RHS 40x40x4
RHS1-T3-EC
EN 1.4301
18/EC
899
RHS 30x30x3
RHS2-N1-EC
EN 1.4301
37/EC
20
RHS 30x30x3
RHS2-T1-EC
EN 1.4301
25/EC
604
RHS 30x30x3
RHS2-T2-EC
EN 1.4301
16/EC
747
RHS 30x30x3
RHS2-T3-EC
EN 1.4301
9/EC
871
Concentric load In the following (Figure 3), the comparison between the resistance from concentric compression tests and design resistance is performed. In the calculations, the temperature is assumed to be uniformly distributed throughout the cross-section and along the column. The yield strength used (0.2%-proof stress) at normal temperature was determined for cold-formed material with tensile tests. The modulus of elasticity is assumed to be 170 000 N/mm^ (0.85 E) at normal temperature. The load-bearing capacities were calculated with modified slendemess corresponding to the temperature in question. The
528 mechanical properties (modulus of elasticity and yield strength) are reduced at elevated temperatures according to Table 1. In Figure 3, the design values and test values are shown as relative values. The relative values of test values were determined by dividing the test value by design resistance at normal temperature. A comparison of the fire test results with the design resistance values shows that for columns with low slenderness ( ^20 = 1,1 - 1,17), the calculation method underestimates the resistance. All the test results are on the safe side except one (circular hollow section). Furthermore the test results of columns of material EN 1.4571 are notably on the safe side. When the gas temperature is assumed to rise according to the standard time-temperature curve ISO 834 (1975), the gas temperature after 30 minutes of fire exposure is 842 °C, and the temperature of thin walled cross-sections approaches the gas temperature. It can be seen in Figure 3 that a temperature of 842 °C corresponds to load levels 0.22...0.28, and that according to the test results the load level is higher. In particular, for RHS 40x40x4 of material 1.4571 the load level according to the test results is over 0.40.
Temperature [C] CHS RHS -RHS -RHS
30x30x3 30x30x3 40x40x4 40x40x4
Test result Test result Ti Design resistance Design resistance
RHS 40x40x4 Ti Test result - RHS 30x30x3 Design resistance - CHS 33x2 Design resistance RHS 40x40x4 Test result
Figure 3: The relative values of fire resistance of concentric loaded columns. The relative values of test values are determined by dividing the test value by design resistance at normal temperature Eccentric load Respectively, the comparison between the resistance from eccentric compression tests and design resistance is performed and the reliability of the calculation method can be estimated based on Figure 4. The design values and test values are shown as relative values. The relative values of the test values were determined by dividing the test value by the calculated value at normal temperature. Eccentric loaded tests were carried out for RHS 40x40x4 ( X2,, = 1,17) and RHS 30x30x3 ( ho = 1,52) of material EN 1.4301. The design method also underestimates the load capacities for eccentric loaded columns, when modified slenderness is low.
529 m
j
^
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I
I
I
1
I
1
I
I
I
I
I
i
0.6 J J
•j
i
I
I
I
I
I
I
1
i
i
I
I
1
1
'-'A''^- - L
-^
I
I
I
I
T ' - - - ^ xi I
I
I I
I.
I ^
I ^
_I
I ^
I ^
^^""•^. I
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I I
I I
I I
I I
I I
I I
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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
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w
if
/
1
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-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\.<^-\-/-\._/
—Lru\j-u^n,r\r^r
Z-ECVG
Z-CC+B/G
Z-B/G
Eccentric compression at restrained flange
Gravity load
Centric compression and gravity load
Centric compression
iiiiiiiiiiimii L/~w-w^L/'\.rwLru
Z-EC/U
Z-CC+B/U
Z-B/U
Eccentric compression at free flange
&
Uplift load
Centric compression and uplift load
Bearing
1}
1 ]) 1
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tj J"
fl ])
d
C J)
? J
1
Steel plate
Timber block 20 X 40 X 150 mm
q
Figure 3: Schematic view of the test set-up for bending with uplift load
534
The axial load was transmitted through knife-edges placed at the ends of the purlins. Warping was restrained by stiff end-plates connected to the sections with pre-tightened bolt connections. To obtain the uniform uplift loading, the test set-up shown in Figure 3 was used. In pressure loading, the specimen was placed upside down and the load was transmitted by eight instead of four points. An example of the deformations and measured deflections is shown in Figure 4.
5
10 15 20 25 30 35 40 Displacement (mm)
Figure 4: Bending with gravity load: Final deformations and measured displacements at the mid-span TESTS ON WELDED I AND CHS BEAMS, COLUMNS AND BEAM-COLUMNS The nominal cross-sectional dimensions of the sections are shown in Table 3. Two materials were used. A complete test series was performed for the I-members made from EN 1.4301 (AISI 304). The 1-160x160 members were also welded from EN 1.4462 (Duplex 2205). The beams were welded by a continuous submerged arc welding (SAW) process. The CHS tube sizes 140x2 and 140x3 were of EN1.4435 (AISI 316L) and the 140x4 size was of EN1.4541 (AISI 321). For each thickness of circular hollow section, one bending resistance test was performed. Flexural buckling tests were carried out for column lengths of 2 200, 3 300 and 4 400 mm. Then, flexural buckling reduces the compression resistance by factors of 0.90, 0.70 and 0.50. The beam-column tests, for checking the interaction of bending and compression, were carried out for the same lengths as for the column tests, but the load was adjusted on the centreline of the wall of the CHS section (tests ECl, EC2 and EC3). TABLE 3 NOMINAL CROSS-SECTIONAL GEOMETRY OF THE SPECIMENS
1 i^
® T~ tf H tw
1-160x80 1-160x160 I - 160x160 Duplex |l-320x160
H (mm)
(mm)
B (mm)
(mm)
160 160 160 320
6 6 6 6
80 160 160 160
10 10 10 10
tw
tf
0
CHS -140x2 CHS - 140x3 CHS - 140x4
D (mm)
t (mm)
139.7 139.7 139.7
2 3 4
535
The spans and test arrangement of the beams were determined such that, in the case of a short beam, the lateral-torsional buckling did not reduce the resistance. The other two nominal spans were 1 000 mm and 2 500 mm for 1-160x80, and 2 500 and 5 000 mm for cross-sections 1-160x160 and I320x160. Then, the reduction factor for lateral-torsional buckling is about 0.4 - 0.5 for the shorter and 0.7 - 0.85 for the longer spans. For Duplex steel, only an 1-160x160 section was tested. A special test set-up, where each individual degree of freedom was arranged separately, was made for the lateral buckling tests. An example of a buckling test and the measured deflections is shown in Figure 5. 800 700 600 500 400
300 4J 200 I 100
0 10
20
30
40
50
Displacement (mm)
Figure 5: An example of lateral buckling test and measured horizontal displacements of upper flange at mid-span Flexural buckling tests were carried out for three different column heights, which were chosen from the series: 600, 1 200, 2 000, 3 300 and 5 000 mm (tests CI, C2 and C3). Tests for both buckling in the strong and weak directions were performed (buckling about the major and minor axis of stiffness). For Duplex steel, only 1-160x160 was tested in the strong direction. The approximated reduction factors for flexural buckling of the tested columns are shown in Table 4. The hinged boundary condition was arranged by a triangular bar positioned in a groove, and the weak-axis and torsional buckling was prevented by a special support system, as shown in Figure 6. Figure 6 also shows an example of the measured deflections. TABLE 4 TARGET REDUCTION FACTORS FOR FLEXURAL BUCKLING
Cross-section 1-160x80 1-160x160 I - 160x160 Duplex
Strong direction L3 L2 LI 0.50 0.85 0.70 0.55 0.80 0.70 0.40 0.80 0.60
Weak direction | L3 L2 LI 0.35 0.85 0.60 0.90 0.45 0.70
-
-
-1
Stub-column tests were performed for determining the resistance and average material properties of the cross-sections. The column had fixed boundary conditions. The ends of the columns were milled flat to allow proper seating of the ends to the rigid plates of the testing machine. Three or four linear strain gauges were attached longitudinally to the mid-length of the columns. The columns were then free to all cross-sectional deformations. An example of a stub-column test and a measured stress-strain curve is shown in Figure 7.
536
20 40 60 Displacement (mm)
Figure 6: An example offlexural buckling tests and measured horizontal displacements at mid-span
0,2
0,4 0,6 Strain (%)
0,8
1
Figure 7: Stub column test for CHS 140x4 and measured stress-strain curves of three strain gauges LOADING RATES During testing, constant loading rates were set to a target failure at 25 - 45 min. Up to roughly 75% of the predicted maximum load Fpred (approx. 15 min.), the load increment was about 5% Fpred/min. and thereafter 2.5% Fpred/min. (approx. 10 - 30 min.). The loads were applied by hydraulic actuators. The load in the sheeting and Z-sections tests was increased using load control, except in the intermediate support tests where displacement control was used. The CHS and I-section and stub-column tests were performed using displacement control. When displacement control was used, the test was continued until the applied load had decreased by 10 - 15% of its peak value or the maximum displacement was more than the specimen length divided by 50. All the data from the load cells and displacements were recorded at 10-second intervals. The loading rates and data recording were performed as above for sheetings. In the case of bending and axial compression, the applied loads were based on the capacities determined in the centric axial compression, gravity and uplift load tests. The axial compression and uniform load were increased simultaneously. The loading velocities were such that, if the axial compression or bending tests were
537
performed separately, the times to collapse would be equal. This ensured that roughly half the capacity was due to axial force and half due to bending force. MATERIAL TESTS The actual properties of the sheetings were determined from unloaded specimens. Because all the test profiles were manufactured from the same coil of raw material, only three test coupons with different transverse positions were cut longitudinally from each type of profile. One test piece was taken from each test of the Z-sections. The piece was cut longitudinally from the first Z-profile to have failed in the test. The test specimens (totally 7 pieces) were cut from the end of the beam and from the middle of the web. Three additional test pieces from the web of an unloaded section were taken to check the effect of loading. From each sample of CHS source material bar, two material specimens were taken, one from the side opposite the weld, and one from the side 90 degrees to the weld. From each sample of source material, three material specimens in longitudinal direction were taken, one from the middle of the plate, and two from 50 mm from the edges of the plate. The material tests on the specimens were performed according to EN 10002-1. Specimens taken from the profiles were machined into the shape prescribed in EN 10002-1, Annex A. Stainless steels are sensitive to strain rate, therefore the tests were carried out under strain control with a more specific strain rate than given in EN 1002-1. The test was started with a constant strain rate of 6%/min. until Rpi.o was exceeded and thereafter a strain rate of 48%/min was used. The mechanical properties, Rpo.oi, Rpo.2, Rm, E, A5 and n, were determined from the measured stressstrain behaviour. The modulus of elasticity was approximated manually from the origin of the stressstrain curve. The parameter n, determined as described in ENV 1993-1-4 (1996), indicates the curvature of the stress-strain curve. CROSS-SECTIONAL DIMENSIONS AND INITIAL IMPERFECTIONS The dimensions of the sheetings shown in Figure 8 were measured digitally from specimens of length 500 mm. The specimens had the same average width as the sheeting used for the member tests. As the dimensions were determined from the up-side, so the results show the shape of the upper surface of the sheeting.
RAN-113
Figure 8: Dimensions determined from sheetings
538
For Z-sections restrained by sheeting, the cross-sectional properties and imperfections shown in Figure 9 were measured from the middle of each Z-section. In addition, the straightness of the flat parts was checked visually, with the naked eye.
Figure 9: Scheme for checking the cross-sectional dimensions and straightness of a channel section For welded CHS and I-sections, the measured cross-sectional dimensions and initial imperfections shown in Figure 10 were measured from the mid-length of each specimen. The straightness of the CHS members was measured in the direction of measured Di and D3.
Figure 10: Scheme for checking the cross-sectional dimensions and initial imperfections of I-sections ACKNOWLEDGEMENTS The following organizations sponsored the tests described in this paper: Avesta Sheffield AB Research Foundation, Centro Sviluppo Materiali spa, European Coal and Steel Community, Studiengesellshaft Stahlanwendung eV, Ugine SA, Outokumpu Polarit Oy and The Steel Construction Institute. Their support is gratefully acknowledged. REFERENCES Burgan B.A., Baddoo N.R. and Gilsenan, K.A. (1998). Structural design of stainless steel in UK: Backgound to current pratice and new. Stainless Steel In Structures - An Experts Seminar, London 2122 September 1998. The Steel Construction Institute, 24 p. ENV 1993-1-3. (1996). Eurocode 3: Design of steel structures. Part 1.3: General rules. Supplementary rules for cold formed thin gauge members and sheeting. European Committee for Standardization (CEN). 128 p. ENV 1993-1-4. (1996). Eurocode 3: Design of steel structures. Part 1.4: General rules. Supplementary rules for stainless steel. European Committee for Standardization (CEN). 55 p. Kolari K. and Talja A. (1994). Design of cold-formed beam-columns restrained by sheeting. Technical Research Centre of Finland, Espoo, Finland. 45 p.+ app. 46 p.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
539
MODELLING OF THE CYCLIC BEHAVIOUR OF BOLTED TEE-STUBS C. Faella, V. Piluso, G. Rizzano Department of Civil Engineering, University of Salerno, Italy
ABSTRACT The behaviour of beam-to-column joints plays an important role in the performance of steel frames. For this reason, the criteria and the formulations for predicting the rotational stiffness and the flexural resistance of the most common connection typologies have been recently codified by Eurocode 3 in its Annex J (CEN, 1997). The codified method for predicting the rotational behaviour of beam-to-column joints under static loading conditions is based on the so-called component approach whose potentialities have been widely investigated. However, its application is basically limited to monotonic loading conditions. In order to extend the component approach to seismic applications, the modelling of the cyclic response of the joint components is necessary. Starting from the observation that the main sources of deformability and plastic deformation capacity of bolted connections can be modelled by means of an equivalent T-Stub, on the basis of the preliminary results of an experimental program devoted to the cyclic response of the most important component of bolted connections, a first attempt to model the cyclic force-displacement curve of bolted T-Stub is performed in this paper. Therefore, this work represents a first step towards the prediction of the cyclic behaviour of bolted joints starting from the knowledge of their geometrical and mechanical properties. KEYWORDS Cyclic behaviour, bolted connections, energy dissipation capacity, joint modelling, degradation laws. INTRODUCTION Eurocode 3 has opened the door to the analysis of steel frames taking into account the rotational behaviour of beam-to-column joints. In particular, on the basis of the component approach, the rules to predict the rotational stiffness and the flexural resistance of welded and bolted beam-to-column connections are codified. The provisions given in Eurocode 3 allow the design of semirigid steel frames, under monotonic loading conditions, taking into account the actual rotational behaviour of beam-to-column joint. On the contrary, regarding cycUc loading conditions, such as those occurring under seismic motion, the state of knowledge is still not sufficient to allow the prediction of the beam-to-column joint behaviour starting from their geometrical and mechanical properties. Regarding beam-to-column joints under cyclic actions, it is useful to point out that Eurocode 8
540
recommends the use of full-strength joints characterized by a flexural resistance greater than 1.2 times the plastic resistance of the connected beams. The use of partial-strength joints is not forbidden, but it is strongly discouraged because the experimental analysis of the adopted structural details is required. However, some considerations have to be accounted for. The above design requirement can lead to expensive structural details requiring the use of appropriate stiffening elements. On the contrary, the use of semirigid beam-to-column connections could represent, from the economical point of view, a more convenient structural solution, with respect to the rigid one, provided that serviceability requirements are not determinant (Guisse, 1993; Colson and Bjorhovde, 1992; Anderson et al.,1995; Faella et al., 1997). In addition, an appropriate joint semirigid design can lead to a plastic rotation supply compatible with the plastic rotation demand under seismic motion. In fact, it can be recognized that the overall joint ductility and energy dissipation capacity is provided by the contribution of all the components engaged in plastic range (Gobhorah et al., 1992). It is clear that the knowledge of the joint cyclic behaviour and its modelhng represent a fundamental point when the frame design is based on the dissipation of the seismic input energy in the connecting elements. Many research programs on the cyclic behaviour of beam-to-column connections have been carried out worldwide (Bernuzzi et al., 1996; Calado, 1995; Pekcan et al., 1995; De Martino et al., 1981; Ballio et al., 1986) aiming at the identification of the behavioural parameters governing the cyclic response and at the modelling of the hysterethic behaviour (De Martino et al., 1984; De Stefano et al., 1994; Bernuzzi et al, 1996; Bernuzzi and Serafini, 1997). In addition, many efforts have been spent to analyse the low cycle fatigue (Krawinkler et al., 1971, Engelhardt and husain, 1992, 1993; Mander et al., 1994; Bernuzzi et al., 1997b; Calado et al., 1998; Castiglioni et al., 1996; Bursi and Galvani, 1997; Faella et al., 1998). Most of the mentioned works deals with the overall joint behaviour and its modelling. This approach does not allow the quantitative identification of the contribution of each component and, as a consequence, of the role played by the geometrical and mechanical parameters. A different approach can be based on the observation that the cyclic behaviour of beam-to-column joints can be predicted by properly combining the cyclic response of its basic components. This approach represents the extension to the cyclic behaviour of the philosophy adopted in Eurocode 3 with reference to monotonic loading conditions. A first study based on this approach has been performed by Madas and Elnashai (1992) with reference to bolted connections, but starting from a simpUfied modelling of the cyclic behaviour of the basic components which does not include any rule to account for the degradation of stiffness and strength due to low cycle fatigue. An accurate prediction of the joint rotational behaviour under cyclic loads, based on the component approach, requires the prehminary characterization of the cyclic behaviour of the joint components. For this reason, an experimental program devoted to the analysis of the cyclic behaviour of the most important component of bolted connections has been planned. In this work, the first results of this experimental program are presented. In particular, the analysis is focused on the behaviour of bolted T-Stubs which are commonly used for the modelling of the most important components of bolted joints. EXPERIMENTAL RESULTS The specimens are constituted by the coupling of T-stub elements which have been obtained from rolled profiles of HEA series, steel grade Fe430, by cutting along the web plane. These T-stubs are connected through the flanges by means of two high strength bolts (class 10.9). The bolt diameter is equal to 20 mm (Fig.l). The tightening of the bolts has been performed by means of a calibrated wrench. The applied torque gives rise to an axial force in the bolts equal to 0.80 times the bolt yield resistance. The experimental program has required the testing of 12 specimens, 6 derived from an HEA 160 profile and 6 from an HEA220 profile. With reference to the specimens made of HEA 160, 1 monotonic test and 5 cyclic tests under constant amplitude have been carried
Section X-X
Plan
Lateral view
r ^
m
*
x -•X
f
"4-
Figure 1: geometrical parameters of specimens
541 Table 1: Geometrical and mechanical properties of tested specimens SPECIMEN HEA160-M1 HEA160-C1 HEA160-C2 HEA160-C3 HEA160-C4 HEA160-C5 HEA220-M1 HEA220-M2 HEA220-C1 HEA220-C2 HEA220-C3 HEA220-C4
dh (mm)
tf (mm)
20 20 20 20 20 20 20 20 20 20 20 20
9.10 9.05 9.17 9.02 9.12 9.17 10.85 10.98 10.95 11.00 10.95 10.88
m (mm) 30.40 30.15 29.55 31.60 29.00 30.10 32.60 29.40 33.45 41.25 40.90 32.40
n (mm) 34.10 34.60 35.00 32.90 36.00 34.65 58.50 61.95 57.65 49.85 49.95 58.20
r(mm) 15.0 15.0 15.0 15.0 15.0 15.0 18.0 18.0 18.0 18.0 18.0 18.0
^(mm) 190.0 190.0 190.0 190.0 190.0 190.0 190.0 190.0 190.0 190.0 190.0 190.0
fy (N/mm^) 340.1 340.1 340.1 340.1 340.1 340.1 293.1 293.1 293.1 293.1 293.1 293.1
/•„(N/mm') 473.1 473.1 473.1 473.1 473.1 473.1 453.8 453.8 453.8 453.8 453.8 453.8
out. Regarding the specimens derived from HEA220 profile, 2 monotonic tests and 4 cyclic tests under constant amplitude have been performed. With reference to the notation of Fig. 1, the measured values of the geometrical properties of the specimens are given in Table 1. The experimental tests have been carried out at the Material and Structure Laboratory of the Department of Civil Engineering of Salerno University. Under displacement control, all the specimens have been subjected to a tensile axial force which is applied to the webs tightened by the jaws of the testing machine, a Schenck Hydropuls S56 (maximum test load 630 kN, piston stroke ± 125 mm). In addition, coupon tensile tests have been performed to establish the mechanical properties of the material. The values of the yield stress/v and of the ultimate strength/,, are given in Tab.l. The main purpose of the monotonic tests is the evaluation of the plastic deformation capacity of the specimens whose value has been used to establish the range of the amphtude values to be adopted in cyclic tests. The results of the monotonic tests are presented in Fig.2 both for HEA160-M1 and for HEA220-M1 specimens. The failure mode of HEA160-M1 specimen was characterized by significant yielding of the flanges with yield lines developing similarly to the non-circular pattern of Eurocode 3. This gave rise to great plastic deformations. The attainment of the maximum load carrying capacity and the subsequent failure were practically due to the penetration of the bolt head and the bolt nut in the flange hole. In addition, the cracking of the flange was observed close to toe of the flange-to-web fillet in the central part of the flange close to the bolts. Even though both specimens were designed, on the base of their nominal properties, to fail involving the flanges only, in the case of HEA220-M1 specimen the premature failure of one bolt prevented the complete development of the non-circular pattern of yielding. The same failure mode occurred in a second monotonic test confirming that the ultimate plastic deformation of this specimen is less than that of HEA160 profile due to the overstrength of the flange which was not accounted for in the design. 350 Regarding the cyclic tests, all specimens HEA220^-«^' exhibited the same failure mode independently 300 y^ ' of the imposed displacement amplitude. Crac».^^^^HEA160 ^250 king of flanges initially developed in the central / \ y ^ ^ ^ ^ part of the flange at the flange-to-web connec=-200 / >< 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«
V(!lnilcl«l: IT-S,TO-2
(amplitude = 20.0 mm)
HEA160-C5
(amplitude = 30.0 mm)
'
DISPLACEMENT (mm)
DISHACEMENT (mm)
DISPl-ACEMENT (mm)
HEA160-C4
^^\\
S -100
,
HEA220-C1
(amplitude = 7.0 mm)
'""""'":'*°
i. '°\.^^^^^^^1^'°°
l^^^^r ' c, Irtdcplctfd: 0-5, i0.2n....-50. 75-1 («)•...-300. 305.30(S,307.30«.3<».3in
DISPI.ACEMErn" (mm)
HEA220-C2
(amplitudes 10.0mm)
DISP1.ACEMENT (mm)
HEA220-C3
(amplitude = 16.0 m m)
HEA220-C4
(amplitude = 24.0 mm)
c,cl«.oc™ple.,rr,c.«r,= 14
complelef
\\^
• 'il^^fit^'v cycles depictnl: 0-S, 6-R-...-38
DISPI-ACEMENT (mm)
DISPI.ACEMENT (mm)
DISPIACEMENT (mm)
:--^^?^-:-" .,'.,.,.,.,.,. DISPI.ACEMENT (mm)
Figure 3: Cyclic tests of bolted T-stubs a progressive deterioration up to failure of axial strength, stiffness and energy dissipation capacity, as it is testified in Fig.3. ENERGY DISSIPATION CAPACITY With reference to the identification of the failure condition, there is not a criterion universally recognized. Generally reference is made to the deterioration of strength, stiffness or energy dissipation capacity. The influence of the failure criterion on the low cycle fatigue curve of bolted T-stubs has been analysed in a previous work by the same authors (Faella et al., 1998). Herein, a degradation of the energy dissipation capacity equal to 50% of the energy dissipated during the first cycle has been assumed as failure condition. The adopted level of degradation corresponds to an abrupt increase of the velocity of the hysteretic energy degradation (Castiglioni and Calado, 1996). It can be observed that the prediction of the T-Stub energy dissipation capacity under cyclic loads cannot be based on the energy dissipated in monotonic tests. This is due to the fact that, as underlined in the previous section, the failure mode under monotonic loading conditions can be different from that occurring under cyclic loads. For this reason, the energy dissipation corresponding to the conventional failure condition has been related to the energy Ea dissipated in the monotonic test up to the achievement of a displacement amplitude corresponding to that of the cyclic test. As the failure under cyclic loads is always characterised by the complete fracture of one of the T-Stub flanges, the corresponding monotonic failure mode is a collapse mechanism involving the T-Stub flanges only. With reference to this mechanism, it has been recognized (Faella et al., 1997) that the theoretical value of the ultimate plastic displacement of a couple of bolted T-stubs is given by:
543 (1)
S/,.r/, =
where m is the distance between the bolt axis and the section corresponding to the flange-to-web connection (Fig.l), tf is the flange thickness and C is a constant depending on the true stress-true strain curve of the material. Starting from the results of the coupon tensile tests, the values C = 0.309 and C = 0.355 have been computed for HEA160 and HEA220 specimens, respectively. The plastic part 6;, of the displacement amplitude 5 of the experimental tests can be properly expressed in nondimensional form by means of the parameter: (2) S = A. C Regarding the energy dissipation capacity Ecc exhibited under constant amplitude cyclic tests, it can be properly nondimensionalised considering the parameter: ^
Ecc
(3)
On the base of these considerations and by means of a regression analysis of the experimental results, with a coefficient of correlation equal to 0.95, the following relationship has been obtained (Fig.4): F ^
ft = 5.263
^ V^-^' - ^
W
BEHAVIOURAL PARAMETERS AND MODEL The aim of the model presented in this paper is the prediction of the cyclic behaviour of bolted T-stubs starting from the knowledge of their geometrical and mechanical properties. To this scope the monotonic force-displacement curve has to be preliminarly predicted. Such prediction can be developed by means of a theoretical approach presented by the same authors in a previous work (Faella et al., 1997). As soon as the monotonic behaviour has been predicted, the cyclic behaviour of bolted T-stubs can be modelled provided that the rules for strength and stiffness degradation and for the pinching of the hysteresis loops are derived. It can be observed from experimental results (Fig.3) that the point corresponding to the load inversion remains practically unchanged during the loading process. These points (A and D in Fig. 5) can be identified starting from the maximum load achieved in the first cycle and by the initial stiffness. Therefore, the unloading branch is strictly identified for all the cycles, provided that the load degradation law is known. On the basis of a regression analysis of the experimental results, for each cycle the load degradation has been related to the corresponding cumulated energy and displacement amplitude. The following
Fj. ''max \ O
:
o HEA220 specimens • HEA160 specimens
\o :
\ B _
Fy
Mo /i
\C
1Xl/D f^/ r'
W 40
1\
\ a
/ \\/A
-~--~-J_o 0.4
0.6 t f 5P Cm2
Figure 4: Nondimensional relationship between the energy dissipation
/
r°
-Fy Pmax
Figure 5: Cyclic model
5 max c
544 relationship has been obtained: Fi
= 1
f^
( Sm
a\ 2 6,
(5)
E t Ecc
where E,c is the energy cumulated up to the i-th cycle and the coefficients au ai and as are given by: ai = 0.046 , ai = 1.027 , a^ = 2.379 (6) The displacement amplitude 5^ corresponds to the Umit of the elastic range and it is equal to the ratio Fy/Ko between the force corresponding to first yielding and the initial stiffness without bolt preloading which is exhibited in the experimental curve during unloading. Eqn.(5) is characterized by a coefficient of correlation equal to 0.90. It can be observed that the stiffness degradation and the pinching phenomenon are promoted by the detachment of the flanges at the bolt axis due to the plastic flexural and extensional deformation of the bolts. The reloading branch can be approximated by means of two straight lines with a different slope (Fig.5). The point C, corresponding to the intersection of the two straight lines, is approximately lined up with the point A which corresponds to the inversion of the load sign and with the point B corresponding to the limit of the conventional elastic range in monotonic loading conditions (Fig.5). Therefore, the slope of the straigth line connecting the above mentioned points (A, B and C) can be assumed equal to: t
a
=
^max
(7)
Omax ~" 2 i^max/Ao
In order to verify the reliability of Eqn.(7), which allows the reduction of the number of involved parameters, the values Oexp minimizing the scatter between the energy dissipated in the experimental test and the one obtained from the simplified model based on a bilinear reloading branch have been computed and compared with the a values provided by Eqn.(7). In Fig.6 such comparison is depicted showing an acceptable degree of accuracy. In addition, in order to completely describe the pinching phenomenon, the knowledge of the stiffness Kj of the first part of the reloading branch is necessary (Fig.5). On the basis of a regression analysis of the experimental results, the following relationship has been derived:
,
, (B^J' fE/^,^'" \
Ko = I - bi
(8)
2 6v
where the coefficients b\,b2 and bs are: bi = 0.695 , fc = 0.125
(9)
, bs = 0.110
It is useful to observe that 2 5v is the threshold amplitude of 5 beyond which degradation phenomena begin (Fig.5). Eqn.(8) is characterized by a coefficient of correlation equal to 0.85.
HEA160-C4 1
(amplitude=20mm)
experimental test
200 f UJ
O tc
/
100
x^^^^^^^^^^M
'
n
0
o u. -100
^^^^^^^^
-200 -300-5
Figure 6: Reliability of the parameter a
[ 0
.
[ . \ . [ , 5 10 15 DISPLACEMENT (mm)
20
Figure 7: Model cyclic behaviour
25
545
: 5 = 24 mrti //
WJi
/
^
wO.6 / / /
8 = 7mm
" 0.4
"'^
L /
.i;;^^i^
[T^S-^H^^ 5
;
. 5 = 20innJ
HEA220 model experimental
,
10 20 50 NUMBER OF CYCLES
u ^ 1
100
1_
200
. ,
^,,, — -- '
.--;>^ ;
"—-'' :
\ 3 6 r ^.'-'
. ^
8 = '3 mm
N^/
/
5 = ^ mm
\ HEA160 model experimental
,.„..
500
1
2
5 10 20 NUMBER OF CYCLES
Figure 8: Energy dissipation capacity: Figure 9: Energy dissipation capacity: HEA220 specimens HEA160 specimens As a conclusion, the modelling of the cyclic behaviour of bolted T-stubs requires the following steps: • prediction of the monotonic force-displacement curve (Faella et al., 1997); • computation of the energy Eo dissipated under monotonic conditions up to the displacement 5max; • estimation, through Eqn. (4), of the energy dissipation capacity under cyclic action for the imposed displacement amplitude 5,nax; • computation of the force F,nax corresponding on the monotonic F - 6 curve to the displacement amplitude Smax of the imposed cyclic action; • definition of the strength degradation rule by means of Eqn. (5); • definition of the stiffness degradation rule and of the pinching phenomenon by means of Eqn.(8) and of the parameter a given by Eqn.(7). It can be recognized that the empirical part of the above modelling of the T-stub cyclic behaviour is constituted only by the strength and stiffness degradation rules given by Eqns. (5) and (8). The semi-cycle, i.e. the parameters Ko, Fmax and a, can be theoretically predicted starting from the theoretical prediction of the monotonic F - 6 curve (Faella et al., 1997). In order to verify the reUability of the proposed model, the comparison with the experimental results has been performed. From the qualitative point of view, Fig.7 shows the degree of accuracy of the model in predicting the cyclic behaviour. However, the reliability of the model can be better verified by means of a comparison in terms of energy dissipation. In Figs. 8 and 9, the results of this comparison are shown. It can be observed that, for all the experimental tests, the scatters between the experimental values of the energy dissipation and the ones predicted by means of the proposed model, are not particularly significant and always on the safe side. CONCLUSIONS In this work, the first results of an experimental program devoted to the analysis and modelling of the cyclic behaviour of beam-to-column joints are presented. In particular, on the basis of 12 experimental tests on T-Stub assemblages (9 in cychc loading conditions and 3 in monotonic loading conditions), the stiffness and strength degradation laws have been derived. In addition, as the failure mode under monotonic loading conditions can be different from that occurring under cyclic loads (complete fracture of T-Stub flanges) the correlafion between the energy dissipation corresponding to the failure condition and the energy dissipated in monotonic conditions up to a displacement amplitude equal to that of the cyclic test has been provided. On the basis of the above analysis, a model for predicting the cyclic behaviour of the T-Stub assemblages starting from their geometrical and mechanical properties has been developed. Finally, the reliability of the model has been testified by the good agreement with the experimental results in terms of energy dissipation capacity. The above results are encouraging regarding the possibility of predicting the cyclic behaviour of bolted beam-to-column joints, starting from their geometrical and mechanical properties, by means of a component based approach. To this scope, additional experimental tests regarding both single components and complete beam-to-column joints have been planned.
546 REFERENCES Anderson, D., Colson, A. and Jaspart, J.P. (1995): "Connections and Frames Design for Economy", Costruzioni Metalliche, No. 4. Ballio, G., Calado, L., De Martino A., Faella, C. and Mazzolani, P.M. (1986):" Steel Beam-to-Column Joints under Cyclic Loads: Experimental and Numerical Approach", European Conference on Earthquake Engineering, Lisbona, 7-12, July. Bernuzzi, C , Calado, L. and Castiglioni, C.A. (1997): "Steel Beam-to-Column Joints: Failure Criteria and Cumulative Damage Models", Second International Conference on Behaviour of Steel Structures in Seismic Areas, Kyoto, Japan, 4-7 August. Bernuzzi, C. and Serafini, N. (1997):" A Mechanical Model for the Prediction of the Cyclic Responce of Top-and Seat Cleated Steel Connections", CTA, Italian Conference on Steel Construction, Ancona, 2-5, October. Bernuzzi, C , Zandonini, R. and Zanon, P. (1996):"Experimental Analysis and Modelling of Semi-Rigid Steel Joints under Cyclic Reversal Loading", Journal of Constructional Steel Research, Vol. 38, No. 2, pp. 95-123. Bursi, O.S. and Galvani, M. (1997):"Low Cycle Behaviour of Isolated Bolted Tee-Stubs and Extended End Plate Connections", CTA, Italian Conference on Steel Construction, Ancona, 2-5, October. Calado, L. (1995):"Experimental Research and Analytical Modelling of the Cyclic Behaviour of Bolted SemiRigid Connections", Steel Structures, Eurosteel '95, Kounadis ed., Balkema. Calado, L., Castiglioni, C.A., Barbaglia, P. and Bernuzzi, C. (1998):"Seismic Design Criteria Based on Cumulative Damage Concept", 11th European Conference on Earthquake Engineering, Paris, 6-11 September. Castiglioni, C.A. and Calado, L. (1996):"Low-Cycle Fatigue Behaviour and Damage Assessment of Semi-Rigid Beam-to-Column Connections in Steel", International Colloquium on Semi-Rigid Structural Connections, lABSE, Instanbul, 25-27 September. Colson, A. and Bjorhovde, R. (I992):"lnteret Economique des Assemblages Semi-Rigides", Construction Metallique. No. 2. De Martino, A., Faella, C. and Mazzolani, F.M. (1984): "Simulation of Be^m-to-Column Joint Behaviour under Cyclic Loads", Costruzioni Metalliche, No. 6. De Martino, G., Sanpaolesi, L., Biolzi, L., Caramelli, S., Tacchi, R. (1981):"Indagine Sperimentale sulla Resistenza e Duttilita di Collegamenti Strutturali", Ricerca Italsider - Comunita Europea, Monografia No. 5. De Stefano, M., De Luca, A. and Astaneh, A. (1994): "Modelling of Cyclic Moment-Rotation Response of Double-Angle Connections", Journal of Structural Engineering, ASCE, Vol. 120, No. 1. Engelhardt, M.D. and Husain, A.S. (1992):"Cyclic Tests on Large Scale Steel Moment Connections", Tenth World Conference on Earthquake Engineering, Madrid, Balkema Rotterdam, pp. 2885-2890. Engelhardt, M.D. and Husain, A.S. (1993):"Cyclic-Loading Performance of Welded Flange-Bolted Web Connections", Journal of Structural Engineering, ASCE, Vol. 119, pp. 3537-3550, N. 12, December. Faella, C , Piluso, V. and Rizzano, G. (1997): "A New Method to Design Extended End Plate Connections and Semirigid Brased Frames", Journal of Constructional Steel Research, Vol. 41, No. 1, pp. 61-97. Faella, C , Piluso, V. and Rizzano, G. (1997): "Plastic Deformation Capacity of Bolted T-Stubs", Second International Conference on Behaviour of Steel Structures in Seismic Areas, Kyoto, Japan, 4-7 August. Faella, C , Piluso, V. and Rizzano, G. (1998): "Cyclic Behaviour of Bolted Joint Components", Journal of Constructional Steel Research, Vol.46, No.1-3, paper number 129. Ghoborah, A., Korol, R.M. and Osman, A. (1992):"Cyclic Behaviour of Extended End Plate Joints", Journal of Structural Engineering, ASCE, Vol. 118, No. 5, pp. 1333-1353. Guisse, S. (1993):"Quelle Economic Attendre de la Mise en Oeuvre de Noeuds Semi-Rigides?", Construction Metallique, No. 3. Krawinkler, H., Bertero, V.V. and Popov, E.P. (1971):" InelasticBehaviour of Steel Beam-to-Column Subassemblages". Report N. UBC/EERC-71/7, Earthquake Engineering Research Center, University of California, Berkeley. Madas, P.J. and Elnashai, A.S. (1992):" A Component-Based Model for Beam-to-Column Connections", Earthquake Engineering, Thenth World Conference, ed. Balkema, Rotterdam. Mander, J.B., Stuart, S.S. and Pekcan, G.:"Low-Cycle Fatigue Behaviour of Semi-Rigid Top-and-Seat Angle Connections", Engineering Journal, American Institute of Steel Construction, Third quarter, pp. 111 -122. Pekcan, G., Mander, J.B. and Chen, S.S. (1995):"Experimental and Analytical Study of Low-Cycle Fatigue Behaviour of Semi-Rigid Top-and-Seat Angle Connections", Technical report NCEER-95-0002, State University of New York at Buffalo, January.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
547
A NEW METHOD FOR THE CHARACTERISATION OF THE FIRE PROTECTION MATERIALS J. S. Myllymaki and D. Baroudi VTT Building Technology, Fire Technology, P.O. Box 1803, FIN-02044 VTT, FINLAND
ABSTRACT A numerical method is presented to determine specific heat and thermal conductivity of fire protection materials using temperature measurements. In this inverse analysis the direct problem consists of semidiscretized variational form of the heat conduction problem. The inverse problem is solved by means of two different techniques of constrained minimization. It is shown by a numerical examples that the methods can be potentially used to the solution of both the thermal conductivity and specific heat. The convergence of the predicted conductivity is examined from an illustrated 1-D case of a fire protected steel plate. Results of the simulated case show that good estimations on the thermal conductivity can be obtained. Comparisons with the accuracy of inverse methods used in the method NT FIRE 021 of fire protected steel structures are made. KEYWORDS: inverse methods, heat conduction, thermal conductivity, fire protection, steel structures
INTRODUCTION When submitted to a fire, steel and aluminium structures get overheated and lose their load bearing capacity. In order to ensure sufficient resistance times, it is usually necessary to provide steel structures with protection. In Nordic countries fire protections are studied according to the Nordtest method NT FIRE 021 (1985). This method allows, on the basis of the fire resistance test results of two loaded beams and several unloaded columns, to work out so called effective thermal conductivity of the protection. In this method validation of the thermal conductivity is based on the inverse solution of a one simplified one-dimensional ordinary differential equation (ODE). The obtained effective thermal conductivity cannot be used in FEM analyses. NT FIRE 021 method is limited only to the solution of conductivity and specific heat cannot be determined. To date, various advanced methods have been developed for the inverse heat conduction problem involving the determination of thermal conductivity from measured temperatures inside the material. Usually the inverse solution is based on the minimization of the errors between measured and calculated temperatures in a least squares sense. This method is called Output Least Squares Method (OLS). Existing methods of inverse analysis (Tervola 1989, Lam &Yeung 1995, Dhima 1986) require
548 multiple spatial temperature measurements inside the material. To date only a few studies have concentrated on the inverse determination of thermal conductivity from boundary measurements (Kohn & Vogelius 1984, Lin & Cheng 1997). In this paper two minimization methods for the inverse problems are presented. Both Regularized Output Least Squares Method (RLS) and minimization of the smoothness of the thermal property with constrains on the residual error were used. The methods were applied to the inverse solution of direct problem obtained by Finite Element Method. It is shown by a numerical examples of fire protected aluminum plate that the method can be used to the solution of both the thermal conductivity and specific heat. Dhima (1986) has shown the same with OLS method. To study the convergence RLS method in estimation of thermal conductivity from boundary temperature measurements an illustrated 1-D example of a fire protected steel sheet is examined.
Steel section
Fire protection
Figure 1: Cross-section of a protected steel structure and FEM discretization. FORMULATION OF THE DIRECT HEAT CONDUCTION PROBLEM Consider a metal structure divided into one dimensional (ID) elements as shown in Fig. 1. Variational form of the heat conduction problem (Eriksson et al. 1996) with the temperature field approximated by linear basis functions 7'(jc,0 = N ' ( x ) T ' ( 0 produces the non-linear initial value problem C(r,T)t(0 = f (r,T) - K(r,T)T(0 ,t>0 with T(0) = T^, t = 0
(1)
with elementary conductivity and capacity matrices K%j =
j A{T{x)) N,,{x)
Nj^,{x) A{x) dx,
C,j = J p{T{x)) C{T{X))N,(X) NJ{X) Aix) dx,
and force vector f^i = -
ij = l 2
(2)
i,j = l, 2
(3)
, ^ , respectively. Here T(t) is the global vector of the un-
known temperatures. The elementary volume is dQ.= A{x) dx. The length of the element e is a' =xl-xl and A(jc) is the perimeter at x. Conductivity matrices and force vectors are integrated using Gauss-Legendre scheme. Capacity matrices were integrated with Newton-Cote scheme using nodes as integration points in order to get a diagonal capacity matrix.
549 FORMULATION OF THE INVERSE HEAT CONDUCTION PROBLEM (IHCP) Consider determination of temperature dependent thermal properties in an initial value problem (1) using measured temperature. The vector a= a(T) = [^{T) C^(T) ) of unknown thermal conductivity and specific heat is discretized with respect to the temperature using piece-wise linear basis functions. Vector a contains nodal values of the unknown thermal properties at certain temperatures. A realistic initial value is given for a. Using the regularized output least squares method (RLS) a is solved from the minimization problem (Groetsch 1993)
inin-^^||r^£(^;x;0--7;i^,«(^,0|r+«||L alP with respect to a,
(4)
where vector Tp^{d\x\t) = ^(x)T{t) is the solution of the initial value problem (1). Vector T^^^ contains the measured temperatures. Euclidean norms are calculated in the temperature measurement 2
points X at temperature sampling time t as ||/(x; r)|| = X X K ^•^'' ^ ^\ '
Since the inverse problem is ill-posed, i.e. small variations in the temperature measurements cause large scatter in the inversion results, it has to be regularized. In RLS method one seeks a minimum for the functional (4) where cir(>0)is a small regularization parameter and or differential operator L = \, d a/dT ov d^aIdT^. Parameter a controls how much weight is given to the residual norm \Tpp {X\ x\ t) - Tj^^^ (x, 0 enforcing the consistency relative to the norm L al enforcing stability of the solution. The problem is the appropriate choose of the parameter a so that we can distinguish the real signal from the measurement errors, the noise. Perhaps clearest rule to choose the regularization parameter is discrepancy principle (Groetsch 1993) where the residual norm is set equal to upper bound -
ii2
r^£(A;Jc;0-^a,a(^,0
tmax
^/? ^^, where <5^ is measure of error during time J
~ J
*^
dt and ^
0
is the error at certain time. Promising formulation of the inverse problem that avoids the laborous work of finding an optimal value of a is to minimize
mm,.
L all ^ with respect to a using constraint J^^(X;x; t) - T^^^^ (3c,0
(5)
DETERMINATION OF THERMAL PROPERTIES OF GYPSUM BOARD Following study is an example of the use of different types of constraints in the estimation of thermal properties of gypsum board. Thermal conductivity and specific heat may be regarded as competing parameters. Optimality of the objective function may be reached with different combinations of thermal conductivity and specific heat. This means that a unique solution for the heat conductivity and the heat capacity may not necessary exist. On the other hand as we use the whole time history of the recorded temperature one expects that this restricts considerably the set of physically admissible solutions, since the obtained properties should retrace the whole observed history.
550 Several tests of different types would be needed for reliable determination of the thermal properties. As a preliminary investigation we limit ourselves to two cone calorimeter tests. In the first test a gypsum board specimen (Ai = 100 mm x 100 mm) of 12.5 mm thickness and 720 k g W density lay on a 30 mm thick layer of mineral wool. The gypsum board was exposed to heat flux qcone = 25 kW/m^ . In the second test there was additionally a 10 mm thick aluminium plate under the gypsum board. Temperature on the upper surface of the gypsum board was measured with infrared device and inside the specimens with thermocouples. Specific heat of the gypsum is first determined using a lumped thermal capacity model applied to the data of the first test. Conductivity of the gypsum board was then calculated using the data of the second test using conduction model with Fourier heat conduction law.
400 jj300 >- 200 Gyps-data Gyps-calc.
100
10
20 Time (min)
30
50
100
150
200
250
300
Temperature (oC)
Figure 2: a). Measured surface temperature (bold lines), the calculated temperature of the gypsum (thin line), b) calculated specific heat of the gypsum board. Solution of heat capacity. Formulation of direct and inverse problem. Assuming constant temperature field in the gypsum plate and using equations (1-3) we got ODE (6)
-^f(T(x;t),t)
T(t) = ^pc(T)V
where integration was carried using Newton-Cotes quadrature in order to obtain a lumped specific heat capacity. The specific heat capacity c = c(T) was found as the solution of the constrained minimization problem (with regularisation parameter a = 10' ) ^in ^T^ta{t)-f^^i,{t;c)f-\-a\\cf
} with constraint W{C)G[16 %, 18 % ]
(7)
where the moisture content w of the effective C^ is bounded within experimental range of 16 - 20 %. Specific heat was discretized using piece-wise linear basis function N with respect to the temperatures (T) = ^Nj
Cj(T) = N'C where the vector c = (c(Ti) c(T2) ...C(TMC)).
Determination of heat conductivity. Formulation of direct and inverse problem. Here the test with 10 mm thick aluminium plate directly under the gypsum board was considered. Temperature inside the gypsum board was approximated with linear element. Aluminium temperature was taken as constant. Conductivity matrix was integrated using one-point Gauss-Legendre scheme.
551
Used diagonal capacity obtained by Newton-Cotes scheme captures the peak in the specific heat of the water around the region of temperature 100 °C. The semiscretization leads to equation:
tith-
\K/d.
(7;-T)=O,
1 P,c,{Tj
V, ^
2 P.c^Tj
V,
/+-
(8)
where conductivity is calculated at the centre of the gypsum board A^X^ ~ —/l^
T.+T^
AA^-0.5)
and heat capacity of aluminium as Q = p^A^d^c^[T^)
0.25
—o—cond.
20000
- - - • • - - cond. RBttersson 0.20
O 0.15 o
1
—o—cap.
1
\
15000
^"^"^SJ*
10000
:^o.io 5000
0.05
0.00* 1000
1500
Time (s)
y-
^V^^^^>--0-^M>~o-120
170
0
220
Temperature (oC)
Figure 3: a) Measured (Al, thick line) and calculated temperatures, b) Calculated conductivity of gypsum(thick line) compared to values of Pettersson & Odeen (1978). Heat capacity is also shown. Thermal conductivity of the gypsum board was discretized using a number of (Mx, - 1) piece-wise linear basis functions with respect to the temperature stored in the vector a with components a. = X{T.), i = 1... Mx of the unknown vector a. Thermal conductivity was found as the solution of the next constrained non-linear optimisation problem: tin I (7{T) - [ L a\[with constraint ||7;,,a(0-^ca/c.(^;a)|| ^ A r V A ^ ' I ^ n u n ' ^ m a x ] }
(9)
where Ndata is the number of sampling times. Values Smin (=5 °C) and Smax (=10 °C) were minimum and maximum of the standard deviation of the measured temperature.Value of conductivity at normal temperature X (20 °C) was held fixed. Also the positivity of X was used as a constraint. The smooth weight function (T(T) was equal to one away from 100 °C and equal to zero for a narrow region around 100 °C Use of weight function allows possible discontinuities of the derivatives of the thermal conductivity around 100 °C. The operator L was the central difference approximation of d^AydT^.
CONVERGENCE OF INVERSE SOLUTION OF THERMAL CONDUCTIVITY To study convergence of the present numerical method, a case of an insulated steel plate was considered. Following material properties were used; density of steel p^ =7850 kg/m^, specific heat of steel c^ = 540 J/kgK, density of fire protection p = 220 kg/m^, specific heat of fire protection c = 1000 J/kgK.
552 Thickness of the steel plate was d^= 0.0041 m and thickness of the fire protection was d = 0.02 m. Thermal conductivity of the steel Z^ = 50 W/mK was applied. Function of the thermal conductivity of /" rp\P>.\
the fire protection was X (T) = X^ 1 +
, where A,= 0.0251 W/mK, 7;^= 411 K, p^ =2.404
yT,j
and T (K) is absolute temperature of the protection. Fictitious temperature data of the steel was calculated using equation
Tr'{h) = T^^'\t,) + k,5T', t,=n^At,,
(10)
where TJ^^ is the temperature of the steel calculated with ABAQUS (1994) and S T represents amplitude of simulated temperature noise and A: is a generated random variable in the range from -1 to 1. In the ABAQUS solution 13 one dimensional 2-noded elements DC1D2 with three elements in steel and 10 elements in insulation were used. Boundary condition at x=0 was adiabatic. The temperature at the boundary of the fire protection x = d^-\-dp followed the temperature-time curve of standard ISO 834.
I n
1
n
Protection
Temperature approximation
Fig. 4: One dimensional case of an insulated steel plate. Fire protection divided into two elements. In the inverse solution the steel plate was divided into one element with constant temperature field causing that the steel is only acting by specific heat. The fire protection was divided into linear elements of equal lengths. Thermal conductivity was obtained by minimizing l|2
^^iJV
'au,)-r
(r,)
II
-*||2
(11)
+JLA
using the fictitious temperature of the steel calculated from Eqn. (10) with 6 T = \^ C. Value of conductivity at normal temperature A, (20 °C) was fixed. Unknown vector Xj
consisted of the thermal
conductivity at temperatures 50 °C, 100 °C, 150 °C...etc. Condition Xp{T) > 0 was used as restriction to the admissible solution. Initial value for each element of vector Xj was 20 °C conductivity. The effect of the h-discretization to the convergence of the inverse solution was studied. Figure 5a shows that the error between the real thermal conductivity and the inverse solution with one element at temperatures 50 °C- 350 °C is large. In Figure 5a the error of the thermal conductivity is plotted as a function of the number of elements. Error of the thermal conductivity has been calculated using 100
following relative error measure Er{%) =
^}\KAT)-K(T)\
J
j
j
dT, where the subscripts es and
ex denote the estimated and exact values, respectively. The inverse solution converges to the correct
553 solution as the number of elements is increased as seen in Fig. 6a and in Table 1. This was observed also by Lin and Cheng (1997) using several control volumes . In the method NT FIRE 021 thermal conductivity is calculated directly from inverse equation of one ODE using the measured steel and gas temperatures. Thermal conductivity calculated by method NT FIRE 021 is presented in Fig. 6b and error measure in Table 1. TABLE 1. AVERAGE RELATIVE ERROR E R (%) OF A DEPENDING ON THE NUMBER OF ELEMENTS N e
Ne
350-900 °C 1 2 3 4 5
NT FIRE 021
w
I,
E c
E 0.35
' ^
^
A1
3 C
jrH
^
31.91 2.97 1.77 1.63 1.73 77.98
5"
/
/^ v^
20-900 "C
8.25 2.49 1.88 1.56 0.84 18.81
0.2
1"
% 0.05
Temperature (oC)
Temperature (°C)
b)
a)
Fig. 5. Inverse solution of the thermal conductivity (line with balls) compared to the exact thermal conductivity (solid line). Inverse solutions a) one element b) two elements.
-•-CtrriLClivityaO-gOOoC
^ ^ 0.4
1
E 0.35
Hrh-QaTlJClivity3eO-900oC \r
[\
^
0.3
;> 0.25 O 3 0.2
\ , ^ r--
O 0.15
1
I ^ L IA1
_,ir
J>
4^
_g 0.05 0
Temperature (°C)
log(N^
a)
b)
Figure 6: a) Inverse solution of the thermal conductivity as a function of the number of elements N^, b) Solution of the thermal conductivity (line with squares) by using the method NT FIRE 021.
554 CONCLUSIONS Numerical method involving direct problem obtained by finite element method combined with two regularized inverse methods was presented. Method were used in determination temperature-dependent thermal properties; thermal conductivity and specific heat, of fire protection material from temperature measurements. Convergence of the inverse solution was studied. It was shown that the thermal conductivity of the insulation can be inversely solved in 1-dimensional case with very good accuracy using only temperature measurements at the boundaries; at steel and at the surface of the insulation. It is shown that accuracy of the presented method is better than the accuracy of the standard methods used for fire protected steel structures. In this paper the method is used for 1-dimensional cases and can be applied to steel plates insulated symmetrically with the fire protection. The mathematical method can be extended to two-dimensional cases using two dimensional elements or axis-symmetric elements. The tests for fire protected steel structures are conducted in large scale furnace tests. The application of the presented method would be more useful after it is developed further to twodimensional cases. When this is achieved the presented method can be applied for example to the assessment of heavily insulated steel structures. REFERENCES ABAQUS (1994). Standard User^s Manual Volume 1 Version 5.4, Hibbit, Karlsson & Sorensen, Inc. Dhima D. (1986). Contribution a la characterisation thermique de materiaux de protection de profiles metalliques soumis a un incendie. These de doctorat de I'Universite Pierre et Marie Curie. Eriksson, K., Estep, D., Hansbo P. and Johnson C. (1996) Computational Differential Equations. Studentlitteratur, Lund. Groetsch, C.W.(1993). Inverse Problems in the Mathematical Sciences, Wieweg Mathematics for Scientists and Engineers, Vieweg, Braunschweig/Wiesbaden. Kohn, R. and Vogelius, M. (1984). Determining Conductivity by Boundary Measurements, Commun. Pure Appl. Math. 37, 289-298, Lam, T.T. and Yeung, W.K. (1995). Inverse Determination of Thermal Conductivity for One-Dimensional Problems, /. Thermophys. Heat Transfer 9, 335-344. Lin J.H. and Cheng T.F. (1997) Numerical Estimation of Thermal Conductivity from Boundary Temperature Measurements. Numer. Heat Transfer Part A 32,187-203. NT FIRE 021. 1985. Nordtest method, NORDTEST 1985. Pettersson O. and Odeen K. (1978). Brandteknisk dimensionering, Liberforlag, Stockholm, Sweden. Tervola P. A (1989). Method to Determine the Thermal Conductivity from Measured Temperature Profiles. Int. J. Heat Mass Transfer 32, 1425-1430. Wickstrom, U. (1985). Temperature Analysis of Heavily-insulated Steel Stmctures Exposed to Fire, Fire Safety Journal 9, 2S\-2S5.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
BLOATING FLAME-RETARDANT COATINGS ON THE BASIS OF VERMICULITE FOR STEEL BUILDINGS CONSTRUCTION P.P.Gedeonov^ and T.P.Gedeonova^ ^"CARBON Co ltd.", box 3296, Culture street, 38, Chelyabinsk, 454081, Russia. ABSTRACT To improve fire-resistance and reliability of buildings and structures has been and remains the vital problem of researchers dealing with the development of the efficient means of protection in case of fire. Lately the most interesting and promising problem has been the development of bloating flame-retardant coatings, which are studied in detail in given paper. New conception of the development of efficient and rehable bloating flameretardant coatings on the basis of hydrated micas, finely dispersed hydrated micas powders and minerals associated with hydrated micas as well as tehnological solution of complex application of all the components extracted together with hydrated micas is suggested for the first time. Information on the development instructions on ''Vermiculite fire-resistant coatings for steel". It has allowed to rate relatively lamellar, eflfectiv3e and is reasonably cheap protection of building designs against a fire during a fire. The coating is reliable, efficient and comparatively cheap. Fire-resistance tests were carried out in Fire Protection Scientific Research Institute of Russian ministry of Home Affairs. The results of the tests and the resolution on the application of the coatings in the components of buildings and structures already put into operation and being a new are favourable. KEYWORDS. Bloating, fire-resistance coating, steel construction, micas, hydrated micas, vermiculite.
555
556 The protection of metallic construction against a fire durung a fire providing fulfilment with them of the functionality is represented by the relavent problem of building both customary, and unique structures. The special notice is given on the creation of lamellar foamed fireproof covers ( FFC ), executed from high-efFicient materials.
Actuality of creation FFC is dictated by that being
relatively lamellar, they do not result to designs
and provide
reasonably
significant
increase the weigh of
building,
high fireresistance to building designs and are executed
by a mechanised way. To such covers the special requirements are presented. They should have good heatprotection properties with a low factor heatconductivity, to display protective
properties at the
expense of chemical decomposition of substance, dehydratation, allocation of inert gases, decreased concentration of a oxygen at a surface of cover, sublimation, ray-radiation and etc.. The these requirements
concentrates hydrated micas - vermiculite, hydrobiotite and hydromuscovite
answered. Availability in compositions FFC hydromicas results to a lot which are
majority of
displayed
at
a
flame
effect
during a fire.
of
positive
effects,
There are such attributes as
foamed ( FFC ) with absorption of a plenty is warm and simultaneous dehydratation, sharp decrease heatconductivity and increase of protective properties.
In
conventional
action from a fire, reflective ability and many of other
non-foaming
vermiculites
fireproof
covers
is usually
used
foamed vermiculite, which executes a role only effective heat-insulator. Introduction micas in FFC, for the first time carried out by us [ 1 ] , permits largely to transmit listed above properties fireproof to cover. By a distinctive feature FFC on the basis vermiculite is their ability at
heating foaming and to be increased in volume. Thus in foamed cover pro-
cesses, ensuring protective effects occur, about which speech higher went. The increase of protective action FFC is reached by the introduction also foam vermiculite - dehydratated and rehydratated. At creation compositions for FFC certain interest presents foaming at temperature up to vermiculite, capable to rehydratation, that
is to repeated restoration in structure of a water as a result of sorption
from atmosphere in quantity close to initial. [3 ] . Vermiculites, burned at temperature up to 800...900°C dehydratated completely and lose ability to repeated hydratation. At creation FFC it should take into account unique properties used vermiculite. Rehydratated vermiculite, having properties dehydratated small heatconductivity, noncombustion, low volumetric weight and etc. - requires, smaller costs of manufacture and restores in to structure a water. After a fiirnacing of hidromica it results in increase of protective effect fire-resistant compositions, containing in structure
this
component,
as
was
confirmed by us as a result of conducted work [2]. The increasefire-resistantand improvement physico-mehanical, technological properties of mixes and covers executed from them as process of their drawing, operation and at fire effect during a fire, is reached by the introduction in composition fine-graned powders ( FGP ) burned and nonburned vermiculite's, as well as FGP and small-sized fracions accompanying hydromicas minerals, going away wear out at presentin dump and being is properly wastes of manufacture. [3,4,5,6].
557
To it concern, hombleude, pyroxene, feldspar, calcite, apatite, magnetite and other minerals, in smaller quantities - sphene, aegerite, amphibole, sungulite, nepheline and other. The introduction in structures FFC whole spectrum minerals, extracted at development vermiculites deposits up concentrates hydromicas to a complex accompanying to them minerals, permits to create relatively lamellar, reasonably cheap and effective foamed fire-resistant covers for reliable protection of buildings and structures fi-om a flame during a fire. Our work on creation FFC have shown, that fi-actional structure vermiculite, entered in fireresistant composition, as a concentrate of micas so, as foaming ( rehydratated and dehydratated ) and contents hydrated making influence on fireresistance of covers. So, at increase of the size fi-actions (see Figure 1) for considered vermiculites fi*om 0 - 0,15 up to 0,6 - 1,25 mm fi-eresistance is increased on 18...26 %, and then drops, coming nearer on significance to fireresistance ofcompositions on vermiculites most small-sized fi-actions. Tf,min 60\
0-0,15 0,15-0,3 0,3-0,8 0,8-1,25 1,25-2,5
fraction sizes, mm Figure. 1: Dependencefireresistancefi-omfi-actioncontent of grane filler infireproofcovers 1,2, 3,6 and 7 - contents hydromicas respectively (in %) - 97: 80; 50; 20 and 15;4 - rehydratated vermiculite; 5 • dehydratated vermiculite The
maximum
fireresistance
0,6 -1,25 mm. Using
observed
vermiculite of
covers and bared of a protected
on fi"actions by the size of 0,3 - 0,6 mm and
larger fi-actions makes
more
surface, that makes so them hardly applicable as the basis in
lamellar foamed on height upto 50 ...60 mm fireproof is covered by a mix hydrated making is
dehydratated. Fireproof effective
increase of the contents in
observed essential ( up to three times ) growth fireresistance of
cover. So, at the contents in a mix of below, than at 96 %. Use
intensive destruction of
15 % hydrated making by
it about in three times
rehydratated vermiculite in a mix more effectively, in comparison with covers based on the rehydratated
vermiculite more them 16 %
then covers based dehydratated vermiculite . By minimum quantity hydrated vermiculite
in the mix should be considered 20 %, with its reduction as a result of fast warm up is sharply reduced fireresistance of cover and the blanket is destroyed, of volumetric weight, dispersion, the losses of weight
at tempere,
sorption
of
properties,
heatconductivity
at
high
temperatures
and
558 other properties small sized fractions and fine-graned of powders non burned and burned hydromicas and accompanying to them minerals have shown on a
opportunity of
application
them
for
increase of efficiency FFC. On the basis of conducted researches a series of structures for
fireresistant compositions is
developed. By these structures were protected samples, which were subjected fire to tests in Russian Scientific
Research Institute of fire defence from Balashiha. For this purpose were prepared steel
samples (columns and plate), which were protected by mentioned structures. The columns by section 200 X 200 mm and height of The sizes
1700 mm were executed from sheet have become thickness of 16 mm.
of sheets made 600 x 600 mm of various thickness 4, 7 and 20 mm. The thickness
of a
layer of cover in all cases was same - 10 - 12 mm. The process of manufacturing samples was reduced to clearing of a surface of a metal from rust, scale, dust. Then in chess the order were put anckormarker. The area under anckor-marker has made 12 % from common, being a subject fireresistance. The sizes anckor-marker have made 3 x 3 cm in Samples with covers before
the plan and height of 0,8 thickness of cover.
test up to a equilibrium condition during two months were maintained.
The plates were tested in a horizontal situation,
on two samples
of each structure. Average
significance of humidity of materials of covers in a day of tests on structures has made accordingly for I - 1 7 % ,
II - 14 %, ni -19 %. The achieved limits fireresistance are submitted in
the
table 1. TABLE 1 LIMITS FIRERESISTANCE OF STEEL PLATES DEPENDING ON THEIR THICKNESS Thickness of steel ofaolate. mm 4 7 20
Limits fireiresistance, mines (hr\ fo r structures ni I II* 74(1,23) 65 (1,08) 71 (1,18) 86 (1,43) 86(1,43) 85 ( 1,41) 109 (1,81) 96 ( 1,60) 99(1,55) * - layered cover
With the purpose of revealing of influence of thickness of a metal on fireproof ability of cover at one significance of thickness of a layer 10...12mmsteels plates, protected are tested are same by structures. The covers
on steel
plates
at fire effect
behaved
similarly
to
covers
on
columns. As a resuh of conducted tests [ 7 ] , we find of first dependence ( figure 2 ), which permits established limitfireresistanceof steel designs, protected foamed covers by thickness of 10 ...12mm vermiculites graned
filler.
In the basis of construction given curve criterion of heating
ofprotected steel Designs up to critical temperature, at which in conditions of a fire occurs. Abscise axis
loss of
its bearing ability
of the schedule characterises a limit
fireresistance of steel design, axis of ordinate sindicated thickness of a metal in
designs.
The
559 limit
fireresistance
of columns,
beams,
farms
is defined from the schedule on size of
parameter. Indicated thickness of a metal define under Egn. 1. 10 F d= where
i
,
(1)
F - Area of cross section, cm^; i - Heated part of perimeter, cm. O, mm r
20,
' '
' 1
Id
/
12
(0 6 6 I,
/ 2 O i \ 40 60 80 100 Tf,min Figure.2. Dependence fireresistance limit Tf,iiiin of steel bearing designs from indicated At steel farms settlement indicated thickness of a metal define (determine) on the least significance for all loaded elements. In the case probe up reinforced concrete plates ( floors ) on bottom beh of farms or
the
bottom
shelves of
beams
defined determined on indicated thickness of a
a
limit fireresistance metal
the
of
beams need to be
bottom belt ( shelf). The
received
dependence limit fireresistance of steel bearing designs from indicated thickness of a metal (see Figure 2) permits more precisely to take
into account thickness of a protected element of
designs by that to nominate more precisely ( excepting a overexpenditure of a material ) of
cover, that increase
of
efficiency of cover as
a whole
provides.
thickness
560 Tests one and polylayess of covers have shown, that for layered variant growth of a limit fireresistance
on is observed 10 - 13 %. In comparison with singlelayess, executed from same
compositions and at equal common thickness of a layer. Stated testifies to necessity of the account ( recordkeeping ) of properties of types vermiculite, enabling to make the most of unique properties hydrated micas, and those most to increase pro tective effect of cover, economy a material and other. Results of tests
underlie in the basis of the developed project standards and instruction
on structures, manufacturing and technology of a device foamed of fire resistant covers
on
the
basis vermiculite for fireproofing of metal designs [ 8 ]. For fireprotection of a steelworks raised buildings as in under construction conditions, and immediately on a steel construction plant. Alongside with it the concept of creation foamed fireproof covers based on the hydrated micas and
accompanying to them minerals
is
developed [ 9,
10]. References l.GEDEONOV P.P.and other (1976). Fireproof a raw mix The copyright certificate on the inventionN 509563. USSR 04 B 43 /10, A 04 B 1 / 94 . Opening. Inventions 13 2. GEDEONOV P.P.(1977).Influence of a type andfractionalof a structure vermiculite on fireresistance of covers. Technology and property build ding minerals and concrete on the basis wastes of industrial manufacture and way products. Cheyabinsk. UralSRI of building project, 3. GEDEONOV P.P.. (1990) Hydromicas and accompanying to them minerals in complex compositions foamedfireproofcovers. The thesis of the reports to a technological conference " Scientists of Ijevsck Mechanical Institute to industry ". Ijevsck . 4. GEDEONOV P.P., BRONSKIY B.A. (1976).About some main properties and areas ofapplication fine-graned hydromicas. Building materials and products on the basis wastes of a industry and vermiculite. - Cheyabinsk : UralSRI of building project. 5. GEDEONOV P.P. (1991). Foamed fireproof covers based on the vermiculite Structural materials. N7. 6. GEDEONOV P.P., (1992). Compositions for foamedfireproofcovers based on the vermiculite. Structural materials. N6. 7. GEDEONOV P.P. (1981).Heat& technology testing based on foamed fireproof vermiculites covers. Heat resisting concrete, Materials and designs. UralSRI of building project, Cheyabinsk 8. GEDEONOV P.P. (1978).Instruction on structures, manufacturing and technology of a device foamed offire-resistantcovers on the basis vermiculite forfireproofingof building metal designs. UralSRI of building project, Chelyabinsk 9.GEDEONOV P.P, GEDEONOV S.P. (1992). Composition for fireproof of cover. The copyright certificate on the invention N 1723068 USSR 04 B 28 / 26 ( USSR ). Opening. Inventions, 12 10. GEDEONOV P.P. The concept of creation fireproof of covers on the basis hydrated micas and accompanying to them minerals // Interinstitute the collection "The problems of rational use of resources, perfec tion of technology and methods of account in building" N 1. Ijevsck . 1991.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
561
FIRE RESISTIVITY OF STEEL AND ALUMINIUM CONSTRUCTIONS PROTECTED BY A BLOATED COATING Y. Orlowsky^ K. Orlowska\ T. Shnal^ ^' ^ Department of Motor Ways, "Lvivska Politekhnica" State University, 12 Stepan Bandera Str., Lviv, 290046 , Ukraine. ^ Department of Building Constructions and Bridges, "Lvivska Politekhnica" State University, 12 Stepan Bandera Str., Lviv, 290046 , Ukraine
ABSTRACT A composition of the effective bloated coating for the protection of steel and aluminium constructions from fire was developed. Box section columns made of steel as well flat and shaped (finned) covers and platting (spons) used in civil engineering and shipbuilding were tested. The components of the construction were made of the St.3 steel and AMg-62T and 848-41 aluminium alloys. The steel were L..25 mm stick. The testing was being carried out in the fire chamber complying with the ISO 834 International standards requirements. The limit of fire resistively for steel elements amounted to 1.5 hours and for those made of aluminium it equaled one hours. A technique for calculating the limit of fire resistivity for steel and aluminium constructions protected by a bloated coating is suggested.
KEYWORDS Fire, aluminium, fireproof, fire resistivity limit, tests, shipbuilding, building industry.
INTRODUCTION The application of steel and specifically aluminium constructions in civil engineering and shipbuilding has proved to considerably decrease their fire resistivity. In the event of fire such constructions tend to rapidly lose their physical-mechanical properties which is bringing about their substantial damage. Already in first thirty seconds the temperature of 3...4 mm aluminium alloy plates soars to 400...500°C. At this temperature the aluminium alloy is known to reach the critical value of the burning limit. When there is a fire the average temperature inside ships and industrial premises reaches 700...900°C. At such temperatures the carrying construction element as well as body parts of ships not only lose their strength but even start to melt.
562 Fire protection of metal constructions consists in placing a low thermal conductivity layer between the fire source and the material proper. In this case the heating speed will be dependent upon the robustness of the construction and its protective layer thickness as well as upon the thermalphysical properties of the latter. Neglecting the influence of the coating surface upon heat radiation and dripping convection the protective qualities of the coating are determined by thermal conductivity, mass thermal capacity and density. Falling back on comprehensive physical-mechanical studies as well as heating and firing tests a fireproof coating for metal and aluminium alloy constructions capable of enhancing fire resistivity limit in comparison with that of the existing ones has been worked out. The mechanism of bloating the composition developed consists in the following. The temperature effect is responsible for the softening of the carbamide resin and its bloating on the part of incombustible gases released at the fireretardant hydrolysis. At the same time the formation of the carbon layer proceeds due to dehydrolisis of the polyhydrate compounds by the developed acids. The right quotas of the respective components have made it possible to minimize the shrinkage of the coating during the process of its hardening and drying. They also exclude the slide of the coating from the protected surfaces caused by its softening when heated. The introduction of boric anhydride solved in glycerine alongside of the expanded water-repellent pearlite sand and chopped glass fibre allows for the structuring of a foam and prevents its burning. This enhances the flame resistance limit of metal construction protected by the coating in question The mechanism of the reaction of bloated coating on fire is quite complex. The application of any of the known systems of differential equations for the thermal field calculus does not seem to be feasible. No way of determing the heating pattern of the bloated surface constructions has so far been suggested. Each coating composition calls for extensive fire testing which might enable one to set the relevant parameters correlating the coating thickness with the fire resistivity limit, the solidness of the construction and the critical temperature for the given metal. The aim of the study lay in conduction fire test of natural size civil engineering and shipbuilding construction elements protected by the developed coating as well as in working out suggestions for the optimal coating thickness.
MATERIALS, ELEMENTS AND TECHNIQUE OF TESTING Flame protection properties of the basic coating on the carbamide resin were tested by applying fire to the following construction elements: 600x600x3...25 mm steel plates, 1...8 mm thick aluminium sheets, 520x520 fins girdled box elements, 200x200x11 box section steel columns 1700 mm high, 600x600x5...8 mm aluminium alloy plates, fm fire-resisting bulkheads. Steel elements were made of St.3 steel and aluminium ones of AMg-62T and 848-4T sheet alloys. Testing was being carried out as suggested by the experts of the Fire Research Institute in Moscow in complianlians with ST CMEA 1000-88 standards requirements: T = 3451ogl0[8t + 1] + 20 (1) where t is heating time. Fire tests can be conducted also on the samples of smaller size which is in agreement with the accepted guidelines for testing fire refractoriness of building constructions. The testing was carried out in the fire chamber of the Fire Safety Research Institute (Balashikha, Moscow). The dimensions of elements were chosen to exceed the fire chamber opening by 10 mm on average on each side. The metal
563 foundation of samples was coated on one side with the flame protecting layer. The flames were applied by torches put ablaze by the ignition of liquid fuel sprayed into the chamber through a long burner with air blowing. The temperature in the fire chamber was monitored by the TXA-XP thermocouples and that of the heating of test samples was measured by chrome-aliminium alloy thermocouples the readings of which were recorded by the recording potentiometer. As stipulated in p.2.1.3 and 4.1 ST CMEA 1000-88 fire tests of metal constructions with the fire protecting layer can be conducted without loading and their critical temperature is taken for the limit state. The original temperature of the foundation in the process of standard fire testing at its isolation on one or both sides can be exceeded by 550°C for the steel elements, 200°C for the load carrying aluminium alloy elements and 250°C for the fencing elements. One-two mm thick B48-4T aluminium alloy sheets used for the outer plating of hydrofoil boats were isolated on one side and subjected to fire testing on the coated side
THE RESULTS OF TESTING Steel and aluminium plates heating curves are shown in figure 1. In the process of fire testing of elements the following changes were observed in the basic composition coating. After 3 minutes the coating turned black, after 4...5 minutes it became bloated all over by separate blisters. After 7...8 minutes the surface was covered by a bloated layer of foaming viscoelastic mass with a slow bloating and smoothing throughout. After 15 minutes the thickness of the bloated layer found itself on the increase and after 30...35 minutes it was equal to 60 mm. A thin 2...3 mm thick slagged layer was formed on the surface. The bloated layer became stable after 30...45 minutes. After 50...55 minutes the bloated layer surface became porous. Fibrous particles fluctuations could be observed. After 60...65 minutes the coating surface developed fractures which were getting exposed. The surface peeled off the burnt organic flakes. The bloated layer remained till the end of the fire test. IZKJV
1
1
•""
"•
'•' '
6j
lAAA IVKAJ
Ron ouu ACif\ OUU
1 :
kIV J
^
ACU\ *fUU
Ji
-^ 0
!*Hf:
20
40
60
1
3
1 ll< 1II
80
100 120 t , min Figure 1. Heating curves for coated steel and aluminium plates : 1- 3 mm thick steel plate ; 210 mm thick steel plate ; 3- 25 mm thick steel plate ; 4 - 5 mm thick aluminium plate ; 5- 8 mm thick aluminium plate ; 6- standard temperature / time curve.
The plates were being tested till the critical temperature on the pad surface was reached. It was 500°C for steel constructions and 250°C for those of aluminium alloy. The fire resisitivity limit for steel plates was found to be within 71... 105 minutes and that for the aluminium alloys within 60...72 minutes. After the samples had been completely cooled the coating on the surface was mostly preserved however it was covered with blisters and in some places there were considerable 5... 10 mm blisters.
564 The curves of the critical temperatures of the heated plates make it possible to conclude that an increase in the layer thickness leads to the increased fire resistivity limit (figure 2). 30
5, mm 1
25 20 15 10
/ll
1 ./
5
40
60
100 120 t ,mins Figure 2. The influence of plate thickness on the precritical temperature heating when coated : 1- steel plates protected with the coating developed ; 2- aluminium plates protected with the coating developed. 80
The introduction of metal constructions into the building industry is hampered by their low fire resistivity limit. Fire resistivity testing of steel and aluminium alloys constructions protected with a bloated coating is presented. The coating developed on the basis of carbamide resin allows for an increase in the fire resistivity limit of metal constructions. The experience of implementing this coating has proved highly effective . To prove the effectiveness of the suggested layer composition more fire tests were conducted on the steel column elements made of the even sided angle bar N 20 with the shelf thickness of 11 mm. The critical temperature for the columns was reached after 79... 83 minutes. The heating curves for the steel columns protected by the suggested layer are shown on figure 3.
e:c 1200
p I '
|»
MM«MM
1//I
400
r/
200
/ M
600
^ \
ll
1
1
1*11.1
Z'
X
1 Jim 1 Ll 1 • 1 ^•111
•
i^*-
1000 800
•II
1
0 0
20
40
60
80 100 120 t, mins Figure 3. Heating curves for the box section steel columns: 1-without any coating; 2 - standard temperature / time curve ; 3- a column coated by the protective layer developed. The critical temperature of aluminium alloy constructions is twice as low as that of the steel elements. That is why their protection with thermoinsulatory materials is indispensible even at a 15 min. fire resistivity limit. Fire testing on deck fins and bulkheads revealed that fire resistivity limit for the unprotected elements was as low as 60 minutes. This is the chief reason for recommending the
565 suggested coating composition for the protection of light load carrying and fencing elements and constructions in shipbuilding and building industry.
DISSCUSION AND SUGGESTIONS The results of the study have shown the critical heat resistivity of the coating (R^d/X) in the course of fire to be changing according to a couplete curve (fig. 4) depending on the temperature and the bloated layer thickness. The R=f(0) curve has 5 distinct areas: The ab area is responsible for the increase in heat resistivity without changing the thickness of coating. Within the temperature range of 20... 170°C as a result of the rise in the coating temperature and a drop of the heat conductivity coefficient /I; the be area revealing a sharp drop in /? as a result of foaming coating and a subsequent decrease of 1; the cd is revealing a gradual increase of R which as accompanying by the bloating of the coated surface and a slight decrease of A; the de area revealing a sharp increasin R resulting from an intense bloating which becomes stable at the maximum temperature heat of 820°C at which X begins to rise with the increasing of the diameter; the ef area in which R begins to drop because the bloated layer thickness is decreasing and X keeps on increasing.
R, m^hour ^C/ccal 0.6
60
0.4
40
0.2
A
20
[AM 200
400
600
800
1000 ^.°C
10 203^40^^90
Figure 4. Dependency heat resistivity coating (1) on temperature, time and density of the bloated layer (2) at fire test standard temperature / time curve. The received dependences of R=f(h 9, dp, X) made it possible to construct a nomogram (figure 5) for detecting the thickness of the layer providing for fire resistivity of aluminium alloys in the course of 60 min. The nomogram in questions built for the 40-400 m"^ order of elements robustness . The experimental 16,5x9x0,5 cm pad plates were corresponding to the 200 m"^ order of robustness. The remaining dimensions were taken proceeding from the aluminium alloys rolled stock standards. The temperature AOat was calculated by the formula: A^. = — ^ • ^ • - ^ • ( ^ , - ^ > /
1.^ k
X
^aPa
(2)
566 where dj/ X is coating heat resistivity( m^hour °C), Ap/V the element robustness coefficient which as a ratio of the surface Ap inchned towards the fire source and the metal volume, m'^; c^ pa are mass thermal capacity and metal density respectively, ccal/(kg °C) and kg/m^; 0 ^ ^m are the temperature of the medium andthatof the metal in °C; At, AOat stand for an increase of the heating time and metal temperature; K is the complete heat transfer factor established using ^ = 20 + 3 * 1 0 - ^ ( ^ / + ^ „ ^ ) * ( ^ , + ^ J (3) where 0^ and 9^ are the temperature values of the medium and the metal, °K.
600
h, m m
400
200
100
Figure 5. Nomogram temperature.
200
300
400 ApA/, m-^
for detecting the thickness of layer providing for section factor
and
CONCLUSIONS The developed bloating layer raises the fire resistivity limit of steel elements to 1.5 hours and that of aluminium alloys to 1 hour. Fire resistivity limit of construction elements is dependent on the section thickness. An increase in the thickness of steel elements from 3 to 25 mm prolongs the refractoriness limit by 40 minutes. A similar though slighter procedure for aluminium alloys of the order of 1-3 mm would prolong the respective time by 15 minutes.This coating is recommended for increasing fire resistivity of load carrying building constructions made of steel such as columns, cross-bars, undercharge bars, etc. and also those made of aluminium alloys (light plating, fencing walls with sandwich like warmth-keeping lagging, membrane type constructions and also in shipbuilding (load carrying and fencing elements of ship bulkheads, horizontal span of plating and beam sets, carlings etc. in ship hulls). Determining the layers thickness is feasible basing upon the extend of construction robustness and the metal critical temperatures falling back on the nomograms built on the fire test taking into account the changing thermal-physical characteristics of coating in the course of being bloated by temperature according to the standard temperature / time curve
Session A8 CONNECTIONS
This Page Intentionally Left Blank
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
569
New Design Provisions for Cold-Formed Steel Bolted Connections
R. A. LaBoube^ and W. W. Yu* ^Department of Civil Engineering, University of Missouri-Rolla Rolla, MO 65409, USA
ABSTRACT The 1996 Edition of the American Iron and Steel Institute (AJSI) Specification for the Design of Cold-Formed Steel Structural Members does not address the limit states of shear lag, staggered holes in a tension member connection, and limited elongation of a bolt hole prior to ultimate bearing capacity. Research at the University of Missouri-Rolla has examined the behavior of bolted connections in cold-formed steel flat sheet, angles and channel members subject to tension load. Based on the findings of the UMR research, additional design provisions were developed. The UMR equations provide for a design approach for cold-formed steel members and connections that is similar to the design approach used for hot-rolled steel design in the United States.
KEYWORDS Cold-formed steel, connections, bolts, shear lag, tension member, staggered holes, design strength. INTRODUCTION Design provisions contained in the 1996 Edition of the American Iron and Steel Institute's Specification for the Design of Cold-Formed Steel Structural Members (Specification, 1996) do not address the following limit states: influence of shear lag, staggered holes in a tension member connection, and limited elongation of a bolt hole prior to ultimate bearing capacity. Researchers at the University of Missouri-Rolla (UMR) have examined the behavior of bolted connections in cold-formed steel flat sheet, angles, and channel members subjected to tension
570
load. The intent of the research was to develop design guidelines for the above limit states. Based on the findings of the UMR research, design equations were developedfi*oadoption by the AISI Specification. This paper will summarize the three year UMR research effort and its findings. BEHAVIOR OF FLAT SHEET CONNECTIONS The behavior of connections of flat sheets were investigated for the limit states of both bearing andfiracturein the net section. The bearing investigation was focused defining the serviceability limit of a bolted connection. Tensile coupons tests were conducted to obtain the mechanical properties of the steel sheets. Table 1 summarizes the measured thicknesses and mechanical properties of the sheet steels used in this study. TABLE 1 Material Properties lickness (mm) 1.02 1.78 3.05
Fy (MPa) 247 221 253
Fu (MPa) 385 362 366
Fu/Fy 1.56 1.64 1.45
Elongation (%) 50 50 44
Each test assembly consisted of two flat sheet specimens of like thickness (Fig. 1). All test assemblies used 13.7 mm (Vi in.) diameter A325T bolts in 14.29 mm (9/16 in.) diameter punched holes. The bolts were installed snug tight to simulate the bolt tightening procedure used in field applications. Carril et al. (1994) provided details regarding the seventy-five tests that were conducted. The elongation of the bolted connections was measured using an LVDT that was attached to the test assembly, as shown in Fig. 2. The applied load and elongation readings of the connection were recorded at one second intervals using a computer acquisition system. Typical Ibaddeflection curves are presented by Carril et al. (1994). Deformation At Bolt Hole When deformation at a bolt hole is a design consideration, the present Specification provisions (1996) will overestimate the connection capacity. Based on the research of Carril (Carril, 1994; LaBoube et al., 1996), the following design equation was proposed: P„=1.93dtF,
(1)
where d = bolt diameter, t = bare sheet thickness, and F^ = tensile strength of the sheet.
571 Although Eq. 1 is similar in format to the AISC (1993) equation, ?„ = 2.4 d t F^, for the same limit state, Eq. 1 would create a discontinuity in connection strength for sheet thickness of 4.796 mm (3/16 in.). Therefore, the following transition equation was developed (LaBoube and Yu, 1995) and has been adopted for the next edition of the AISI Specification: P„ = (0.183t+1.53)dtF,
(2)
where SI units must be used. Fracture in Net Section The AISI (Specification, 1996) and AISC (Load, 1993) design approaches differ for determining the net section strength of a flat sheet bolted connection. AISc has applied a shear lag factor for the net section strength evaluation, where as AISI has based its' design on only net area. For example, for bolted connection without washers under the bolt head or the nut, the AISI nominal strength would be determined by Eq. 3, whereas, AISC would apply Eq. 4: P„ = ( l - r + 2.5rd/s) F, < F,
(3)
Pn = A,F,
(4)
where A^ = effective net area, A^ = net area of connected part, F^ = tensile strength of connected part, r = force transmitted by the bolt or bolts at the section considered, divided by the tension force in the member at that section, s = spacing of the bolts perpendicular to the line of stress, or gross width of the sheet for a single line of bolts. For flat sheet connections, A^ = A^. Based on the results of 30 connection assembly tests (Fig. 1), fracture in the net section strength was studied (Carril, 1994). A comparison of the tested to computed capacities, Eqs. 3 and 4, indicated that the AISI Specification was marginally more conservative than the AISC Specification. Table 2 summarizes the statistical findings. TABLE 2 Comparison of AISI and AISC for Fracture in the Net Section
Mean COV
AISI, Eq. 3 1.135 0.061
AISC, Eq. 4 0.952 0.056
Staggered Bolt Holes For staggered hole failure paths, the AISC Specification appUes the following equation for determination of net area:
A„ = Ag-nAt + (Es'V4g)t
(5)
572
where Ag = gross area, iib = number of bolt holes, dh= diameter of hole, s' = longitudinal centerto-center spacing of consecutive holes, g = transverse center-to-center spacing between fastener gage lines. The diameter of the hole, dj,, is taken as the nominal bolt diameter plus 1.59 mm for a standard hole. Based on a limited test program, Holcomb et al. (1995) determined that the AISC (Load, 1993) design approach for evaluating the connection strength, when a staggered failure pattern is possible, is marginally unconservative. Holcomb determined that when using Eq. 5, the computed load capacity was, on the average, 11 percent greater than the tested load capacity. Therefore, the AISI Specification has adopted the following equation for the evaluation of the net section of a tension member with staggered holes: A„ = 0.9[Ag-nAt + (Zs'2/4g)t]
(6)
The nominal strength is determined by using Eq. 4 with A^, defined by Eq. 6.
BEHAVIOR OF OTHER THAN FLAT SHEET CONNECTIONS The behavior of connections of angles and channel sections were investigated for the limit states of both bearing andfiracturein the net section. Table 1 hsts the measured thicknesses and mechanical properties for the sheets used to fabricate the sections. Both angle and chaimel sections were subjected to a tensile load parallel to their longitudinal axis. Each test assembly consisted of two sections connected back-to-back forming a single shear connection. All test assembhes used 13.7 mm (Vi in.) diameter A325T bolts in 14.29 mm (9/16 in.) punched holes. The bolts were installed snug tight. Fracture in Net Section For both angle and Channel sections, shear lag may have a negative influence on the tension capacity. The Specification for the Design of Cold-Formed Steel Structural Members (1996) does not provide shear lag design guidance. AISC (Load, 1993) stipulates the shear lag influence by use of the following area reduction factor equation: U = 1 - x/L ^ 0.9
(7)
where x = distancefiromthe shear plane to the centroid of the section and L = length of the connection. The study of Holcomb et al. (1995) developed similar design relationships for cold-formed steel angles and channels. Holcomb, in his study, conducted 54 tests for angle sections and 51 tests using channels. Based on additional analysis of Holcomb's data, LaBoube and Yu (1995; 1996) proposed the following design relationships which subsequently have been adopted for the next edition of the AISI Specification:
573 U = 1.0 for members when the load is transmitted directly to all of the cross-sectional elements. Otherwise, the reduction coefficient U is determined as follows: (a) For angle members having two or more bolts in the line of force U = 1.0-1.20 x/L <0.9 (8) but U shall not be less than 0.4 (b) For Channel member having two or more bolts in the line of force U = 1.0-0.36 x/L <0.9 (9) but U shall not be less than 0.5 The above equations are valid when the distance along the line of forcefi-omthe edge of the connected part to the center of the nearest hole is greater than or equal to 1.5 d, and the distance along the line of force between centers of adjacent holes are greater than or equal to 3d, where d is the bolt diameter. The nominal tensile strength is computed by the following: P„ = AgF, Pn = AeF,
(10) (11)
Where Ag = gross area of cross section, Fy = design yield stress, and A^ = U A„ ,effective net area. The net area, A^, is computed by using Eq. 6. The research also showed that for single bolt connections, bearing controlled the nominal strength of the connection.
SUMMARY Based on the results of UMR studies of bolted connections for flat sheets, and angles, and channels cold-formed fi-om flat sheet, significant changes have been adopted for bolted connection design in the AISI Specification. These changes affect both bearing andfi*acturein the net section computations.
ACKNOWLEDGMENTS The authors wish to express their appreciation to former graduate students, J. L. Carril and B. D. Holcomb for their diligence. Also, the fimding provided by the American Iron and Steel Institute is acknowledged. Technical guidance was provided by the AISI Connections Subcommittee (M. Golovin, chairman).
REFERENCES
574
Carril, J. L., LaBoube, R. A., and Yu, W. W. (1994), "Tensile and Bearing Capacities of Bolted Connections," First Summary Report. Civil Engineering Study 94-1, Cold-Formed Steel Series, Department of Civil Engineering, Center for Cold-Formed Steel Structures, University of Missouri-Rolla Holcomb, B. D., LaBoube, R. A., and Yu, W. W. (1995), "Tensile and Bearing Capacities of Bolted Connections," Second Summary Report. Civil Engineering Study 95-1, Cold-Formed Steel Series, Department of Civil Engineering, Center for Cold-Formed Steel Structures, University of Missouri-Rolla LaBoube, R. A., and Yu, W. W., (1996), "Additional Design Considerations for Bolted Connections," Proceedings of the Thirteenth International Specialty Conference on ColdFormed Steel Structures, University of Missouri-Rolla, RoUa, MO LaBoube, R. A., Yu, W. W., and Carril, J. L. (1996), "Serviceability Limit State for ColdFormed Steel Bolted Connections," Proceedings of the Third International Workshop on Connections in Steel Structures III, Elsevier Science Ltd Load and Resistance Factor Design Specification for Structural Steel Buildings (1993), American Institute of Steel Construction, Chicago, IL Specification for the Design of Cold-Formed Steel Structural Members (1996), American Iron and Steel Institute, Washington, D.C.
575
-Specimen
-Specimen
Side View
Front View Fig. 1 Typical Test Assembly
576
Side View
Front View Fig. 2 LVDT Attachment of Test Assembly
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
577
LOAD-SHARING OF PRESS-JOINTS IN THIN-WALLED STEEL STRUCTURES KariKolari VTT Building Technology P.O. Box 1807,02044 VTT, Finland
ABSTRACT This paper summarizes a series of connection tests on cold-formed steel plates using mechanical fasteners. The paper covers clinched, screwed and riveted connections subjected to both static and cyclic loading. The results of multiple fastener connection tests demonstrated that, in spite of the low deformation capacity of press-joints, the capacity of a multiple-fastener connection almost reached the sum of an individual fastener's capacity. Therefore, it appears that press-joints do share load efficiently. The results of the cyclic loading tests revealed the gratifyingly good performance of the press-joints in cyclic loading. KEYWORDS Steel structures, press-join, clinch, rivet, screw, shear strength, cyclic loading, connection, joint INTRODUCTION Press-joining (clinching) and self-pierce riveting are relatively new fastening techniques used in joining sheet metal. The methods are increasingly being used in sheet metal applications that require high strength and high-speed joining. The self-pierce riveting process was initially developed to eliminate the need for pre-drilled corrosion prone holes in metal components (King et al. 1995). Punch
llD IPD Figure 1: The principle of press-joining. Insertion depth ID and penetration depth PD, (Fig. by Axel Schulte) Clinching is a combination of drawing and forming that locks together two layers of sheet metal. Therefore there is no need to buy, sort or feed separate fasteners. According to Liebig & Beyer (1987), "the background of press-joining is to be found in the German Standard: DIN 8593 part 5 under the
578 process group. Shear insertion joining". The principle of press-joining is shown in Figure 1. The number of published documents on the behaviour of press-joining in building industry applications remains rather low. Published test results are mainly oriented towards the behaviour and use of press-joins in the sheet metal work industry, rather than towards the critical areas of building structures, as also pointed out by Davies et al. (1996). This paper describes a series of tests on cold-formed steel plates connected using mechanical fasteners and covers the testing of clinched, screwed and riveted connections to study their behaviour in both static and cyclic loading. This paper is based on the work reported by Kolari & Hakka-Ronnholm (1998). TEST PROGRAMME A series of lap shear tests were carried out for static loading of single and multiple fastener tests. A total of 19 cyclic loading tests were also conducted. Materials The selection of the two steel sheet materials and self-tapping screws was based on those likely to be used in the Nordic countries. The materials were continuously hot-dip zinc-coated to 275 glvc^. Tensile tests were performed at Helsinki University of Technology according to SFS-EN 10 (X)2-l (1990) by Lu et al. (1998). The results are summarized in Table 1. TABLE 1 MATERIAL PROPERTIES OF S 3 5 0 G D + Z 2 7 5 (LU ET AL. 1998) Nominal Sheet Thickness,
Observed Thickness,
Yield Strength,
Tensile Strength,
mm
Re, N/mm^
Rm, N/mm^
mm
Test Direction Relative to Rolling Direction, mm
LO L5 L5
Longitudinal Longitudinal Transverse
0.96 1.42 1.42
380 378 403
484 492 495
Percentage Elongation After Fracture, A[%] 23.8 25.3 16.5
Fasteners The objective of the research programme was to study the behaviour of two different press-joint types: circular and rectangular. The objective, however, was not to reveal differences between the brand names, such as Attexor and Trumpf. Besides the press-joints, a few screwed and riveted connections were subjected to testing for reference. At the time of manufacturing, appropriate tools for Trumpf pliers were not available for the total thickness range of 2 to 3 mm for the present study. The tool used with the pliers was intended for connections of total thickness of 3.1 to 3.5 mm. Therefore, the connections in question may have properties inferior to those that are manufactured with appropriate tools. The tool and fastener details are given in Tables 2 and 3.
579 TABLE 2 PRESS-JOINT DETAILS
Type / Total connection thickness, mm
Pliers
Tools
Circular / 2.0
Attexor Spot Clinch 0302 AS Attexor Spot Clinch 0302 AS TrumpfTF 350-0
SR504 10 45 20 (punch 0 4.5 mm) SR504 10 42 20 (punch 0 4.2 mm) No 3. 128748*
Circular/3.0 Rectangular / 2.0 / 3.0 ±
,
Directive bottom thickness, mm 0.65 1.0 not adjustable
,n,
Note! The tool is intended for connections with total thickness of 3.1-3.5 mm.
TABLE 3 SCREWED AND RIVETED CONNECTIONS Fastener
Type
Diameter
Length
Self-pierce rivet
HENROB, R50743CM03/C05
5.0 mm
7.0 mm
Self-tapping screw
SPEDEC SL4-F-4,8 x 16
4.8 mm
16 mm
Each fastener and its location were measured prior to the shear tests. The bottom thickness and the width of the punch side shown in Figure 2 were measured for all the press-joints. Micrographs were taken for the development of design resistance calculations (Figure 3).
Circular press-joint
Rectangular press-joint:
Screw diameter
.... (C3)
f
Figure 2: Fastener dimension measurements
Figure 3: Micrographs of Circular and Rectangular press-joint, sheet thickness t=1.0 mm Sample instrumentation for static loading Shear tests were performed by VTT Building Technology using an Instron testing machine. A gauge length extensometer was attached directly to the sample and the data was recorded by a PC, as shown
580 in Figure 4. The loading rate (mm/minute) in all tests was controlled to ensure the test was completed within a time frame of 30 to 240 seconds, as given in the standard, AS/NZS 4600 (1996).
Figure 4: Shear test arrangement in static loading tests of multiple-fastener connections. TEST RESULTS Static loading The objective of the multiple fastener connection tests was to establish the load-sharing between the fasteners. The capacity of multiple fastener connections is usually less than the sum of a single fastener connection capacity, if the fastener's behaviour is brittle-like. The multiple fastener connection capacity depends on the load-sharing between the fasteners and the load sharing in turn depends on the deformation capacity of a single fastener. It was known that the deformation capacity of single press-joints is lower than that of screws (Macindoe & Hanks 1994, Davies et al. 1996, Kesti et al. 1997). Therefore, load-sharing tests were regarded as essential before press-joining could be used in building industry applications, as also pointed out by Macindoe & Hanks. 5 fasteners, t s 1,5mm Rivet
35 30 25
Srrfw
i—'—"~:^-i-—-r" mii-——'^ r'
5
t^^'^
'
/ Z^'^'*—1
1 20
10
\ ^
1 //^>/_J_^^«— 1r\ '^^-^^^
1
\
BITILOING TECHNOLOGY
diu .«• n-T-
r^
^^
1
i
,
.
1
1
\
^-
4 5 6 Displacement, mm
1
\
;
1
:. — i
>• - | -
: e n c 3 e n C3 C3
3
^ i
! ^ °0 °0 0 1 1 — • • 1 r r r 1—^-: i DD D D D 1
|f-i^7^-.,v-:--£ I
1
\
\
\
\-
7
Figure 5: Load displacement curves when number of press-joints N = 5, t = 1.5 mm Load displacement curves for multiple fastener connections showed that the scatter of results in all the tests remained low. Typical load displacement curves for a sheet thickness of 1.5 mm are shown in
581 Figure 5. Summaries of 154 tests for multiple fastener connections are given in Tables 4 and 5. Regardless of the low deformation capacity of press-joints, the capacity of a multiple fastener connection closely approaches the sum of a single fastener's capacity. TABLE 4 SUMMARY OF MULTIPLE FASTENER CONNECTIONS TESTS FOR SHEET TfflCKNESS 1.0 n m i Fastener Orientation, Type
N(e)
Length-wise Length-wise Length-wise Transverse Transverse Transverse Circular Circular Circular Rivet Rivet Screw Screw
3 4 5 3 4 5 3 4 5 3 5 3 5
RA Average Peak Load, [kN]
RN = R A / N Load/Fastener, [kN]
Rl Average Single Fastener Peak Load, [kN]^"^
3.07 4.15 5.13 4.31 5.89 7.43 6.67 9.43 11.73 14.14 20.92^*'^ 11.11 17.65
1.02 1.04 1.03 1.44 1.47 1.49 2.22 2.36 2.35 4.71 4.18 3.70 3.53
1.05 1.05 1.05 1.41 1.41 1.41 2.32 2.32 2.32 5.01 5.01 3.57 3.57 Average^*^^ Stand, dev.^'^
RN/ /^i 0.97 0.99 0.98 1.02 1.04 1.05 0.96 1.02 1.01 0.94 0.84 1.04 0.99 1.000 0.048
At a l / ' 1 Q Q » \ W c;„„ ^"^ Tension failure of the sheet, Oave= 20 920/(0.96*(50-5)) = 484 N/mm ''"^ Results indicated with '^^^' have been excluded N equals the number of fasteners per specimen
TABLE 5 SUMMARY OF MULTIPLE FASTENER CONNECTIONS TESTS FOR SHEET THICKNESS 1.5 m m
V 7^1
Fastener Orientation, Type
N
RA Average Peak Load, [kN]
RN Load/Fastener, [kN]
Rl Average Single Fastener Peak Load, [kN]^'^
Length-wise Length-wise Length-wise Transverse Transverse Transverse Circular Circular Circular Rivet Rivet Screw Screw
3 4 5 3 4 5 3 4 5 3 5 3 5
5.37 7.12 8.88 7.90 10.66 13.05 7.42 10.35 12.36 24.12 31.99^**^ 19.88 29.40
1.79 1.78 1.78 2.63 2.67 2,61 2.47 2.59 2.47 8.04 6.40 6.63 5.88
1.82 1.82 1.82 2.76 2.76 2.76 2.62 2.62 2.62 8.36 8.36
0.99 0.98 0.98 0.95 0.97 0.95 0.94 0.99 0.94 0.96 0.77
-
-
Average^*^^ Stand, dev.^'^
0.964 0.017
Single fastener shear tests performed at Helsinki University of Technology ^^^ Tension failure of the sheet, Oave= 31 990/(1.42*(50-5)) = 501 N/mm ^^ '^ Results indicated with ^''^ have been excluded
582 Cyclic loading In total, 20 tests were performed to establish the effect of cyclic loading on the shear strength of single fastener connections. The procedure for shear tests was the following: 1. Load the sample with 100 x (5 + 30 + 150) = 18 500 cycles, as shown in Figure 6. 2. Carry out the static shear strength test of samples after cyclic loading.
Figure 6: Loading history in cyclic loading tests The objective of cyclic loading tests was to simulate the loading history of a typical frame structure over 100 years. The test values of variable actions were based on the following assumptions (Figure 6): a) Characteristic snow load acts 4 to 6 times in a year, b) Characteristic wind load acts 20 to 30 times in a year, c) Apart from the peak loads, some smaller variable loads also act on a structure. The load values are given in table 6. Cyclic loading had only a negligible effect on the peak load of press-joints, as shown in Table 7. TABLE 6 LOAD VALUES FOR CYCLIC LOADING TESTS (MATERIAL S 3 5 0 G D + Z , THICKNESS 1.0 m m ) Fastener
Fo
Circular Rectangular / Transverse Rectangular/longitudinal Screw
0.386 0.235 0.175 0.489
L543 0.941 0.701 1.956
L286 0.784 0.584 1.630
0.772 0.471 0.351 0.978
2.315 1.412 1.052 3.573
TABLE 7. AVERAGE PEAK LOAD VALUES FOR SINGLE FASTENER SPECIMENS AFTER CYCLIC LOADING Fastener
Shear Load, kN After Cyclic Loading
Static
Kcyc
Circular Rectangular / Transverse Rectangular / longitudinal Screw
Mean
Stdev.
Mean
Stdev.
Kyc/ y^stat
2.386 1.412 0.997 3.805
0.045 0.067 0.058 0.116
2.315 1.412 1.052 3.573
0.069 0.034 0.026 0.173
1.031 1.000 0.948 1.065
Average Stand, dev.
1.011 0.050
583 SUMMARY AND CONCLUSIONS The performances of press-joined, screwed and riveted connections were studied experimentally in static and cyclic shear using thin cold-formed steel. Special attention was paid to the behaviour of multiple fastener connections, although some corrosion tests and cyclic loading tests for single fastener connections were carried out for reference. The results of the multiple fastener connection tests showed that, in spite of the low deformation capacity of press-joints, the capacity of a multiple fastener connection closely approaches the sum of an single fastener's capacity. Therefore, it appears that press-joints are able to share load efficiently. The corrosion tests indicate adequate performance of single fastener connections. No severe signs of local corrosion (crevice corrosion or pitting corrosion) could be observed. The rectangular press-joints and the screwed connections exhibited some indications of corrosion but the degree of corrosion remained lower in the crevice than on the upper side of the panels tested. The circular connections showed no signs of corrosion. The following conclusions can be drawn from the tests: 1. The rivets exhibited about 70 to 360% greater peak load than the press-joints. 2.
When the orientation of the rectangular press-joint was changed from length-wise to transverse, the peak load increased by approx. 45%.
3.
The press-joints exhibited 1/10 to 1/3 of the deformation capacity of the rivets.
4.
In the tests of Davies et al. (1996), deformation capacity was higher, very near the value of 3 mm. Therefore it is obvious that deformation capacity is a function of the press-joint tools used.
5.
Press-joints were able to share load efficiently.
6.
The circular press-joints were sensitive to manufacturing parameters. There was a trend of decreasing peak load with an increase in the bottom thickness of the joint
7.
Cyclic loading had no effect on the subsequent shear behaviour of screwed and pressjoined connections.
8. The shear behaviour of press-joints after corrosion tests was good. 9. The tool used with the pliers was intended for connections with a total thickness of 3.1 to 3.5 mm, while the studied thicknesses were 2.0 and 3.0 mm. Therefore, the connections in question may have properties inferior to those that are manufactured with appropriate tools. 10. Deformation capacities of the tested press-joints were inadequate (0.4...3.0 nmi) to pass the requirements given for a fastener's deformation capacity in ECCS (1983). The ECCS's limit of 3 nmi is shown in Figure 7. According to ECCS, "a deflection less than 3 mm can also be sufficient, but then secondary forces and forces caused by temperature variation must be considered''. The background of deformation capacity requirements should be studied more to ensure the applicability of press-joints in building industry applications.
584 Load
Pr: remaining strength Pd: design strength 3 mm
Deflection
Figure 7: Requirements to ensure sufficient deformation capacity (ECCS 1983) ACKNOWLEDGEMENTS The work reported here was sponsored by the Technology Development Centre (TEKES), Rautaruukki Oyj and VTT Building Technology. REFERENCES AS/NZS 4600. (1996). Cold formed steel structures. Homebush: Australian/New Zealand Standard. ECCS. (1983). The Design and Testing of Connections in Steel Sheeting and Sections. European Recommendations for Steel Construction, Publication No 21, ECCS Conmiittee TC7, Working group TWO 7.2. 176 p. Davies, R., Pedreschi, R. and Sinha, B.P. (1996). The Shear Behaviour of Press-Joining in ColdFormed Steel Structures. Thin-Walled Structures 25:3, 153 - 170. King, R. P., CSullivan, J. M., Spurgeon, D. & Bentley, P. (1995). Setting load requirements and fastening strength in the self-pierce riveting process. In: Taylor & Francis (Ed) Advances in manufacturing technology IX, Proc. of the 11th Nat. Conf. on manuf. Res., Leicester, GB. pp. 57 - 61. Kolari, K. & Hakka-Ronnholm, E. (1998). Chnched Connections in Thin-Walled Steel Structures Shear Strength and Corrosion. Espoo: Technical Research Centre of Finland (VTT), VTT Building Technology. 30 p. + app. 30 p. (Report RTE38-IR-16/1998). Kesti, J., Makelainen, P. & Piirainen, A. (1997). Puristusliitosten leikkauskokeet (shear tests for press joints). Tutkimusselostus TeRT-97-08. Teknillinen Korkeakoulu, Terasrakennetekniikan Laboratorio. Liebig, H. P. & Beyer, R. (1987). Press joining of especially coated steel and aluminium sheets. In: Advanced Technology of plasticity. 1987. Vol. E, Stutgart, FRG, 24- 28 Aug 1987. Berlin: SpringerVerlag. pp. 933-940 Lu, W., Kesti, J. & Makelainen, P. (1998). Shear and Cross-Tension Tests for Press-Joins. Espoo: Helsinki University of Technology, Laboratory of Steel Structures. (Report 7.) Macindoe, L. & Hanks, P. (1994). Standard Tests for Cold-Formed Single Fastener Connections. In: Australasian Structural Engineering Conference, Sydney 21-23 September 1994, vol 1. pp 253 - 257. SFS-EN 10 002-1. (1990). Metallien vetokoe. Osa LMenetelma (Metallic materials. Tensile testing. Part 1: Method of test.) 33 p.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
585
STATIC AND CYCLIC SHEAR BEHAVIOUR ANALYSIS OF THE ROSETTE-JOINT P.Makelainen\ J.Kesti\ W.Lu\ H.Pastemak^ and S.Komann^ ^Laboratory of Steel Structures, Helsinki University of Technology, P.O.Box 2100, FIN-02015 HUT, Finland ^Chair of Steel Structures, Brandenburg Technical University Cottbus, P.O.Box 101344, D-03013 Cottbus, Germany
ABSTRACT In this paper, the results of shear tests and Finite Element (FE) analysis under static and cyclic loads for a completely new joint named ROSETTE-joint are presented. The test set-up is introduced and the failure modes and the test results are presented. With NISA/DISPLAY III program, the FE-model is validated by the test results and can be used to analyse the effect of the varying sheet thickness on the shear strength of the ROSETTE-joint. In addition, a thermoelastic stress analysis enables a possibility to get a quick overview of the stress distribution in the critical areas and to find out the progression of the failure mechanisms. The results of the infra red measurements and the FE-analysis using ABAQUS program are presented.
KEYWORDS ROSETTE-joint, shear strength, finite element analysis, thermovision technique, infra red measurement
INTRODUCTION ROSETTE-joining is a completely new press-joining technique for cold-formed steel structures (Makelainen et al (1998a)). ROSETTE-joining has several advantages over other common joining methods used in steel construction, such as riveting, bolting and welding. The joint is formed using the parent metal of the sections to be connected; thus there are no additional fixings. Nor is there need for heating, which may cause damage to protective coatings. The ROSETTE-technology was developed for fully automated, integrated processing of strip coil material directly to any kind of light gauge steel frame components for structural applications such as stud wall panels or roof trusses (Kaitila (1998)). The integrated production system makes prefabricated and dimensioned frame components, making possible the just-in-time (JIT) assembly of frame panels or trusses without further measurements or jigs.
586 The ROSETTE-joint is formed in pairs between prefabricated holes in one jointed part and collared holes in the other part. Firstly, the collars are snapped into holes. Secondly, the ROSETTE-tool head penetrates the hole at the connection point, expands and is pulled back with a hydraulic force. The expanded tool head crimps the collar part against the hole part. Torque is enhanced by multiple teeth in the connection perimeter. The joining process is illustrated in Figure 1 and the finished ROSETTE-joint is shown in Figure 2.
4 i ^
[ffl n D
• ^
i
Figure 1: ROSETTE joining process
Figure 2: ROSETTE-joint
A large amount of tests for estimating the shear strength of the ROSETTE joint under static load was carried out in the Laboratory of Steel Structures at HUT (Makelainen et al (1998b), Kesti et al (1998)). An extensive FE-analysis on the behaviour of the ROSETTE joint was also performed (Lu et al (1998)). The test results of varying the steel sheet thickness were presented. In addition, using NISA/DISPLAY III program, the FE-model was validated by the test results. When dealing with geometrically complicated profiles, it is necessary to control the FE-analyses with experimental research. For this purpose, a number of tests were performed, where the thermoelastic stress analysis was done using the Thermovision technique. The research performed at BTU Cottbus has shown that this method enables a possibility to get a quick overview of the stress distribution in the critical areas and find out the progression of the failure mechanisms. The results of the infra red measurements and the FE-analysis are presented.
SHEAR BEHAVIOUR OF ROSETTE-JOINT UNDER STATIC LOADS Static Shear Load Tests Shear tests were carried out for both flat specimens and folded specimens. The tested specimen was aligned between gripping devices of the Roell & Korthaus testing machine. A maximum straining rate of 1.0 mm/min was used in all tests. The test was terminated when a deformation of about 3 mm was reached. Dimensions of the test specimens are shown in Figure 3. The test set-up was shown in Figure 4. The failure modes are described in Table 1 and shown in Figure 5. Finite Element Analysis The FE-analysis was carried out using EMRC:s NISA / DISPLAY III applications in the Laboratory of Steel Structures at HUT. NISA is a la^ e scale general purpose finite element program with diversified capabilities for the analysis of a wide range of engineering applications.
587 59
45 P2( •fet
?r
12.5 12.5
U¥
m
020 ,
ra
2.5==*n,5
B
'A A .. 35
hli
116.5
B-B
I collar part
hole part
(a) flat specimen
collar part
75
hole part
(b) folded specimen
Figure 3: Dimensions of the test specimens TABLE 1 FAILURE MODES OF ALL SPECIMENS Specimens
Failure Modes Flat Specimens Folded Specimen Hp-1.0 Local buckling of the sheet along the Local buckling of the sheet along the Cp-l.O compressed edge of the hole compressed edge of the hole Hp-1.0 Local buckling of the sheet along the Local buckling of the sheet along the Cp-1.5 compressed edge of the hole compressed edge of the hole Hp-1.5 Local buckling of the sheet along the Failure in collar Cp-l.O compressed edge of the hole and failure in collar Hp-1.5 Local buckling of the sheet along the Local buckling of the sheet along the Cp-1.5 compressed edge of the hole compressed edge of the hole and failure in collar Hp: hole part Cp: collar part 1.0 and 1.5: thickness of steel sheet [mm]
u
@
Ig : Gauge length for measuring the joint displacement /^: Undamped length of the specimen Figure 4: Test set-up
Figure 5: Failure modes of the specimens
| |
1 1 1 J
588 The quadrilateral isoparametric 8-node 3-D general shell element (NKTP=20,NORDR=2) with 6 degrees of freedom per node was chosen for modelling the sheet. The springs were introduced to model the collar. The 2-node uniaxial massless spring in three dimension (NKTP=17, N0RDR=1) with three translation degrees of freedom per node was chosen for modelling the collar. The springs were arranged in the same plane of the hole part sheet and connected to the collar part sheet using the simplest form of MPC equation i.e. coupled displacements (*CPDISP). The symmetry of the specimen was introduced. The linear piecewise model for the material was established according to the results of the test. Modulus of elasticity in all directions was 205 000 N/mm^ and Poisson's ratio was 0.3. Von Mises /Hyushin yield criterion was used with initial yield stress of 270 N/mm^. The load-deflection curve for the ^-analysis and the test results for flat specimen and that for folded specimen are shown in Figure 6. COMPARISON OF TEST RESULTS AND FE ANALYSIS(FLAT SPECIMEN) -1:test 10(1.5mm) -2:test5(1.5mm) 3:FEM(1.5mm) -4:test 10(1.0mm) -5:test4(1.0mm) 6:FEM(1.0mm)
COMPARISON OF TEST RESULTS AND FEM ANALYSIS (FOLDED SPECIMEN)
Displacement [mm]
Figure 6: Validation of FE-model From Figure 6, it can be seen that the FE-analysis results fit the test results very well in the increasing part of load-deflection curves. The comparison of the test results and those of the FE-analysis is shown in Table 2. TABLE 2 COMPARISON OF THE MAXIMUM SHEAR LOADS OF TEST RESULTS AND FE-ANALYSIS Specimens
Thickness [mm] Test results [kN] Hole Collar Maximum Minimum part part value value Flat Hp-l.OCp-1.0 1.0 1.0 8.07 6.18 10.64 Hp-1.5Cp-1.5 1.5 1.5 12.26 1.0 8.32 Folded Hp-l.OCp-1.0 1.0 8.58 12.11 Hp-1.5Cp-1.5 1.5 1.5 12.91 Design value was calculated according to Eurocode 3.
Mean value 6.89 11.39 8.39 12.45
Design values [kN] 4.44 7.49 6.33 8.66
"FEresults [kN] 7.39 12.09 8.01 11.83
"I
589 SHEAR BEHAVIOUR OF ROSETTE-JOINT UNDER CYCLIC LOADS The Principle of the Thermovision Technique The purpose of the thermoelastic stress analysis is to describe thermoelastic effects. When temperature changes occur in a body the stress distribution changes. The change in temperature AT in a homogenous isotropic material under cyclic stress variations and adiabatic conditions can be calculated from the classic equation first introduced by William Thompson (Lord Kelvin, 1853): M = ——•T^^{a,+a^^a^) cp where T a c p A (o 1 + o 2 + o 3)
(1)
is the absolute temperature [T] is the linear thermal expansion coefficient is the heat capacity is the density of the material is the change in the sum of the main stresses
With a given temperature change, Eqn.l. will give the change in the sum of the main stresses. Areas under a pure shear stress state remain unidentified. If the stresses are greater than the yield stress, the temperature becomes considerably greater and is no longer proportional to the changes in stress.
The changes in the temperature should be made without contact with the body. The AGEMA 900 Thermovision system is used as shown in Figure 7 to measure and evaluate these changes. The detector uses long wave frequencies in the infra red area and nitrogen cooling. With each loading step the photo sequence is recorded. From each sequence, a subtraction between maximum and minimum values is calculated and thus the temperature change during one load cycle is obtained. The different pictures are treated using editing software in order to separate the different temperature areas from which the stresses may then be calculated. In the end, a stress map is obtained, which shows the changes in the sums of the main stresses in selected test piece areas.
Figure 7: The AGMEA 900 Thermovision system
Figure 8: Force controlled testing machine
590 All the cyclic tests were carried out with a force directed hydraulic testing machine as shown in Figure 8. In the tests, the cycle load was gradually increased in steps of 1 kN in the beginning, and later, when close to failure, in steps of 0.2 - 0.4 kN. Cyclic Shear Load Tests The tests were done on flat specimens with a measured yield stress of fy = 410 N/mm^, ultimate stress of fu = 470 N/mm^ and thickness t = 1.5 mm. The first test was done with monotonic loading in order to see the general behaviour and the ultimate load reached, which was F = 12.50 kN. The first cyclic test reached a failure load of 12.55 kN after a total of approx. 1240 cycles as shown in Figure 9. Figure 10 shows the thermovision image with a tension force of 8.00 kN. For comparison. Figure 11 shows the corresponding FE-analysis image obtained with ABAQUS program. In the thermovision image, all stresses over the yield limit have the yield value. The greatest compression stresses can be found in the upper part of the brim of the hole. The greatest tension stresses are on the left and right sides of the hole.
Figure 9: Failure mode after 1240 cycles
Figure 10: Thermovision image with a 8.00 kN force
«. Campoiwnb /USECTI0NJ>0INT^(JNV/mANT-l1
L Figure 11: FE- analysis image obtained with ABAQUS program
591 The second cyclic test was principally the same as the first one, but the loading speed was greater. The failure load here was 12.00 kN as shown in Figure 12. Figure 13 and Figure 14 show the comparison between the thermovision and the FE-analysis (under load 9.80 kN) results.
Figure 12: Failure mode under 12.00 kN
Figure 13: Thermovision image with a 9.80 kN force
The last cyclic test was carried out with a constant load level of 8.25 kN which corresponds to approximately 70 % of the failure load and thus a normal usage level. After 7000 cycles, no problems had occurred and the test was terminated. ABAQU5/Pr«Varsion5.S-1 2S-Jtil-9« t&49:16 FRIN GE: S W i c . SlBp2.Tattmma-1.0665: Strass.
Al S E C T 1 0 N _ P 0 I N T _ 3 (INVARIANT-1) -ABACXIS
XS7l9i .000931 I
Y
Figure 14: FE-analysis image obtained with ABAQUS program
592 CONCLUSIONS The static shear tests gave the design values for the ROSETTE-joint in accordance with Eurocode 3, which are 6.33 kN for 1.0-mm thickness and 8.66 kN for 1.5-mm thickness. In addition, the failure modes for the specimens with 1.0-mm thickness collar part and 1.5-mm thickness hole part are due to the collar failure and differ from those of other specimens. The FEmodel was validated by the test results and could be used for further analysis. The cyclic shear test results show that the thermoelastic stress analysis using an infra red camera can be successfully applied to the experimental analysis of steel structures. The stress distribution in a specimen can be quickly obtained with this method already at load levels considerably under the failure load. The correspondence of the test results with the FE-analysis was satisfactory in general. Therefore, it can be concluded that the method is useful in the verification of FE-models.
REFERENCES Kaitila O. (1998). Design of Cold-Formed Steel Roof Trusses Using ROSETTE-Connections. Master's Thesis, Laboratory of Steel Structures, Helsinki University of Technology, Espoo, Finland Kesti J., Lu W. and Makelainen P. (1998). Shear Tests for ROSETTE-joint. Helsinki University of Technology, Laboratory of Steel Structures, Research Report TeRT-98-03, Espoo, Finland Lu W., Segaro P., Kesti J. and Makelainen P. (1998). Study on the Shear Strength of a SingleLap ROSETTE-Joint. Laboratory of Steel Structures, Helsinki University of Technology, TKKTeRT-8, Espoo, Finland Makelainen P., Kesti J. and Piirainen A. (1997). Shear Tests for ROSETTE-joints. Laboratory of Steel Structures, Helsinki University of Technology, Research Report N:o TeRT-97-05, Espoo, Finland Makelainen P., Kesti J. and Sahramaa K.J. (1998a). ROSETTE-Joint: A New Technique for Steel Sheet Joining. Sustainable Steel: The International Conference on Steel in Green Building Construction, Orlando, Fla, U.S.A. Makelainen P., Kesti J. and Kaitila O. (1998b). Advanced Method for Light-Weight Steel Truss Joining. Nordic Steel Construction Conference 98, Proceedings, Volume 7, Bergen, Norway Makelainen P., Kesti J., Kaitila O. and Sahramaa K.J. (1998c). Study on Light-Gauge Steel Roof Trusses with ROSETTE Connection. Recent Research and Developments in Cold-Formed Steel Design and Construction, 14^^ International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Missouri, U.S.A. Makelainen P., Kesti J. and Kaitila O. (1998d). Advanced Method for Light-Weight Steel Truss Jointing. International Conference on Lightweight Structures in Civil Engineering, Warsaw Pasternak H. and Horvath L. (1997). Untersuchung zyklisch beanspruchter Stahlbauteile mit Hilfe der Thermovision. Der Stahlbau, 66:3, 127-135. Pasternak H. and Horvath L. (1996). Stress Pattern Analysis by Thermovision - A Case Study. European Workshop on Thin-Walled Structures, Krzyzowa, Poland User Manual for NISA II Numerically Integrated Elements for System Analysis Version 6.0. (1995), Engineering Mechanics Research Corporation, Michigan 48083 U.S.A. User Manual for Display III™, Version 6.0. (1995), Engineering Mechanics Research Corporation, Michigan 48083 U.S.A.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
593
STUDY ON THE BEHAVIOUR OF A NEW LIGHT-WEIGHT STEEL ROOF TRUSS p. Makelainen and O. Kaitila Laboratory of Steel Structures, Helsinki University of Technology, P.O.Box 2100, FIN-02015 HUT, Finland
ABSTRACT The ROSETTE thin-walled steel truss system presents a new fully integrated premanufactured alternative to light-weight roof truss structures. The trusses will be built up on special industrial production lines from modified top hat sections used as top and bottom chords and channel sections used as webs which are jointed together with the ROSETTE press-joining technique to form a completed structure easy to transport and install. The trusses can be case-dependently designed in order to efficiently respond to different structural forms and loading conditions. A single web section is used when sufficient and can be strengthened by double-nesting two separate sections or by using two or several lateral profiles where greater compressive axial forces are met. All system components have industrial-quality cut details. The profiles may be formed using pre-coated steel for special architectural effects when so desired. A series of laboratory tests have been carried out in order to verify the ROSETTE truss system in practice, hi addition to compression tests on individual sections of different lengths, tests are also done on small structural assemblies, e.g. the eaves section, and on actual full-scale trusses of 10 metre span. Design calculations have been performed on selected roof truss geometries based on the test results, FE-Analysis and on the Eurocode 3, U.S.(AISI) and Australian / New Zealand (AS) design codes.
KEYWORDS Rosette-joint, truss testing, light-weight steel, roof truss, cold-formed steel, steel sheet joining
INTRODUCTION For a description of the Rosette-joint, please refer to Makelainen et al. (1999). This paper presents the first extended test programme performed on the ROSETTE light-weight steel roof truss system. Results of tests on individual members and full scale roof trusses are presented and analysed.
594 DESCRIPTION OF THE ROSETTE - ROOF TRUSS SYSTEM Rosette - trusses are assembled on special industrial production lines from modified hat sections used as top and bottom chords and channel sections used as webs, as portrayed in Figure 1, which are joined together with the Rosette press-joining technique to form a completed structure easy to transport and install. The profiles are manufactured in two size groups using strip coil material of various thicknesses (from 1.0 to 1.5 mm). A single web section is used when sufficient, but it can be strengthened by double-nesting two separate sections and/or by using two or several lateral profiles where greater axial loads are met. At the present time, the application of the Rosette truss system is being examined in the 6 to 12 metre span range. 89 nn Chord section Veb section
Double-nested web sections
Figure 1: Cross-sections of the 89 mm Rosette chord and 38 mm web members
TESTS ON INDIVIDUAL MEMBERS Tests on Web Members Axial compression tests were carried out on four differently arranged sets of 38 mm web sections of cross-sectional thickness 0.94 mm in order to verify their actual failure mode and load. The specimens in groups 1 to 3 were prepared for testing by casting each end in concrete, thus providing unhinged end conditions. All specimens, including group 4, were placed firmly on solid smooth surfaces and the compressive force was applied centrally on the gravitational centroid of the members. The test results are summarized in Table 1. In test groups 1 and 2, they are quite consistent with analytical values determined according to EC 3 , Part 1.3, when rigid end conditions are assumed. Group 3 consists of two specimens of web members with two profiles freely nested one inside the other. The analytical compression capacity was obtained by simply multiplying the capacity for a single profile by two. The average maximum load from the tests was approximately three-fold the test value for a single profile. This high value is due to the greater capability of the nested profiles to resist torsion compared to single profiles. Test-group 4 differs from the first three groups in its overall arrangement and motives. The idea was to examine the way the joints connecting the web profile to the adjacent chord profiles in the actual structure perform under axial loading, and how much rotational support they give to the web profile that has been initially considered hinged at both ends. Each of the three test specimens consisted of a 1 060 mm long web profile element connected by Rosette-joints at each of its ends to a 400 mm long
595 piece of chord profile. The length of the specimens was chosen great enough to prevent the failure of the joints before buckling occured. The distance between the midpoints of the joints was then 1 003 mm for all three specimens. The average maximum test load value was approximately 39 % larger than the analytical value calculated with an effective buckling length reduction factor of ^^ = 1.0. The test load value corresponds to an analytical buckle half-wavelength of 780 mm (Kb = 0.78). This indicates that it would be safe to use an effective buckling length reduction factor of Kb = 0.9, as is quite usual practice in roof truss structures. TABLE 1 3 8 MM WEB COMPRESSION TEST RESULTS (T: : TORSIONAL BUCKLING, F = FLEXURAL BUCKLING, D = DISTORTIONAL BUCKLING) Test Test piece Group number
Total length after setup
#
#
1
1 2 3
mm 660 660 660
Theoretical Buckle Half-wavelength mm 330 330 330
2
4 5 6
1061 1060 1060
530.5 530 530
3
7 8
1063 1061
531.5 530.5
4
11 12 13
1000 1000 1000
1000 1000 1000
Analytical Conpression Capacity kN 33.44 33.44 33.44 Average: 25.56 25.56 25.56 Average: 45.14 45.14 Average: 9.27 9.27 9.27 Average:
Test Result kN 34.24 36.02 36.80 35.69 23.(H 25.06 26.94 25.01 74.72 75.61 75.17 13.21 13.24 12.20 12.88
Ratio between test result and analytical result 1.02 1.08 1.10 1.07 0.90 0.98 1.05 0.98 1.66 1.68 1.67 1.42 1.43 1.32 1.39
Failure Mode
T +D T +D
T+DI T T T F+T F+T
T 1 T
T 1
Tests on Chord Members Similar compression tests to those carried out on individual web profiles (test-groups 1 and 2) have been performed on chord profiles. The actual structure will include continuous chord members that are connected to web members at different intervals and laterally supported by braces every 600 mm. TABLE 2 8 9 MM CHORD COMPRESSION TEST RESULTS (TF = TORSIONAL-FLEXURAL BUCKLING MODE) Test Group
Test piece number
Total length after setup
#
#
1
1 2 3
mm 1258 1255 1255
Theoretical Buckle Half-wavelength mm 629 627.5 627.5
2
4 5 6
1754 1751 1755
877 875.5 877.5
Analytical Compression Capacity kN 52.68 52.68 52.68 Average: St. deviation:
Test Result
32.95 32.95 32.95 Average: St. deviation:
34.65 34.54 34.37 34.52 0.14
kN 47.28 46.92 49.85 48.02 1.60
Ratio between test result and analytical result 0.90 0.89 0.95 0.91
Failure Mode
1.05 1.05 1.04 1.05
TF TF TF
TF TF TF
596 It can be concluded that the design procedure used for the evaluation of the compression capacities is quite compatible with the test results. The analytical calculations and FE-analyses performed predicted a torsional-flexural buckling mode with a stronger deflection in the };-direction and the test results supported this prediction. Also the maximum loads observed in the tests comply with the analytical values to an acceptable degree.
TESTS ON FULL-SCALE TRUSSES The Test Truss General Two full scale 10 metre span trusses have been tested according to the testing procedure described in Eurocode 3: Part 1.3 Appendix A4. The first truss passed the first phase of testing, i.e. the 'Acceptance Test', but failed during the load increase phase of the next test round, i.e. the 'Strength Test'. This failure was due to manufacturing difficulties and insufficient detail design of the truss (Kaitila 1998a). The information received from the first test was analysed and used to improve the details of the second truss while preserving the original basic geometry. The different phases and the results of the second truss test are given in the present chapter. Test set-up The test truss was manufactured from steel plate with cross-sectional wall thickness tots = 0.95 mm (+ zinc coating), yield stress/y,^^^ = 368 N/mm , and modulus of elasticity £ = 189 430 N/mm^ (all values taken for steel in the direction of cold-forming). The profiles used were a modified 89 mm chord and a new 29 mm web profile, as shown in Figure 2. The vertical web profiles on the supports were designed so that they lean against the bottom flange of the bottom chord and could thus directly transmit the load from the structure onto the support as compression, without the chord member having to support unneccessary shear force which would cause strong distortion in the lower part of the chord member, as observed in the tests on eaves members. 89 nn Chord section A
h
.
:3
29 nr\ Web Section z
1^
2
^>
-p '-
1
1
n ( c e n t r e l i n e dinT*Sh 29 nn ( a c t u a l g xp) X.
-"
V 1
JL
16
IP 1
|.12,
,
E e CD CD
4
16 .
62 nn < c e n t r e line dirlension) total 63
Figure 2: The profiles used in the second truss test.
597 The nominal geometry of the tested truss is outhned in Figure 3. The truss was symmetrical about its centre line with a top chord inclination of 18 degrees. The height at the support was approximately 490 mm, which gave the truss a total height of about 2100 mm. The top chords were connected to each other at mid-span using a short web member and specially manufactured jointing plates. The total mass of the actual truss was 75.5 kg.
4 9Rn 9 9(Sn
Figure 3: Nominal geometry of test truss with load cylinders The truss was supported at the ends of the bottom chord with pinned supports. All horizontal displacements were prevented at the lefthand support and free in the plane of the structure at the righthand support. The support plates were long enough to allow for a sufficient support area for both web members at the support. The lateral supports were made at the top chord every 600 mm by simply bolting the top flange of the chord to the c 600 loading rig. The load cylinders were hinged in the plane of the structure but fixed in the plane perpendicular to that of the truss. The dimensions of the actual truss differed quite little from the nominal values. The actual dimensions of the manufactured profiles differed from the nominal cross-sections by less than 5 %. The production of the joints was done successfully this time without the problems that occurred in the manufacturing of the first test truss. Outline of Test Procedure The testing was performed according to the procedure described in Eurocode 3 Part 1.3 Appendix A4: Tests on Structures and Portions of Structures. This method includes three distinct phases, an 'Acceptance Test', a 'Strength Test' and a 'Prototype Failure Test'. The loading was applied at eighteen distinct points (nine on each side of the truss's midline) with c600 mm space between them so, that at mid-point there was no load and thus the space between the two middle load cylinders was 1 200 mm. Only symmetrical evenly distributed loading was considered in this test. The load was pumped into a hydrostatic pressure cylinder using a handpump and subsequently evenly divided between all 18 load cylinders. Each load cylinder had a 420 mm long loading pad which transmitted the load from the cylinder onto the structure. The loading pad is 80 mm wide which made it possible to place the 63 mm wide top chord profile centrally under the pad and leave a minimum space of approximately 8 mm for distortional or other deformation of the cross-section on both sides of the profile. Vertical deflections
598 were measured with displacement bulbs at the mid-point and the quarter points of the bottom chord, and at the ends and the mid-point of the top chords. Horizontal displacement of the supports was also measured. Computer model of test truss A STAAD Ill-analysis was performed for the design of the truss. The material values used for the model were: • wall thickness t = 0.96 mm • yield strength/^ =fyt = 350 N/mm^ • modulus of elasticity E = 2\0 000 N/mm^ The connection (i.e. two joints) capacity used in the analysis was taken as Fc,conn = 10.8 kN. Progression and results of the full scale truss test The second test truss successfully passed all phases of testing and the maximum load reached was 48.5 kN. The course of the test can be most simply explained with the aid of the diagram given in Figure 4 showing the deflection of the truss at mid-span measured from the bottom chord. The graph is complemented with numbers showing the different phases of testing. 1. The test was begun at zero load and the load was steadily increased up to 25.16 kN, where it was held for one hour. The unlinearities in the curve during load increase were caused by the movement in the joints due to production tolerances. Point 1 marks the beginning of the one hour period. During load increase or decrease, displacement values were taken at 5 second intervals. During the constant load phases, they were recorded every 30 seconds.
8
\ 1 '^o
u^ ?
4
•o
.s
I
2
1^
S 30
1
\
,fi
\
^
^-^
My
3
^^7
Displacement at mid-point of truss [mm]
Figure 4: Deflection at mid-span of truss (see text for notes) 2. Point 2 marks the end of the one hour period. The maximum deflection at this stage was 11.28 mm or L / 850. The load was then gradually taken off. 3. The residual deflection after the 'Acceptance Test' phase was 1.74 mm (15 % of the maximum recorded). The allowable value is 20 %, so the truss passed this first phase successfully. Because
599 the vertical webs were initially extended all the way to the bottom of the chord on the support, no local deformation of any importance occurred during this phase. The behaviour of the truss was very good during this first phase. The test load was initially evaluated as 32.0 kN due to a miscalculation. Therefore a quick decision was made at the beginning of the one hour period of this second phase of testing, to increase the test load by 10 % up to 35.2 kN. Point 4 marks the small escalation caused by this mistake before the 10 % increase. Line 5 shows the beginning of the one hour period of the 'Strength Test' phase at load value 35.2 kN. 6. Point 6 marks the end of this one hour period. The maximum deflection recorded at this stage was 18.06 mm or L/550. Point 7 marks the residual deflection at mid-span after the removal of the load. This total residual deflection was 4.51 mm, i.e. the deflection was decreased by 75 %, much more than the 20 % needed at this stage. No actual tear was observed, but a slight beginning of local deformations could be seen in the chord members in the area of the most heavily loaded joints, i.e. beginning shear deformations like the ones portrayed in Figure 5 were starting to appear, but in a much smaller scale than in the photographs.
Figure 5: Deformations at the left side support area of the top chord just before failure (left) and after failure (right). The free edges of the top chord deformed into slight sine-shaped curves under loading, as expected, but this deformation was elastic and the original shape of the members was regained after the removal of the load. The deformation happened in such a way, that consecutive portions separated by web members were deformed in opposite directions, i.e. the first one towards the inside, the second one towards the outside etc. A similar deformation occurred in the bottom chord, although this part of the structure should primarily be under tensile stress. The effect of bending moment caused the deformation of the free edges of the bottom chord profiles. The individual web members did not show indication of insufficiency. After the truss had satisfactorily passed the 'Strength Test'-phase, the last stage with loading up to failure was begun. During the increase of the load, the longer webs were considerably deformed in torsion and flexure. Nevertheless, the final failure did not occur directly due to this but to the joints in the first tension webs counting from outside, as expected from the computer analysis. The failure load was 48.5 kN, although it can be argued that the functional capacity of the truss was reached around a total load value of 46 kN, because of the strong torsional-flexural deformations of the longer web members.
600 CONCLUSIONS This paper presents the general results of the first analysis including a test programme on the Rosette steel roof truss system and individual members. The behaviour of the truss was linear and predictable throughout the testing procedure. The structure successfully passed the first and second stages of the EC 3 testing procedure, 'Acceptance Test' and 'Strength Test', respectively. The manufacturing of the truss was carried out with a much better standard of quality than in the first test, where several imperfections caused the truss's early failure (Kaitila 1998a). The individual members acted well in this test. There was no significant plastic deformation before the last stages prior to failure. The safety factors for the joints are considerably larger than those used for the members (7= 1.25 compared with 7 = 1.1, respectively). Therefore it is not surprising that it is the joints that tend to become critical in truss design. Furthermore, because the chord members did not cause any problems in this test, it might be concluded that the chord profile has unnecessary extra capacity and reasons for reducing the chord profile in size might exist. However, it is perhaps too early to draw such a conclusion, since the effects of this type of change need to be examined on the level of a complete structure. The connection technique used to join together the top chords at mid-span should be studied and designed in a more efficient manner with an analysis extending to the effect of a suggested solution on the behaviour of the complete structure. The truss passed the requirements set by the European design standard. Further optimization and more detailed design is needed for the application of the Rosette system to high-quantity production, but a strong confidence in the abilities of the system can be justified by this test.
ACKNOWLEDGEMENTS The authors would like to acknowledge Mr. Kimmo J. Sahramaa (FUSA Tech Inc., Reston, VA, USA), the innovator of the Rosette-joint technology, and Mr. Juha Arola (Rosette Systems Ltd, Kauniainen, Finland) for the initiation and support of this research project.
REFERENCES Kaitila O. (1998a). Design of Cold-Formed Steel Roof Trusses Using Rosette - Connections, Master's Thesis, Helsinki University of Technology, Espoo, Finland Kaitila O. (1998b). Second Full Scale Truss Test on a Rosette - Joined Roof Truss, Research Report TeRT-98-04, Helsinki University of Technology, Espoo, Finland Kesti J., Lu W., Makelainen P. (1998). Shear Tests for ROSETTE Connection, Research Report TeRT98-03, Helsinki University of Technology, Espoo, Finland Makelainen P., Kesti J., Kaitila O., Sahramaa K.J. (1998a). Study on Light-Gauge Roof Trusses with Rosette Connections, 14^^ International Specialty Conference on Cold-Formed Steel Structures, St.Louis, Missouri, USA Makelainen P., Kesti J., Kaitila O. (1998b). Advanced Method for Light-Weight Steel Truss Joining, Nordic Steel Construction Conference 98, Bergen, Norway Makelainen P., Kesti J., Lu W., Kaitila O., Pasternak H. and Komann S. (1999). Static and Cyclic Shear Behaviour Analysis of the Rosette - Joint, Proceedings of the Fourth International Conference on Steel and Aluminium Structures (ICSAS '99), Espoo, Finland
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
601
BEARING DESIGN OF COLD FORMED STEEL BOLTED CONNECTIONS C.A. Rogers^ and G.J. Hancock^ ^PhD Research Student, Department of Civil Engineering, University of Sydney, Sydney NSW, 2006 Australia ^BHP Professor of Steel Structures, Department of Civil Engineering, University of Sydney, Sydney NSW, 2006 Australia
ABSTRACT The results of recently completed bolted connection shear tests indicate that the current provisions set out in the AS/NZS 4600, AISI and Eurocode cold formed steel design standards cannot be used to accurately predict the failure mode of connections that are fabricated from thin G550 and G300 sheet steels. Furthermore, these design standards cannot be used to accurately determine the bearing resistance of bolted specimens based on a failure criterion for predicted loads. The measured variation in bearing resistance between connection tests of thin 0.42 mm G550 sheet steels and more typical 1.0 mm and thicker sheet steels has been used to develop a gradated bearing coefficient method, that is dependent on the thickness of the connected materials and the size of the bolt(s) used in the connection. It is recommended that the gradated bearing coefficient formulation, the unreduced net section resistance, and the Eurocode design method for end pull-out be used in the design of bolted connections.
KEYWORDS Bolted connection. Sheet steel, Bearing design. Failure mode.
INTRODUCTION Cold formed structural members are fabricated from sheet steels which must meet the material requirements prescribed in applicable national design standards. The Australian / New Zealand standard for cold-formed steel structures (AS/NZS 4600) (SA/SNZ, 1996) allows for the use of thin (t < 0.9 mm), high strength (/y = 550 MPa) sheet steels in all structural sections. However, due to the low ductility exhibited by sheet steels that are cold reduced to thickness, engineers are required by design standards to use a yield stress and ultimate strength reduced to 75% of the minimum specified values. The American Iron and Steel Institute (AISI) Specification (AISI, 1997a) further limits the use of thin, high strength steels to roofing, siding and floor decking panels. Sheet steels are required to have a minimum elongation capability to ensure that members and connections can undergo small displacements without a loss in structural performance, and to reduce the harmful effects of stress concentrations. The ductility criterion specified in the Australian / New
602 Zealand and North American (AISI, 1997a, CSA, 1994) design standards is based on an investigation of sheet steels by Dhalla and Winter (1974a,b) which did not include the thin, high strength G550 sheet steels that are available today (see AS 1397 (1993)). The analysis of previously completed bolted connection tests that were fabricated from 0.42 and 0.60 mm G550 and G300 sheet steels (Rogers and Hancock 1997b, 1998a) provides two significant results. Firstly, the low material ductility measured in coupon tests (Rogers and Hancock, 1996, 1997a) does not influence the net section fracture mode of failure. Secondly, the capacity of connections that are composed of thin G550 and G300 sheet steels is overestimated when current design standards are used to predict bearing loads. Additional bolted connection specimens that were fabricated from 0.80 and 1.00 mm G550 and G300 sheet steels have been tested and are described in detail in Rogers and Hancock (1998b). All of these test specimens have been used to develop a gradated bearing coefficient method that is dependent on the thickness of the connected materials and the diameter of the bolt(s) used in the connection. The measured variation in bearing resistance between the thin 0.42 mm G550 sheet steels and the typical 1.0 mm and thicker sheet steels is incorporated into a general bearing formulation. The additional bolted connections were dimensioned such that only bearing failure would occur, with test specimens milled from the longitudinal, transverse and diagonal directions of the sheet. Background Information on G550 and G300 Sheet Steels The steels investigated as a part of this paper were produced using a process called cold reduction, which can be used to increase the strength and hardness, as well as produce an accurate thickness for sheet steels and other steel products. This process causes the grain stmcture of cold reduced steels to elongate in the rolling direction, which produces a directional increase in material strength and a decrease in material ductility. The effects of cold working are cumulative, i.e. grain distortion increases with further cold working as a result of an increase in total dislocation density, however, it is possible to change the distorted grain stmcture and to control the steel properties through subsequent heat treatment. Various types of heat treatment exist and are used for different steel products. Both G300 and G550 sheet steels are stress relief annealed, i.e. the total dislocation density is reduced by annealing, although recrystallisation does not occur. G300 sheet steels are annealed to a greater extent in comparison with G550 sheet steels (BHP, 1992). This procedure results in near isotropic material properties for mild sheet steels (G300), although some preferred grain orientation remains.
BOLTED CONNECTION TESTS AND RESULTS General A total of 18 additional single overlap bolted connection specimens (Rogers and Hancock, 1998b) were tested at the University of Sydney, to complement the 158 bolted connection tests that are presented in Rogers and Hancock (1997b, 1998a). The main objective of this experimental testing phase was to develop a gradated bearing failure design provision for bolted connections that are fabricated from thin G550 and G300 sheet steels. Three different sheet steels were tested, including both G550 and G300 grades, i.e. 0.80 mm G550, 1.0 mm G550 and 0.80 mm G300, and used as a basis for comparison with the current design equations specified in the Australian / New Zealand, North American and European (7996) cold formed steel design standards. Testing of these steels was necessary to determine the variation in bearing coefficient with thickness. All steels were cold reduced to thickness, with an aluminum/zinc alloy (zincalume-AZ) coating and obtained from standard coils during normal rolling operations. Comparison of Ultimate Test-to-Design Standard Predicted Loads Dynamic ultimate test loads. Put, were used in comparison with the predicted ultimate connection strengths, Pup, determined using the relevant design standards without the 0.75/y and 0.75/u reduction. The lowest
603 calculated load from the various connection equations within any one design standard is defined as the predicted mode of failure. Conclusions regarding the adequacy of design formulations based on a comparison of test-to-predicted ratios where the actual and predicted mode of failure (see Figure 1) do not match are invalid. Hence, detailed statistical information of the test-to-predicted ratios for the various design standards is not provided, however, overall values for the bolted connection specimens that are presented in Rogers and Hancock (1997b, 1998a,b) can be found in Table 1. Only the CSA-S136 design standard can be used to adequately predict the failure modes of the different bolted connection test specimens. The ratio of correct-to-incorrect failure mode prediction for the AS/NZS 4600 and AISI design standards is 92-84, where the majority of incorrect predictions were defined as net section failure when a bearing failure occurred in the test specimen. The error in predicted failure mode can be attributed to design equations which overestimate and underestimate the bearing and net section fracture resistance, respectively. Bearing resistance equations are based on a large array of data which does not include a significant number of specimens with thickness less than 0.6 mm. The lack of specimens in this range has allowed the AS/NZS 4600 and AISI design standards to overlook the influence of thickness on bearing capacity. Test results from this research also show that it is not necessary to reduce the net section fracture capacity at connections as a function of the number of bolts and width of the specimen. Development of the net section stress reduction formulation contained in the AS/NZS 4600 and AISI design standards was originally based on studies of bolted connections in which test specimens that failed by net section fracture, comer pull out, end pull-out, as well as bearing were included (Popowich, 1969; Winter, 1956). It is likely that the net section stress reduction equation models the bearing behaviour of bolted connections, due to the possible misidentification of bearing failures in specimens (see Rogers and Hancock (1998c)). Initial crack at centre of originally drilled bolt hole
^ ,
Specimen necks over the width
-Piling of the sheet steel in front of the bolt
- Initial tear at edge of piled sheet steel
Tear extends through cross-section
CM 'CO Edge of washer Piling of the -^ sheet steel in front of the bolt Bearing with End Cnriing Restrained
Bearing wttli End Cnriing
FIGURE 1 BOLTED CONNECTION FAILURE PATTERNS
Use of the CSA-S136 design standard provides a ratio of correct-to-incorrect failure mode prediction of 1706. The six incorrectly predicted specimens were double bolted G3(X) tests for which net section failure occurred instead of the predicted bearing failure. The coefficient of C = 2 used in the CSA-S 136 bearing equation for connections where dit > \5 may be overly conservative for mild sheet steels. Use of the Eurocode Standard gives a ratio of correct-to-incorrect failure mode prediction of 149-27. The large number of incorrect failure mode predictions is due to an overestimated bearing capacity and an underestimated net section fracture capacity, similar to that observed for the ASINZS 4600 and AISI design standards.
604 TABLE 1 DESIGN STANDARD P^t / /'up STATISTICAL DATA FOR ALL TESTS (FULL/u USED) Design Standard
* ut / * up
Failure Mode Design Standard Prediction
* ut / ^up
Failure Mode Prediction
Eurocode
AS/NZS4600&AIS1 Mean No. S.D. Co.V.
0.880 176 0.198 0.227
Correct = 92 Incorrect = 84
Mean No. S.D. Co.V.
1.115 176 0.269 0.243
Correct = 170 Incorrect = 6
Mean No. S.D Co.V.
0.959 176 0.178 0.187
Correct = 149 Incorrect = 27
Mean No. S.D. Co.V.
1.077 176 0.149 0.139
Correct = 167 Incorrect = 9
Proposed Method
CSA-S136
Comparison of Ultimate Test-to-Failure Criterion Predicted Loads The bolted specimens that were tested for this research were divided into separate categories according to the recorded failure mode, Le. the three ultimate limit states that were observed; end pull-out, bearing and net section fracture (see Figure 1). Thus, the predicted connection capacity that was used in comparison with the ultimate load obtained for a bolted connection was calculated using the design equation developed for that failure mode, e.g. all specimens which failed by bearing were compared with the predicted bearing capacity. Statistical results for all of the bolted connection test specimens that are contained in Rogers and Hancock (1997b, 1998a,b) can be found in Table 2. TABLE2 FAILURE BASED CRITERION Put / /'up STATISTICAL DATA FOR ALL TESTS (FULL/u USED) Design Standard
AS/NZS4600&AISI Mean No. S.D. Co.V.
Bearing End Pull-Out
Net Design Section Standard
Bearing End Pull-Out
Put 1 Pup
Put 1 Pup
Put / ^up
Put 1 Pup
Put 1 Pup
Put 1 Pup
0.894 36 0.131 0.150
0.726 99 0.149 0.207
1.137 26 0.054 0.050
Mean No. S.D. Co.V
1.072 36 0.157 0.150
0.871 99 0.179 0.207
1.075 26 0.050 0.048
1.313 36 0.413 0.324
1.074 99 0.217 0.204
1.007 26 0.048 0.050
Proposed Method Mean No. S.D. Co.V
1.072 36 0.157 0.150
1.089 99 0.163 0.151
1.007 26 0.048 0.050
Eurocode
CSA-S136 Mean No. S.D. Co.V
Net Section
The overall failure based criterion results indicate that the AS/NZS 4600 and AISI design standards can both be used to conservatively predict the net section failure loads of bolted connections. However, the calculated end pull-out resistance is found to be unconservative with a mean test-to-predicted ratio of 0.894. The connection resistance of bolted specimens which failed in bearing is inaccurately modelled by the existing AS/NZS 4600 and AISI design provisions. The resulting mean test-to-predicted ratios are significantly unconservative, with a mean Put / Pup value of 0.726. The ultimate connection resistance of bolted specimens determined using the Eurocode design standard can be more accurately calculated in comparison with the AS/NZS 4600 and AISI design standards. End pull-out failure can be conservatively predicted based on the results provided in Table 2. Net section fracture
605 prediction behaviour remains conservative, although not to the extent exhibited by the AS/NZS 4600 and AISI design standards. However, the bearing resistance formulation remains significantly unconservative with a mean test-to-predicted ratio of 0.871, determined using all of the bearing failure test specimens. The CSA-S136 design standard provides overly conservative predictions of the end pull-out capacity for the sheet steels that were tested. Net section fracture connection resistance can be accurately modelled using the net cross-sectional area and the ultimate material strength without a stress reduction factor. More importantly, a dramatic improvement in the ability to predict the ultimate bearing resistance of the sheet steels that were tested for this research project occurs. The mean test-to-predicted ratio increases to a conservative value of 1.074. The results of the additional bolted connection specimens (Rogers and Hancock, 1998b) reveal that the AS/NZS 4600 and AISI design standards remain unconservative when used to predict the bearing failure loads of the 0.80 and 1.00 mm G550 sheet steels. However, the bearing resistance of the 0.80 mm G300 test specimens can be accurately calculated; which is most likely due to the use of mild sheet steel data in the development of the current bearing design expressions. The CSA-S136 and Eurocode design standards can be used to conservatively predict the load carrying capacity of bolted connections that were composed of the 0.80 and 1.00 mm sheet steels. The previously tested 0.42 and 0.60 mm G550 sheet steel bolted connections show that the bearing resistance is dependent on the thickness of the connected materials. The thinnest sheet steels that were tested provide the most unconservative test-to-predicted ratios for bearing failure, e.g. Put / Pup = 0.591 for the transverse 0.42 mm G550 test specimens that failed in bearing (see Rogers and Hancock (1998b)). In addition, the increased test-to-predicted ratios for the mild G300 sheet steels indicate that the bearing resistance coefficient may also be dependent on the material properties of the sheet steels.
PROPOSED DESIGN PROVISIONS FOR BOLTED CONNECTIONS Significantly unconservative predictions of the load resistance obtained for certain bolted connection test specimens have demonstrated a need for a gradated bearing coefficient which is dependent on the stability of the edge of the bolt hole. Unconservative predictions of connection bearing capacity have been recorded for the bolted test specimens where thin sheet steels are connected and loaded in shear, as shown for the failure based criterion test-to-predicted results calculated using the AS/NZS 4600, AISI and Eurocode design standards for the 0.42 mm G550, 0.60mm G550, 0.60 mm G300 test specimens, as well as the 0.80 mm G550 and 1.0 mm G550 test specimens (which all failed in bearing) (Rogers and Hancock, 1998b). A proposed method to accommodate for the change in bearing behaviour, that relies on the ratio of bolt diameter to sheet thickness, dlt, is presented. This proposed method includes the gross yielding, Eq. 1, and the net section fracture, Eq. 2, failure provisions that are contained in the CSA-S136 design standard, i.e. no stress reduction factor is used. Calculation of the end pull-out resistance follows the procedure given in the Eurocode design standard, Eq. 3. The recommended equations for gross yielding failure, net section fracture and end pull-out failure are as follows: M=Ag/y
(1)
where Ag is the area of the gross cross-section and^ is the yield or 0.2% proof stress. M=An/u
(2)
where An is the area of the net cross-section and/u is the ultimate strength. V, = tefJ\.2
(3)
where t is the thickness of the thinnest connected part and e is the distance measured parallel to the direction of ^plied force from the centre of a standard hole to the nearest edge of an adjacent hole or to the end of the connected part.
606 Modification of the existing bolted connection design provisions was made to the bearing formulation. Bearing stress ratios, /bu / /u, for some of the bolted connection test specimens that failed by bearing are illustrated in Figure 2. The bearing stress ratios for these specimens decrease as the thickness decreases, hence, a formulation to calculate a bearing coefficient, which is similar to that recommended in the CSAS136 design standard, is proposed. 4.0 3.5 i.O l.b 9
1.5 1.0-
OEndPull-Out AS/NZS4600.AISI
U Bearing / End Pull-Out
AS/NZS4600.AISI
Eurocode
A Bearing
jgK 4fe —A-1 1 g ^=f=
ACSAS136 ^Proi^
A^
42.0 1.5
a
0.5
^
ProDosed. Eurocode CSAS136
\y^ 2
3
4
e/d
e/d
0.42 mm G550
0.80 mm G550
FIGURE 2 BEARING STRESS RATIOS FOR VARIOUS SHEET STEELS (FULL/. USED) 4.0
^ 3.5 \ & 3.0 •S* ^ 2.5 1 2.0 1 e^ 1.5 \ bf) S
S
<3
T
L
;j
/-Proposed d/t < 10 :C = 3.0 / lO 22 iC^l.S /
Eurocode-^ C = 2.5
^ AS/NZS 4600. AISI
'^^^^
/
}
3.0 CSA-S136-' d/t<\0:C = C = 30t/d \015:C =
10 0.5 0.0
25
15
d/t FIGURE 3 EXISTING AND PROPOSED BEARING COEFFICIENTS FOR BOLTED CONNECTIONS
At present, the bearing coefficient that is contained in the AS/NZS 4600 and AJSI design standards is a constant C = 3.0 for single shear bolted connections with washers under the head and nut. The Eurocode design standard also specifies a constant bearing coefficient of C = 2.5 for bolted connections. The CSA-S136 design standard requires that the bearing coefficient vary depending on the ratio of d/t, as shown in Figure 3. The proposed method contains a gradated bearing coefficient which is also dependent on d/t, however, the minimum possible value is lowered to 1.8 and the rate of change of the bearing coefficient is modified accordingly. The proposed bearing formulation specifies that for a single shear connection the nominal bearing capacity is calculated as follows, Vb = Ctdfu
(4)
where t and/u are the thickness and tensile strength of the member, and C is the variable bearing coefficient as shown in Table 3. TABLE 3 PROPOSED FACTOR C, FOR BEARING RESISTANCE
d/t d/t < 10 I0 22
3.0 4.0-0.\ d/t 1.8
607 Comparison of the Proposed Method with Existing Design Standards Overall statistical information that was calculated using the existing design standards, as well as the proposed method, for the bolted connection tests that were completed for this research project can be found in Tables 12. A distinct improvement in the mean values of the test-to-predicted ratios, comparing the proposed method with the AS/NZS 4600, AISI and Eurocode design standards, is evident for the specimens where thin sheet steels are connected. In the case of the 0.42 mm G550 test specimens the mean Put / Pup ratios for the longitudinal, transverse and diagonal directions improve from 0.661, 0.591 and 0.622 for the AS/NZS 4600 and AISI design standards to 1.101, 0.986 and 1.036 for the proposed method (see Rogers and Hancock (1998b)). A dramatic improvement in the mean test-to-predicted ratios also occurs for the 0.60 mm G550 test specimens, where for the AS/NZS 4600 and AISI design standards Put / ^up ratios of 0.718, 0.636 and 0.693 were calculated for the longitudinal, transverse and diagonal specimens, respectively, and for the proposed method corresponding ratios of 1.096,0.971 and 1.058 were determined (see Rogers and Hancock (1998b)). An improvement in the mean Put / Pup ratios also occurs for the 0.60 mm G300 test specimens, however, the resulting test-to-predicted ratios rise above 1.0 due to the dependence of bearing resistance on material properties as well as the d/t ratio. The resulting Put / Pup ratios for the G300 sheet steel bolted connection specimens, using the proposed gradated bearing resistance method, are conservative in comparison with the ratios for similar thickness G550 sheet steels. This result is caused by the dependence of the bearing coefficient on the material properties, as well as the thickness of the connected sheet steels. An attempt to model the relationship between the bearing resistance and the material properties, along with other variables, has been completed by Zadanfarrokh and Bryan (1992), as well as Bryan (1993). The authors of this paper have not included a material property dependence in the gradated bearing formulation to limit the complexity of the recommended formulation. The bearing resistance of bolted connections that are constructed of G300 and other mild sheet steels can be conservatively predicted using the recommended formulation, hence, design predictions of connection capacity will be safe. It is the unconservative behaviour observed for bolted connections that are constructed of thin G550 sheet steels that needs to be accommodated for in design. AISI Calibrated Resistance Factors, (I) The resistance equations for end pull-out, bearing and net section failure that are contained in the proposed method for bolted connection design were calibrated according to the procedure specified in the AISI commentary (1997b) with a j8b of 3.5. Calibration of the proposed method for end pull-out, bearing and net section failure was completed using both the full value of the ultimate strength, /u, and the reduced value specified for thin G550 sheet steels, 0.75/u. Resistance factors determined using the Australian, New Zealand and USA dead and live load factors with Put / Pup ratios calculated using all of the bolted connections specimens that were tested for this research exceed the required 0 = 0.60 for the end pull-out and bearing failure equations. Similarly, use of the proposed equation for net section fracture yields resistance factors which exceed the required 0 = 0.765 (0.75 USA) for j3o = 3.5. Calculated resistance factors for the Canadian and European dead and live load factors exceed the required (j) = 0.75 and (j) = 0.80, respectively. If the 0.75/u reduction factor is applied to all of the test data the calculated resistance factors increase to values far above those factors currently used in the design of bolted connections.
CONCLUSIONS The results of the bolted connection tests that were completed for this research indicate that the current connection provisions set out in the AS/NZS 4600, AISI and Eurocode design standards cannot be used to accurately predict the failure mode of bolted connections that are fabricated from thin G550 and G300 sheet steels. Furthermore, these design standards cannot be used to accurately determine the bearing resistance of bolted specimens based on a failure criterion for predicted loads. It is necessary to incorporate a gradated
608 bearing resistance equation which is dependent on the thickness of the connected material, similar to that found in the CSA-S136 design standard. The net section fracture of 0.42 and 0.60 mm, G550 and G300 sheet steels at connections can be accurately and reliably predicted without the use of a stress reduction factor based on the configuration of bolts and specimen width. The net section fracture resistance of a bolted connection calculated following the CSA-S136 design standard procedure, where the net cross-sectional area and the ultimate material strength are used, is adequate. The proposed design method for bolted connections that are loaded in shear can be used to improve the accuracy of the predicted load resistance when thin sheet steels are joined. It is recommended that the gradated bearing coefficient formulation, the unreduced net section resistance, and the Eurocode method for end pull-out be used in the design of bolted connections. Calibration of the proposed bolted connection design provisions using the full ultimate strength, /u, reveals that the end pull-out, bearing and net section fracture design equations yield resistance factors which exceed currently specified values. The calculated resistance factors increase to values far above those currendy used in the design of bolted connections if the 0.75/u reduction factor is applied to all of the test data.
ACKNOWLEDGEMENTS The authors would like to thank the Australian Research Council and BHP Coated Steel Division for their financial support. The first author is supported by a joint Commonwealth of Australia and Centre for Advanced Structural Engineering Scholarship.
REFERENCES American Iron and Steel Instittite. (1997a)." 1996 Edition of the Specification for the Design of Cold-Formed Steel Structural Members", Washington, DC, USA. American Iron and Steel Institute. (1997b). "Commentary on the 1996 Edition of the Specification for the Design of Cold-Formed Steel Structural Members", Washington, EXT, USA. BHP. (1992). "The Making of Iron and Steel", Seventh Edition, BHP Steel Group, Melboume Vic, Australia. Bryan, E.R.. (1993). "The Design of Bolted Joints in Cold-Formed Steel Sections", Thin-Walled Structures, Vol. 16,239-262. Canadian Standards Association, S136. (1994). "Cold Formed Steel Structural Members", Etobicoke, OnL, Canada. Dhalla, A.K., Winter, G.. (1974a). "Steel Ductility Measurements", J. Struct. Div., ASCE, Vol. 100, No. ST2, All-AAA. Dhalla, A.K., Winter, G.. (1974b). "Suggested Steel Ductility Requirements", J. Struct Div., ASCE, Vol. 100, No. ST2,445^62. European Committee for Standardisation, Eurocode 3. (1996). "Design of steel structures. Part 1.3 General rules. Supplementary rules for cold formed thin gauge members and sheeting", Brussels, Belgium. Popowich, D.W.. (1969). "Tension Capacity of Bolted Connections in Light Gage Cold-Formed Steel", Thesis presented to the Graduate School of Cornell University for the Degree of Master of Science, School of Civil Engineering. Rogers, C.A., Hancock, G.J.. (1996). "Ductility of G550 Sheet Steels in Tension - Elongation Measurements and Perforated Tests", Research Report No. R735, Centre for Advanced Structural Engineering, University of Sydney, Sydney, NSW, Australia. Rogers, C.A., Hancock, GJ.. (1997a). "Ductility of G550 Sheet Steels in Tension", J. Struct. Engrg.,ASCE, Vol. 123, No. 12,1586-1594. Rogers, C.A., Hancock, G.J.. (1997b). "Bolted Connection Tests of Thin G550 and G300 Sheet Steels", Research Report No. R749, Centre for Advanced Structural Engineering, University of Sydney, Sydney, NSW, Australia. Rogers, C.A., Hancock, G.J.. (1998a). "Bolted Connection Tests of Thin G550 and G300 Sheet Steels", J. Struct. Engrg., ASCE, Vol. 124, No. 7,798-808. Rogers, C.A., Hancock, G.J.. (1998b). "New Bolted Connection Design Formulae for G550 and G300 Sheet Steels Less Than 1.0 mm Thick", Research Report No. R769, Centre for Advanced Structural Engineering, University of Sydney, Sydney, NSW, Australia. Rogers, C.A., Hancock, G.J. (1998c) "Failure Modes of Bolted Sheet Steel Connections Loaded in Shear", Research Report No. R772, Centre for Advanced Structural Engineering, Department of Civil Engineering, University of Sydney, Sydney, Australia. Standards Australia/ Standards New Zealand. (1996). "Cold-formed steel structures - AS/NZS 4600", Sydney, NSW, Australia. Standards Australia (1993). "Steel sheet and strip - Hot-dipped zinc-coatedOTaluminium/zinc coated - AS 1397", Sydney, NSW, Australia Winter, G.. (1956). "Tests on Bolted Connections in Light Gage Steel", J. Struct. Div., ASCE, Vol. 82, No. ST2,920-1 - 920-25. Zadanfarrokh, F., Bryan, E.R.. (1992). 'Testing and Design of Bolted Connections in Cold Formed Steel Sections", Proceedings of the 11* Intemational Speciality Conference on Cold-Formed Steel Structures, University of Missouri-Rolla, Rolla, MO, USA, 625-662.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
609
PULL-OVER STRENGTH OF TRAPEZOIDAL STEEL CLADDINGS R.B. Tang^ and M. Mahendran^ 1
Eindhoven University of Technology, Dep. of Architecture, Building & Planning, The Netherlands ^ Physical Infrastructure Centre, Queensland University of Technology, Brisbane, Australia
ABSTRACT When crest-fixed thin steel roof claddings are subjected to wind uplift, local pull-over/dimpling failures occur prematurely at their screwed connections because of the large stress concentrations in sheeting under the screw heads. When high strength steel sheeting with reduced ductility is used, a pull-over failure associated with splitting occurs whereas a local dimpling failure occurs for mild steel sheeting with greater ductility. During high wind events such as storms and cyclones, both of these localised failures can then lead to severe damage to the buildings and their contents. This paper presents the details of an analytical investigation of the pull-over failure associated with local dimpling that occurs in mild steel trapezoidal claddings. It includes details of the finite element model used, and a simple design formula.
KEYWORDS Crest-fixed trapezoidal steel cladding systems, wind uplift, local dimpling and pull-over failures, finite element modelling, design formula
1.
INTRODUCTION
The profiled steel roof and wall cladding systems in Australia (Figs, la, b) are commonly made of very thin steels and are crest-fixed with screw fasteners (Fig. Ic) whilst in the USA and Europe the claddings are made of thicker steels and are valley-fixed. Furthermore, in the USA/Europe scenario, snow loading often dominates the design of these buildings. In contrast, in Australia and its neighbouring countries, wind uplift/suction loading dominates the design of low-rise buildings, which suffer severe damage during high wind events such as cyclones and storms. Such damage has often been initiated by the local failures of the screwed connections in steel cladding systems under wind uplift/suction loading (Mahendran, 1994). There are two common local failures, namely pull-out and pull-over failures. The pull-out failure occurs when the screw fastener pulls out of the batten or purlin whereas the pull-over failure occurs when the screw head pulls over by splitting the steel sheeting (Fig.2a) due to the presence of large stress concentrations around the fastener holes under wind uplift loading (Mahendran, 1994). However, a localised dimpling failure (Fig. 2b) also occurs instead of splitting for corrugated roofing and some shapes of trapezoidal cladding and when claddings are made of steels with greater ductility. The localised dimpling failure has also been referred to as pull-over failure in this paper because the steel sheeting survives only a few cycles of storm/cyclone loading once a localised dimpling failure has
610 occurred (Mahendran, 1990a). Figure 2c shows a pull-over failure following local dimpling and less than 100 cycles of loading (low cycle fatigue). The pull-out failure has abeady been investigated by Mahendran and Tang (1998). Both types of pull-over failures (splitting and local dimpling) have also been mvestigated previously (Mahendran, 1990a,b, 1994, 1995; Xu, 1994), however, further research data is needed in order to develop reliable design formulae that can be used in design practice without the need for further testing. Current design formulae and test methods (AISI, 1989, Eurocode, 1992, SA, 1996) have been found to be inadequate for crest-fixed steel cladding systems (Mahendran, 1995 and Mahendran and Tang, 1998). Thus an investigation was carried out to study the pull-over failures in profiled steel cladding systems under static wind uplift/suction conditions using both experiments and fmite element analyses. This paper presents the details of the analytical investigation relating to the pullover failures in one of the trapezoidal steel claddings (Type A in Fig. la) and the results.
Type A Type B (a) Trapezoidal (b) Corrugated (c) Crest-fixing Figure 1. Standard Profiled Steel Cladding Systems used in Australia
m^M kmmM (a) Static Pull-over : splitting (b) Static Pull-over : Dunpling (c) Fatigue Pull-over Figure 2. Typical Local Pull-over and Dimpling Failures of Profiled Steel Claddings
»M«S^^^M*:^s(A*iii£^'*s^%i!t«(ii.-'
FINITE ELEMENT ANALYSIS 2.1
Model Geometry
In the analysis and testing of multi-span continuous steel claddings, the use of a two-span steel cladding assembly with simply supported ends (Fig.3a) has been considered adequate to model the critical regions of a multi-span cladding system under a uniform wind uplift pressure (Mahendran, 1990a,b, 1994). The steel claddings of one or two sheet width and crest-fixed to battens/purlins using screw fasteners have been used in these experiments. The uniform wind uplift pressure was simulated by either using air bags or layers of bricks until the local failures occurred at the critical central support fasteners that had the largest force. The measured central support reaction force was used to determine the local pull-over failure load per fastener. The same two-span cladding assembly under a uniform wind uplift pressure was considered in this investigation on trapezoidal (Type A) cladding using a fmite element analysis (FEA) program ABAQUS. However, the symmetry of geometry and loading present in a two-span cladding system allowed modelling of only half of the two-span cladding system. Therefore, a model of one-span long (900 mm) and an appropriate width equal to half the pitch of the steel profile (95 mm for trapezoidal Type A cladding) was considered to be adequate in representing the steel cladding in the FEA (Fig.3b). This satisfactorily reduces the size of the fmite element model and hence solution times. This modelling approach is similar to that used by Mahendran (1994) and Xu (1994).
611
(a) Two-span cladding (b) Equivalent Model Figure 3. Modelling of Two-span Trapezoidal Type A Cladding 2,2
Analysis Type and Finite Elements
The intermittently crest-fixed, thin steel sheeting subjected to wind uplift loading undergoes large crosssectional distortion followed by localised yielding around the fastener holes. In the case of trapezoidal Type A sheeting, the screwed ribs are separated by wide pans, which allows both longitudinal and traverse bending actions in the sheeting and the resulting cross-sectional distortion (see Fig.4a). This distortion leads to large stress concentrations at the fastener holes, leading to a premature localised failure with or without splitting (see Fig.4b). In order to take into account the large deformations of the sheeting and localised yielding and deformations at the fastener hole, a non-linear static analysis including both geometry and material effects was used.
(a) Cross-sectional distortion of sheeting (b) Large localised deformations Figure 4. Deformations in Steel Cladding under Wind Uplift In order to precisely model the observed behaviour of steel cladding, the following considerations were given to the selection of element type from the ABAQUS element library: a) Since the cladding is subjected to a combined effect of in-plane membrane and bending actions, the elements must be able to represent such behaviour and deformations within the thin sheeting. b) The elements must allow all distortional effects as large cross-sectional distortion was observed in the full scale cladding experiments (see Fig. 4) c) The elements must be capable of handling large displacements and elasto-plastic deformations. In the ABAQUS element library, the shell elements generally satisfy all the above criteria. Therefore large strain quadrilateral shell elements (S4R) were used whenever possible whilst triangular large strain shell elements (STRI3) were used for filling the intersections between fine and coarse meshes and for modifying S4R elements that failed distortion check. When the available computer resources were limited, the more general shell elements, S4R5 and STRI35 were used. The difference in results when using the two fypes of elements was negligible in most cases (<8%). 2.3
Loading and Support Conditions
ABAQUS permits applications of specifying pressure loads on shell elements. A uniform surface pressure across all shell elements was considered to be suitable to represent the static wind uplift/suction pressure loading adequately.
612 Figures 3b and 6b show the boundary conditions used due to the presence of symmetry conditions. The nodes on the plane of symmetry were constrained from translation perpendicular to the plane of symmetry and rotation about the in-plane axis. In addition to these, the constraint conditions between the steel sheeting and the screw head at the critical central support must be modelled adequately, ie., the presence of a 2 mm neoprene washer between the screw head and steel sheeting had to be modelled (see Fig.5a). For this purpose, linear spring elements (SPRINGA) were used between the screw head (solid elements) and steel sheeting (shell elements) as shown in Fig. 5b. In this manner, all the nodes within the area of the neoprene washer were constrained against z-translation. The screw head was constrained against all three translations. Experiments showed that the nodes at the edge of screw fastener hole were moving away from the screw shaft. Therefore, both x- and y-translations of these nodes were assumed to be free.
(a) Physical Model (b) Finite Element Model Figure 5. Modelling the Support Provided by Screw Fastener The FEA using linear spring elements requires an elastic stif&iess value. The elastic compressive stiffness of the neoprene washer was first determined using a simple test method and was then used in the input data for the spring elements in the FEA. The compressive load-deflection curve of the neoprene washer obtained from tests was found to be nonlinear under higher loading, however, the use of the measured non-linear response with non-linear springs in the FEA was not possible due to convergent problems. The use of linear springs with the measured initial elastic compressive stif&iess was found to be simple and easy while providing reasonably accurate results. Figure 5 shows the details of the modelling of the support/restraints provided by the screw fastener using linear SPRINGA elements whereas Figure 6b shows the overall finite element mesh used in this investigation for trapezoidal Type A cladding. As seen in Figures 6a and b, a very fine mesh was used in the vicinity of the central support screw fastener hole to accurately model the local deformations and the high membrane stress concentrations. The material properties of the steel were determined from tensile testing for use in the non-linear FEA. Tensile test specimens from a lower grade steel (Grade 250 with min. yield stress of 250 MPa) and the commonly used higher grade steel (G550 steel with a min yield stress of 550 MPa) were tested. The lower grade steels exhibited a well-defined yield point (300 MPa), strain hardening and a high percentage of elongation at failure. However, the higher grade steels, particularly the 0.42 mm G550 steel had no yield plateau and very little strain hardening. The 0.2% proof stress was 690 MPa and the strain at failure was only about 2%. In all the analyses, the material properties were idealised to be perfectly-elastic-plastic. The modulus of elasticity (E) and Poisson's ratio ( v) were taken as 200,000 MPa and 0.3, respectively. This investigation ignored the small curvature at the comers of the trapezoidal claddings as the comer radius was quite small. The effects due to the presence of initial geometric imperfection and residual stresses due to cold-forming were also not included in the FEA as they were considered to be negligible. 2.4
Model Validation
All convergence studies and sensitivity analyses were carried out before the detailed analyses. It was assumed that the chosen finite element model would produce accurate results for the pull-over failures m
613 profiled steel cladding systems. A series of preliminary analyses was first conducted in order to validate the accuracy of the finite element model. Preliminary analyses confirmed that significant cross-sectional distortion occurred in the cladding with the centre of the pan deflecting upwards quite significantly (Fig. 6b). They also showed the localised deformations around the fastener hole, which are very similar to those observed in two-span cladding experiments (Fig. 6a).
Origiiuii Profile
(a) Localised Deformation around the fastener hole (b) Overall Deformation Figure 6. Deformations of Two-span Trapezoidal Type A Cladding Failure Load (Nff) 1400 1200 1000 800
1 t 1 T
/ / / y^ / / / / / // / / / // / / / / / / / / / / /
Bcperiment: US Bcperiment: S Bcperiment: U FB^: US-Central Support UhScrewed Ran FE^: S-MdSpan Screwed aest FBK: U-MdSJjan UhScrewed ftin
Deflection (mm)
W/ 0
5
10
15
20
25
30
35
40
45
Figure 7. Comparison of Load-deflection Curves Following the verification of deformation shapes (Fig. 6), comparisons of the load-deflection curves were made with available two-span experimental results for trapezoidal Type A cladding made of 0.42 mm G550 steel. Figure 7 shows good agreement between the load-deflection curves obtained from the FEA and two-span experiments, which further validates the accuracy of the fmite element model used in this study. However, the steel claddings made of high tensile steels with reduced ductility such as G550 steel undergoes pull-over failures initiated by splitting, and the FEA is not capable of predicting this pull-over failure load as it does not include an adequate splitting criterion. In the case of 0.42 mm G550 steel cladding, the FEA predicted a failure load per fastener of 1405 N, which is in fact the maximum load the cladding could carry, provided there was no splitting prior to this. The corresponding pressure was 6.86 kPa. It is possible the true pull-over load caused by splitting is lower than this value as the FEA is not capable of detecting splitting that was observed in the experiments. Nevertheless, the FEA failure load results for the commonly used 0.42 mm 0550 steel trapezoidal Type A cladding agreed reasonably well with the experimental results (Failure load per fastener 1365 N and pressure 6.7 kPa). However, this agreement cannot be expected for other trapezoidal profiles with different geometry.
614 The high tensile steel used (G550) has significantly reduced ductility with a strain at failure during tensile tests being only 2%. The use of measured stress-strain curve with a steep unloading curve to lower stress values was found to be inadequate in modelling the splitting around the fastener hole (Tang and Mahendran, 1998). Since the tensile coupons split at 2% strain level, it was considered that transverse splitting around the fastener hole occurred when the longitudinal membrane tensile strain at the edge of the fastener hole reached 0.02. The FEA clearly showed the presence of large membrane tensile strains in the longitudinal direction at the fastener hole. Large longitudinal membrane tensile strains at the edge of the hole might have initiated the transverse splitting at the hole. Results from a number of FEA were investigated to determine whether the failure load predicted by FEA coincided with the load at 2% longitudinal membrane tensile strain. It was found that there was little correlation between these two results. Although longitudinal membrane tensile strains dominate and cause the splitting, the strain level at which this splitting will occur is not known under the combined membrane and bending strains in the region. Therefore the results from the FEA are not applicable to claddings made of high tensile steel such as G550 steel unless an accurate failure criterion for splitting is developed for these steels under combined membrane tension and bending actions. The use of 2% strain at failure obtained from simple tension tests may not be applicable here. Research is currently under way to determine a splitting criterion for the high strength steels used in Australia. However, ABAQUS was able to model the local dimpling and pull-over behaviour adequately in the trapezoidal Type A sheeting made of steels with greater ductility. Therefore the finite element model developed in this investigation was used to determine the pull-over failure loads associated with local dimpling only, ie. for the trapezoidal claddings made of lower grade steels with greater ductility, for example, G250 steel. Results from these analyses were analysed and appropriate design formulae were developed for these claddings.
3.
RESULTS AND DISCUSSION
The finite element model discussed in the previous section was used to determine the local dimpling loads of a range of trapezoidal Type A steel claddings made of steels with greater ductility (strain at failure > 10%). The following parameters were varied in this study: base metal thickness of steel t of 0.30, 0.40, 0.42, 0.48, 0.50 and 0.60 mm; steel yield stress fy of 300, 400, 500, 600 and 690 MPa; diameter of screw head or washer d of 6.5, 8.0, 10.4, 14.5, 17.5, 20.0 and 22.0 mm; and diameter of screw shaft (hole size) do of 5.2 and 9.0 mm. Selected results agreed well with previous results obtained by Mahendran (1994) and hence confirm the accuracy of the FEA model used in this study. The results were then grouped on the basis of steel thickness, and diameter of the screw head or washer, analysed and comparisons made on the basis of these groups. The screw shaft/hole size do does not affect the pull-over strength of crest-fixed steel cladding systems. In the analysis of results, failure loads corresponding to a do of 5.2 mm were used. Currently the American (AISI, 1989), Australian (SA, 1996) and European (Eurocode, 1992) design provisions recommend the following design formulae for the pull-over strength: AISI (1989), SA (1996) Fov = 1.51 d fu (la) Eurocode 3 (1992) Fov = 1.11 dw fy (lb) where d = larger value of the screw head or the washer diameter, but limited to 12.7 mm t = thickness of the cladding material dw = the washer diameter fu = ultimate tensile strength of steel fy = yield stress of steel These formulae were found to be inadequate for crest-fixed steel cladding systems (Tang and Mahendran, 1998). Mahendran (1994) developed a formula (Eq. 2) for pull-over strength of crest-fixed steel cladding systems based on limited results from FEA. By using the term fu^^^, this equation elimmates the need for the use of 75% of the specified minimum strength for G550 steels with a thickness less than 0.9 mm. Fov = ct^f;^'
(2)
615 where c = 0.74 for Trapezoidal-Type A Claddings and 0.66 for Trapezoidal-type B claddings. However Eq.2 does not include the effects of screw fastener or washer diameter in predicting the pullover failure loads. Hence it must be considered inadequate. Therefore a new formula is proposed as follows by including the screw head/washer diameter d. Equation 3a was first considered as a simple extension of Mahendran's design formula by including d, but keeping the same t^ and fu*^^ terms. Subsequently, Equation 3b was developed as a general equation with different power coefficients for t and fu. Unlike Mahendran's (1994) method, the constants of c, a, p, and x were determined by considering all the parameters simultaneously. The "Solver" in Microsoft Excel, which is based on the method of least squares, was used to obtain the best equation that fits the test data with minimum errors. These constants are given in Table 1. It is considered unnecessary to impose an upper limit for d as in the current design formula. Hence, only the actual values of diameter d were used in Table 1.
where c, a, p, x = coefficients
Fov = cd"t^fy^^^ Fov = cd"tPfy'' and fy = yield strength of steel in MPa
(3a) (3b)
In this case, the yield strength of steel fy is used instead of the ultimate tensile strength fu as this study deals with claddings made of ductile steels. Table 1 results show that local dimpling failure loads were predicted well by both Equations 3a and 3b using the properties of the steel and screw fasteners used in the FEA. The improved Equation 3b appears to provide more uniform and less conservative mean (1.00 to 1.10) and coefficient of variation values (mostly less than 0.10) in all groups than Equation 3a, and is therefore recommended. However, in order to reduce the improved design formulae to a single equation for all groups, the parameters a, p and % were then forced to be 0.25, 2.30 and 0.65, respectively. These changes were still able to provide uniform mean values (1.03 to 1.05) and COV values (less than 0.10). Therefore Equation 3b is recommended with the following parameters: c = 0.08, a = 0.25, P = 2.30, and X=0.65. It must be noted than in the above equation, t and d are in mm and fy in MPa. TABLE 1 . FEA to Predicted Values Based on Improved Formulae for Trapezoidal-Type A Claddings
1
Steel Screw fy (MPa) t(mm) d(mm) All All < 12.7 All >12.7 <0.45 All >0.45
C 0.40 0.08 0.25 0.55 0.48
Equation 3a a Mean 0.30 1.00 1.00 1.09 0.45 1.03 0.15 1.02 0.45 1.01
COV 0.13 0.15 0.13 0.09 0.13
c 0.08 0.85 0.08 0.30 0.05
a 0.25 0.45 0.35 0.15 0.35
Equation 3b P X Mean 2.30 0.65 1.00 2.70 0.25 1.05 2.35 0.60 1.10 2.40 0.50 1.01 2.05 0.65 1.08
COV 0.07 0.19 0.04 0.04 0.08
Equation 4 | Mean COV 1.04 0.07 1.05 0.10 1.03 0.05 1.04 0.07 1.03 0.07
Although Equation 3b provides accurate local dimpling strength values, it does not include the geometrical parameters of the cladding profile. This implies that it is only applicable to profiles that are identical or similar to the commonly used cladding profiles (Fig. la). As the local dimpling strength of steel cladding systems greatly depends on the geometry of the profile, this investigation considered the effects of five geometrical parameters, the crest width b, crest height H, pitch P, trough width B, and pan height h. The preliminary FEA confirmed that within the small range of pan height values used in practice (4 to 8 mm), the cladding strength changed only marginally. Therefore, it was decided not to include h in further analyses. The remaining four geometrical parameters were varied in this investigation and the results (273 of them) were used to develop a suitable formula in the form of Equation 4 for trapezoidalType A cladding. Fov= c d" t^ fv^ b'' rf P"^ B" (4) d, t, and fy as defined in Equations 3 a and 3 b where c, a, P, x, k, I wi? and n are constants
616 The constants in Equation 4 were determined using the same procedure described earlier for Equation 3b as a = 0.25, P = 2.30, % = 0.65, and c = 0.30, k = 0.15,1 = 0.55, m = -0.85, and n = 0.20. The recommended Equations 3b and 4 are based on average local dimpling and pull-over strengths from limited number of test data and analyses. The actual pull-over strength of a real connection can be considerably less than the value predicted by these equations because of the expected variations in material, fabrication and loading effects. Therefore a capacity reduction factor commonly used in design codes should be recommended for the pull-over strength predicted by these equations. Tang (1998) has determined these factors using a statistical model recommended by AISI (1989). Based on these calculations, a capacity reduction factor of 0.60 is recommended for use with Equations 3b and 4. Although steel claddings and screw fasteners used in this investigation were obtained from particular manufacturers, results should be equally applicable to other claddings and screw fasteners provided they comply with the respective specifications for the grades of steels and fasteners used in this investigation.
4. CONCLUSIONS An analytical investigation into the pull-over failures of screwed connections in crest-fixed trapezoidal steel claddings has been described in this paper. Both types of local pull-over failures (splitting and dimpling) were investigated in detail and reasons for their occurrence given. Finite element analyses were unable to model the splitting type pull-over failures observed in high tensile steel claddings with reduced ductility. However, they were used to model the local dimpling failures that occurred for trapezoidal claddings made of ductile steels. A design formula including not only the thickness and yield strength of steel and the diameter of screw fasteners or washers used, but also the geometrical parameters of the profiled cladding has been developed for one of the trapezoidal claddings.
5. REFERENCES American Iron and Steel Institute (AISI) (1989), Specification for the Design of Cold-formed Steel Structural Members, 1986 Edition with 1989 Addendum, AISI, Washington, DC. Eurocode 3 (1992), Design of Steel Structures, Part 1.3 - Cold-formed Thin-gauge Members and Sheeting, Draft document CEN/TC250/SC3 - PTl A, Commission of the European Communities. Mahendran, M. (1990a), Fatigue Behaviour of Corrugated Roofing under Simulated Wind Loading, Civil Eng. Trans., I.E.Aust., Vol.32, No.4, pp.219-226 MsJiendran, M. (1990b), Static Behaviour of Corrugated Roofing under Simulated Wind Loading, Civil Eng. Trans., I.E.Aust., Vol.32, No.4, pp.211-218 Mahendran, M. (1994), Behaviour and Design of Crest-fixed Profiled Steel Roof Claddings under High Wind Forces, Engng. Struct., Vol.16, No.5. Mahendran, M. (1995), Test Methods for Determination of Pull-through Strength of Screwed Connections in Profiled Steel Claddmgs, Civil Eng Trans., lEAust., Vol.CE37, No.3, pp.219-227. Standards Australia (SAA) (1996), AS4600 Steel Structures Code. Mahendran, M. and Tang, R.B. (1998), Pull-out Strength of Steel Roof and Wall Cladding Systems, J. of Structural Engineering, ASCE, Vol.124, No. 10, pp.1192-1201. Standards Australia (SA) (1996) Cold-formed Steel Structures Tang, R.B. and Mahendran, M. (1998), Pull-through Strength of Profiled Steel Cladding Systems, Research Monograph 98-6, Physical Infrastructure Centre, Queensland Univ. of Technology, Brisbane. Tang, R.B. (1997), Local Failures of Steel Cladding Systems under Wind Uplift, M.Eng. Thesis, School of Civil Engineering, QUT. Xu, Y.L. (1994) Behaviour of Different Profiled Roofing Sheets Subject to Simulated Wind Uplift, Technical Report No. 37, James Cock University Cyclone Testing Station, Townsville.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
617
INVESTIGATION OF THE CAUSES OF THE COLLAPSE OF A LARGE SPAN STRUCTURE R H. Fakury\ F. A. de Paula\ R. M. Gon^alves^, R M. da Silva^ ^Engineering School of Federal University of Minas Gerais, Belo Horizonte - Minas Gerais - Brazil ^Engineering School at Sao Carlos of University of Sao Paulo, Sao Carlos - Sao Paulo - Brazil
ABSTRACT This paper presents the investigation results concerning the causes of the collapse of a large span steel structure occurred in Brazil. The structure was a space truss with height of 3m, span of 110m and length of 128m, supported by reinforced concrete columns. Support appliances that allow rotations along roof structure span direction only did the connections of the space truss to the concrete columns. The steel members were made of tubular sections with diameters varying from 76mm to 324mm. The space truss nodes were composed of flat steel plates welded to each other and spatially distributed in suitable directions. In order to obtain the failure reasons, it has been assessed the design and the structural conception, the detailing, the manufacture and assembly of the structure, the structural stability and the need of nonlinear and dynamic analysis. A set of tests in selected members of the steel structure has been also made.
KEYWORDS Steel structure, structural collapse, joint connection, structural stability, space truss, space frame.
INTRODUCTION Not long ago a great steel roof structure built in Brazil failed. It was a recently inaugurated construction that had been used few times, with capacity to shelter about 35 thousand people in popular events. Fortunately no event was happening and there were no victims in the accident. At the moment of the collapse it was raining softly and it has not been noticed the occurrence of significant winds.
DESCRIPTION OF THE STRUCTURE - GENERAL CONCEPTION A steel space truss supported by reinforced concrete columns constituted the failed structure. It also sheltered bleachers of reinforced concrete and was conceived as a pyramidal mesh, with 3 m of height
618 and rectangular base of sides of approximately 4m and 4.3m. In the transversal direction it presented span of 110m, described by a curve formed by the agreement of three arches with different rays. Longitudinally it had length of 128m, with a central part of 60m corresponding to the section supported by the concrete columns and two 34m cantilevers. The steel roof had under the main mesh two straight longitudinal catwalks that contributed to support the 34m cantilevers. It still had above the central part of the main mesh another mesh of 3m of height that, besides contributing for the rigidity of the structure, was used as the monitor framing. The reinforced concrete columns were in number of twelve, six on each side of the pyramidal mesh, spaced to each other of 12m. They had length of about 6m and transversal rectangular section with constant width of 70cm and height varying from 300cm at the base to 200cm at the top. The covering was constituted of galvanized self-supported steel sheet covered by a special product for thermal and acoustic isolation and to make the roof waterproof The figure 1 supplies a wide view of the structure.
Figure 1 - General view of the structure Structural Members The members that formed the steel mesh, in number greater then 10000, were constituted by tubular sections arranged in the following groups: • tubes of external diameter of 76.2mm and 88.9mm, thickness of 2.25mm and 2.65mm respectively, with cold formed straight ends for connection (figure 2-a); • tubes of external diameter between 101.6mm and 218.1mm, thickness between 3.00mm and 12.7mm, with welded plate ends for connections (figure 2-b); • tubes of external diameter of 323.8mm and thickness of 12.7mm, with external ring of reinforcement and welded plate ends for connections (figure 2-c). The tubes with diameter up to 165mm were cold formed and longitudinally welded and the others were hot rolled.
619
3-
/"
---I---
/
BEE: (a) diameter of 76.2mm and 88.9mm
-^--
(b) diameter between 101.6mm and 218.1mm
]E
-^
^
(c) diameter of 323.8mm Figure 2 - Structural members Connections The members of the steel mesh were connected to each other by means of joints composed by welded plates designed in the appropriate directions (figure 3-a), and the connections between the member ends and the joints were bolted. The appliances that supported the steel mesh behaved as mechanical hinges, with rotation freedom in the transversal plans, assembled on the top of the concrete columns (figure 3-b).
(b) support appliance Figure 3 - Connection elements
620 STRUCTURAL ANALYSIS AND DESIGN An evaluation of the original design data showed that it has been made a linear elastic analysis of the steel mesh, considered as a space truss linked to the columns of reinforced concrete. It has been considered the action of dead load, live load and wind load. The dead load was constituted by the self-weight of the structural elements, sidings, installations and fixed equipment. The live load involved the action due to the movable equipment and the consideration of a distributed load equal to 0,25 kN/m^. The wind was supposed acting in 0°, 45°, 90° and 180° orientations, with basic speed of 31 m/s and pressure coefficients obtained in an aerodynamic test, Blessmann (1993). The design of the mesh steel members, including its connection plates and bolts, was made following the Brazilian standard for design of steel structures, NBR 8800 (1986). This standard presents a process to determine the strength of bars submitted to axial force similar to the Eurocode 3 (1992).
STRUCTURAL DESIGN EVALUATION To evaluate the structural analysis, the design process and still to obtain more accurate conclusions about the steel pyramidal mesh behavior, trying to identify the causes of the structural collapse, several studies were made. They involved the verification of the need of a 2^^^ order theory analysis and/or a dynamic analysis, the evaluation of the structure global stability, evaluation of the stability of the structure members, considering the joints influence, and the pertinence of the space truss model used. Structural 2^ Order Theory Analysis "knd
The results obtained in an elastic structural 2° order theory analysis, considering large displacements, didn't present significant differences (3% maximum variation) in comparison to the analysis in a 1^^ order analysis, not only for displacements but also for normal forces in the steel members. Dynamic Analysis A modal dynamic analysis was made to determine the structure natural frequencies. The first four natural frequencies obtained were 0,60Hz, 0,83Hz, l,07Hz and l,27Hz. As the first two frequencies are smaller than the unit, it would be advisable to realize a dynamic analysis to assess the wind effects. Figures 4 and 5 show the vibration modes corresponding to the first two natural frequencies.
Figure 4 - Vibration mode corresponding to the T^ natural frequency
621
I I I i Figure 5 - Vibration mode corresponding to the 2°^ natural frequency Global Stability It was obtained the elastic critical load of the structure equal to 35 times the nominal dead load value, with the buckling mode characterized by vertical displacements together with of a lateral movement. This buckling mode is similar to vibration mode corresponding to the T* natural frequency (figure 4). It is a quite high value, which allows concluding that the structure presented stable form. Stability of Bars and Joints As mentioned before, the members under compression force were designed in agreement with NBR 8800 (1986). These bars were considered with constant moment of inertia, equal to the moment of inertia of the tubular section, from node to node (as the model used in the structural analysis was of a space truss, the bars were considered with pinned ends). This consideration do not lead to a safe procedure because it doesn't take into account that the bars possess moment of inertia that varies considerably along the length (from node to node). There are segments with very small moment of inertia and segments with quite high moment of inertia close to the member ends, as can be seen in the figure 6. joint plate central region of the joint
joint plate + tube connection plates
Y
^
tube
Figure 6 - Bars with variable moment of inertia
The exact consideration of the variable moment of inertia increases the slendemess ratio, which can reach 100% increment in bars with larger diameter tubes, reducing its resistance to compression
622 forces. As the bars are very close designed to its ultimate strength, many of them certainly would not have enough strength if the moment of inertia variation were properly considered. In table 1, for bars with tubes of all diameters, it can be seen: • values of the relationship between the effective length of the tubular member and the length of the n o d e - t o - n o d e bar (Ltube/Lnode-node);
•
values of the slendemess ratios considering the bar with variable moment of inertia (hv) and with constant moment of inertia equal to the moment of inertia of the tube (A.ic); • values of the design strength for these two cases ((t)Niv and (t)Nic) and the relationship among these strengths. It can be noticed that the strength of members with a variable moment of inertia is closer to the physical actual behavior than the situation with constant moment of inertia. TABLE! VALUES OF PRISMATIC MEMBERS AND VARIABLE MEMBERS
Tube D jc t (mm) 76,2 jc 2,25 88,9x2,65 1 101,6x3,00 114,3x4,25 139,7x4,75 165,1x4,75 168,3x8,74 219,1x12,7 323,8 X 12,7
(|)Niv l.ytube/L'node-node
A,iv
A-ic
0,83 0,82 0,88 0,88 0,88 0,88 0,83 0,82 0,75
178 165 121 115 96 86 87 78 72
158 136 119 111 89 74 77 53 36
(kN) 27 43 94 162 283 375 680 1391 2188
(t)Nic (kN) 34 61 97 171 307 416 746 1579 2627
(l)Niv/(t)Nic 0,79 0,70 0,97 0,95 1 0,92 0,90 0,91 0,88 0,83
To find the slenderness ratio of the bars with variable moment of inertia (Xi^,), a 2" order theory analysis program was used for obtaining the tension of elastic buckling in the segment constituted by the tubular member (fe), and used the equation: " ^
(1)
where E is the modulus of elasticity of steel. This procedure is recommended by Eurocode 3 (1992), since the bar with variable moment of inertia doesn't possess initial curvature greater than a thousandth of its length (fi-om node to node), and that its buckling mode approaches a sine-curve. A set of 12 tests was also carried out in bars with similar connections to the ones used in the collapsed structure, Gon^alves et al. (1996). Four of them were for tubes of diameter of 76,2mm, thickness of 2,25mm and cold formed straight ends, four to tubes of diameter of 101,6mm, thickness of 3mm and extremities with welded plates and four to tubes of diameter of 114,3mm, thickness of 4,25 mm and extremities also with welded plates. The lengths, from node to node, were of 4162mm to the first two sections and of 4354mm for the last section. The obtained resuhs showed that in the tube of cold formed straight ends the nominal compression strength was on the average 44% smaller than the theoretical calculated value, considering the variation of existent moment of inertia, and in the bars with extremities with welded steel plates this difference was a little larger. In the first case, the results of the tests can be justified by the sensibility of this node type for the imperfections in the manufacture of the bars and eccentricities in the line of application of the load.
623 Validity of the Space Truss Model As the proposed steel pyramidal mesh presented a member-to-member connection where the joints had great relative length, including a region with high rigidity, one felt convenient to study the validity of the space truss adopted in the structural analysis. For this, the mesh was analyzed as a space frame, without moment freedom in the bar ends, which were considered with variable moment of inertia, as described in the precedent item. Comparing the displacements between the two models, a maximum difference about 15% was verified. Regarding the member axial forces, no significant difference was detected, except for some not very requested bars, with axial force smaller than 10 kN, that were designed with great safety. It was observed however, in the frame model, the occurrence of relatively high bending moments in the direction of the small moment of inertia axis of the joint plate (figure 3-a). These moments in some case result in stresses of lOOMPa and many of these connection plates would not have enough strength to support the solicitations if such stresses were included in the design. Another problem detected when using the space frame model refers to the link of the steel structure to the support appliances. As the top chord didn't converge for the work point of those appliances, the corresponding eccentricities result in a high bending moment in the steel plate of the support appliance used to connect the bottom chord (tube of diameter of 323,8 mm - figure 7). This bending moment had enough intensity to cause the formation of a plastic hinge in the steel plate. ate to connect the botton chord
plate to connect the top chord
eccentncity Figure 7 - Eccentricity of the support loads It can be said that the model of space truss used in the structural analysis supplies good results for axial member forces and displacements. However, considering the used joint types, the space frame model can be considered more appropriate. This model defines in a suitable way the behavior of the mesh, allowing a more correct design of joint plates, tubes connection plates and support appliances.
FAILURE MODE The failure mode of the structure observed in the survey of the wreckage can be considered compatible with the evaluation done in the precedent item and is illustrated in the figure 8. It was characterized by the rupture of the steel plate used in the support appliance to connect the bottom chord, in all the six concrete columns localized in one side of the structure. With that, the structure loosened from the supports and dropped to its external side, with the formation of two longitudinal plastic hinges close to the monitor framing. The plastic hinges occurred with the excessive deformation of joint plates and segments where the joint plate and the tube connection plate work together.
624
Figure 8 - Failure mode CONCLUSIONS Based in the obtained results, it can be concluded that the adopted joint type had contributed to the collapse, mainly because of its great length in relation to the effective length of the tubular members. This aspect caused a significant reduction of the strength to the axial compression of the bars. In the case of the bars with cold-formed straight ends, the strength reduction is still greater, as could be attested by the results of the mentioned tests. The axis eccentricity of the top chords in relation to the point of work of the support appliance certainly also influenced in the collapse. It was verified that a dynamic analysis must have been done in the phase of the structure design. However, it can be affirmed that the dynamic effects didn't have influence in the collapse that happened without atmospheric turbulence or base accelerations due to rhythmic activities. Finally, it can be affirmed that the space frame model would be more suitable for the structural analysis, instead of the used space truss model. The space frame model would define better the behavior of the mesh and it would allow a more correct design of joint plates, tube connection plates and support appliances.
REFERENCES Blesmann, J. (1993). Aerodynamic Study, FUNDATEC-UFRGS, Porto Alegre, Brazil. Eurocode 3 (1992). ENV1993-1-1: Design of Steel Structures-General Rules and Rules for Buildings. European Committee for Standardization, Brussels, Belgium. Gongalves, R.M., Fakury, R.H. and Magalhaes, J.R.M. (1996). Performance of Tubular Steel Sections Subjected to Compression-Theoretical Experimental Analysis, In; Proceedings of the 5th International Colloquium on Structural Stability, 439-449, Rio de Janeiro, Brazil. NBR 8800 (1986). Design of Building Steel Structures. ABNT - Brazilian Association of Technical Standards, Rio de Janeiro, Brazil.
Session B2 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.
627
COLUMN CURVES FOR STAINLESS STEEL ALLOYS and J. RondaP ' Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia ^ MSM Institute of Civil Engineering, University of Liege, Quai Banning 6, 4000 Liege, Belgium
ABSTRACT The paper describes a column curve formulation capable of producing strength curves for cold-formed stainless steel columns. The formulation uses a nonlinear expression for the imperfection parameter but is otherwise identical to the Perry-Robertson equation used in EurocodeS, Parts 1.1, 1.3 and 1.4. It is shown that several column curves are necessary for accurately describing the strength of austenitic and austenitic-ferritic stainless steel alloy columns and two curves are proposed. One of these is close to the curve for cold-formed sections currently used in the draft Eurocode3, Part 1.4. A new column curve is also proposed for ferritic alloys and 12% Chromium weldable structural steels.
KEYWORDS Stainless steel. Columns, Flexural buckling, Design, Multiple column curves, Eurocode 3 Part 1.4.
INTRODUCTION The draft European Standard for stainless steel structures, Eurocode3, Part 1.4, (Eurocode3 1996a) was developed in the early 1990s largely by the British Steel Construction Institute (SCI). Initially the recommendations were published as the Euro Inox Design Manual (EURO INOX, 1994) featuring design recommendations, a commentary and worked examples. The current draft of Eurocode3, Part 1.4, hereon referred to as "EC3, 1.4," was published in 1996 (Eurocode3, 1996a) following the initial draft in 1992 and a revision in 1994. The Euro Inox Design Manual was based on the test data available in the early 1990s. Most tests had been conducted on structural members cold-formed from annealed austenitic alloys. For this reason, the draft standard only applies to austenitic and austeniticferritic (duplex) alloys and not to ferritic alloys. The latter alloys can, however, be designed using the informative Annex D of EC3, 1.4, but as will be shown in this paper, the design approach described in Annex D of EC3, 1.4, is very conservative for compression members. The draft EC3, 1.4, specifies two strength curves for columns failing by flexural buckling and uses a Perry-Robertson curve with a linear imperfection parameter, r] = a{X-X,)
(1)
628 For cold-formed open and rolled tubular sections, the imperfection parameter is defined by a = 0.49 and ^ = 0.4. For welded sections, the values of a = 0.76 and XQ = 0.2 apply. In specifying these values, no regard is made to the fact that the mechanical properties of the alloys covered by ECS, 1.4, are different and consequently, the corresponding strength curves are different. The differences arise partly because the chemical compositions are different and partly because different degrees of cold-working may be used in the forming process. The effects of chemical compositions and cold-working are recognised by use of strength classes (S220, S240, S290, S350, S480) and cold-worked strength grades (C700, C850, etc) respectively. The numerals refer to the 0.2% proof stress and tensile strength for the Sxxx and Cxxx strength classes/grades respectively. However, despite the differences in 0.2% proof stress of the alloys pertaining to these strength classes and cold-worked grades, a single column curve is specified in the current draft of EC3, 1.4, for cold-formed open and rolled tubular sections. It will be shown in this paper that there are significant differences in the strength curves of the various strength classes and cold-worked grades, and that several column curves therefore should be used. The column design provisions of the ASCE/ANSI Standard for Stainless Steel Cold-formed Members (ASCE 1991) uses a tangent modulus approach and thus specifies different column curves for alloys with different mechanical properties. While this approach is more general and accurate, it is implicit and requires iteration. The EC3, 1.4, approach is explicit but lacks accuracy. It also lacks a strength curve for ferritic alloys. While the simplicity of the EC3, 1.4, approach makes it preferable from the designer's viewpoint, additional strength curves to the curve presently used in EC3, 1.4, for coldformed sections are required. The objective of this paper is to present an accurate column curve formulation capable of taking into account the different mechanical properties applicable to the stainless steel alloys covered by EC3, 1.4. The formulation is the same as that specified in EC3, 1.4, except that the imperfection parameter takes the form,
Jl = a({X-\y-\)
(2)
rather than the linear form of Eqn. 1. The formulation can produce accurate strength curves for austenitic and austenitic-ferritic (duplex) alloys, as well as ferritic alloys and 12% Chromium weldable structural steels. Using the formulation, two column curves are proposed in place of the current strength curve of EC3, 1.4, and a new column curve is proposed for ferritic alloys and 12% Chromium weldable structural steels. The scope of the paper is limited to cold-formed sections failing by flexural buckling.
THE IMPERFECTION PARAMETER The column curve formulation employed in this paper is described in detail in Rasmussen & Rondal (1997). It is generally applicable to metallic columns with non-linear stress-strain curves such as stainless steel and aluminium columns. In developing the method, advanced finite element analyses were carried out on columns with different slendemess for a wide range of mechanical properties, represented by the Ramberg-Osgood parameters EQ, ao.2 and n. {EQ is the initial Young's modulus, ao.2 the 0.2% proof stress and n an exponent which controls the sharpness of the knee of the stress-strain curve). The values of imperfection parameter (r|), which provided exact agreement between the Perry equation and the finite element strengths, were then back-calculated and an analytic expression for the imperfection parameter (Eqn. 2) was proposed. Figure 1 shows the exact values of r| (referred to as "target values" in Fig. 1) and the fit to these values using Eqn. 2. The Ti-curves are shown for e =
629 OQ.2/EO = 0.001 using n = 3, 5, 10 and 100. The latter value represents bi-linear materials such as ordinary carbon structural steel. 1.75
\ ^ = 0.001
Fig. 1: Exact and approximate r|-values The main conclusion drawn from Fig. 1 is that the target r|-curve is nearly linear in the case of « = 100 but not for lower values of n. For this reason, the linear form of the imperfection parameter, whilst appropriate for carbon structural steels, cannot be expected to provide accurate predictions for nonlinear materials with low values of n, such as stainless steel and aluminium alloys. Typically, stainless steel alloys have n-values between 3 and 10, while aluminium structural alloys have n-values between 8 and 40. As shown in Rasmussen & Rondal (1997), the nonlinear form of the imperfection parameter given by Eqn. 2 is capable of providing close fits to finite element strength curves for a wide range of mechanical properties that includes all stainless steel alloys used in structural practice.
COLUMN CURVE FORMULATION In using the procedure described in Rasmussen & Rondal (1997), the mechanical properties are assumed to be defined in terms of the Ramberg-Osgood parameters (EQ, ao.2, n). These are assumed to have been obtained from curve fits to measured stress-strain curves of stub column tests or compression coupon tests of the finished product. A Perry curve is adopted as strength curve by modifying the imperfection parameter to be expressed by Eqn. 2 where the constants a, p, ^ and X\ are expressed in terms of the Ramberg-Osgood parameters (EQ, ao.2, «)• Thus, the nondimensional column strength is calculated using, X=
r = ^
(3) (4)
where the imperfection parameter (r|) is given by Eqn. 2 and the constants (a, P, ^ , A,i) are expressed in terms of the parameters e = OQ.I/EQ and n as detailed in Rasmussen and Rondal (1997).
630 In Eqns 3-4, % and X are defined as (5) (6) (7)
{Llrf where GU, L and r are the ultimate stress, effective length and radius of gyration respectively.
Figure 2 shows a comparison between the general column curve formulation (%) and finite element strengths {s = au/ao.2) for e = 0.001 and « = 3, 5 and 10. As shown in Rasmussen & Rondal (1997), the general formulation is highly accurate with a maximum discrepancy of 5.8% for the entire parameter range. This discrepancy was, in fact, encountered at ^ = 1.25 for the case (n, e) = (10, 0.001) which is included in Fig. 2. The general column curve formulation has been verified against established solutions and tests on aluminium columns (Bernard et al., 1973), as shown in Rasmussen and Rondal (1998a). Figure 2 demonstrates the influence of the n-parameter on the strength curves. In particular, the strength curve has single curvature for «=3 but triple curvature for n = 10. It is evident that a single column curve cannot be expected to be accurate for the complete range of stainless steel alloys with values of n between 3 and 10. 1.4 e= -^- =0.001
X.s
Fig. 2: Comparison of general column curve formulation with advanced finite element strengths
COLUMN CURVES Cold-formed austenitic and austenitic-ferritic (duplex) alloys EC3, 1.4, defines values of £0, cyo.2 and n for strength classes S220, S240 and S480, as shown in Table 1. Using these values and the expressions for a, (3, XQ and X\ given in Rasmussen & Rondal (1997), the
631 values of a, p, XK) and Xi shown in Table 1 have been obtained. These define the column curves for the three strength classes, as shown in Fig. 3. The column curves for strength classes S220 and S240 (austenitic alloys) are close as should be expected because of the slight differences in their values of ao.2 and n. However, the column curve for strength class S480 (duplex alloys) is significantly higher than those for classes S220 and S240. The reason is that the proof stress is much higher, and this raises the strength curve because the column strength is sensitive to changes in e = CQ.I^EQ for low values of 71, (while it is not for high values of n, as shown in Rasmussen & Rondal (1997)). Figure 3 also shows the EC3, 1.4, strength curve for cold-formed sections. It appears the EC3, 1.4, curve is fairly close to the column curve for strength class S480 obtained using the general column curve formulation, although the former has double curvature while the latter has single curvature.
TABLE 1 a, P, Xo AND Xi VALUES FOR S220, S240 AND S480 STRENGTH CLASSES Strength class S220 S240 S480
Eo (GPa) 200 200 200
Cyo.2
n
a
P
Xo
X,
(MPa) 220 240 480
5.5 6 4
1.24 1.14 1.31
0.18 0.16 0.18
0.55 0.56 0.67
0.30 0.30 0.37
1.4
1
r
1.2 1.0 0.8 Euler (EQ) 0.6 I0.4 0.2 0 ' 0
S220 S240 S480
0.25
220 5.5 240 6 480 4
0.5
0.75
LO X
L25
1.5
1.75
2.0
Fig. 3: Column curves for S220, S240 and S480 strength classes Austenitic and duplex alloys are susceptible to cold-working much more so than carbon steel. The mechanical properties can be enhanced by cold-reducing the thickness of the sheet or strip before forming. Alternatively, the properties can be enhanced by roll-forming strip into hollow sections. To demonstrate the effect of cold-working, the values of ao.2 and n given in the ASCE Specification for Cold-formed Stainless Steel Structural Members (ASCE 1991) have been used to determine the values of a, P, Xo and Xi shown in Table 2. The parameters are shown for annealed, V4-hard and Vi-hard AISI 304 and 316 alloys (1.4301 and 1.4401 in EN10088 (1995) respectively). The annealed alloys pertain to strength classes S220 and S240 respectively. In EC3, 1.4, different values of 0.2% proof stress are
632 assigned to annealed AISI 304 and 316 alloys, while the same and somewhat lower value (193 MPa versus 220 MPa and 240 MPa) is used in the ASCE Specification. Table 2 also shows values of ao.2 and n for hollow sections cold-rolled from annealed AISI 304 and 304L alloys (1.4301 and 1.4306 in EN10088 respectively), as reported in Talja & Salmi (1995) and Rasmussen & Hancock (1993) respectively. The values of ao.2 and n were determined as the averages of the those values given for SHS in Rasmussen & Hancock (1993) and Series RHS-3 in Talja & Salmi (1995). The values were obtained from stub column tests. The hollow sections would fall in the coldworked strength grades C700 and C850 of EC3, 1.4, as far as proof stress is concerned. TABLE 2 a , p , %Q AND ^ 1 VALUES FOR COLD-WORKED AUSTENITIC ALLOYS
Alloy (AISI) 304,316 304,316 304,316 304, 304L
Hardness annealed V4-hard i/2-hard Cold-rolled
1.4
1
T
(GPa) 200 200 200 200
1
1
Cyo.2
n
a
P
Xo
X,
(MPa) 193 345 448 505
4.1 4.58 4.22 3.2
1.57 1.31 1.29 1.45
0.28 0.16 0.16 0.28
0.55 0.62 0.66 0.68
0.20 0.35 0.38 0.31
I
1"' ' —r'"
I -"
1.2 1.0
\ \V.V^EC3,1.4\
0.8 / Euler (EQ) 0.6 0.4 0.2 0
L AISI 304, 316 ann AISI 304, 316 1/4-hard AISI 304, 316 1/2-hard r Cold-formed SHS&RHS AISI304
1 0.25
1
0.5
^0.2 193 345 448 505
^^^2^^^^«V^ 4.1 7 / / ^ ^ 4.58// 4.22'/ 3.2 '
1 0.75
L 1.0 X
A
J
1
1
1.25
1.5
1.75
2.0
Fig. 4: Strength curves for work-hardened AISI 304 and 316 stainless steel columns
The strength curves obtained using the general column curve formulation and the values of a, P, ^ and X\ given in Table 2 are shown in Fig. 4. The significant difference between the curves is mainly a result of the different values of 0.2% proof stress, since the n-values are close. This emphasises the point made above that the strength curve is sensitive to changes in proof stress for low values of n. The EC3, 1.4, column curve for cold-formed sections is also included in the figure. Generally it lies between the upper two curves obtained using the general formulation.
633 It should be clear from Figs 3 and 4 that a single column curve cannot be expected to accurately predict the strength of all strength classes (S220, S240, S290, S350, S480) and cold-worked strength grades (C700, C850, etc) included in EC3, 1.4.
Ferritic alloys The informative Annex D of EC3, 1.4, allows the strength of ferritic alloy columns to be determined using Eurocode3, Part 1.1, (Eurocode3, 1992) or Eurocode3, Part 1.3, (Eurocode3, 1996b) by replacing the initial Young's modulus (£"0) by the secant modulus {E^) determined at the 0.2% proof stress. This leads to a scaling of the normalised slendemess such that Xf replaces X in the strength equations, where (8)
''-'IT
Using the Ramberg Osgood expression for the stress-strain curve, the term (-^IEO/ES(CO2)) is derived as, ' ^ -
I + O.OO2A.
(9)
The expression for ^EQ/ES is independent of n because the secant modulus is determined at ao.2. The present draft of EC3, 1.4, does not include mechanical properties for ferritic alloys. In this paper, the properties for AISI409 and 430 alloys (1.4512 and 1.4016 in EN 10088 (1995) respectively) given in the ASCE Specification will be used. Furthermore, the properties of 12% Chromium weldable structural steels given in the South African Code (SABS 1997) for stainless steel structures will be used. 12% Chromium weldable structural steels, popularly known as 3Crl2 steels, are corrosion resistant but do stain under normal atmospheric conditions. Strictly speaking, they are not a stainless steel alloy, however, their mechanical properties are similar to those of ferritic alloys. Table 3 summarises the values of ao.2 and n given in the ASCE Specification and the South African Code for AISI 409 and 430 alloys (1.4512 and 1.4016 in EN10088) and 12% Chromium weldable structural steels (3Crl2). TABLE 3 a, p, Xo AND Xi VALUES FOR AISI 409, 430 ALLOYS AND 3Crl2 STEELS Alloy (AISI) 409 430 3Crl2
EO
CTo.2
n
a
P
Xo
X,
(GPa) 200 200 200
(MPa) 207 276 258
9.5 6.25 7.4
0.76 1.08 0.94
0.18 0.14 0.15
0.51 0.58 0.56
0.19 0.32 0.27
Table 3 also lists the values of a, P, XQ and ^i obtained using the values of (£"0, ao.2, n) and the expressions given in Rasmussen & Rondal (1997). The general column curves obtained using the values of a, P, Xo and X\ are shown in Fig. 5. The curves are generally close, although some discrepancy is observed at low slendemess values.
634 1.4 1.2 1.0 0.8 Euler(Eo) 0.6 0.4 AISI409 AISI430 3Crl2
0.2 0
0.25
0.5
0.75
1.0
1.25
1.5
1.75
2.0
Fig. 5: Strength curves for ferritic stainless steel alloys Figure 5 also shows the strength curves obtained using Annex D of EC3, 1.4. The curves are based on the b-curve of EC3, 1.1 and 1.3, defined by (a, XQ) = (0.34, 0.2). Using the values of EQ and ao.2 shown in Table 3 and Eqns 8-9, the normalised slendemess values were calculated as Xf=\.l\X,Xf= 1.56X, and Xf = 1.59X, for AISI 409, AISI 430 and 3Crl2 alloys respectively. It appears that the column curves resulting from the procedure specified in Annex D of EC3, 1.4, are very conservative.
Proposed column curves The preceding sections have demonstrated that for sections cold-formed from austenitic and duplex alloys, a single column curve cannot provide accurate strength predictions for all strength classes and cold-worked strength grades included in EC3, 1.4. Furthermore, a column curve for ferritic alloys and 12% Chromium weldable structural steels is required in lieu of the present conservative approach set out in Annex D of the code. This calls for the use of multiple column curves. It is proposed in this paper to use two column curves for cold-formed austenitic and duplex alloys and a third curve for coldformed ferritic alloys and 12% Chromium weldable structural steels (3Crl2). The proposed classification is set out in Table 4. TABLE 4 a, p, Xo AND Xi VALUES FOR PROPOSED COLUMN CURVES, Eo=200 GPa Column curve a-curve b-curve c-curve
Alloy austenitic, duplex austenitic, duplex ferritic, 3Crl2
Strength class/grade
CJ0.2
n
a
P
Xo
h
S350, S480, C700, C850
(MPa) 480
4
1.31
0.18
0.67
0.37
S220, S240, S290
240
4
1.53
0.25
0.58
0.23
250
7.5
0.93
0.15
0.56
0.26
635 Table 4 also lists representative values of ao.2 and n for each column curve. Using these values and assuming EQ = 200 GPa, the values of a, p, XQ and X\ shown in Table 4 were obtained using the expressions given in Rasmussen & Rondal (1997). The column curves defined by the values of a, P, XQ and X\ are shown in Fig. 6. The figure also contains the current ECS, 1.4, column curve for coldformed sections, which is closest to the proposed a-curve.
1.4 1.2 1.0 0.8 Euler (EQ) 0.6 0.4 0.2
0
0.25
0.5
0.75
1.0 X
1.25
1.5
1.75
2.0
Fig. 6: Proposed column curves for stainless steel columns
CONCLUSIONS The general column curve formulation described in Rasmussen & Rondal (1997) has been applied to stainless steel alloys. Two strength curves are proposed to cover the strength classes (S220, S240, S290, S350, S480) and cold-worked strength grades (C700, C850) included in EurocodeS, Part 1.4. These strength classes and grades apply to austenitic and austenitic-ferritic alloys. One further curve is proposed for ferritic alloys and 12% Chromium weldable structural steels (3Crl2). The column curves apply to cold-formed open and hollow sections, for which the membrane residual stresses are generally small. Columns fabricated by welding will have different strength curves because of higher levels of membrane residual stress. The strength curves for fabricated sections will depend on the strength class, as for cold-formed sections. However, at present, a single curve is specified in EC3, 1.4. Evidently, this curve can be expressed by the general formulation using (a, p, Xo, X\) = (0.76, 1, 0.2, 0). The column curve formulation applied in this paper is the same as that currently used in EC3, 1.4, except that a nonlinear expression is used for the imperfection parameter in stead of the linear expression adopted in EC3, 1.4. The formulation has been shown also to produce accurate strength curves for aluminium columns (Rasmussen & Rondal 1998a, Rasmussen & Rondal 1999). Furthermore, the nonlinear expression for the imperfection parameter simplifies to the linear form used in EC3, 1.1 and 1.3, for bi-linear materials. Thus, it can be concluded that the formulation offers a generalisation of the imperfection parameter expression currently used in the Eurocodes, and that it is
636 capable of unifying the European rules for carbon steel, stainless steel and aluminium columns while maintaining high accuracy, (see also Rasmussen & Rondal (1998b)).
REFERENCES ASCE, 1991, Specification for Cold-formed Stainless Steel Structural Members, American Society of Civil Engineers, ANSI/ASCE-8-90, New York. Bernard, A., Frey, P., Janss, J. & Massonnet, C , (1973), "Recherches sur le Comportement au Flambement de Barres en Aluminium", lABSE Memoires, Vol. 33-1, Zurich, pp 1-32. EN 10088 (1995), Stainless Steels - Part 1: List of Stainless Steels, EN 10088-1, European Committee for Standardisation (CEN), Brussels. Eurocode3 (1992), Design of Steel Structures, Part 1.1: General Rules and Rules for Buildings, ENV 1993-1.1, European Committee for Standardisation (CEN), Brussels. Eurocode3 (1996a), Design of Steel Structures, Part 1.4: General Rules - Supplementary Rules for Stainless Steels , ENV 1993-1.4, European Committee for Standardisation (CEN), Brussels. Eurocode3 (1996b), Design of Steel Structures, Part 1.3: General Rules - Cold-formed Thin Gauge Members and Sheeting, ENV 1993-1.3, European Committee for Standardisation (CEN), Brussels. EURO INOX, (1994), Design Manual for Structural Stainless Steel, European Stainless Steel Development Information Group (EURO INOX), Nickel Development Institute, Toronto, Canada. Rasmussen, K.J.R. & Hancock, G.J., (1993), "Stainless Steel Tubular Members. I: Columns", Journal of Structural Engineering, American Society of Civil Engineers, Vol. 119, No. 8, pp 2349-2367. Rasmussen, K.J.R.'& Rondal, J., (1997), "Strength curves for metal columns", Journal of Structural Engineering, American Society of Civil Engineers, Vol. 123, No. 6, pp 721-728. Rasmussen, K.J.R. & Rondal, J., (1998a), "Strength curves for aluminium alloy columns". Research Report No. R771, Department of Civil Engineering, University of Sydney. Rasmussen, K.J.R. & Rondal, J., (1998b), "A Unified Approach to Column Design", Journal of Constructional Steel Research, Vol. 46, Nos 1-3, 1998, pp 127-128, Paper No. 85. Rasmussen, K.J.R. & Rondal, J., (1999), "Column Curve Formulation for Aluminium Alloys", Proceedings, 4* International Conference on Steel and Aluminium Structures, ICSAS'99, Helsinki. SABS, (1997), Structural Use of Steel, Part4: The design of Cold-formed Stainless Steel Structural Members, SABS 0162-4, South African Bureau of Standards, Pretoria. Talja, A., & Salmi, P. (1995), "Design of Stainless Steel RHS Beams, Columns and Beam-columns", VTT Research Notes No. 1619, Technical Research Centre of Finland (VTT), Espoo.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
637
A PARAMETRIC STUDY ON THE ROTATIONAL CAPACITY OF ALUMINIUM BEAMS USING NON-LINEAR FEM G. De Matteis^ L. A. Moen^, O.S. Hopperstad^ R. Landolfo^ M. Langseth^, P.M. Mazzolani^ ^ Department of Structural Analysis and Design, University of Naples Federico II, Naples, ITALY ^ Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, NORWAY ^ DSSAR, University of Chieti, Pescara, ITALY
ABSTRACT Within a general research project developed in co-operation between the Norwegian University of Science and Technology and the University of Naples Federico II, a numerical study to assess the rotational capacity of aluminium alloy beams is performed in this paper. A parametric analysis has been carried out by using the implicit FEM code ABAQUS/Standard, aiming at investigating the effects of governing parameters on the non-linear response of aluminium beams in bending. Hence, with reference to hollow rectangular cross-sections, the influence of flange slendemess, strain hardening of the material, shape factor of the section and web stiffness have been focused for two of the most common alloys. The obtained results show the importance of such parameters on both buckling strength and rotational capacity of the member, demanding for the improvement of Eurocode 9 guidelines in cross-sectional classification of aluminium alloy beams.
KEYWORDS Aluminium alloy beams. Strain hardening. Rotational capacity. Cross-sectional classification. Local buckling. Numerical analysis.
INTRODUCTION The evaluation of rotational capacity represents an extensively studied subject, particularly in the field of metal structures (Kuhlmann, 1989; Mazzolani and Piluso, 1997). Fundamental methods for the plastic design of structures are in fact based upon the assumption that plastic hinges supply sufficient rotational capacity in order to assure the redistribution of bending moments until a complete plastic mechanism is developed. Usually, due to strain hardening of the material, the section moment capacity
638 exceeds the plastic moment before exhibiting the unstable behaviour, the failure being due to flange local buckling, web local buckling or member lateral torsional buckling (Mazzolani and Piluso, 1996). The "ductility" of the section is then defined as a function of the amount of plastic deformation until the moment capacity falls again below the plastic moment. Design codes for steel structures provide simplified rules assessing the capability of cross sections to behave inelastically without loosing their bending strength capacity. Classification systems for cross sections are therefore essentially based upon fixing suitable slendemess limits for plate elements composing the section corresponding to the stage where local buckling occurs. In particular, class 1 and class 2 label cross-sections where buckling occurs after the fiilly plastic moment is attained, the former assuring the capability to develop plastic hinges with high rotational capacity (ductile sections), while the latter providing reduced inelastic deformations only (compact section). For aluminium structures, Eurocode 9 (1998) is the first code allowing for a complete inelastic analysis. Based upon experimental results, a classification of cross-sections has been proposed according to the slendemess of the individual elements composing the section. In particular, the widthto-thickness ratio h/t and the conventional elastic stress limit ^0.2 are recognised as the only governing parameters, following an approach qualitatively similar to the one used for steel. Hence, no consideration is given to the hardening of the material, the moment gradient in the beam as well as for the interaction between the individual plates composing the section. Moreover, no particular attention is paid for the possibility of premature tensile failure due to the reduced material ductility and no provision is made about the influence of plastic anisotropy that aluminium alloys may exhibit. Aiming at improved knowledge of inelastic beam behaviour for cross-sectional classification, recently, a general research project has been undertaken at Norwegian University of Science and Technology developed in cooperation with University of Naples Federico II. An experimental investigation concerning several extruded aluminium profiles subjected to a moment gradient loading has been carried out, mainly pointing out the importance of strain hardening on the rotational capacity of aluminium beams (Moen et al. 1998-a). A suitable numerical model developed by means of the implicit non-linear finite element code ABAQUS has been set up, showing its capability to interpret the observed behaviour of aluminium members in bending, provided that both anisotropy and strain hardening of the material are well described (Moen et al. 1998-b). The current paper focuses on the influence of each governing parameter on both buckling strength and unstable behaviour of aluminium cross-sections. A parametrical analysis is presented for unwelded rectangular hollow profiles in case of both heat-treated and non-heat-treated alloys. The study is carried out by considering the effect of each parameter acting alone and in interaction.
DESCRIPTION OF THE STUDY General Aluminium beams considered in this study are unwelded profiles with rectangular hollow crosssection. They are simply supported and vertically loaded at the mid-span. The analysed scheme is the same used for experimental tests presented in Moen et al (1998-a). In order to avoid the use of transversal welded stiffeners, which could strongly affect the failure behaviour of the specimens, the load was applied on top of the beam section through a compact steel block with total width of 150 mm. Besides, the beam length was extended approximately 10% beyond the end supports, so as to avoid crippling phenomena due to concentrated forces at the support.
639 Based on such experimental results, a suitable numerical model of rotational capacity in aluminium beams was established, using ABAQUS/Standard. It was presented in Moen et al (1998-b). The comparison of results has shown the capability of the proposed model to accurately predict the loaddeflection behaviour of aluminium cross-sections belonging to class 1 and 2, where the assessment of rotational capacity truly represents an interesting target. By accounting for the plastic anisotropy and the actual uniaxial stress-strain behaviour of the material, the inelastic response of the beam was, in fact, satisfactory described in both stable and unstable parts. Numerical Modelling The geometry of the adopted model is depicted in Figure 1. Symmetry is assumed in both longitudinal and transversal vertical planes, so that only one quarter of the box-section beam has been modelled. 4node general shell elements with reduced integration (S4R) have been used, by adopting a mesh refinement varying along the longitudinal direction as shown in the figure. A contact algorithm between the rigid master surface, simulating the loading device, and the slave node set consisting of the part of the beam anticipating contact is employed as stated in Hibbitt et al (1997). part 1 (2 el.
part 2 (20 elements)
part 3 (16 elApart 4 (32gU Rigid klading s
r
A
fl^
111111111111111 "jOn^\
700 mm
Cross-section (24 elements)
••••»*
% • • • • i
^ 150 mm ^L 150 mm ^ b/2=40 mm
1=1000 mm
'' Figure 1: The adopted numerical model As far as the material idealisation is concerned, the following simplified version of the multicomponent stress-strain curve proposed by Hopperstad (1993), has been adopted to describe the uniaxial true stress-plastic strain behaviour:
G = Yo+Q[\-Qxp{-CSp)]
(1)
where YQ is the stress at proportionality limit in a uniaxial test, conventionally assumed equal to the elastic limit strength fo2, Sp is the plastic strain and the material parameters Q and C determine the magnitude of the strain hardening and the shape of the stress-strain curve, respectively. According to available experimental tests, a suitable value for C is 10, whereas factor Q varies depending on the alloy, normally ranging between 80 N/mm^ (low hardening) and 200 N/mm^ (high hardening). In order to account for the experimentally observed plastic anisotropy (Moen et al, 1998c), the Hill (1952) yield criterion has been considered. The stress ratios ry defining the yield surface weighted to the isotropic surface have been determined as described in Moen et al (1998b), by assuming the average of values found for the tested alloys. In particular, if the direction of extrusion (xx) is assumed as the reference direction (r^c =1), for each plate element composing the cross-section, the transversal stress ratio (ryy) and the shear stress ratio (r^y) have been assumed equal to 0.95 and 0.84, respectively, while all stress ratios throughout thickness directions (r^, r^x, r^y) have been fixed equal to 1.
640 The Examined Cases The current study mainly deals with the influence of flange local buckling phenomena on the loadbearing capacity of aluminium alloy members. Other failure modes are not considered in the current study. It is expected that tensile failure could limit the rotational capacity of aluminium beams. In case of flange buckling, the web is also deformed providing a rotational restraint to the flange, as a flinction of its bending stifftiess. Moreover, such a rotational restraint is influenced by the stress distribution in the cross-section at local buckling, due to the combined effect of material strain hardening and section geometry. As a consequence of the above considerations the investigated parameters to rotational capacity have been chosen. The slendemess of the flange hEis been accounted for through the normalised slendemess X = 7/o.2he = 0.52{b/t)-ylfQ2/E, where a^ is the elastic stress at local buckling for a simply supported plate element, being b the flange width, t the flange thickness, E the elastic modulus of the material (^=70000 N/mm^). The web restraint action has been normalised respect to the parameter k^=s^lh, being s and h the thickness and the depth of the web element, respectively. Such a parameter represents the elastic contribution to bending stiffness of the web element considered simplified as a beam. The shape factor of the section has been accounted for by the ratio a = htjhs. All the members have been analysed with reference to the same moment gradient loading. It was experimentally observed that the amount of moment gradient essentially influences the unstable part of moment-rotation curve, the corresponding governing parameter being the ratio L/b, where L is the half span of the beam. Within the current investigation, the above ratio is fixed equal to Z/Z?=l 000/80. As far as the material is concerned, the AA 6082 alloy temper T4 and T6 have been considered. The former is a material that is naturally aged at room temperature, while the latter is fully heat-treated and artificially aged. As a consequence of the different thermal state, they exhibit different features in terms of strength, strain hardening and ductility. As a rule, temper T6 has higher strength, but a less pronounced strain hardening and a reduced ductility. Based on tensile tests on specimens reported in Moen et al. (1998-a),/o.2 =160 N/mm^ Q=200 N/mm^ and/0.2 =320 N/mm^ Q=80 N/mm^ have been fixed for temper T4 and T6, respectively. Moreover, in order to additionally investigate the effect of the strain hardening on the inelastic response of aluminium members, the parameter Q has been varied in a larger range, i.e. Q=50-300 N/mm^, keeping constant all the other mechanical and geometrical parameters. The study has been carried out by assuming the basic values of the above parameters as follows: a =1, A^=2.7 mm^, while X =0.331 and 0.469 for temper T6 and T4, respectively. Such a choice resulted in assuming the same geometric slendemess ratios b/t=h/s=l3.3 for both flange and web plate elements. It should be observed that the chosen values for the slendemess result in class 2, according to EC9.
THE OBTAINED RESULTS The response of the stmcture has been presented in terms of normalised moment-rotation curves {M/MQ2 -0/OO.2)» whereM = F(L-c?)/2 is the moment at the mid-span, being F the applied vertical load and d the half length of the loading rigid surface, and 0 is the rotation of the beam. The elastic limit moment A/0.2 has been determined by considering the approximate elastic section modulus Wg = bth + 5/1^/3, while, by integrating the elastic curvatures, the rotation at limit elastic state has been expressed as 60.2 = /o.2(^ + d)/Eh.
641 According to Mazzolani and Piluso (1997), the stable part (po) and the global rotational capacity (P) are defined as follows (Figvire 2): (2)
"0.2
(3)
6n7
where the rotations 0„ and 0c correspond to the attainment of the ultimate moment (M„) and the collapse moment, respectively, the latter conventionally defined at the intersection with the elastic moment line.
"0.2
^(/%.2
Figure 2: Definition of rotational capacity The normalised moment-rotation curves for the examined cases are depicted in Figure 3, where the range of variation of each parameter is also reported. It is to be noted that, in order to reduce the difference in the shape among curves related to different shape factor values, the graphs corresponding to this investigation have been normalised with respect to the fiilly plastic moment (Mp) and to the corresponding elastic rotation (0^). In all the other cases, the level corresponding to Mp is indicated by a full line. This allows for verifying that the considered sections are able to exploit the fully plastic bending resistance (class 2 sections). Besides, in Figure 4, obtained values for rotational capacity (p) and its stable part (po) are presented. For the same b/t ratio, temper T4 always exhibits a more ductile behaviour than temper T6, due to the difference in slendemess and hardening between the two considered tempers. As was expected, the slendemess appears to be the most influential parameter, producing a large variation in available rotational capacity for both tempers (Figures 3a,b and 4a,b). Both web restraint and shape factor seem to be able to noticeably affect the inelastic response of the member, especially in case of temper T4 (Figures 4d,J). In such a case, the variation of kw and a parameters within realistic ranges leads the section to behave like a ductile section, even if it was conceived as class 2 according to EC9. Moreover, Figures 34/^ show that neither k^ nor a seriously influence the post-buckling behaviour of the section, the relevant normalised M-9 curves resulting in similar slopes in this range. For the sake of comparison, in Figures 4a, b the slendemess limits according to EC9 provisions are reported for both unwelded heat-treated and non-heat-treated alloys. As far as heat-treated alloys
642 TEMPER T6
TEMPER T4
(Basic values: X=0A7, k,,=2.7 mm^, a=l)
a)
(Basic values: X = 0.33, ky,=2.1 mm^, a =1)
1.8
\
1.6 •
\
1.4 • 1.2 •H'W/'^o.aJ-^ 1.0 0.8 0.6
1 / •
:
:
^
:
:
:
:
0.4 0 . 2 •1 / 0.0 •
1=0.29, 0.33, 0.37, 0.41, 0.44, 0.47, 0.5 i
'
i
'
1
'
1
10
0
2
4
-pi
6
8
10
12
14
16
-'^pl
Figure 3: Normalised moment-rotation curves (temper T6) are concerned, if allowance is made for the stable part ofM-6 curves only, such limits seem to give values corresponding to a po value equal to about 4, which is in accordance with usually accepted limits for steel members (Daali and Korol, 1995). On the other side, such a limit seems to be far on the safe side, if the post-buckling behaviour is accounted for determining the available rotational capacity. Anyway, one must mention that the obtained P versus X curve is referred to a given value of the L/h ratio. On the contrary EC9 limits appears to be too safe in case temper T4, which is naturally aged material and therefore to be considered as not-heat-treated. As a matter of the fact, po=4 is obtained at the class 2-to-3 slenderness limit. At the given class l-to-2 slendemess limit the rotational
643 capacity is much higher. This is due to the fact that the code takes into account the effect of hardening through the distinction between heat-treated and non-heat-treated alloys, providing boundary limits that follow a trend opposite to the one found in this study. The effect purely due to hardening is shown in Figure 5 in case of the slendemess value previously referred to temper T6. Both the Mt/Mo.2 ratio and rotational capacity values increase when the strain hardening increases. Such a variation is not negligible and should be investigated at different slendemess levels. Moreover the obtained results point out the influence of the hardening on the inelastic response of the member, demanding for a more comprehensive approach in classifying aluminium cross-sections. TEMPER T6 (Basic values: X = 0.47, ky,=l.l mm^ a =1) a)
1 ^ 1
'^ \
\
TEMPER T4 (Basic values: A, = 0.33, ^=2.7 mm^, a =1) b)
1
C
"-^
^
0.5
2.0
\\
(0
"(05 tt ^
\ 1
Po>P
l
CI ^
^'
II
1 *^ 1 JO
^1 ^1
o> O UJ.
\\
\
\
V) ^\
|4 Ic^ 1 i2 2 is iQ oO >
\\
1 1
l\x
L
3.5
5.0
k^ [mm ]
6.5
8.0
0.5
1.0
1.5
2.0
2.5
K [^^]
Figure 4: The obtained values of rotational capacities. Filled circles: pQ, unfilled circles: p.
3.0
644
200
Q [N/mm^]
Figure 5: The influence of strain hardening (X = 0.47, ^=2.7 mm^ a =1)
INTERACTION AMONG PARAMETERS The previous study has shown that the rotational capacity of aluminium cross-sections is affected by several parameters. Anyway, the combined influence of investigated factors has not yet been stated. In order to quantify the main effect of each factor as well as their interactive effect a statistical two level factorial design procedure has been applied, according to the method presented by Box et al (1978). According to Table 1, two different levels (low value and high value) for every parameter (Q, A,, ^^, a) have been fixed. Therefore, 2"*= 16 numerical analyses have been run, aiming at the £issessment of the response in terms of M/M).2, P and Po for all the possible combinations of variables and relative levels. Through the averages of the obtained results, it is possible to get the main effect of each parameter over all conditions of the other factors. Besides, a simple statistical elaboration allows measuring the effect of two-factor, three-factor and four-factor interactions. TABLE 1 HIGH AND LOW VALUES OF INTERACTING PARAMETERS
Parameter Hardening Flange slendemess
X=
Symbol
Units
Low value
High value
Q
N/mm^
80
200
0.331
0.469
mm^
1.0
2.7
-
1.0
2.0
0.52{b/tyfojE
Web restraint
K^s^lh
Shape factor
a = bt/hs
The results are summarised in Figure 6, where each symbol denotes the participation of the corresponding parameter to the obtained effect. It is quite evident that the main effects are predominant if compared with the ones related to factor interactions. In particular, the most important factors are slendemess and strain hardening. The former is able to provide an average variation ofMyMo.i and Po equal to about 0.15 and 3.75, respectively, while the latter gives 0.15 and 2, respectively. As expected, the shape factor is another influencing parameter for the ratio Mt/Mo2.
645 As far as parameter interaction is concerned, the main outline is that the rotational capacity is differently influenced by each factor depending on the level of slendemess value (see the interaction effects of X to other parameters in Fig. 6). Truly, this result was predictable, due to the strong difference in rotational capacity corresponding to different slendemess levels. Among other factor interactions, the only significant appears to be the web restraint-to-shape factor one. This can be explained by considering that the web contribution to restrain flange buckling depends on the stress distribution in the cross-section and therefore on the section geometry. Finally, with reference to the ratio MJMQ2, the combined Q-Xo-X effect confirms the important influence of the strain hardening also on the inelastic pre-buckling behaviour of aluminium alloy crosssections.
Three factor effects
' fj^^^^^
Figure 6: Factorial design analysis results
CONCLUSIONS AND FURTHER DEVELOPMENTS A numerical investigation concerning the assessment of rotational capacity of unwelded aluminium rectangular hollow cross-sections was performed. In particular, for both heat-treated and non-heattreated alloys, the influence of the main parameters on the inelastic bearing capacity of member under moment gradient was carried out. The obtained results emphasise that the simplified classification system proposed by EC9, since no allowance is made for all governing parameters, is remarkably conservative, especially in case of non-heat-treated alloys if the same rotational capacity requirements as for steel are adopted. In fact, the effect of the hardening seems to be not suitably accounted for. This has been observed especially in case of temper T4, which actually is a not-heat-treated alloy but behaving better that fiilly heat-treated temper T6. Additionally, extensive unaccounted rotational capacity is availablefi-omthe post-buckling behaviour of the cross-sections.
646 A study on the interaction effect among parameters was accomplished, too. It outlines that the main effects due to each single parameter are much important than the ones given by two-, three- or fourfactor interaction. The above outcome assures that the variables chosen in the current study are suitable as governing parameters of the investigated problem. As a matter of fact, according to the current codification, the flange slendemess appears to be the most important factor, strongly influencing the amount of plastic deformation before collapse. On the other side, the influence of strain hardening is remarkable too, being able to provide variations of rotational capacity in a similar range. Finally, it is to be remembered that the performed analysis did not study several influential factors. Further developments of this subject are therefore desirable. In particular, the study on the effect due to L/b ratio could definitively state how, in classifying cross-sections, it is possible to account for the unstable part of the rotational capacity. Besides, additional investigations should be carried out aiming at assessing the susceptibility to tensile failure in limiting inelastic excursion of aluminium beams in bending.
References Box, G.E.P., Hunter, W.G., Hunter, G.S., (1978). Statistics for Experimenters. An Introduction to Design Data Analysis, and Model Building, John Wiley & Sons, Inc, New York. Daali, M.L., Korol, R.M., (1995). Prediction of Local Buckling and Rotation Capacity at Maximum Moment. Journal of Construe. Steel Research, Elsevier Science Publishers Ltd., 32, Pp 1-13. ENV 1999 (Eurocode 9) (1998). Design of aluminium structures, CEN. Hibbitt, Karlsson & Sorensen, Inc. (1997). ABAQUS/Standard Theory Manual, Version5.7. Hill, R. (1950). The Mathematical Theory of Plasticity, Clarendon Press, Oxford. Hopperstad, O.S. (1993). Modelling of Cyclic Plasticity with Application to Steel and Aluminium Structures, PhD Dissertation, Division of Structural Engineering, Norwegian University of Science and Technology. Kuhlmann, U. (1989). Definition of Flange Slendemess Limits on the Basis of Rotation Capacity Values. Journal of Construe. Steel Research, Elsevier Science Publishers Ltd., 14, Pp 21-40. Mazzolani, F.M., Piluso, V. (1996). Theory and Design of Seismic Resistant Steel Frames, E & FN SPON, London. Mazzolani, F.M., Piluso, V. (1997). Prediction of the Rotation Capacity of Aluminium Alloy Beams. Thin-walled structures, Elsevier Science Ltd., Vol. 27:1, Pp. 103-16. Moen, L., Hopperstad, O.S, Langseth, M. (1998-a). Rotational capacity of aluminium beams subjected to non-uniform bending - Part I: Experiments. Submittedfor journal publication. Moen, L., De Matteis, G., Hopperstad, O.S, Langseth, M., Landolfo, R. Mazzolani, F.M. (1998-b). Rotational capacity of aluminium beams subjected to non-uniform bending - Part II: Numerical Model. Submittedfor journal publication. Moen, L., Langseth, M., Hopperstad, O.S. (1998-c). Elastoplastic buckling of anisotropic aluminium plate elements. Journal of Structural Engineering, ASCE, Vol. 124:6, Pp. 712-19.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
647
ALUMINIUM TUBES FLATTENED (STAMPED) ENDS SUBJECTED TO COMPRESSION - A THEORETICAL AND EXPERIMENTAL ANALYSIS
R.M. Gon9alves; M. Malite; JJ. Sales Department of Structural Engineering, University of Sao Paulo at the Sao Carlos campus Av. Dr. Carlos Botelho, 1465 - 13560-250 - Sao Carlos, SP, Brazil (e.mail: [email protected])
ABSTRACT Aluminium tubes are widely used in space structures. This paper presents and discusses the results of critical load buckling observed in a theoretical and experimental study of aluminium tubes with several types of stamping at their extremities, subjected to compression. These bars are used in Brazilian space structures. The results analyzed were obtained from bars with 110mm diameter and 2.5mm thickness, consisting of 24 prototypes with three types of stamping at the extremities of the bars and three different lengths: 3,333mm; 2,750mm and 1,800mm, corresponding to approximate slendemess ratios of 86, 71 and 46, respectively. Measurements were taken of applied forces using load cells, lateral displacements in the two main directions were measured using LVDTs, and the strain in the central section of the bar and at some points in the area of connection to the support device was measured using strain gages. Comparisons are made between the experimental results and the result forecast from the usual designing models, evaluating the influence of variable stif&iess, as well as the results compared against those recommended by the technical code. KEYWORDS Alximinium structures, space structures, experimental analysis, bars, compression, tests, L INTRODUCTION Space structures have been in use in Brazil for some time, principally in roof designs involving large open spaces with few supports. These structures can be defined as reticulated and consisting of noncoplanar bars connected by nodes.
648 The systems commonly employed in Brazil involve the use of steel or aluminium bars with stamped ends, that may be connected to steel nodes or with single bolt. To date, these systems have not been sufiBciently studied, particularly as regards the structural behaviour of their nodes and the stamping of their bar ends, and this lack has resulted in the partial or even total collapse of some structures. Among several such cases, some of the most recent major accidents involving structures of this type are the collapse of the roofs of an aquatic park, a sports gymnasium in Sao Paulo (using aluminium bars), and a convention center in the city of Manaus. Other accidents involving smaller structures have also been recorded. Stamping of the bar end leads to a significant reduction of stiffiiess in this region, which implies a reduction of the bar's capacity. This effect is usually disregarded in space truss designs (flat, dome or cylindrical), which may lead to highly unfavorable and, therefore, unsafe conditions. This work consists of a theoretical and experimental study of the structural performance of bars subjected to axial compression, considering the variation of bar stifi&iess at the nodes. The study involved testing aluminium bars with 110mm diameter x 2,5mm thickness, consisting of 24 models with 3 types of stamping at the ends of the bars (model types A, B and C), in three different lengths: 3,333mm, 2,750mm and 1,800mm, corresponding the slendemess ratio(A,) equal to 86, 71 and 46, respectively. The geometric properties of the transversal section are gross area: Ag = 8,5 cm^ and radius of gyration: 3,89 cm. Fig. 1 illustrates the stamping types used at the ends of the tested models. MODEL A
•4-f MODEL B
-^-^ UOGEL C
s^
* ^ Fig. 1 - Detail of the ends of the tested models. The models were all takenfi*omthe same lot of material of ASTM 6351-T4 aluminium alloy (ABNT 6351) and aged after stamping until they reached T6 temper. The nominal values of the mechanical properties are listed in Table 1. Table 1 Mechanical properties of the ASTM 6351 alloy - Nominal values 1
Temper fy (MPa) ASTM 6351-T4 119 [ ASTM 635J_- T6 260 fy = yield stress 1^ = tensile strength 4sm ^ bearing strength (contact) A = elongation (in 50mm)
f„ (MPa)
fesm (MPa)
190 295
380 590
A(%) 1 18
10
1
649 The models were tested to determine the ultimate normal strength of isolated bars and to carry out a preliminary analysis of the structural performance of the different stamping types. 2. MECHANICAL CHARACTERIZATION TESTS AND BEHAVIOUR OF THE STAMPED REGIONS The aluminium alloy was mechanically characterized by means of tension tests on specimens, according to the ASTM E 8M-94b specification - Standard test methods for tension testing of metallic materials [metric]. The sample consisted of two tube segments, from which four specimens were taken in diametrically opposite positions. Of the four specimens, two were instrumented with a clip gage, and two with strain gages stuck to opposite sides and joined in series. Table 2 presents the results of these tests. Table 2 Results of tension tests fy (MPa)
fu (MPa)
Et (MPa)
Temper T4) 252 66,728 154 317 67,957 1 Temper T6) 291 fy = yield stress (offset 0,2% - machine stroke, Lo = 80mm) fu = tensile strength E = modulus of elasticity (evaluated by the glued-on strain gages) A = elongation (in 50mm)
A(%) 23.2
13.6 1
Axial compression tests were also carried out on 4 tube segment specimens of 250mm height, with the mid-section instrumented with two strain gages placed diametrically opposite each other. An MTS 815 machine was used, with constant testing speed equal to 1 kN/s. The failure mode in these tests consisted of local bending located in the wall of the tube at the extremity. Table 3 shows the average results obtained from these tests. Table 3 Results of tube segment tests Media Temper T6 Pmax = maximum load in test fmax - maximum stress Ec = modulus of elasticity
Pa..x(kN) 234
f„.a, (MPa) 275
Ec(MPa) 67,084
1
Specimens of small bar lengths (total length equal to 600mm) were also tested, with one of their ends stamped and connected to a steel supporting device by means of three (j) 19mm nominal diameter bolts. Figure 2 illustrates the main dimensions of these models. Four specimens were prepared for each type of stamping. Photos 1 and 2 show the position of the specimens for testing of the stamped regions, carried out with an MTS - 815 machine at a loading speed up to collapse of IkN/s.
650
TUBO^ 110x2,5
Fig. 2: - Dimensions
Photo 2 - Front view of test
Photo 3 - Side view of tested specimen
3. EXPERIMENTAL ANALYSIS The bar models in three lengths were tested in a vertical position, using a frame, and were subjected to axial loads applied in steps by means of a hydraulic actuator, with the applied load measured using a load cell installed at one of the ends of the model (support 1). The models were connected in the following manner: at one end (support 2), the supporting device was placed directly against the reaction frame, providing partial restriction of rotation, while at the other end (support 1), the support device was placed against a spherical knee coupled to the load cell, providing pinned connection. The measuring instruments used in these tests were 8 displacement transducers placed along the bar (5 in the direction of least inertia, i.e. at the ends, at mid-length, and at each quarter of the length; and 3 transducers in an orthogonal direction to the bar, i.e. at the ends and at mid-length), from which the displacements transversal to the longitudinal axis were obtained. Four strain gages were placed at the central transversal section of each model, with the main purpose of observing the evolution of bending during all the loading phases.
Photo 3 - General view of the model
Photo 4 - Testing model
651 During the tests, the results of displacement (transducers), strain (strain gages), and applied load (load cell) were collected using an automatic data collecting system. No measurements were taken of residual tensions and the geometric imperfections of the models were less than L/1000 in the center of the bar.
4. EXPERIMENTAL RESULTS Table 4 shows the maximum applied loads (ultimate test loads) for the 24 tested models, which were assumed as the normal critical load. Table 4 Maximum test load for the models. Critical Load (kN) | Model B Model C 1 Model A 2 2 2 Slenderness 1 3 1 3 1 3 + -1+ 90 90 80 97 88 90 X = 86 63 95 82 50 90 93 86 80 x=n 78 70 75 120 117 79 85 80 1 A, = 46 Note: The results indicated by (-) are unreliable; those indicated by (+) were not performed The collapse mode for all the models tested corresponded to the formation of plastic hinges at the end of the bar at support 1 (upper) in relation to the axis of least inertia. These hinges always occurred at support 1, since rotation was partially restricted at support 2 (lower). Some results, of significant models, are presented in the graphs below. Graphs 1, 2 and 3 show the results of displacement in the direction of least inertia for the three different types of stamping, with bar lengths equal to 1,800mm ( X=46). 120
120
100
100
100 H
80
80 J
60
.•^*^-. ^ A I I - S O
40
— • — INF SUPPORT 1 — • — MIDDLE SPAN A SUP SUPPORT 1
u. . 5
10
15
20
/ INF SUPPORT MIDDLE SPAN
20
Graph 1 - Load vs Displacement Model A
•4
1
1
-5
S(mm)
Graph 2 - Load vs Displacement Model B
X
I
SUP SUPPORT
25
5(mm)
i
— • — INF SUPPORT
''
— • — MIDDLE SPAN
0
5
*
1
SUP SUPPORT 1
10
15
20
25
5 (mm)
Graph 3 - Load vs Displacement Model C
It can be noted that the C type stamping showed higher normal compression load values since this type of stamping increases the stifi&iess in the region of the node in relation to the others. Graphs 4, 5 and 6 illustrate the behaviour of applied load versus displacements for the three lengths of bars tested, considering the C type stamping.
652 100 n
. ••
80
^
. . ^
1:
f;
I.
i:
60-
u u
40-
Q.
d
INF SUPPORT MIDDLE SPAN SUP SUPPORT
— • — INF SUPPORT
i I 1
20-
— • — MIDDLE SPAN A
SUP SUPPORT
—•— INF SUPPORT —•— MIDDLE SPAN * SUP SUPPORT
1
0-
—,—1
—1—1—1—1—I—1—1—1—1—1—1—
10
15
10
20
S(mm)
15
5
5 (mm)
Graph 4 - Load vs Displacement (Model C - X,=86)
10
15
5 (mm)
Graph 5 - Load vs Displacement (Model C - X=71)
Graph 6 - Load vs Displacement (Model C - >.=46)
(jraphs 7, 8 and 9 show the strains observed in the transversal section of the mid-span for the C type stamping, with varying bar lengths ( X= 86, X,=71 e >.=46).
80-
. • / • • 60-
Uu Q.
*0-\
lit:
iL w
—"— H S.G. 1 — — HS.G.2 " HS.G.3 —»— H S.G. 4
z ST 40-
''
//
—-— S.G. 1 —•— S.G. 2 * S.G.3
20-
- » — S.G. 4 1
'
1
0-
r—
2000
8 (HE)
Graph 7 - Load vs Strain Load vs Strain (Model C - A,=86)
\
—'^—r—'—
1000
—— —•— * —y—
1 / 1000
1500
8(H8)
z(\iz)
Graph 8 - Load vs Strain Load vs Strain (Model C-A,=71)
S.G. 1 S.G. 2 S.G.3 S.G. 4
Graph 9 - Load vs Strain (Model C - A,=46)
The above graphs are examples of the results obtained for the different models tested and highlight the type of stamping for lower values of slendemess and testing conditions that lead to a support condition where the end connected to the hydraulic actuator is effectively pinned while the other end presents restricted gyration. It should be pointed out that the slendemess index indicated does not consider the variation of inertia at the bar ends. 5. THEORETICAL ANALYSIS The purpose of this work is not to engage in a discussion of the theoretical concepts of bar instability and buckling curves adopted by the codes but rather, is limited to making comparisons between the test results and the calculations based on the recommendations of technical codes.
653 Based on the recommendations of the American Standard, the limit of slendemess ratio for ASTM 6351 - T6 alloy is C2 = 63, which results in the following limits for the bending curve: -forX< 19
/,,=/
=26^^
- for 19
f„=29,6-0,l9A\^mkN/^^]
_^^]Omi
-forA,>63
(1)
cm
kN/ ]
(2)
(3)
Thus, for the geometrical properties of the 100 x 2,5 (mm) tube, using E= 70,000 MPa and the conventional yield point of 260 MPa, it is possible to obtain the buckling curve for these tubes and to compare the results obtained. Graph 10 portrays the theoretical buckling curve for the ASTM - T6 aluminium alloy, as well as the results obtained from the 24 models. The limit yield fy = 26 kN/cm^ corresponds to the conventional value of the aluminium alloy used in the tube tests.
30 n
• • T
40
60
-Theoretical Curve 1 Model A Model C Model C
80
100
Slenderness A. Graph 10 - Buckling curve and experimental results
6. CONCLUSIONS The experimental analysis developed in this study corresponded to axial compression tests on isolated bars, with the basic objective of comparing the structural performance of aluminium bars with stanped ends, varying the stamping details herein called A, B and C and the slendemess of the bars, using the values of A, = 46, A = 71 e A = 86.
654 The results of the tests on specimens representing the node region demonstrated a reduction of the ultimate capacity of bars with stamped ends, indicating that the collapse mode in this region clearly depends on the stamped end. In the bar tests, the lower the slendemess, the higher the dispersion of ultimate load values. This resulted from the stronger influence of the detail of the bar end and the imperfections inherent in isolated bar tests. The bars whose slendemess ratio correspond to elastic behaviour presented results that were compatible with those obtained theoretically. However, the results for the inelastic bars clearly indicate the influence of the stamped ends, with results far below those obtained theoretically. It must be kept in mind that the theoretical values were calculated based on the condition of articulated supports, (KL = L), while in the tests, one of the bar ends was supported directly on the reaction frame, i.e., a flat surface, which entails some restriction to rotation. To conclude, it should be pointed that axial compressions tests carried out on isolated bars probably do not represent the behaviour of the latter in a structure. The variability of results is mostly attributable to initial imperfections of the bars themselves and to imperfections in setting up the tests, culminating in initial eccentricities that can cause premature failure. It is, therefore, important to carry out an experimental analysis of models for a more in-depth and significant analysis. ACKNOWLEDGEMENTS The authors acknowledge the financial support of FAPESP (State of Sao Paulo Foundation for Research Support). Special thanks are extended to the staff of ALUSUD Eng. e Ind. de Constru9ao Espacial Ltda. for their unfailing support and interest in research of space structures.
BIBLIOGRAFIA AMERICAN SOCIETY FOR TESTING AND MATERIALS (1994). E8M - Standard Test Methods and Definitions for Mechanical Testing of Steel Products. Philadelphia. ASSOCIAgAO BRASILEIRA DE NORMAS TECNICAS (1986). NBR-8800 - Projeto e Execu9ao de Estruturas de A90 de Edificios: Metodo dos Estados Limites. Rio de Janeiro EL-SHEIKH, A.I.; McCONNEL, R.E. Experimental Study of Behavior of Composite Space Trusses. Journal of the Structural Engineering, Vol.119, No 3, p. 747-766, march, 1993. EUROPEAN COMMITTEE FOR STANDARDIZATION (CEN), Eurocode 3: Design of Steel Structures. Part 1.1: General Rules and Rules for Buildings, Brussels, 1992. GONQALVES, R.M.; FAKURY,R.H.; MAGALHAES,J.R.M. Performance of Tubular Steel Sections Subjected to Compression: Theoretical - Experimental Analysis. In: STABILITY PROBLEMS IN DESIGNING, CONSTRUCTION AND REHABILITATION OF METAL STRUCTURES Ed. by Ronaldo C. Batista, Eduardo de M. Batista and Michele S. Pfeil. COPPE/UFRJ, Rio de Janeiro, August 1996, p.439-449 KISSEL J.R., FERRY R.L. (1995) Aluminium Structures - A guide to their specifications and design, John WHey & Sons, INC, 417p. MAGALHAES, J.R.M.; MALITE, M. Alguns Aspectos Relativos ao Projeto e a Constru9ao de Estruturas Metalicas Espaciais. In: Congresso de Engenharia Civil - Universidade Federal de Juiz de Fora, 2. Juiz de Fora, 1996. Anais. Juiz de Fora, UFJF, 1996. p.282-291. SALES, J.J. et al. Ensaios de Compressao em Corpos-de-prova e Barras de Aluminio com Extremidades Estampadas. Sao Carlos, abril 1996. (Relatorio Tecnico).
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
655
INTERPRETATIVE MODELS FOR ALUMINIUM ALLOY CONNECTIONS G. De Matteis \ A. Mandara ^, F. M. Mazzolani ^ ^ Department of Structural Analysis and Design University of Naples Federico II, Naples, ITALY ^ Department of Civil Engineering Second University of Naples, Aversa (CE), ITALY
ABSTRACT A concentrated plasticity approach is presented in this paper, aiming at a more accurate evaluation of the inelastic behaviour of aluminium alloy joints. A new model is proposed, based on an energy equivalence with the actual spread plasticity structure. It is able to account for both the actual strainhardening feature of the material and the available ductility. In addition, it allows a proper set up of the simplified approach based on the concept of equivalent perfectly plastic hinge, as used in the codification on steel joints. The current method also provides an adjustment of the correction factors given in EC9 for the inelastic analysis of aluminium structures. Finally, a suitable procedure for the evaluation of the ductility demand in relation to the moment redistribution, is proposed.
KEYWORDS Joints, Aluminium alloys. Plastic hinge. Strain hardening. Ductility, Concentrated plasticity INTRODUCTION No specific calculation method is presently available for the prediction of the behaviour of aluminium joints up to failure. A simplified approach could follow the same way as in the EC3 Annex J (1998) for steel joints, but this provision is clearly unable to comply with the actual features of aluminium alloys, mainly for being the evaluation of strain hardening and ductility effects impossible. The prediction of aluminium joint behaviour is a matter which has mainly to be framed within the plastic design of structures made of hardening material. It is well known, in fact, that the calculation in the inelastic range of these structures cannot be performed by means of the usual methods adopted for steel structures. For this reason these methods should be adequately modified in order to keep into account all the influencing material properties. This would lead to more sophisticated procedures which, inevitably, result in a higher degree of practical difficulty (Mazzolani et al, 1996). The possible calculation methods for aluminium joints are listed in decreasing order of accuracy as follows:
656 (1) (2) (3) (4) (5)
FEM simulation using solid elements; FEM simulation using shell elements; Concentrated plasticity models with equivalent non-linear plastic hinges; Concentrated plasticity models with equivalent linear hardening plastic hinges; Concentrated plasticity models with equivalent perfectly plastic hinges.
Of the above methods, the first two only permit to thoroughly investigate the structural inelastic behaviour, by accounting for the actual available ductility and hardening effect. In particular, method (1) has been applied by De Matteis et al (1998) for the prediction of aluminium T-stub behaviour and for the interpretation of the relevant collapse mechanisms. In order to achieve a good accuracy, the procedure has been calibrated on the basis of existing experimental tests. Refined methods based on non-linear FEM simulation could be used anyway, but they are often rather cumbersome, especially when used by practising engineers not familiar vAih the use of advanced calculation codes. As a cheaper alternative to spread plasticity models, methods based on concentrated plasticity approaches can guarantee a sufficient reliability in predicting the structural response, provided that a suitable procedure for checking material available ductility against structural ductility demand is defined. At the same time, in order to predict a possible overstrength in the post-elastic range, the material hardening should be also accounted for in these simplified procedures. Among these methods, the ones at point (3) and (4) are more accurate than the traditional plastic hinge method (5). An approach based on concentrated plasticity models with linear hardening plastic hinges has been already presented by De Matteis & Mandara (1997). Nevertheless, the application of such methods inevitably involves the use of non-linear iterative procedures, which can not be conveniently applied for practical purposes. Therefore, in this case, the most suitable approach would remain the use of the plastic hinge method as used for steel, that is based on the concept of perfectly plastic hinge. The possibility to follow this way was firstly analysed at the end of the Seventies (Mazzolani, 1995) and more recently by Mandara & Mazzolani (1995). The results obtained led to the proposal of a correction factor r| to be applied in evaluating the plastic moment of cross sections. This factor was matched by fitting results obtained v^th the application of usual plastic hinge method to the ones coming fi-om non-linear FEM analyses for a given strain limit. Because of its simplicity, this approach has been shared also by EC9 Part 1-1, Annex D "Methods of Global Analysis" (1998). The main purpose of the present paper is to set up a concentrated plasticity model with non-linear plastic hinges (3) based on an energetic equivalence between the actual spread plasticity structure and the idealised one. Such a model could be used for simplified analyses interpreting the inelastic response of monodimensional aluminium alloy structural components, such as for example the T-stub components. This model also allows tiie linear plastic hinge (4) and the perfectly plastic hinge (5) methods to be properly derived as simplified cases. In particular, unlike previous studies (Mandara & Mazzolani, 1995), where the correction factor r| was evaluated on the basis of the global bearing capacity, the energy based equivalence should lead to a more accurate evaluation of the actual inelastic behaviour, in particular when strongly hardening alloys are involved.
NON-LINEAR PLASTIC HINGE IDEALISATION The Structural Model The majority of the structural models adopted for the representation of joint behaviour, including the methods provided by codification (EVN 1993-1-1, Annex J, 1998), is based on simplified monodimensional idealisations, which allows the overall response to be interpreted through elementary mechanical models. For instance, the simple tensioned T-stub is usually represented by means of a continuous beam with rectangular section on four supports (Figure 1). Many connection typologies, such as end plate and angle cleat joints, can be represented by this model, which provides an acceptable degree of accuracy in the prediction of the joint response, at least for steel. Thus, a suitable concentrated plasticity model may be defined startingfi*omthe continuous beam and considering the elementary beam portion comprised between the section of maximum bending moment, assumed as the
657 location of the concentrated plastic hinge, and the section where bending moment is zero. The behaviour of the resulting cantilever structural scheme is shown in Figure 2a. If the presence of concentrated forces only is assumed - as a rule for joints - a linear distribution of bending moment is obtained. When the element is stressed in elastic-plastic range, the distribution of bending curvature is shown in Figure 2b. For aluminium structures, elastic limit moment M0.2 may be conventionally assumed as the one producing a maximum strain in the section equal to fo2^E, fo2 being the conventional yielding stress of the material and E the elastic modulus. For higher values of the moment, the curvature increases according to the hardening features of the material until a strain limit (8//;„), corresponding to the ultimate deformation, is reached.
^^=>^ ^
A
B X I iv
/ ^ /
D -^
E ^—n^
c ^
Figure 1: Typical T-stub structural schemes The concentrated plasticity model involves an elastic behaviour along the whole beam length, except in the clamped section, where a concentrated hinge is assumed to form when a given value of the bending moment is attained. The plastic hinge will be, therefore, characterised by an ad hoc rigid-inelastic behaviour, being activated when the conventional elastic moment M0.2 is exceeded. It is clear that an equivalent concentrated plasticity model cannot keep all the features of the actual curvature diagram. This means that the proposed simplified approach will be able to provide either the same rotation or the same deflection at the loaded end but not both of them, depending on whether the curvature diagram is assumed to have the same area or the same centroid as the actual diagram, respectively. An alternative approach can assume that the actual and the simplified approach to be equivalent in terms of absorbed energy. This leads to write the following condition:
J[Mxdz =
M^ - + (t).M„ 2EI
(1)
where x is the curvature of the spread plasticity model and ^ is the plastic concentrated rotation at the plastic hinge. Based on this assumption, in case of rectangular cross-section, the M-cj) curves characterising the behaviour of the concentrated hinge have been assessed. They are shown in Figure 3, for several values of the hardening parameter n of the Ramberg-Osgood law. The evaluation of the term {M^dz has been made through a numerical procedure, where the Ramberg-Osgood material model has been assumed for the determining the true curvature distribution along the beam span. Lines corresponding to the attainment on the actual structure of deformation limits equal to 5, 8, 10, 12 and 14 times the elastic limit strain {^0.2^/02 IE) are also indicated, by means of the corresponding limit curvature values (X5 > •••»Xi4)- The plastic rotation ^ has been normalised with respect to ^0.2 - M).2 ^ I2EI, corresponding to the elasticfi-eeend point rotation of cantilever beam at the attainment of the elastic limit moment. As one may expect, the equivalence in terms of energy does not ensure a perfect correspondence between the two models, in terms of neither end rotation nor end deflection. Nevertheless, the
658 discrepancy as respect to the spread plasticity model is small, as shown in Figures 4a, where the error in terms of both end rotation (j)^ /^ and deflection v^ /v is plotted as a function of MM0.2, for a value of the hardening parameter n = 5. Such a value corresponds to the highest strain hardening effect possible in practice and, thus, is expected to produce the greatest error magnitude. For the sake of comparison, in the same figure, the error is also plotted when the equivalence is expressed in terms of displacement (subscript v) and rotation (subscript ^) of the loaded section.
MmJEI Figure 2: Cantilever scheme assumed for the definition of equivalent concentrated plasticity model From the global point of view, the approach based on the energy equivalence seems to be more appropriate for predicting the ultimate structural response compared with an equivalence based on displacements or rotations. On the basis of the energy equivalence, the resulting errors in displacement and rotation are plotted in Figure 4b for several value of the hardening parameter n. Error magnitude is within 9% for displacements and 6% for rotations.
0.0
1.0
2.0
3.0
/0.2 4.0
Figure 3: Characterisation of non-linear concentrated plastic hinge The Characterisation of Non-linear Plastic Hinge for Redundant Structural Schemes The above outlined approach is sufficient for the definition of an equivalent linear hardening or even a perfectly plastic hinge for statically determined structures only. When redundant schemes are considered, the moment redistribution involves a change (6) in the position of the zero-moment points along the beam span, in order to move from the elastic distribution of bending moments to the inelastic one (Figure 5). The problem is therefore complicated by the fact that the equivalent cantilever length is not constant and, consequently, a trial and error procedure would be necessary to update step-by-step the hinge M-^ relationship to the current value. In particular, M-^ curves will be affected by the redistribution factor p=Mmax^Mmm, being the ratio of zero point shifting to the beam length an increasing function of p, vanishing when p approaches the unity.
659 1 Error... _
* *" •'." * • '\-^'>.,
1 —^ 1 -) — ( J , I
n=20
•*••>;
Jl=5
^ • ^ '"•'•"5
n=20
^>^^L)
\n=5 M/Mo.2
0
1.20
1.40
1.60
1.80
2.00
Figure 4: Error due to equivalent non-linear plastic hinge idealisation As an example, Figure 6 shows the M-cj) relationship for the 4-support beam of Figure 1, with p = 3, « = 10, Z = Im and^oi = 200 N/mm^. The lower curve (lece/) represents the equivalent M-^ relationship for the plastic hinge forming in section C, relative to beam portion BC, evaluated by assuming a cantilever length equal to that corresponding to the elastic bending moment distribution (see figure 5). In the same way, the upper curve (IAB^/A) represents the equivalent M-cj) relationship for the plastic hinge forming in section A, relative to beam portion AB, evaluated by assuming a cantilever length equal to that corresponding to the attainment of the limit elastic moment in section A. In addition, the intermediate curve (LBC - LAB) represents an ideal fully plastic situation, when no further redistribution occurs and the factor p reaches the unity. Since the final moment distribution will tend to this condition, the actual trend of the M-cj) relationship for both beam portions LBC and IAB will approach the curve corresponding to p = 1 as far as the value of moment in the hinges increases. Such a situation is shown in the same figure by means of thicker curves, which represent the actual characterisation of plastic hinges all over the loading process, accounting for the shifting of zero bending moment point B.
Figure 5: Shift of zero-moment points
Figure 6: Plastic hinge characterisations
It is interesting to observe that the greater the factor p, the wider the gap between these limit curves characterising the plastic hinge behaviour. Besides, the tendency of the actual curve to leave the initial curve and to approach the final one (Z^BC^^AB) is faster in case of low-hardening materials (high n values). The determination of these curves is not easy and should be done by means of special techniques, by matching the reference length of equivalent cantilever beams at each load level. Anyway, nondimensional curves could be provided in case of several n and p values, allowing the actual non-linear Af-cj) curves to be determined for all cases interesting for practical applications.
660 LINEAR HARDENING PLASTIC HINGE IDEALISATION Curves of Figure 6 may be also used for defining a simplified equivalent linear hardening plastic hinge, whose behaviour can be set as a function of a given deformation limit corresponding to the available ductility for the considered alloy. A linear behaviour can be therefore defined by imposing bending moment levels corresponding to such deformation limits on the basis of the actual M-^ curves defined through the above non-linear model and therefore accounting for the effective spread of plasticity along the beam span. For aluminium alloys it is customary to refer to deformation limits equal to 5 or 10 times the deformation at elastic limit state (Mazzolani, 1995). For each of these limits and for a given value of p it is possible to define an equivalent linear hardening hinge. An example is shovm in Figure 6. By assuming such an idealisation for hinges, it is possible to get a good accuracy in the interpretation of the structural behaviour, as shown in Figure 7, where force-deflection relationship is plotted for p = 3 and « = 10. The reference curve (FEM) considers spread plasticity along beam span, while curves labelled as xs and xio consider an elastic beam and concentrated linear plastic hinges at maximum and minimum moment sections, defined as above stated and referred to the corresponding deformation limits ixiim). 3.5 3.0
\F/F
12
Xiim
2.5
Xlim
FEM
2.0 1.5 1.0 0.5 0.0
= Xio
....-;;;
Xs
J^ p = 3; A? = 1 0
1
\
V/ V m ' •' 0.2 1
Figure 7: Results of linear plastic hinge method
Figure 8: Definition of perfectly plastic hinges
The application of this approach, conventionally referred to as the "Hardening plastic hinge method", has been already proposed by De Matteis & Mandara (1997), with the difference that the properties of the plastic hinge were derived from an equivalent beam section with a length equal to the beam depth, but without keeping into account a priori the effect of redundancy. The authors also provided a simple example of closed form formulation for the interpretation of the behaviour of a continuous beam on elastic supports. The method presented herein allows a closer representation of the actual structural behaviour, together with a definition of the M-^ relationship for the plastic hinge independent of the length assumed for the hinge itself ELASTIC-PERFECTLY PLASTIC HINGE IDEALISATION The concept of linear hardening plastic hinge is useful to provide a degree of accuracy not far from that obtainable with the more sophisticated spread plasticity approach. Nevertheless, it should be pointed out that its application to practical cases still remains rather cumbersome. This is mainly a consequence of both the step by step calculation, always required in this kind of approach, and the difficulty to define the plastic hinge properties at all possible hinge locations. On the other hand, simple formulations existing for steel joints (ECS - Aimex J), based on the limit analysis assumptions, are not applicable tout-court. This may represent a strong limitation for some aluminium alloys, whose behaviour in terms of available ductility allows the limit analysis to be successfully applied, in spite of a not negligible hardening. On the contrary, a special care should be taken for those alloys whose behaviour may result not enough ductile to comply with the plastic mechanism requirements. These observations lead to consider the traditional perfectly plastic hinge method as the most suitable approach for the inelastic analysis of aluminium joints, provided that some appropriate modifications
661 are introduced to take into account the particular features of the material. The correction factor r\ considered in Eurocode 9 (ENV 1999, 1998) represents a first attempt to the appHcation of this procedure. The method is quite easy to be appUed, a simple correction of the conventional yield stress /o.2 being necessary, according to the hardening and ductility features of the alloy. The adjustment of r| was made on the basis of an equivalence, in terms of load bearing capacity for a given ductility level, between the concentrated plasticity model and the spread plasticity one. Since this assumption could yield unconservative results, in particular when strongly hardening alloys are considered, a modification has been proposed in this paper, by means of which the correction factor r\ has been evaluated via an energy equivalence with the non-linear plastic hinge model (Figure 8). For an assumed value of the ultimate curvature and for the corresponding value of the concentrated plastic rotation, it is possible to defme the value of the bending moment plateau involving the same strain energy as the nonlinear model at the given deformation Hmit. From the physical point of view, this means that the higher energy level of the perfectly plastic hinge model in the first part of plastic rotation (section ABC of the shaded area in Figure 8) is offset by a lower energy level in the range of higher rotation values (section BDE). This position is particularly important for strongly hardening materials, for which comparatively lower values of r| are obtained as compared with those provided by EC9. This is shown in figure 9, where the values of r| versus the hardening parameter n are plotted for different values of the redistribution factor p. For higher values of n (low-hardening materials), the values of r\ obtained via the energy equivalence are quite similar to those given by EC9. Figure 9 also allows evidencing that r\ is practically independent of p, as already shown by Mandara & Mazzolani (1995). 1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.90 0.85
Tn
'
*\^
V i:
A
**»
^p=l,00 ^p=l,67 -^p=3,00 -^p=3,67 •••EC9
= "fv « ^ "
A/.U.4i
1 1 [ [ [ 1
1.30 1.25 1.20 1.15 1.10 1.05 1.00
j
0.95 0.90 0.85
-.... n 12
16
20
24
28
^
^/.: ='/nv.^
[^\
A///m
^
S
^
^ p = l , Tol — p=l,67 [ — p=3,00 f ^ p=3,67 [ •*EC9 r
• " A^.V.Z
l
n 12
16
20
24
28
Figure 9: Comparison of r| factor via energetic approach and EC9 EVALUATION OF DUCTILITY DEMAND The above calculations can be profitably used for the evaluation of the ductility demand of the structural scheme as a function of the redistribution parameter p. Referring to a T-stub joint, (Figure 1), the required ductihty properties may be evaluated starting from the ultimate load F„ calculated according to the plastic hinge method and by assuming a completely developed collapse mechanism, that is in the same way assumed in ECS-Annex J for the evaluation of the ultimate load capacity of steel joints. Under this assumption, the ultimate load F„ for the beam is given by SMJL, where M„ is the fiilly plastic bending moment for the cross section. M„ can be evaluated by accounting for different values of the r| factor, in such a way to assess the ductility demand according to different ways to use the equivalent perfectly plastic hinge method. For a given p, once F« is determined, it is possible to obtain fi-om the solution of the spread plasticity model the value of the corresponding maximum curvature Xmax. The comparison between the available ductility, which can be expressed in terms of the curvature limit value Xiim, and the ductility demand allows for assessing whether the scheme ductility requirements are compatible with the alloy maximum strain or not. Figure 10 shows such a comparison for xiim = 5x0.2 and lOxo.2, in case of hardening parameter « = 10 and 20. Values of r| according to both the proposed perfectly plastic hinge method based upon energy equivalence and EC9 have been considered. In addition, for the sake of comparison, the case r| = 1 has been examined. The comparison evidences that, as expected, the ductility demand is an increasing function of p. In particular, it is
662 clearly shown that when some p values are exceeded, the application of concentrated plastic hinge method is unconservative and therefore a step-by-step procedure with ductility control is required. CONCLUSIONS The prediction of joint behaviour represents one of the aspect not yet covered by the existing codification on aluminium alloy structures. The appUcation of the existing rules for steel is an attractive possibihty in order to keep as much as possible some existing design rules which are very common among the practising engineers. Nevertheless, such a chance is strongly limited by the aluminium features, which make the above mentioned rules hardly appUcable. In this paper an attempt to use the common method of plastic hinge with a suitable correction accounting for hardening and available ductility has been proposed. This has led to assess the range of applicability of the concentrated plasticity approach in relation to the redistribution needings and, hence, to the ductility demand of the structural model. The results obtained suggest that such a generalisation is feasible, provided that suitable correction factors are used and a special care is needed in the selection of the alloys to be used.
Figure 10: Evaluation of ductility demand for different r| factor evaluations REFERENCES De Matteis, G. and Mandara, A. (1997). A Concentrated Plasticity Model for Hardening Material Structures. In Proceedings ofXVI Congresso C.T.A., Ancona, 281-92. De Matteis, G. Mandara, A. and Mazzolani, F.M. (1998). Numerical Analysis for T-stub Aluminium Joints. Inproc. of the 4*^ Int Conf. on "Computational Structures Technology", CST'98, Edinburgh. Eurocode 3 (ENV 1993-1.1) - Annex J, (1998), Joints in Building Frames, CEN/TC250/SC3-PT9. Eurocode 9, (ENV 1999-1.1), (1998), Design ofAluminium Alloy Structures, CEN/TC250/SC9. Mandara, A and Mazzolani, F.M. (1995). Behavioural Aspects and Ductility Demand of Aluminium Alloy Structures. InProc. of ICSAS'95 Conference, Istanbul. Mazzolani F.M. (1995), Aluminium Alloy Structures", E & FN SPON, London. Mazzolani, F.M., De Matteis, G. and Mandara, A. (1996). Classification System for Aluminium Alloy Connections, inProc. of lABSE Int. Colloq. on "Semi-Rigid Structural Connections", Istanbul. ACKNOWLEDGEMENTS This work has been developed in the framework of activity of the research project "Methods of Behaviour Prediction for Aluminium Alloy Joints", sponsored by the ItaUan Ministry of University and Scientific and Technological Research (MURST). The helpful co-operation of the student Luigi Marano during the preparation of his diploma thesis is also gratefully acknowledged.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
663
LOCAL BUCKLING OF ALUMINIUM CHANNELS UNDER UNIFORM COMPRESSION: EXPERIMENTAL ANALYSIS 1
2
2
2
P.M. Mazzolani , C. Paella , V. Piluso , G. Rizzano Department of Structural Analysis and Design, University of Naples, Pederico II, Italy Department of Civil Engineering, University of Salerno, Italy
ABSTRACT This paper deals with the preliminary results of an extensive experimental program devoted to the evaluation of the ultimate resistance of aluminium alloy channels subjected to local buckling under uniform compression. In particular, the results of 34 stub column tests performed on aluminium channels made of 6000 series alloy are presented. Both the effective width approach and the effective thickness approach are applied for predicting the local buckling resistance of channels under uniform compression. The accuracy of the two approaches is discussed on the basis of the experimental evidence. The Eurocode 9 provisions for local buckling of both flat internal and flat outstanding elements in thin walled sections are examined and compared with the obtained results. KEYWORDS Lx)cal buckling, stub column test, aluminium channels, Eurocode 9, effective width, effective thickness, experimental tests. INTRODUCTION It is well known that both the ultimate resistance and the plastic deformation capacity of metal members are strongly affected by local buckling phenomena, whose occurrence is governed by the width-to-thickness ratios of the plate elements constituting the member section. In particular, depending on the magnitude of the width-to-thickness ratios, local buckling can occur in elastic or in plastic range. Therefore, the occurrence of local buckling can prevent the complete exploitment of the plastic reserves of the section, so that it has to be conveniently delayed by properly Umiting the width-to-thickness ratios. Por this reason, modem codes divide cross-sections into different behavioural classes. With reference to Eurocodes (Eurocode 3 on "Steel Structures" and Eurocode 9 on "Aluminium Structures"), four classes are considered. Ductile sections (first class) are characterized by the ability to develop the whole plastic resistance with high plastic deformation capacity. The whole plastic resistance can be attained also in the case of compact sections (second class), but with a hmited plastic deformation capacity. Semi-compact sections
664 (third class) locally buckle before the complete development of the plastic reserves, so that the plastic deformation capacity is very limited. Finally, in case of slender sections (fourth class), local buckling occurs in elastic range. Within the activities of CEN-TC250/SC9, i.e. the european committee charged of the preparation of Eurocode 9 on "Aluminium Structures", an extensive and exhaustive experimental program dealing with the influence of the width-to-thickness ratios on the local buckling of heat treated aluminium alloys has been planned, aiming to the evaluation of the relationship between stress and strain corresponding to the occurrence of local buckling, and the slendemess parameters of the plate elements composing the section. This goal can be achieved by analysing the results obtained from a great number of stub column tests carried out on specimens of different cross-sectional shapes having width-to-thickness ratios covering the whole range of variability of the commonly extruded profiles. In particular, the investigation of different cross-sectional shapes is necessary, because the occurrence of local buckling depends also on the influence of the edge restraining conditions of the plate elements composing the sections. Under this point of view, two types of plate elements can be recognised, namely flat internal elements and flat outstanding elements. The prediction of the ultimate resistance due to the occurrence of local buckling is quite complex even in the most simple case of members under uniform compression. In fact, it is affected by the following issues: - maximum local slendemess parameter; - slendemess parameter interaction; - restraining conditions of plate elements. In particular, the second issue is not considered even in modem codes such as Eurocode 3 for steel structures and Eurocode 9 for aluminium structures whose classification criterion is based on the maximum slendemess parameter only, without taking into account that flange is restrained by the web and viceversa. In other words, Eurocodes do not account for the interaction between the slendemess parameters of the plate elements constituting the member sections. Starting from the identification of the three issues mentioned above, the whole experimental program has been divided into three parts. The first part has been devoted to the investigation of SHS (square hollow section) members subjected to local buckling under uniform compression. This is the case of sections constituted by flat internal elements only. In addition, the occurrence of local buckling is governed by a single local slendemess parameter. This part of the experimental program has led to the formulation of a classification criterion for flat internal elements (Mazzolani et al., 1996), based on four classes, which has been introduced in Eurocode 9. The second part has been devoted to RHS (rectangular hollow section) members which are still constituted by flat internal elemets only, but require two slendemess parameters for defining the conditions corresponding to the occurrence of local buckling. This part of the experimental program has led to the setting up of an empirical formulation for predicting the strain corresponding to the occurrence of local buckling starting from the values of the two slendemess parameters characterizing the sections (Mazzolani et aL, 1997). As a consequence, a new classification criterion accounting for the interaction between the slendemess parameters of the plate elements constituting the member section has been proposed (Mazzolani et al., 1998). Finally, the third part of the experimental program deals with the influence of the edge restraining conditions of the section plate elements. To this scope, channels and angles subjected to local buckling under uniform compression will be investigated. In fact, channels are composed both by flat internal elements and by flat outstanding elements, while angles are constituted by flat outstanding elements only. As a result, the influence of the restraining conditions will be outlined. In this paper, the preliminary test results on aluminium channels subjected to local buckling under uniform compression are presented. The experimental tests have been performed at the Material and Stmcture Laboratory of the Department of Civil Engineering of Salerno University. The obtained test results are compared with the numerical predictions of the ultimate load carrying capacity based both on the effective width approach and on the effective thickness approach recently codified in Eurocode 9. CROSS SECTIONAL AND MATERIAL PROPERTIES OF TESTED PROFILES The preliminary testing results presented in this paper are based on specimens obtained by cutting the flanges of stub members made of RHS and SHS profiles along their longitudinal axis, so that stub
665 TABLE 1 GEOMETRICAL PROPERTIES OF TESTED CHANNELS AND ULTIMATE AXIAL LOAD 1 TEST CIA CIB C2A C2B C3A C3B C4A C4B C5A C5B C6A C6B C7A C7B C8A C8B C9A C9B ClOA ClOB CllA CllB C12A C12B C13A C13B C14A C14B C15A C15B C16A C16B C17A C17B
a (mm) 39.8 39.8 99.9 99.9 50.1 50.1 40.1 40.1 60.0 60.0 40.3 40.0 70.0 70.0 49.8 50.0 34.0 34.0 40.1 40.1 80.2 80.2 39.7 39.8 40.0 40.0 80.0 79.7 80.2 80.1 100.0 100.0 120.3 120.1
h (mm) 47.6 48.2 17.3 18.5 48.8 47.1 28.9 27.1 17.1 18.9 37.5 38.6 31.8 34.2 23.2 22.6 28.6 27.7 27.6 27.6 18.9 17.2 21.6 21.6 18.1 18.0 37.8 37.7 38.4 37.9 57.9 59.3 47.4 47.8
c (mm) la (mm) 48.1 4.10 47.3 3.90 17.1 3.80 3.95 18.6 3.90 48.8 47.2 4.00 27.1 2.50 28.8 2.50 17.2 2.60 18.9 2.50 37.6 3.90 38.6 3.90 31.9 4.00 34.4 3.90 23.2 3.00 22.6 3.00 28.4 2.90 27.4 3.00 28.5 2.00 28.6 2.00 3.90 18.9 17.1 3.80 21.0 2.40 21.0 2.50 18.1 4.10 18.0 4.10 38.0 4.20 37.9 4.00 38.3 1.90 37.7 1.90 57.1 6.60 58.7 6.60 48.5 4.50 49.0 i 4.50
h (mm) 4.00 4.00 4.10 4.00 3.90 4.00 2.60 2.50 2.50 2.60 3.90 3.90 4.30 3.90 2.90 3.00 3.00 3.10 2.00 2.00 3.90 3.90 2.40 2.50 4.00 3.90 4.00 4.20 1.90 1.90 4.60 4.40 6.50 6.50
tc (mm) \A (mm^) h (mm) TyVcx.(kN) 504.38 3.80 96.5 1 106.65 1 497.34 3.80 96.5 109.80 486.84 96.4 3.90 103.85 96.4 4.00 511.40 107.05 3.90 545.61 97.6 124.05 3.90 541.28 97.6 123.15 230.39 97.2 59.72 2.50 97.2 227.50 2.50 58.98 97.2 228.75 2.50 58.00 233.64 97.2 2.50 59.28 419.64 3.90 72.6 96.95 3.90 426.66 72.6 99.10 505.56 68.2 3.80 91.60 516.22 68.2 4.10 89.50 3.00 268.58 72.2 61.54 3.00 267.60 72.2 60.82 3.00 252.20 67.7 62.90 2.90 249.33 67.7 61.90 2.00 184.40 71.7 43.18 2.00 184.60 71.7 41.66 3.90 429.78 72.8 105.80 3.90 408.89 72.8 100.50 2.50 187.86 72.0 54.06 2.40 191.65 72.0 55.88 3.90 274.60 68.5 79.00 4.00 273.81 68.5 77.25 4.20 612.36 77.5 131.10 595.94 4.00 77.5 130.70 2.00 294.53 50.34 77.5 2.00 292.19 77.5 49.38 4.40 1064.53 146.8 333.40 4.60 1077.89 146.8 333.00 6.50 1052.55 146.0 338.00 6.50 1057.50 146.0 319.80 1
channels are obtained. The measured geometrical properties of the prepared stub channels are given in Table 1. In this table, a is the web depth, while b and c are the width of the flanges. The corresponding values of the thickness are ta, W and tc, respectively. As result of the measured geometrical properties, the area A of the cross-section is obtained. Two kind of channels have been examined: C with sharp corners and C with rounded comers. Specimens CIA to CI5B have sharp comers, while specimens C16A to C17B have rounded comers. In this latter case, the external radius of the comer is equal to 15 mm, while the internal one is equal to 10 mm. Tensile tests have been carried out to evaluate the material properties. The testing machine is a Schenck Hydropuls S56 (maximun test load 630 kN, piston stroke ±125 mm) equipped with an extensometer DSA 25/20N (emax = ± 20%). The measured values of the elastic modulus (E), the conventional elastic limits (/b.2 and/o.i), the ultimate strength {ft) and the exponent n of the RambergOsgood law are presented in Table 2. As usual, the exponent n has been calibrated according to the values of/o.i and/o.2 (Mazzolani, 1995). This means that the constitutive law of the material has been modelled according to the following relationship:
666 TABLE 2 MATERIAL PROPERTIES OF TESTED CHANNELS SPECIMEN
1 1
CI C2 C3 C4 C5 C6 C7 C8 C9 CIO CU C12 C13 C14 C15 C16 C17
ALLOY 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6060 T6 6082 6060 T6 6060 T6 6060 T6 6082 TF 6082 TF
fo.2(N/mm') 216.6 216.6 216.0 234.6 234.6 222.0 175.7 222.5 212.5 234.3 222.0 251.3 223.6 194.2 186.7 323.0 323.0
E (N/mm^) 70203 70203 68504 62761 62761 63508 70211 64863 11160 69263 63508 67488 72265 71733 65125 68841 68841
/b.i(N/mm') 213.3 213.3 211.5 229.4 229.4 216.3 169.9 222.5 204.8 230.0 216.3 246.0 223.6 194.2 186.7 315.7 315.7
£ = -^ + 0.002 E fo.2
n 45.2 45.2 33.3 31.3 31.3 26.6 20.6 28.9 18.6 37.5 26.6 32.9 19.1 26.8 27.5 30.3 30.3
f, (N/mm') 242.2 242.2 236.6 258.9 258.9 258.6 202.9 244.8 235.0 253.3 258.6 276.9 244.3 220.3 203.9 342.8
342.8
1 (1)
An accurate evaluation of the ultimate strain e^ has not been carried out, but it has been roughly observed that the nominal value has been always exceeded. STUB COLUMN TESTS The compression tests have been performed with a Schenck RBS4000-E2 testing machine (maximum load 4000 kN, piston stroke ±100 mm). The stub column tests have been carried out under displacement control. The axial displacements have been measured by means of three inductive displacement trasducers (stroke ±10 mm, sensitivity 80 mV/V), The mean value of the three measures has been considered. Neither top nor base rotation of specimens has been observed, because, as expected, all specimens have failed due to pure local buckling without any coupling phenomenon. The test results are summarized in Table 1, where the ultimate resistance A^exp is given. THE EFFECTIVE WIDTH APPROACH It is well known that the critical stress leading to the buckling of an ideal plate element, i.e. free of imperfections, in elastic range is given by: (2)
12(1 -v^)(w/tf where w is the width of the plate element and t is the corresponding thickness. The buckling coefficient k accounts for the edge restraining conditions and the w/h ratio, being h the plate dimension orthogonal to the loaded edge w. With reference to the channels examined in this paper, it results w = a and t=ta for the flat internal element, w = b and t = W for the first flat outstanding element and, finally, w = c and t=tc for the second flat oustanding element. Regarding the buckling coefficient, in case of flat internal elements, it can be assumed:
667 k =4
for
a h Ih "^ la
(3)
- > 1 a for
J
- < 1 a
where h is the height of the specimen; while, in case of flat oustanding elements, it results: (4)
k = 0.456 +
h
where w = ^ or H^ = c depending on the considered element. In case of buckling in elastic-plastic range, the critical stress can be computed by properly modifying the eulerian one, given by Eqn.(2), through a factor Tj depending on the stress-strain relationship of the material and on the stress level. In other words, in case of plastic local buckling, the critical stress can be computed through the following relationship: (5) nocr where the non-dimensional factor T) is dependent on the stress-strain curve of material and on the stress level. Among the formulations for the ii factor proposed in the technical literature by different researchers, whose review has been given by Ghersi and Landolfo (1995), the following relationships have been considered:
Es
[0.50 + 0.50 V0.25 +
0J5Et/Es
(7) (8)
^=¥
where Et is the tangent modulus and Es is the secant modulus. The tangent modulus theory, i.e. the use of Eqn.(5) v^th r| given by Eqn.(6), provides the lowest values of the critical stress in plastic range, while the secant modulus theory, i.e. the use of Eqn.(5) with Tj given by Eqn.(8), gives the highest values. The use of Eqn.(7) leads to intermediate results. It has been proposed by Li and Reid (1992) and provides excellent results in predicting the ultimate resistance of RHS and SHS members subjected to local buckling under uniform compression (Langseth and Hopperstad, 1995). In the post-buckling phase, the in-plane stress distribution becomes non-uniform. Stress concentration occurs close to the restrained edges, while zones far from restrained edges are understressed. Due to this redistribution, the ultimate strength Cu is generally higher than the buckling stress Ocr- The computation of the ultimate load requires a non-linear procedure which can be easily developed by using the effective width concept. This approach consists in substituting the actual non-uniform stress distribution with a constant value a, equal to the maximum stress occurring at the restrained edges, acting on an ideal plate having a reduced width Weff defined as that corresponding to the condition Ocr = a. Therefore, the effective width is given by: Weff = t
kn^E 12(1
\\/2
(9)
v^)a
With reference to the examined case, the effective width of the flat internal element Oeff can be computed by Eqn.(9) with t = ta, while those of the flat outstanding elements beff and Ceff are still computed by Eqn.(9) with t = tb and t = fc, respectively.
668 The buckling coefficient k accounts for the restraining conditions of the plate elements and is computed by means of Eqn.(3) for the flat internal element and by Eqn.(4), with w = Z? or w = c, for the flat outstanding elements. Regarding the Poisson's coefficient, it is assumed dependent on the stress level through the following relationship (Gerard and Becker, 1957; Hopperstad et al., 1996; Moen, 1996): (10)
0.2 E f
= 0.5
Finally, regarding the influence of geometrical inperfections, they have been accounted for by properly reducing the effective width according to the following relationships: 0.11
W . J 9 ^ = Weff
^eff
<
w
(11)
V
for flat internal elements, and: wV=
1.19W,J9^ 1 -
.iM
0.149
(12)
for flat outstanding elements. Eqn.(l 1) is practically the classical Winter formula where, according to Landolfo and Mazzolani (1996), the imperfection coefficient has been assumed equal to 0.11 instead of 0.22 to account for the average imperfections rather than for their maximum values. Similarly, Eqn.( 12) is practically the one suggested by Kalyanaraman et al. (1977) for flat outstanding elements, where the imperfection coefficient has been assumed equal to 0.149 instead of 0.298 to account for average imperfections. Non-linearity arises due to the dependence of the effective width on the stress level. By increasing step-by-step the axial strain, the corresponding values of a and w'eff are computed through equations (1) and (11) or (12), respectively. Therefore, for each step, the effective area and the corresponding axial load can be computed. The ultimate resistance is evaluated as the maximum load obtained in the loading process, while the buckling strain is the corresponding strain value. With reference to the secant modulus theory, i.e. Eqn.(8), the comparison between experimental and theoretical results is given in Fig. 1, where a very good agreement can be observed. In fact, the average value of the ratio Nth/^cxp between the computed value of the ultimate resistance Nth and the experiiriental one A^exp is 0.96 with a standard deviation equal to 0.067. The accuracy of the effective width approach reduces, in the examined case of channels under uniform compression, when Eqns.(6) or (7) are applied. In fact, in thefirstcase, the average value ofNf/z/A^exp is equal to 0.79 and the standard deviation is 0.052, while, in the second case, the corresponding values are 0.93 and 0.072, respectively. The above results show the accuracy of the effective width approach for predicting the ultimate EFFECTIVE WIDTH APPROACH
Nexp^ L
ra
nfl
1
A V E R A G E = 0.96
0.8 H
°'® H f 1
it 1 UHf 1 itiuf
0.4 H 1 f'JI 11 1 r NI 11 ] f ] 1 JI 1 r f
0.2 H
dLnlinfnintJUN
InHy
n y ^
J \\\ n U1J LI L. J td Li mlih UlnlJ inllAI LilJIniJ JldU
o^^ d*-^ o'^''^ o^*^ o^*^ o^^ o^^ o*'^
SPECIMEN
Figure 1: Reliability of the effective width approach for predicting ultimate axial resistance
669 resistance of aluminium alloy members subjected to local buckling, provided that buckling in plastic range is properly accounted for, i.e. by using the secant modulus theory. On the contrary, in case of box sections (Mazzolani et al., 1997; 1998) the formulation suggested bt Lee and Reid (Eqn. 7) is the one providing the best accuracy. THE EFFECTIVE THICKNESS APPROACH OF EUROCODE 9 The effective thickness approach for evaluating the resistance of metal members subjected to local buckling has been recently introduced in Eurocode 9 for aluminium structures (CEN, 1998) With reference to channels subjected to local buckling under uniform compression, the application of the effective thickness approach for computing the axial resistance requires the evaluation of an effective area by properly reducing the thickness of the plate elements constituting the member section. The magnitude of the reduction depends on the width-to-thickness ratio, the edge restraining conditions, the type of alloy and the member fabrication process. Regarding the edge restraining conditions, flat internal elements or flat outstanding elements can be identified. Two families of alloys are considered: heat-treated alloys and non heat-treated alloys. Finally, regarding the member fabrication, welded or unwelded members are identified. The presented experimental tests deal with heat-treated unwelded aluminium channels, so that only the distinction between flat internal elements and flat outstanding elements is of concern. In any case, the reduction coefficient for computing the effective thickness is given by: (13) p = 1 for p/e < (p/e)o
A p/e
52
(P/e)'
for
p/e > (P/£)^
(14)
where p is the slendemess parameter which, for uniform compression stress, is equal to the width-tothickness ratio of the plate element and e = (250//o.2) ' • The numerical coefficients 5i, 52 and (p/e)o define the buckling curve. For flat internal elements of heat-treated unwelded members, they are equal to 32, 220 and 22, respectively; while for outstanding elements of heat-treated unwelded members they are equal to 10, 24 and 6, respectively. As soon as the buckling curves are defined, Eqns. (13) and (14) allow the computation of theeffective thickness of all the plate elements constituting the member section. As a result, the effective area Aeff is obtained and the axial resistance is computed as Aefffy.lWith reference to the presented experimental tests, the results coming from the application of the codified effective thickness approach are compared with the experimental ones (Fig.2). The average value of the ratio NEC9/NQXP between the axial resistance predicted through Eurocode 9 approach and EUROCODE 9 APPROACH NexD
l- nf
1
MVCnMVJiC . = O.OD
PHn
n ifiPi ^ '
INt _hj
kll. J L.I J 1 llilL JLill.ltiil
JlJiJUlJfiiiJ JUL ILJiJLl JknlLl LJLiJ L.ibiLiLJLl 1
o^'^ o-^^ c?'^ o^^ o^'^ o^^ o^'^ o**^ c?^o^^^o'^''^d^'^''o'^'^*^d^''^o'^^^o^''''o<^^ SPECIMEN
Figure 2: Reliability of Eurocode 9 approach for predicting ultimate axial resistance
670 the experimental one is equal to 0.86, while the corresponding standard deviation is equal to 0.081. It means that the codified method based on the effective thickness approach leads to results which are a little more conservative than the ones coming from the effective width approach. CONCLUSIONS The preliminary experimental test results of a research programdevoted to the analysis of aluminium alloy channels subjected to local buckling under uniform compression have been presented and discussed in this paper. In particular, the effective v^dth approach and the effective thickness approach for evaluating the ultimate axial resistance of such members have been applied and compared with the experimental evidence. This comparison has confirmed the high degree of accuracy of the effective width approach, provided that the secant modulus theory is used for predicting the stress level leading to local buckling in plastic range. The accuracy in predicting the axial resistance of member s subjected to local buckling reduces when the effective thickness approach is applied. However, the more conservative results of such approach are fully justified, due to the simplicity of its codified format, which is a very helpful design tool for practical purposes. REFERENCES CEN (European Committee for Standardisation) (1998) ENVprl999: "Eurocode 9: Design of Aluminium Alloy Structures". Gerard, G. and Becker, H. (1957): "Handbook of Structural Stability: Part I - Buckling of Flat Plates", NACA, Tech. Note N.3781. Ghersi, Aand Landolfo, R. (1995): "Thin-Walled Sections in Round-house type Material: ASimulation Model", Italian Conference on Steel Construction, C.T.A., Riva del Garda, 15-18 October. Hopperstad, O.S., Langseth, M. and Tryland, T. (1996): "An Experimental and Theoretical Study on the Stability and Ultimate Strength of Aluminium Alloy Outstands in Compression", Norwegian University of Science and Technology, Department of Structural Engineering, Report N. R-14-95. Kalyanaraman, V., Pekoz, T. and Winter, G. (1977): "Unstiffened Conpression Elements", Journal of Structural Division, ASCE, Vol.103, ST9, pp.1833-1848. Landolfo, R and Mazzolani, P.M. (1997): "The background of EC9 Design Curves for Slender Sections", Volume in honour of Prof. J. Lindner, December 1997. Langseth, M. and Hopperstad, O.S. (1995): "Local Buckling of Square Thin-Walled Aluminium Extrusions", ICSAS '95, Proceedings of the Third International Conference on Steel and Aluminium Structures, Istanbul. Li, S. and Reid, S.R. (1992): "The Plastic Buckling of Axially Compressed Square Tubes", Journal of Applied Mechanics, Vol.59. Mazzolani, P.M. (1995): ''Aluminium Alloy Structures, E&FN Spon, an Imprint of Chapman & Hall. Mazzolani, P.M., Paella, C , Piluso, V. and Rizzano, G. (1996): "Experimental Analysis of Aluminium Alloy SHS-Members Subjected to Local Buckling under Uniform Compression", 5th International Colloquium on Structural Stability, SSRC, Brazilian Session, Rio de Janeiro, August 5-7. Mazzolani, P.M., Paella, C , Piluso, V. and Rizzano, G. (1997): "Local Buckling of Aluminium Alloy RHS-members: Experimental Analysis", XVI Congresso C.T.A, Italian Conference on Steel Construction, Ancona, 2-5 Ottobre. Mazzolani, P.M., Paella, C , Piluso, V. and Rizzano, G. (1998): "Local Buckling of Aluminium Members: Experimental Analysis and Cross-Section Classification", Department of Civil Engineering, University of Salerno, Italy. Moen, L. (1996): "Some Considerations on the Rotational Capacity of Aluminium Alloy Beams", Norwegian University of Science and Technology, Department of Structural Engineering, Report N.R-10-96. ACKNOWLEDGEMENTS The profiles for preparing the tested specimens have been provided by Alures Alumix Group (Italy) (now Alcoa Italia), Alusingen (Germany), Baco Contracts Alloy Extrusions Ltd (England), Hydro Aluminium Structures (Norway) and Pechiney Batiment (Prance) whose support is gratefully acknowledged.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
671
LOCAL IMPACT ON ALUMINIUM PLATING ^::J B. Boon and H. Weijs'
Ship Structures Laboratory, Delft University of Technology, NL
ABSTRACT Collision and grounding scenarios are studied for ferries and tankers because of the risks for humans and environment that such accidents may pose. Similarly for high-speed vessels this research is needed. Important thereon is the criterion dictating first failure (cracking) of aluminium plates. Drop tests with a cone on perpendicular and horizontal plates are a begin of such research.
1
INTRODUCTION
Initiated in response to some serious accidents a fair amount of research recently has been performed on the consequences of collisions and grounding to the structures of tankers and ferries. At the same time in the design efforts were made to reduce the consequences of such accidents by careftiUy selecting structural design principles and construction materials. Collisions and grounding may have very serious consequences for modem high speed ferries because of the large number of passengers aboard, the velocity of these craft and their inherently light (and hence damageable?) construction. Considering these risks it is surprising how little research has been performed on the consequences of collisions and grounding for the structure of this type of vessel. Designers seem to be guided by the damaged stability requirements of authorities and classification societies: "Even if one hull of this catamaran is ripped open from stem to stem, the lives of those on board still are not endangered". An IMO-recommendation exists about maximum deceleration of the passengers in their chairs in order to prevent serious injuries in case of head-on collision at full speed. The results of the work done for tankers and (conventional) ferries is useful for high-speed craft only to a limited extent because of the generally completely different hull materials, i.e. mild steel for the conventional craft and aluminium composites or high-strength steel for the novel ship types. For other vessel types similar investigations have been far less. The Ship Stmctures Laboratory of the Delft University of Technology in The Netherlands have performed some preliminary investigations into impact behaviour of aluminium being one of important hull materials for high speed ships, Weijs (1998). After evaluating published research on the response of aluminium structures it was decided to put emphasis on the failure criteria to be used. Some experiments were carried out, preliminary results of which are reported here. Further research in the field is needed.
^ Presently employed by Lloyd's Register of Shipping London, UK
672 2
PAST ACCIDENTS WITH HIGH SPEED CRAFT
It is a matter of debate whether high-speed craft run a bigger or smaller risk of collision and grounding than more traditional ships. It is obvious, however, that when it occurs the high speed and the inherent lighter structure may lead to more serious consequences. The number of accidents that actually happened seems to be small. This may be because of the still limited number of such ships in operation. It may also be due to better-trained crews, better navigation equipment or better manoeuvrability. The following accidents are the more important ones that happened. In November 1991 the "Sea Cat" travelling at 36 knots in darkness and heavy rainstorms ran into a rock. Two people were killed and 74 injured. Because the collision was not head-on but under an angle the consequences were not more disastrous. Fast Ferry International (1992). In November 1994 the 40m catamaran "Royal Vancouver" collided with the Ro-Ro vessel "Queen of Saanich". Because of the reduced speed of less than 10 knots and the angle of collision injuries to the passengers were limited although the catamaran sustained substantial damage (note: the other vessel had no real damage), Fast Ferry International (1995) The 42m catamaran "Saint-Malo" sailing at about 27 knots in rough seas hit a rock when trying to avoid some fishing marker buoys. The two hulls experienced damage over a large part of their length. Although the depth of the damage was limited, it caused water ingress into 6 of the 13 watertight compartments. The deceleration of the vessel was relatively small so that no passengers were injured. Fast Ferry International (1995). The accidents described suffice to show that they do occur. The consequences were limited due to details of the circumstances of the mishaps. Had they be slightly different, however, the consequences easily could have been far more dramatic.
3
REASONS FOR STUDYING IMPATC OF ALUMINIUM STRUCTURES
The accidents described above gave rise to recommendations to study the behaviour of the structure of high-speed vessels under collision and grounding circumstances. Actually, however, very little attention to these phenomena is given in either research or the actual design of such ships. The international Maritime Organisation (IMO) in their code for high speed craft. Fast Ferry International (1995), give regulations in which the declaration in case of a head-on collision with a solid vertical rock is given in formula form. The deceleration depends on the length of the vessel, its height, its draught, the displacement and the operational speed. The deceleration thus determined is used to design fixing of loose equipment and to determine the extent of damage (x=6^ /2g) in which no accommodation may be located etc. As an alternative to the empirical formulation given, direct calculation of the collision scenario may be used. Note that from the designer's point of view low deceleration is favourable with regard to loads on loose equipment and risk of injury to passengers and crew. It is unfavourable, however, with regard to the extent of damage, hence area of non-use for accommodation and possibly, volume flooded. Notwithstanding the above the requirements of IMO with regard to crashworthiness of high-speed vessels is very limited. With regard to flooding the requirements is expressed in a number of watertight compartments that must be considered damaged. Many designers use this as the sole requirements consider explicit analysis of a collision scenario to be superfluous.
673 However, study of the behaviour of high speed craft with impact loading from collision and grounding allows better evaluation of the risks involved in novel design features. And it also allows optimisation of the designs for safety: a rational choice can be made between structural alternatives that each comply with the existing rules.
4
SET-UP OF THE EXPERIMENTS
The response of a ship structure to impact loads in general develops in three stages subsequent in time. First an elastic response develops. Plasticity does not play a role and after the load has been removed, the structure will regain its original shape. In general this stage lasts only for very small loads. Secondly inelastic response will occur. After release of the load permanent deformations will remain. These deformations may either be due to localised and general yielding and/or buckling in one of its many forms. Experiments and numerical simulations for aluminium plates have been performed, Zhu et al (1994). The simulations were valid till the experimentally determined moment of start of rupture. A method of representing that moment is given by Zhu and Atkins (1998), depending on the strain in the plate material in two directions. Only with minor impact loads the response will be restricted to this stage. The third stage starts at first failure (i.e. cracking) of a plate. From this point on crack propagation (tearing) of plates, stiffeners and welds plays a major role. Of course water may enter the vessel right from the start of this stage if the crack is located below the water line. The initial crack will then develop into a larger hole, the size of which will dictate the rate of water ingress. In case of a double shell and watertight bulkheads the extent of the damage will determine the number of compartment damaged in the final situation. Already from this description the importance of being able to predict the response in this third stage. At the same time, however, the very fact of an initial crack influences the strength of the structure. Once a crack has initiated, membrane stresses, which play an overwhelming role in the second stage, will to a large extent disappear. Extension of the damage will require far less energy than is the case as long as no crack exists. Little is known of the failure criteria for aluminium plates, in particular for the relatively thick plates used in ships. It was decided to concentrate the research on determination of this failure criterion. The experiments performed consist of dropping a round-nosed cone (representing the comer of a container) onto aluminium plates with ftill scale dimensions. Drop velocities correspond to the speed of such vessels. Translated to a real ship situation this criterion may be deemed governing when a high-speed craft collides with a floating object such as a container. Because the sharp bow the angle of contact generally will be between 20 and 25 degrees, Zhu et al (1994). Some of the drop tests therefore were performed on inclined plates.
5
TEST PARTICULARS
Typical thickness for side plating in the bow section of a 70 to 120 m long catamaran is 6-8 mm. A frequenfly used alloy for hull plating 5083-H321 is chosen. Material properties are 5uit=300 MPa, 5o.2=215 MPa, 8uit=12%, p=2.66xl0^ kg/m^ E=71000 MPa, v=0.33. Typical transverse stiffener spacing in the bow section id 700-1000 mm, while the longitudinals are spaced 300-600 mm.
674 A square plate is used to eliminate effects from L/B ratio, no stiffeners and no welds are incorporated in this stage. The indentor is shaped as an axisymmetrical cone with top angle 90 degrees and a 20 mm radius of the nose. The influence of two factors on the material behaviour of the aluminium plate under impact by a cone was investigated: impact velocity and angle of impact. In total four different models are investigated: perpendicular, static; inclined, static; perpendicular, dynamic; inclined, static. Only two angles and two velocities are examined, as the purpose is only to determine whether or not there is a significant difference in material behaviour. For the static tests a servo-hydraulic 100 tons test machine was used. The movement of the head is computer and electronically controlled. The velocity is set to be 6.228 mm/min during the perpendicular and 14.737 mm/min during the inclined test. In this way the velocity perpendicular to the plate is equal in both cases. During the tests were measured: the vertical load, the horizontal load, the stroke v and the strain at some locations. For the test set-up see Fig. 1.
Fig. 1 The static test The dynamic tests were performed using a 6 meter high drop tower (Fig. 2). Measured were the vertical load, the horizontal load, displacements and strains at some locations.
Fig. 2 The dynamic tests in the drop tower In the dynamic test two physical phenomena disturb the measured data when compared to the static situation. The inertial load is needed to accelerate the specimen to the same velocity as that of the indentor, causes a sharp load peak in the beginning followed by decaying oscillations. Harmonic oscillations of the sample arise as a result of the small rise time of the load.
675 PERPENDICULAR STATIC RESULTS The vertical force is plotted versus the displacement of the cone in figure 3. It rises steadily to approximately 50 kN at a displacement of v=26 mm. Then the force drops until about v=35 mm to a steady level of 40kN. The decline of the force is the result of cracking of the plate. This means that the deflection at which the plate starts cracking is only three times the plate thickness.
Fig. 3 Force versus displacement in the perpendicular static test Strains were measured at three different positions. Cracks start growing on the borders of a small square, which centre is the centre of the contact area on the back side of the plate. The sides of the square are parallel to the symmetry axis of the model. The cracks grow outwards through the thickness of the plate. The first through- thickness cracks reach the front side of the plate in the boundary of the contact area. This justifies the conclusion that the crack grows outwards, because the contact area grows steadily. As soon as one crack is through thickness it starts growing quickly outwards with a tendency to follow the symmetry axis. The parallel crack that did not reach through the thickness of the plate will stop growing as the stresses are released as a result of the through thickness cracking of the plate. In the direction perpendicular to the first crack the same process of one of the two cracks growing through thickness in the boundary of the contact areas takes place, it can also be seen that shortly before the crack reaches the front side of the remaining intersection deforms fast.
7
INCLINED STATIC RESULTS
The vertical force in the inclined tests all reach their maximum value at the moment the displacement of the cone is between 65 and 70 mm, the maximum vertical force lies in between 41 and 49 kN. Photos were made on the backside of the plate during the test. Information from these pictures combined with the resulting damage gives a good impression of what happens during the tests. Initially the cone slides over the plate surface, leaving behind a track, which looks as if the material has yielded at the time the cone was coming past. On the back side of the plate the first crack starts growing in front of the cone, perpendicular to the translation vector of the cone. One test has been stopped at this point as a through-thickness crack was expected (also because a snapping sound was heard). After removing the cone it can be seen that on thefi*ontside a crack started on the boundary of the contact area behind the cone. Infi*ontof the cone there is no signfi-oma crack at this stage. In the next step it is seen that the crack infi-ontof the cone only goes halfway the plate thickness. Here a crack starts that runs in the middle of the plate thickness in the y-direction (forward of the cone). At
676 the same time the crack behind the cone runs through the thickness of the plate and cracks start growing parallel to the y-axis on the back side of the plate. Thus a stroke of material at the backside of the plate which is approximately half the plate thickness is bent outwards, while the upper half stays put. Then finally the first crack visible from the back, cracks through thickness, leaving a small block of material behind and giving the cone free way to punch through the plate. The cone moves further down bending outward the stroke of material infi-ontof it and sideward the material to on the side of the cone.
8
PERPENDICULAR DYNAMIC RESULTS
During the perpendicular dynamic tests, it was notified that the indentor bounced back several times. The energy in the second hit is about 3-7% of the energy of the first hit. Though this hit can be recognised in the measured signals it is small compared with the response of the first hit. Therefore it is decided not to take the second hit or the hits thereafter into account. The vertical force history shows two peaks directly after the hit (fig. 4) It also shows that the 0-point probably is estimated wrong and should have been at what is now time=2ms. The peaks in the force curves are 80 and 115 kN with a minimum of 20 kN in between. The peaks thereafter are probably the result of harmonic oscillations.
Fig. 4 Vertical force versus time in the dynamic perpendicular test From the plates can be seen that the indentor did not hit the plate exactly in the centre all the time (average 6.5 mm out of the centre). The through-thickness crack always runs over the edge of the contact area. In two cases three quarters of the edge is cracked, in the other two cases just a quarter. In the last cases the backside of the contact area shows large deformations. The small circles are used to measure strains up to 14% ±3%. Apparently the cracks start more to the centre of the backside of the plate to the symmetry axis. Then it runs through the plate away from the centre to reach the topside of the plate in the edge of the contact area. An example of the crack shape is given in Fig. 5 as photo and as sketch. In the latter the strains measured at a circle around the initial crack. The numbers given indicate strain in radial direction (upper) and in tangential direction (lower).
677
Fig. 5 Photo and sketch with radial and tangential strains for a dynamic perpendicular test
9
INCLINED DYNAMIC RESULTS
The test set-up is shown in Fig. 5. Three tests are performed. The first of the three plates was impacted with a mass 91.5 kg from a height of 5.5m. The damage that resulted from this test was much larger than the damage in the static tests. During the static tests it was decided that further indenting would not provide any relevant information, the same accounts for the dynamic tests and therefore it was decided to diminish the mass to 50.6 kg. During these tests no bouncing back from the indentor took place.
Fig. 6 Test set-up for the dynamic inclined tests The vertical force shows the same maximum for all three tests (50 kN). The vertical force caused by the 91.5 kg drop weight continues after the peak at a level of 30 kN for a while before dropping back to zero, while the force caused by the smaller mass fall back to zero immediately. The horizontal force in the first test reaches its maximum (45kN) at time=10ms, it falls back and oscillates a bit to end at a constant level of 18 kN. For the second and third test the maximum value is reached at 15 ms and is 60 kN. The constant level that is reached after the oscillations are damped is 45 kN.
678 10
DISCUSSION AND CONCLUSIONS
From the tests performed it is clear that fracture behaviour of an aluminium plate even in relatively simple circumstances is complicated. After crack initiation energy is needed not only for propagation of the crack in a tearing mode, but also for bending of the plate sectors (the petals). A reliable criterion for crack initiation of relatively thick plates as used in ships, still has to be developed. Generally it is assumed that aluminium is not very strain rate sensitive. Nevertheless the experiments seem to indicate that in dynamic circumstances more energy is needed for cracking than is needed in static tests. More research is needed to describe the fracture behaviour of aluminium plates. Numerical simulation incorporating a reliable fracture criterion has to be developed in order to be able to include response of aluminium ship structures to impact loads in the design of high-speed vessels.
REFERENCES
"Seacat accident" Fast Ferry International,{April 1992), 31-38 "Canadian accident report". Fast ferry International, (March 1995), 29-32 "Saint-Malo accident" report published Fast Ferry International, (October 1995), 19-28 IMG: "International Code of Safety for High Speed Craft (HSC-code)", London, (1995) McGeorge, D., Echtermeyer, A.T., Hayman, B. (1995), "Local strength of plating in high speed craft", Proc. Third International Conference on Fast Sea Transportation, ed. Kruppa, C.F.L., Schiffbautechnische Gesellschaft, 233 - 244, Berlin, Germany H. Weijs (1998): "Local impact on aluminium ship plating caused by a collision at high speed". Delft University of Technology, report SSL-368 Zhu, L., Faulkner, D. (1994): "Dynamic inelastic behaviour of plates in minor ship collisions ", Int. Journal Impact Engineering, vol. 15, Zhu, L., Atkins, A.G (1998).: "Failure criteria for ship collision and grounding". Practical Design of Ships and Mobile Units, PRADS '98, Elsevier, Amsterdam, The Netherlands
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
679
CREEP BUCKLING OF METAL COLUMNS AT ELEVATED TEMPERATURES J.S. Myllymaki^ and R. Kouhia^ iVTT BuUding Technology, Fire Technology P.O. Box 1803, FIN-02044 VTT, Finland ^Helsinki University of Technology, Laboratory of Structural Mechanics P.O. Box 2100, FIN-02015 HUT, Finland
ABSTRACT Multiplicative viscoplastic law has been used in the analysis of the behaviour of the aluminium alloy AA 6063-T6 at high temperatures. In the identification of the parameters of the viscosityhardening law an inverse solution technique, regularized output least squares method (RLS), has been used. It is shown that multiplicative viscoplasticity law satisfactorily predicts behavior of aluminium alloy AA6063-T6 in transient temperature creep tests and in steady state tests of different strain rates. The viscoplasticity law is implemented into a finite element-code, thus facilitating creep buckling analysis of aluminium structures. Comparisons to simple idealized I section solutions of viscoplastic creep buckling of unprotected aluminium RHS columns in the fire are given. The example shows that idealized I section solution is in simple cases a precise engineering tool.
KEYWORDS Creep, buckling, aluminium, viscoplasticity, inverse methods, high-temperature material data
INTRODUCTION The process of deterioration of metal-supported building elements in fire can be predicted with reasonable accuracy, if the plastic behaviour of metals at elevated temperatures is known. Substantial amount of information is already available concerning various properties of steels at elevated temperatures (Harmathy k Stanzak 1970, Anderberg 1988, Kirby k Preston 1988). Also a lot of investigations have been conducted on the properties of aluminium at elevated temperatures (Engstrom k Sandstrom 1993, Hammad k Ramadan 1989, Voorhees k Freeman 1960). In many cases test results are not available for commercial aluminium alloys and properties of aluminium alloys have been obtained using classical steady-state tensile tests or creep tests. In the Eurocodes the rate-independent thermo-elasto-plastic material models with properties derived from transient temperature creep tests or steady-state tests are used for steels. These material properties correspond to certain fixed temperature rate usually 10 K/min used in the
680 transient tests (Kirby k Preston 1988). In the case when the temperature rate of the metal structure is lower than the rate used in the transient tests elasto-plastic models are non-conservative and rate (creep) dependent viscoplastic models are more preferable (Anderberg 1988). An inherent problem in the viscoplastic material models is that the hardening terms in the equations often cannot be easily identified or directly determined from experimental data. This paper presents a systematic method, regularized output least squares method (RLS), for delineating the hardening terms from test data. The parameters of multiplicative viscoplastic constitutive model have been identified for alumium alloy AA6063-T6 using transient temperature creep tests and constant temperature steady-state tests. Simple idealized I section solutions of viscoplastic creep buckling of pin-ended column are presented. Similar simple solutions have been used by Harmathy &; Stanzak (1970) for creep bending and Forsen (1995) and Bazant k Cedolin (1991) for creep buckling analysis of steel and aluminium columns. A numerical example of an unprotected aluminium RHS column in the fire is given. Comparisons between the simple approach and the FE-results are presented.
VISCOPLASTIC CONSTITUTIVE EQUATIONS The adopted constitutive equation for the inelastic deformation is Norton-Bailey type power law. Only isotropic hardening in the case of small strains (less than about 5 %) is considered. The total strain rate is assumed to be decomposable into elastic, plastic and thermal components e,j = e'ij 4- e?. + e'^ = ^ + i^j + atSij where the elastic component e^j can be obtained from Hooke's law, and a is the coefficient of thermal expansion. The plastic strain rate ef^ is expresessed by equation (Lemaitre & Chaboche 1994):
where Sij is the deviatoric stress tensor and a = J\sijSij evolution equation for the plastic multiplier p is
P-yi^^j^^j
yxp^/^)
is von Mises equivalent stress. The
'
In uniaxial case the following equations are obtained: e = e' -h eP -h a t , a = Ee\
(1)
G = K(eP)'/^(^P)'/^,
where A^ is the viscosity exponent, M is the hardening exponent, K is the coefficient of resistance and E is the modulus of elasticity which all are temperature dependent.
MATERIAL EXPERIMENTS Test material was aluminium alloy AA 6063-T6 manufactured by Nordic Aluminium. Test specimens (Fig. 1) were cut out from as received 20 mm wide and 4 mm thick aluminium alloy
681 TABLE 1 CHEMICAL COMPOSITION (%) OF ALUMINIUM ALLOY AA 6063-T6
Mg 0.48
Si 0.44
Mn Cu 0.0043 0.0038
59.6
Fe 0.22
17.9 h< M
AA 6063-T6
Zn 0.012
40
Ti V Pb Cr Zr Ti+Zr 0.022 0.009 0.009 0.003 0.003 0.025
17.9 •rt- M
/
59.6 Hole d= 6
m
¥SL
Figure 1: Test specimen for tension tests.
100 STRAIN (%)
150 200 250 TEMPERATURE (°C)
300
350
400
Figure 2: (a) Stress-strain curves of aluminium alloy AA 6063-T6 (strain rates 7 x 10 ^ and 7 X 10"^). (b) Measured stress dependent strain in transient tests.
682 (a)
(b) — EC9
— Sandstrdm 2-hour curve • A X a
o Steady-state 0,18 •
Steady-stale 1,8
STP291 Transient tests 1.8mm/min 0.18mm/min
A Transient 10 K/min
100
200
300
400
TEMPERATURE (''C)
100
200
300
400
TEMPERATURE (oC)
Figure 3: (a) The dependence of the 0.2 % yield strength and (b) and UTS on temperature compared to Eurocode 9 (1995) curve, ASTM STP 291 (Voorhees k Freeman 1960) and Engstrom k Sandstrom (1993) 2-hour curve for alloy AA 6063-T6. A A 6063-T6 sheet longitudinally to wrought direction. Chemical composition (%) of the test material measured by spectral analysis is presented in Table 1. Steady-state hardening tests were carried out with two equal tests at strain rate 7 x 10"^ s~^ (crosshead speed 0,18 mm/min) and one test at strain rate 7 x 10"^ s~^ (crosshead speed 1,8 mm/min) at temperatures 50 °C, 100 °C, 150 °C, 200 °C, 225 °C, 250 °C and 300 °C. Three tensile tests were carried out at room temperature. In the transient test the air temperature rate in the test oven was 5, 10 and 30 K/min. Transient tensile tests were carried out with two equal tests at each stress level of 3, 20, 40, 60, 80, 100, 120, 140, 160, 180 and 190 MPa. Thermal strain and coefficient of thermal expansion of the aluminium alloy were determined with three tests at load level of 3 MPa. The stress-strain curves of AA 6063-T6 alloy at strain rates 7 x 10"^ and 7 x 10~^ are shown in Fig. 2a. The following feature is apparent; at the same strain, the higher the strain rate is, the higher will be the stress. In the tests under transient heating conditions stress dependent strain curves were obtained by excluding the thermal strain. Typical curves of heating rate 10 K/min are shown up to 2.5 % strain in Fig. 2b. The depence of of 0.2 % yield strength and the ultimate tensile strength (U.T.S) on the test temperature is shown in Fig 3. In general, the U.T.S and yield strength decrease slowly from room temperature up to 150 °C and decrease rapidly at higher temperatures.
PARAMETER IDENTIFICATION Determination of a non-constant parameters in an initial value problem (1) on the base of the measured uniaxial strain e^ata is considered. This kind of inverse problem is very sensitive to small variations in the measurements i.e it is ill-posed and it has to be regularized. Here regularized output least squares method (RLS) is applied, Groetsch (1993). The whole temperature interval studied is discretized into a chosen number of subintervals [Tj,rj+i]. The parameters tti = {K{Ti)jM{T)i,N{Ti),E{Ti))'^ are approximated as piecewise linear functions of tempera-
683
i^ 1 '''''-
M
1 ^^^ cf
\1 /
1
\\
\
\ \ \
0
^
^\ 50
100
150
200
250
300
350
400
ISO
Temperature (° C)
200
2S0
300
3S0
400
Temperature (°C)
Figure 4: Parameters of the multiplicative viscosity law for aluminium alloy AA6063-T6. ture. The unknown parameters are found by minimizing the functional min(|Ka;0-^dataWll' + r | | £ a f )
(2)
with respect to paramaters o. Coefficient r is a positive regularization parameter depending on the noise level of the data and £ a suitable differential operator. The first term in the problem (2) enforces the consistency of the solution when the second term enforces its stability. An appropriate balance between the need to describe the measurements well and the need to achieve a stable solution is reached by finding an optimal regularization parameter. Strain is computed using explixit time integration n+1 - I f^p\N/M J ^^" ^ ^ny ^n+1 = CTn+l/^n+l-
Parameters are solved at constant temperatures using the steady-state test results at two deformation rates and transient results at oven temperature rate 10 K/min. Figure 4 shows the calculated values of Young's modulus E and the coefficient of resistance K and the viscosity and hardening exponents A^, M. Comparisons of the model to the steady-state and transient test results are shown in Figure 5.
VISCOPLASTIC BUCKLING OF PIN-ENDED COLUMN A simple approximate solution method for a pin-ended column is described, see Fig. 6. Solution is sought in a form v{x,t) = Vi{t) sm{7rx/L) where Vi{t) is an unknown center deflection. The compressive load P has an eccentricity e at both column ends. The stress-strain relation is described by equation (Bazant k Cedolin 1991)
r = |(|)+^(a)+at, in which • = e»'
=
684 (a)
(b) Transient tests 140 MPs 250 200
30K/min
r
j^^^^^^^'^'^^
150
100-
Tensile test 150 "C
•j 50
—Te«tr««ult
10K/min
j
Test 10 K/min Test 30 K/min
1
Model30K/min
"Mullip(«tiv«mo<M
00.50
1
1
1
1
1.00
Figure 5: (a) Steady-state results and computed solution at temperature 150 °C with two crosshead speeds (b) transient test results and calculated solution at stress 140 MPa with temperature rate 10 K/min and 30 K/min.
i/. _ . 5'^
M. . h
H e\
62
(7i
(72
(b) (c) Figure 6: (a) Pin-ended column and (b) assumed strain and (c) stress distribution over cross section (Bazant Sz Cedolin 1991).
685 is a creep function of stress a. Evidently, the stress distribution throughout the cross section is generally nonlinear; however, it is approximated by a linear distribution characterized by stress values di and 0^2 at distance c from the center line of the column (Fig. 6). For the cross section at the midheight of the column, these stresses are calculated as
P , Pfa+e) —j—c.
a,, = --±
Substituting these values into the viscoplastic stress-strain relation (1), the strain rates are computed as
*-^(-^-^^«)-H-^^^')-' dt
Unless the initial curvature of the column is so high that the axial force resultant would be outside the core of the cross section, the entire cross section at first undergoes loading, that is ^1 ^ 62 ^ c. At certain deflection the convex face experiences strain reversal and afterwards unloading. At the moment of reversal, function ^2 must be reset to zero to represent the unloading stress-strain diagram from the point of reversal. Now, the curvature rate at midlength k can be calculated using the difference of the strains « = -v,xx = -ViTT^L^ = - ( e i - 62)/2c.
(4)
Equations (3) and (4) form an ordinary nonlinear first-order differential equation, which can be solved with explicit Euler method.
NUMERICAL EXAMPLE Buckling
of a Pin-Ended
RHS-Column
The viscoplasticity law of Chaboche and Lemaitre has been implemented in a finite element-code. In this chapter one buckling example of a rectangular hollow section (RHS) column 120 x 50 x 5 is presented. The column is pin-ended and free to move in the axial direction at the top as in the preceding example. Length of the column is 3 m. The column is discretized by using 10 EulerBernouUi beam elements. Updated Lagrangian formulation is adopted in the formulation of the discrete equilibrium equations. The temperature of the unprotected column is computed by using convection coefficient he = 25 W/m^K and resultant emissivity e = 0.3. In the structural analysis the column is supported in the direction of the weaker axis. The FEM solution is compared to the solution of simple solution. The initial deflection of the column is 1 mm. In the collocation method the RHS section has been transformed as an idealized massless I-section. Flange area has been assumed to be Af = A/2 and the distance of the centroids of the flanges has been 2ci = 2y/I/A. In this way the area and the bending rigidity of the idealized I-section and the real RHS-section are the same. Results of the computations are presented in Figure 7.
CONCLUSIONS Application of the regularized inverse solution show that the use of creep dependent models as the viscoplastic constitutive model illustrated above makes it possible to couple constant temperature test results obtained from hardening or relaxation tests with transient temperature results when
686 (a) 0.20
0.20-]
Aluminium column 120 x 50 x 5 L=3 m
0.18
0.18
0.16
0.16
Collocation method
0.14
Aluminium column 120 x 50 x 5 L=3 m ==FEM Collocation method
s: 0.14
0.12
0=60 MPa I
0.10 0.08 0.06
Z O H LU if LU ^
0.12 -
0.08
0.04
0.02
0.02 25
50
75 100 125 150 175 200 225 250 275 300 325 TEMPERATURE (°C)
h
0.0025
'•
J
0.06
0.04
0.00
0.10
1 50
1
1
1
1
1
1
1
1
1
\
;
75 100 125 150 175 200 225 250 275 300 325 TEMPERATURE (°C)
Figure 7: Center deflection with (a) load = 60 MPa (b) = 75 MPa. the parameters of the constitutive law are identified. It is shown that multiplicative viscoplasticity law satisfactorily predicts temperature rate hardening behaviour of aluminium alloy AA6063-T6 in transient temperature creep tests and strain rate hardening in constant temperature steady-state tests. It should be remembered that the test results at temperature rates 5 K/min and 30 K/min has not been used in the model identification and the model predicts also them. Simple idealized I section solutions of viscoplastic creep buckling of a pin-ended column is presented. Numerical example of unprotected aluminium RHS column in the fire is given. Results show that the simple method gives reasonably accurate deflections when compared to the solution of FEM analysis.
REFERENCES ANDERBERG Y . (1988). Modelling steel behaviour, Fire Safety Journal 13, 17-26. BAZANT Z.P. and CEDOLIN L. (1991). Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories. Oxford University Press. Oxford. ENGSTROM H . and SANDSTROM R . (1993). Evaluation of high temperature strength values of aluminium alloys. Aluminium 69:11, 1007-1013. EUROCODE 9. (1995). Design of aluminium alloy structures Part 1.2. Structural fire design. Draft prENV 1999 Part 1.2. CEN/TC 250/SC 9. FORSEN N . (1995). Fire resistance In: Mazzolani F.M.(1995). Aluminium Alloy Structures. E & FN Spon. GROETSCH C . W . (1993). Inverse Problems in the Mathematical Sciences, Vieweg. HAMMAD A.M. and RAMADAN K.K. (1989). Mechanical properties of Al-Mg alloys at elevated temperatures. Z. Metallkunde. 80:6. 431-438. HARMATHY T.Z., STANZAK W . W . (1970). Elevated-temperature tensile and creep properties of some structural and prestressing steels, ASTM STP 4^4^ 186-208. KiRBY B.R. and. PRESTON R . R . (1988). High temperature properties of hot-rolled, structural steels for use in fire engineering studies. Fire Safety Journal 13, 27-37. LEMAITRE J and CHABOCHE J.-L. (1994). Mechanics of Solid Materials. Cambridge University Press. Cambridge UK. VOORHEES H.R., FREEMAN J . W . (1960). Report on the elevated-temperature properties of Aluminium and Magnesium alloys. ASTM, STP 291.
Session A9 DESIGN FOR HYGROTHERMAL, VIBRATION AND FIRE EFFECTS
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
689
DESIGN FOR HYGROTHERMAL PERFORMANCE AND DURABILITY OF INSULATED SHEET METAL STRUCTURES G. Johannesson Division of Building Technology, Department of Building Sciences, KTH - The Royal Institute of Technology , Stockholm, S 100 44, Sweden.
ABSTRACT The scope of this paper is to outline some relevant fields of competence that are necessary for the completion of insulated light gauge constructions. Structural mechanics have to be combined with knowledge in building physics, i.e. heat and mass transfer in constructions and the consequences these factors have for energy use, moisture problems, the long term performance and durability. Light g9uge steel structures have been gaining market shares in building types that traditionally were reserved for masonry or wood frame constructions, such as residential buildings. Many companies that promote and produce such constructions, are unproportionally interested in the load bearing properties of their constructions, but paying much less attention to properties such as thermal insulation, air tightness and moisture protection. Steel structures also raise new and in some cases different questions fi"om what we have for other building materials. In the paper a review of the knowledge base for construction analysis is given with some examples on how this affects practical design work. A construction that in the above aspects is performing well, in a cost effective way, is usually an environmentally feasible construction regardless of what materials are used for the construction.
KEYWORDS Heat and mass transfer, light gauge steel structures, thermal insulation, thermal bridges, moisture, air tightness, design, energy, indoor climate, environment.
INTRODUCTION In recent years we have seen that light gauge steel structures have been gaining market shares in building types, that traditionally were reserved for masonry or wood frame constructions, such as residential buildings. The light gauge steel structural technology was originally introduced to the market through industrial buildings with somewhat lower performance specifications. This new technology has therefor had to overcome barriers of technical shortcomings and attitudes that has been a result of this historical inheritage. Typical consequences of such an inheritage is, that many companies that promote
690 and produce such constructions, are unproportionally interested in the load bearing properties of their constructions, but paying much less attention to properties such as thermal insulation, air tightness and moisture production. This often goes hand in hand with the attitudes of the buyer or end user buildings that steel constructions automatically lead to some industrial standard in design and performance. If we however look at the recent development in insulated light gauge steel constructions we see that new building systems and components have occurred that make it possible to produce constructions with good thermal insulation in a rational and competitive way. Looking at single projects however, we often come across anomalies, that have their origin in the fact that the people responsible for the design and construction have their competence within the field of structural mechanics but not in building physics. Here we also have to bear in mind that steel structures raise new and in some cases different questions than we have for other building materials. The material has a high thermal conductivity which means that severe and concentrated thermal bridges easily occur and the light gauge steel structure has in itself no air tightness. Steel is impermeable for vapour which in some cases can be beneficial but in other cases can be fatal. Traditional knowledge within this field has hitherto more often been developed considering traditional wood and masonry constructions. An important aspect is that the competitiveness of structural solutions can never be based on cost efficient structural properties only but also on the compatibility of the structure to other components and systems of the building such as thermal insulation, air tightness, moisture protection, extemal and internal cladding, heating and ventilation etc. Light gauge steel constructions have their largest potential in the dimensional stability the of material which makes it possible to cut and form parts with big accuracy. This in turns opens new possibilities for industrial design with 3D CAD and production of buildings with accurately precut or prefabricated parts. At the end of the day this can lead us to constructions which meet very high performance specifications on thermal insulation, air tightness and moisture protection as long that we can generate the basic solutions.
THERMAL INSULATION In the mid-seventies after the energy crisis the new Swedish building code demanded lower U-values even for industrial buildings. This lead to the development of new construction types to meet the new specified U-values. A very competitive construction, so far, had been the corrugated steel deck with rigid insulation and mechanically fastened roof membrane on top. With the demand for lower U-values the thickness of the relatively expensive rigid insulation had to be increased beyond what was considered to be economical. New constructions therefore appeared with a double steel deck, with steel girders between, making way for using less expensive low density thermal insulation. In some of the first constructions the thermal bridge effect of the girder connecting the two surfaces with high thermal conductivity resulted in extra heat flow that did far more than compensate the reduction in heat loss through the rest of the construction due to the increase in the thickness of the thermal insulation.
691
Figure 1: Different arrangements to reduce thermal bridge loss: a) Thermal break, b) Intermittent web, c) Crossed bars, d) Slotted web e) Insulated core beams There soon appeared different solutions as illustrated in Figure 1 to reduce this thermal bridge effect, of which some are listed below: Thermal break: A rigid material, usually wood or polymeric material was placed between the outer girder flange and the outer deck and fixed with a screw between the deck and the flange. Intermittent web: The girder was built up like a ladder with intermittent web connections Crossed bars: A new set of girders were attached perpendicular to the main girders. Slotted webs: Along the web, rows of longitudinal slots were dislocated in such a way that the length of the heat flow path through the web was increased radically. The equivalent thermal conductance of the web could be reduced to one tenth of the thermal conductivity for a solid web. Stainless steel: The thermal conductivity of stainless steel is only one third or one fourth of the thermal conductivity for normal galvanised cold formed steel. Insulated core beams: Two U-profiles around a core of rigid insulation, attached together with a minimum of mechanical connection between the flanges, forming a beam with a low thermal conductance.
Calculation methods The early calculation methods for non-homogenous constructions were less suitable to deal with constructions consisting of materials with high thermal conductivity such as steel and concrete. For wooden construction a commonly used calculation was the so called ^-value method that was based on
692 weighing the X-values of imhomogenous material layer by area. This method is supposed to give values on the safe side, but for sheet metal constructions very much so, proving to be quite useless. Modelling these constructions by using finite difference or finite difference methods was a possible way even though stability problems could sometimes occur using thin elements with high thermal conductivity. Projects where hot box measurements and numerical calculations have been carried out simultaneously have shown a quite acceptable correspondence between measurements and calculated values. Reproducing accurate measurements from different laboratories using different standards has however in some cases proved to be difficult. Constructions being analysed, had a corrugated metal sheet on both sides and steel girder in between, forming a thermal bridge penetrating the insulation. The first constructions of this type were rather clumsily designed and the resulting thermal bridge effect did more than to compensate the increased insulation thickness compared to the earlier constructions. Finite difference calculations showed that the extra heat flow through the thermal bridge followed well defined paths, that could be treated with standard solutions for simple geometry's. The heat flows, or rather the resistance's to heat flows, could be expressed with simple and more or less analytical expressions. Geometric configurations like the single sided linear or circular cooling fin, a simple rod, heat conduction between an embedded pipe and a surface and between a strip and an infinite surface were configurations that were sufficient to model such constructions. This lead on to a network of resistance's that could be solved with common methods for electrical networks.
Ext surface of corrugated sheet metal n T L T ^ ^ ^ ^ ^ T h e n n a l
Ext surf resistance
block
Sheet metal purlin Thermal insulation V_Z
Int surface of corrugated sheet metal
Figure 2. Transformation of a typical light gauge steel construction to a equivalent thermal network. This concept was developed into a hand calculation method for thermal bridges in sheet metal constructions, Johannesson and Aberg, and later Swedish standard, SS 02 42 30 and Norwegian Standard NS 3034, for calculation of the thermal resistance for sheet metal constructions with thermal bridges. A simple computer code based on this method GFSTEEL is available through the GuUfiber company in Sweden. The inaccuracy of the method is for many construction types lower than 5 % compared to a computer code. The main advantage is however that the user gets a clear view of how heat is transferred through the construction and where the weak points are from a thermal point of view. Also thermal bridges in sheet metal constructions are rather tedious to model with finite difference computer programs and three dimensional configurations like single mechanical fasteners can be more accurately modelled with
693 the network method. In other cases details like slotted girder webs can be calculated by the computer or measured in laboratory to get reliable input into the network method.
The modified X-value method. For decades the methods taught to the students and used by most practitioners has been to either weigh the U-values of different construction parts by areas which always gives a too low U-value or to weight the different thermal conductivity's of inhomogeneous material layers by area which always gives too high U-values. The inaccuracy of these methods has been rather high. For timber framed walls the inaccuracy of the calculated thermal bridge effect can be of the order of magnitude 10 % but for masonry structures and sheet metal constructions 50 - 100 %. In a paper by Mao and Johannesson, it is shown that a slight modification of these methods can give very accurate values for a big variety of insulated constructions. The solution was to use the lateral heat transfer properties of the surface layers to limit the possible influence zone for the thermal bridge. When the influence zone has been decided the method of weighed thermal conductivities by area for inhomogeneous layers is applied for the heat flow within the zone. Then the U-values for the zone can be weighed by area for the influence zone and the rest of the constructions. The inaccuracy of this method has to be found to be rather promising, 1 to 5 % for the thermal bridge effect depending on the type of construction. It is also interesting that this method can be developed further to even include constructions with partition joints and three dimensional constructions.
Design aspects For normal building systems, solutions for the performance of the basic configurations has usually either been tested in the laboratory or calculated by reliable calculation methods. However when it comes to joints the design often becomes individual and situations occur giving rise to unexpected thermal bridges. The slotted web can have thermal properties that are equivalent to a wooden stud, but only as long as its obstructs heat flow perpendicular to the slots. This can be exemplified with unfortimate use of the slotted web girder. The girder protrudes the insulated shell as for instance at eaves. In this case the inner flange acts as a direct thermal bridge between the inside and the outside. The window sills in an insulated wall are formed with U profiles with a slotted web covered with some board. The slotted web is in this case of limited use especially if the window frame is placed towards the outer wall surface. Two girder webs are placed against each other or a slotted web is placed against a material with higher thermal conductivity. The thermal performance of a slotted web is only ensured if it has high performance insulation on both sides. In some cases the so called thermal break for a girder has been put between the inner flange and the inner cladding. From energy point of view this is quite equivalent to placing the thermal break on the outside. However since the metallic profile will be relatively cold and we normally have mechanical fasteners i.e. screws between the inner surface and the flange, this creates small but concentrated thermal bridges.
694 which will give rise to local temperature drops at the inner surface. This can in extreme cases lead to condensation and mould growth, but in most cases this will lead to uneven colouring of the surface over time. Experiments has shown the importance of the compatibility of the structural system and the thermal insulation for an optimum thermal performance of the building construction. Unintentional air cavities reduce the thermal resistance of the construction and can jeopardise the intended function of slotted webs and thermal breaks and locally give rise to severe thermal bridges. It is important that the insulation, in the practical construction work, can be fitted into the structure so that no unintended cavities occur.
MOISTURE PROBLEMS Within the field of building physics moisture signifies water, (H2O) in its different phases, vapour, liquid and ice, appearing in indoor and outdoor air, within the materials in building constructions and on their surfaces. Moisture problems involve situations where the moisture content in air, on surfaces or in materials, or the gradient in moisture content, exceeds the limits for normal biological, chemical and mechanical decay or affects the performance of a construction or the building in some other significant negative way. In the design process, the moisture related functional demands on a building construction are, that it should protect the interior environment fi-om moisture fi-om extemal sources of moisture such as rain, snow and ground water and also that the construction should protect itself against moisture problems and damages. Moisture is transferred in building materials and building constructions mainly by four processes, diffusion, air convection, capillary suction in fine pores, and flow of water due to external forces such as wind pressure and gravity. As long as we only have vapour at low relative humidities, moisture transfer takes place by diffusion and convection. At the state of saturation within materials and on surfaces or just below, capillary and gravity flows occur.
Diffusion By Picks law we can assume that water will be transferred by diffiision fi-om a location with higher concentration to a location with lower vapour concentration. Ideally we can assume that the section of air that can transfer vapour by diffiision in a porous material is reduced to the sum of the cross sections of the pores. This is however not that simple, since liquid water will form in the pores below saturation point and in fibrous materials the size of the pores can depend on the relative humidity. By a good approximation, the diffusion coefficient for vapour in a material can be related to the coefficient of diffiision in air by a constant ratio describing the resistance of the material to water vapour flow. In a cold climate wintertime the vapour concentration on the inside of the construction is normally by far exceeding the vapour concentration on the outside. By Picks law we will therefore have a diffiision of water vapour from the inside to the outside. Normally we therefore try to place material layers with high vapour resistance such as a plastic foil towards the interior surface and use material layers with relatively low vapour resistance towards the exterior surface to keep the relative humidity in the construction below critical limits.
695 Metal sheets have practically infinite resistance which means that when used as exterior cladding they are normally ventilated on the inner side with outside air. In insulated light gauge structure the structure normally covers not more that 10 to 15 % of the construction area which means that the moisture balance in the construction is govemed by other material layers. For certain type of profiles and at joints, cavities can occur which are enclosed to the outside where on can risk higher relative humidities and eventually condensation. Also, as mentioned above, thermal bridges can lead to low temperatures on the inside of the vapour barrier with high relative humidities, mould growth and uneven colouring of the surface. An special case is the double sheet sandwich construction with an expanded rigid foam core where no significant diffusion is supposed to take place through the construction. Experience has shown that when problems occur in such construction it is usually due to insufficient performance of the joints combined with deformation of the elements over time.
Capillary moisture in constructions We have mainly three ways of storing water within materials. At low relative humidities the air in the pores contains some water vapour. Due to forces between the water molecules and the material in the pore walls, a layer of water molecules is adhered to the wall. The bond to the next layer will be weaker with increasing distance so that the amount of moisture stored in this way is limited. When the partial pressure of the water vapour in the surrounding air increases, the adhesion forces at concave surfaces dominate the pressure difference between the saturated surface and the air in the pores. This leads to condensation at the concave surfaces even though the air in the pores is not saturated.
Adhesion forces
Circular pipe in water
Water molecule
Figure 3. Capillary forces in a pipe with a small diameter The molecule next to a concave surface will have a shorter mean distance to the surface and therefore the adhesion forces will be stronger than for a molecule at the same distance from a plane surface. We consider a single pore in a material idealised as a vertical pipe with a representative diameter. The lower end of the pipe is in contact with free water and a water gauge is drawn up into the pipe with capillary forces. The angle Q is dependent upon the properties of the liquid in the pipe and the material of the inner pipe surface. For water and most building materials Q can be assumed equal to 0. The water
696 molecules closest to the wall have a strong bond to the surface. The force that holds up the water gauge is then dependent on the surface tension along the perimeter of the pipe. This upward force can be expressed as JP„P = 2 - ; r r c r c o s 0
(1.)
and the pressure drop over the top surface of the water gauge as F
InracosQ K-r"
2crcos0
/' =Av =
(2.)
r
In sheet metal constructions exposed to intermittent water in form of a driving rain etc. two metallic surfaces with a small gap between them can in the same we keep water by capillary forces which in turn can give rise to local corrosion problems. If the distance between the two plates is b the upwards force per m can be expressed as i^„^ = 2 c j c o s 0
(3.)
And the pressure difference over the surface as _ F _ 2 cTcos© ^~A ~
(4.)
b
or, if a = 0.074 N/m, the height of a vertical water gauge can be expressed as ^
LSOr b
(5.)
When b is in the order of 1 mm or less a significant water gauge can build up. A logical consequence of this is that overlaps where the edges are at intervals exposed to water should either be sealed or the sheets should be kept at least some millimetres apart.
AIR TIGHTNESS AND AIR LEAKAGE IN CONSTRUCTIONS To generate air flow between to volumes we need a pressure difference and some kind of an opening where air can pass. The amount of flow will be given by the pressure difference, the geometry and the surface materials of the openings and the properties of air at the actual conditions. The pressure difference in buildings is usually built up by wind forces on the building, the mechanical ventilation system of the building or the density difference between warm inside air and the colder outdoor air. Often all of these three processes are significant at the same time. When the wind passes a building the air velocity around the building varies greatly around the building. On the windward surface the velocity is slowed down and the static pressure increases, at other part such as at the sides and parts of the roof the air speed is high and the static pressure then becomes low. The calculation of the air flow pattern around a building is difficult but in later years progress has been made in numerical modelling. The usual way of treating this problem until today has been to study the air flow and pressure patterns around small scale house models in a wind tunnel to establish, for a given
697 shape, a relation between the reference wind velocity and the pattern of static pressure on the different surfaces. This relation is given by a parameter called the form factor m relating the static pressure at a given location on the building surface to the static pressure po and velocity UQ in the free air flow at a distance from the building. p = po+|i(pUo^/2)
(6.)
Normally p, ranges between 1.0 and -2.0 Normally the surface area of a building with negative form factors is larger than the surface area with positive form factors and the resulting indoor form factor can be around -0.3. Since the indoor air velocities are small the indoor form factor can be assumed to be constant over the entire interior surfaces of the building constructions. The pressure difference over a building construction (Pa) at a given location can thus be calculated as Ap = (|Lies-lLiis)(pV/2)
(7.)
Pes is the form factor at the exterior surface Pis is the form factor at the interior surface
Thermalforces In wintertime the air temperature within a building is higher than outside. This temperature difference in turn will give rise to density differences. •Pressure difference Pi-Pe
H
T-
Reference level
Figur 4: Pressure difference due to the thermal stack effect.
The pressure difference between inside and outside over the building surface can then be expressed as a frmction of the height above the equal pressure level, and the in and outdoor temperatures as
698 (8.)
P r P e = - H 0.044 (Ti-Te)
The location of the reference level will be depending on the distribution of openings in the building. The thermally generated pressure difference over the building construction is of major importance v^hen assessing the risk for moisture problems in a building construction. If the pressure is higher on the inside, any air leakage will bring warm humid air into the colder part of the construction resulting in high moisture level or condensation. If the pressure is higher on the outside, air leakage will bring cold air into insulated parts of the construction resulting in cold spots or through the construction causing draught on the inside.
Mechanical ventilation In old buildings wind and thermal forces are solely used to create adequate ventilation. In most modem buildings mechanical ventilation has been installed to ensure the flows. We make a distinction between a balanced ventilation system where the flows for inlet air and exhaust air in the building are practically equal and an exhaust air ventilation system where air is being mechanically drawn out of the
® Balanced ventilation system Ap = 0
Exhaust ventilation system Ap = negative
Figure 5: Different ventilation systems influence the indoor pressure differently. building and is replaced by incoming air through various openings in the exterior surface. Pure air inlet systems are not common in Sweden because they would create a constant overpressure indoors which is not fortunate the building components in the long run as we will study later on. A fan in a ventilation system typically generates the required air flow at a pressure difference between 150 and 450 Pa. Even for an exhaust ventilation system most of this pressure gain is lost in the components and ductwork of the ventilation system and the pressure difference over the exterior surfaces is only a small part, typically 1-10 Pa.
The total pressure difference The total pressure difference over a building surface is more ofl;en created by more than one of the above processes at a time. The results are however additive so that the resulting pressure difference over a building construction at a given location can be found as a sum of the pressure differences calculated from wind, thermal forces and the ventilation system. Aptot - Apwind - APthermal " Apmech .vent
(9.)
699 The wind pressure is usually defined as positive fi-om the outside to the inside while the pressure generated by thermal forces and the ventilation system are defined as positive from the inside to the outside.
Airflow through building components The pressure difference as calculated above from various processes is the governing potential which generates air flow through leakage paths in the construction. The leakage paths are seldom desired or planned which implies a high degree of uncertainty concerning the geometry and other properties of the paths. Uncontrolled air leakage through building constructions is usually linked to unwanted consequences such as moisture problems, distribution of odours, energy loss and draught. Important exceptions are fresh air inlets and air gaps for ventilation of the exterior parts of constructions. The air leakage path through a construction is usually of a complex nature. As an example we have air entering from the inside through a crack in the drywall, through an overlap in the vapour barrier, through a porous insulation material to a crack in the wind barrier into the external ventilation gap and out
Figure 6: Possible air flow path through an exterior construction through the ventilation openings in the exterior cladding. The air tightness however is often provided for in one layer in the construction such as the vapour barrier, a concrete layer, a sheet metal deck or a wall layer of aerated concrete. The rest of the construction can then often be considered as non-airtight and the openings in the air tight layer can be studied as connecting directly the inside and the outside.
Design
aspects.
Light gauge steel structures have no natural air tightness like masonry structures. The air tightness is similar to timber frame structures usually provided by a UV resistant, 0.2 mm plastic foil behind the interior cladding. To gain acceptable air tightness the foil has to be applied in as large sheets as available with an overlap of 200 to 300 mm at the joints pressed, together between two rigid surfaces. In the design process a special care has to be taken to plan for continuity of the air tightness at element joints, window sills, structural joints and for penetrating elements such as beams, chimneys, ventilation ducts etc. The general rule is that if the method and means to provide this continuity are not properly specified in the construction drawings the result on the site will be poor.
700
At first it was feared that a plastic foil applied to light gauge steel constructions would be vulnerable to sharp edges and be cut open during the installation. Experience has shown that this does not have to be a problem even if it is advisable to be careful. The worst enemy is the same as for timberframed constructions, namely the electrician, who without mercy cuts open the foil where electrical sockets etc. are installed. For constructions with crossed bars it is possible to place the plastic foil on the inside of the outer frame and leave the inner frame open for installation work. A general rule of thumb is that no more than one third of the thermal resistance should be placed on the inside of the vapour barrier to avoid building up to high relative humidities within the construction. For light gauge steel structures this has to be studied extra carefully due to the large thermal bridge effects that can arise within the construction. In recent years an interesting development has been taking place where light gauge steel constructions have been developed to meet the specifications for partitions between dwellings with respect to fire, sound and vibrations. In this context it has to be emphasised that the light gauge constructions do not naturally gain the same air tightness as the masonry constructions which they are meant to replace. At the same time as the distribution of smell, odours and allergenic emissions is a highly observed factor in the healthy buildings discussion. It is therefore advisable to provide partitions between dwellings with the same standard of air tightness as exterior constructions.
ENVIRONMENTAL IMPACT The discussions on environmental impact of building constructions had fi-om the beginning an unfortunate focus on the material used in the construction and their properties. Many authors have now shown that for an insulated construction the major part of the impact is depending on the performance of the construction, namely thermal insulation, air tightness, durability and impact on the indoor climate. A construction that in the above aspects is performing well in a cost effective way is usually an environmentally feasible construction regardless of what materials are used for the construction. This is exemplified in Johaimesson and Levin 1998.
CONCLUSIONS Insulated light gauge steel construction have a considerable potential in the construction work that normally has been reserved for traditional timber and masonry structures. The focus of the builder and end consumer is at for now shifting over from the building phase to the long term operation of the building. This implies energy use, moisture safety, indoor health, durability and maintenance cost. To succeed on this market the suppliers of materials and products have to maintain competence within this field and provide architects and constructors with reliable solutionsfi-omthe above aspects. It is from what we know and have experienced today no difficult to build light gauge steel constructions that meet the above specifications as long as building physics is implemented in the construction process from the early stage.
701 REFERENCES Bu0 F.H. Lund E. Ensrud E. and Birkeland O. Kuldebroer, energibesparing, byggskader. (Thermal Bridges, Energy Conservation and Building Damages). Norges Byggforsningsinstitut, Working report no. 36, Oslo 1981. Johannesson G. Aberg O. Koldbryggor i Platkonstruktioner. (Thermal bridges in Sheet metal constructions). LTH Building Technology, Lund 1981. Johannesson G. Lectures on Building Physics. Lecture notes. KTH Building Technology, Stockholm 1998. Johannesson G. Levin P. Building Physics - No Way Around it. Porceedings of the 1998 CIB World Building Congrss, Gavle, Sweden 7-12/6 1998. Mao G. Thermal bridges. Efficient Models For Energy Analysis in Buildings, Dissertation, KTH Building Technology, Stockholm 1997. Nevander L.E. and Elmarsson B. Fukthandboken. (Handbook on Moisture) Svensk Byggtjanst 1997. Nylund, PO. Rakna med luftlackningen. Samspel byggnad - ventilation. (Infiltration Counts Interaction Building - Ventilation). Report Rl, The Swedish Council for Building Research, Stockholm 1984. Olsen L. Johannesson G. Vejledning i Beregning av Kuldebroer, (Guidelines for Thermal Bridge Calculations). NKB Report 1996:10. Monilsa Oy, Helsinki 1996. Staelens P. Thermal Bridges, A Non-Computerized Calculation Procedure. LTH Building Technology, Lund 1981.
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
703
LONG-TERM PERFORMANCE OF LIGHTGAUGE STEEL -FRAMED ENVELOPE STRUCTURES J. Nieminen and M. Salonvaara VTT Building Technology P.O.BOX 1804 FIN-02044 VTT Finland
ABSTRACT Hygrothermal performance of a new light-gauge steel framed building envelope system has been analyzed using laboratory testing in a calibrated and guarded hot box (ISO 8990), weather resistance tests for full-sized structures, corrosion tests, field measurements at experimental buildings in Ylojarvi, Central Finland and.3-D thermal simulations, 2-D combined heat, air and moisture transfer simulations. The results show that a modern steel wall structure based on perforated steel studs performs satisfactorily in the cold climate of Finland. The perforations reduce significantly heat loss along the web of the profile. The field measurements show that no condensation has occurred in the fi*ame system. Temperature measurements and infi-ared surveys in demonstration buildings show that temperatures on the inner surface of the wall are sufficiently high to prevent surface condensation or even increased surface humidity that could cause mold growth on the surface. According to the calculations, there are no corrosion risks in the steel fi-ames during their estimated service life in the Finnish climate. The climate, however, has an important effect on the performance, and the structures should be designed with regard to cUmatic conditions.
KEY WORDS Steel structures, heat transfer, thermal performance, durability, moisture, corrosion
INTRODUCTION Hygrothermal performance and durability of a steel frame systems used as a fi*ame structure in exterior walls has been studied in the following major research projects carried out in the 1990's in Finland: - Development of light prefabricated steelfi-amedfacade units: Development and hygrothermal testing of new Ught gauge steel frame systems
704 - Development 3-D heat transfer tools for steel structures: numerical 3-D tool for calculation of temperature and heat flow distribution in steel structures - Long-term durability of steel structures: service life assessment methods for various steel structures, field surveys of steel buildings, long-term corrosion tests - Performance of steel buildings: development of tools for hygrothermal calculations, measuring devices and methods - ECSC Mega5 project Application of steel in urban habitats: Low-energy steel house for a cold climate: development, testing and demonstration of a new steel frame system, concept for energyefficient residential housing based on steel frame structures. The overall aim the research has been to develop steel based components to be used in housing, and to verify their performance and durability. A high status has been given to thermal performance of structures in the building envelope THERMAL PERFORMANCE OF LIGHT-GAUGE STEEL FRAMED STRUCTURES Thermal bridges Just like all frame materials in an insulated structure, a steel member is a thermal bridge. But, since thermal conductivity of steel is high, severe bridge effects are possible. The effect of thermal bridges has been reduced by three methods (Figure 1): • using double frame systems • using exterior insulation systems • using perforated (or slotted) the steel profiles (thermoprofiles)
Figure 1. Cross-sections of a single and double framed wall structure used as exterior wall in Finland and perforation alternatives of the web of a typical thermoprofile. Light-gauge steel framed structures based on double frame system (horizontal and vertical frame. Figure 1) have been used as external walls of office and public buildings. The distance between the frames in both directions is typically 0.6 meters, and thermal insulation is installed into the cavities between the frames. Double frame system improves thermal quality of a wall by 20 - 25 % compared to single frame cavity insulated wall, Table 1.
705
Exterior insulation systems reduce thermal bridging in a wall. The effect depends on the thermal properties of the insulation system. Exterior insulation is very advantageous in terms of moisture behavior, since the frame temperatures increase which in tum reduces moisture risks in the frame. Perforated webs in a light-gauge steel profile give two advantages for the structure. Due to the perforations, the thermal properties of the structure are improved. This, in turn, makes it feasible to use the structure as a single-fi'ame wall system. A light-gauge steelfi"amewith perforated web is termed a thermoprofile. U-shaped thermoprofiles are used in prefabricated facade units for high-rise buildings. The load bearing walls of detached and row houses are composed of vertical C-shaped thermoprofiles. The material thickness of the profiles is typically 1.0 - 1.5 mm. The effect of the perforations or slots in a thermoprofile can be taken into account by introducing an equivalent thermal conductivity for the non-perforated material. Heat transfer in the web can be assumed to be pure heat conduction. The equivalent thermal conductivity can be defined by comparing conduction in the perforated case and in the non-perforated case. The equivalent thermal conductivity depends on the perforation system including the shape and dimensions of perforations, the dimensions of the steel necks between the perforations and the thermal conductivity of the material in the perforations, see Figure 2. TABLE 1 ESTIMATED RELATIVE THERMAL RESISTANCE OF INSULATED STEEL FRAMED WALL COMPARED TO ONE DIMENSIONAL RESISTANCE OF THERMAL INSULATION OF CORRESPONDING THICKNESS. ALL INSULATION MATERIALS MINERAL WOOL, THERMAL CONDUCTIVITY 0 . 0 3 7 W / M K .
Wall structure Mineral wool insulation 175 mm Singlefi-amewall: verticalft-ameC 175 - 50 - 1.2 Double frame wall: vertical frame C 125 - 50 - 1.2 and horizontal frame Z 50 - 50 - 1.0 Single frame wall -i- exterior insulation 125 + 50 mm, vertical frame C 125 - 50 -1.2 Singlefi-amewaU: vertical thermoprofiles C 175 - 50 - 1.2
0
Relative thermal resistance, %
100 45 65
75 80
1 2 3 4 THERMAL CONDUCTIVITY OF INSULATION [W/mK]
5
Figure 2. The effect of thermal insulation on the equivalent thermal conductivity of the typical Finnish thermoprofile.
706 The reduced thermal bridging has a considerable effect on reducing the heat conduction in the thermoprofile relative to solid steel. The perforations perform as thermal breaks for the steel member reducing the heat conduction along the web, Figure 3. According to studies carried out in Finland and Sweden /ref. 1 - 7/, the equivalent thermal conductivity of a thermoprofile can be 5 - 10 W/mK, whereas the thermal conductivity of galvanized steel is 55 W/mK.
0,5 4
STEEL FRAME
0,45
0,35
0,2 0,15 -I 125
2,0 mm 1,5 mm - - 1,2 mm 2,0 mm 1,5 mm •" " 1,2 mm " • ^ ^ 1 , 0 mm 45 mm
THERMOMEASUREMENT: ISO8990
WOODEN FRAME 45 * 150...250 150
175
200
225
INSULATION THICKNESS [mm]
Figure 3. Calculated values of thermal transmittance for steel-framed and wood-framed walls. Air tightness of steel buildings Air tightness of steel building has been tested only in 4 cases. Table 3 shows results from measured cases and reference information from typical Finnish buildings. The locations of air leaks in the steel houses were searched using an infra-red camera. The most of the air leaks were caused by insufficient sealing of electrical and ventilation installations leading through the air/vapor barrier of the envelope. These defects were found systematically in aU of the buildings. TABLE 3 A m TIGHTNESS OF STEEL HOUSES AND TYPICAL FINNISH BUILDINGS.
Building type Prefabricated steel houses (row houses) Site built steel house (row house) Site built wooden detached houses Prefabricated wooden detached houses Prefabricated wooden row houses Massive wooden houses (log houses) 1 Concrete buildings
Air leakage rate at 50 Pa, 1 air changes per hour 1,9-2,5 3 3-4 2-4 3-5 7-15 1-5
Temperature distribution in light gauge steel walls (thermoprofiles) Temperature distributions in the single frame structures have been measured both in a series of full scale laboratory weather tests and in the structures of Ylojarvi steel houses. The temperatures on inner
707 wall surface are sufficiently high to prevent ghosting, surface condensation or even relative humidity high enough to increase the risk of mold growth on the wall surface, Figure 4. Temperature on the inner surface of the wall on top of the frame is 1 - 2 °C lower than the temperature between the frames.
Figure 4. Infra-red image of a steel wall. The temperature of the outer flange of the steel frame depend on the thermal properties of the wall outside the frame, figure5. The outer flange of the profile is considerably warmer than outdoor air temperature due to heat conduction from the interior along the web of the steel profile. Even though the perforations in a thermoprofile reduce heat conduction along the frame, the residual conduction increases temperatures in the outer parts of the frame, thus reducing the condensation risk and increasing the drying potential in case of condensation.
10 O LU
a:
WIND PROOF MINERAL WOOL 100 mm
5+
WIND PROOF MINERAL WOOL 50 mm GYPSUM BOARD 9 mm
a: LU Q. LU
0,5
1
1,5
2
2,5
THERMAL RESISTANCE OUTSIDE THE FRAME [m^k/W] Figure 5. The effect of thermal properties of wind proofing on the temperature of outer parts of the frame.
708 DURABILITY OF LIGHT-GAUGE STEEL FRAMED WALLS Moisture variation in steel framed walls The corrosion of a metal depend on the micro climate on the surface of a component. Continuous corrosion is possible, if relative humidity on the metal surface exceeds 80% at the same time as temperature is above 0°C (ISO 9223). Figure? shows the monthly maximum values of relative humidity of eight measuring points in the outer flange of the steel profiles measured in the Ylojarvi steel houses. The results show that relative humidity in wall 1 has exceeded 80%, but no condensation has occurred. The humidity in wall 2 has not exceeded 80%. confidence of the results. Y L O J A R V I STEEL HOUSES MEASURED MONTHLY RH MAXIMUM IN EXTE RN AL W ALLS
100 <
2
4
M O N T H 1996 - 1997
Figure 7. Monthly maximum of relative humidity on the outer flange of the steel members in the external walls of the Ylojarvi steel buildings (ECSC 1998). Corrosion risks caused by materials in contact with light gauge steel Corrosion risks caused by other building materials in contact with steel are studied in an on-going longterm laboratory test. Insulated steel frames are placed in different climates to see the effect of material and air humidity on corrosion. The materials being studied are cellulose fiber insulation, glass wool insulation, rock wool insulation and impregnated wood. The laboratory test has been going-on for about 2,5 years (20000 hours of time of wetness). Test conditions are given in figure 8. The results show, that the cellulose insulation and impregnated wood promotes zinc corrosion in humid conditions. There are no signs of corrosion in the test pieces insulated with mineral wool products. In the case with continuous condensation, edge corrosion of steel was found in all the specimens. The fire retardant chemicals (borax and boric acid) of the cellulose fiber insulation were not stagnant. The chemicals re-crystallized on the steel surface, which caused stronger edge corrosion in the test specimens insulated with cellulose fiber insulation compared to other specimens.
709
Figure 8. Test climates in corrosion tests. Normal indoor conditions are being used as reference (top). Corrosion tests are carried out in humid air (middle) and in conditions where condensation occurs continuously (bottom). Hygrothermal simulations The hours of wetness were also calculated (Salonvaara & Nieminen 1998) for a single frame wall. The 2-D heat, air and moisture transfer simulation program LATENITE (Salonvaara et al., 1994) was used in the simulations: • Case A: single frame wall (thermoprofile) without exterior insulation and • Case B: single frame wall (thermoprofile) with exterior insulation, additional layer of exterior insulation of 50 mm rigid wind proof mineral wool. The hourly climates of Helsinki, Finland and St.Hubert, Belgium were used as a starting point. The orientation of the walls was north which is considered to be the worst orientation in terms of hygrothermal performance due to low solar radiation absorption. Wind-driven rain was not taken into account in the simulations and the walls were assumed to have a cladding with good cavity ventilation behind the cladding. The initial conditions of the material layers were +20°C and 50% relative humidity. The indoor air temperature +22°C or outdoor air temperature if higher than +22°C was assumed. Indoor air moisture content Xin was outdoor air moisture content Xout + 3 g/kg, but limited to 30% < relative humidity < 80%. The simulations were carried out for a two-year period starting September 1. According to the results, the durability of the walls depends mainly on the outdoor climate and the hygrothermal properties of the wind proofing attached on the outside of the profiles. The hygroscopicity of the gypsum board is fairly low, but when moistened by the outdoor air, it dries out rather slowly. This phenomenon increases the time of wetness on the outer surface of the outer flange of a profile. For the case A structure in the climates of Helsinki and St. Hubert the hours of time of wetness were about 2500 and 6000 in the most critical point of the structure. For the case B structure, the calculations resulted in almost non-existent hours of time of wetness in both climates.
710 CONCLUSIONS Using light-gauge steel frames in residential and commercial construction is rather new in Finland, and there is not yet long-term feedback from realized buildings. However, the analysis carried out so far shows that the new application of perforated light gauge steel frames fulfills the requirements set for performance, durability and further more for components of energy efficient buildings. The thermal properties of a thermoprofile are comparable to other solutions of load-bearing frame structures. The durability and the performance of the structures make them suitable for use in a cold climate. The results from the demonstration project in Ylojarvi, including comprehensive laboratory testing and numerical simulation of the structures show, that there are no major moisture or corrosion risks involved with the structures. Thus the structures can be expected to have a long service life.
REFERENCES Mao,G. 1997. Thermal bridges. Efficient models for energy analysis in buildings. Dissertation. Division of Building Technology, Department of Building Sciences, Kungliga Tekniska Hogskolan, Stockholm. Bulletin No 173. ISSN 0346-5918. Nieminen J.; M. Saari; P. Salmi; and K. Tattari. 1997. Low energy steel house for a cold climate. Proceedings of the International Conference Cold Climate HVAC'97. Reykjavik, Iceland May 1-2, 1997. Iceland Heating, Ventilating and Sanitary Association ICEVAC. Nieminen, J.; Kouhia I.;. Johanneson, G; and Mao, G.. 1995. Design and thermal performance of insulated sheet metal structures. Proceedings of the Nordic Steel Construction Conference'95. Malmo, Sweden, June 19 - 21, 1995. Nieminen, J. 1997. New well insulated light-gauge steel framed wall structure. Proceedings of the 5* International Conference Modem Building Materials, Structures and Techniques. Vilnius, Lithuania May 24 - 26, 1997. Vilnius Geminida Technical University. ECSC 1998. Application of steel in urban habitats. Low-energy steel house for a cold climate. Project n:o 7210-S A/902 - 95 - F6.1. Final Report. Rautaruukki Oy. Salonvaara, M., Nieminen, J. (1998) Hygrothermal performance of a new light-gauge steel-framed envelope system. Joumal of Thermal Envelope & Building Science 22:1998. P 167 Blomberg, T. 1996Heat conduction in two and three dimensions. Computer modeling of building physics appHcations. Lund University. Report TVBH-1008. ISO 9223. 1992. International standard. Corrosion of metals and alloys. Corrosivity of atmospheres. Classification. International Organization for Standardization. Salonvaara M. and A. Karagiozis. 1994. Moisture transport in building envelopes using an approximate factorization solution method. In: Proceedings of the Second Annual Conference of the CFD Society of Canada (Ed. James J. Gottlieb and C. Ross Ethier), Toronto, June 1-3 1994.
711 Detailing, Connections Joining of the sections can be realized by shot nails, screws, bolts etc. The execution of typical joining details such as ridge, dormer, verge, jamb, etc. is as easy as with timber. The amount of cutting work on site is reduced when compared to timber.
REAL EXAMPLES FOR ROOF STRUCTURES In figure 11 a photo is given showing the model of the roof structure made of steel which is in service for expositions etc.
Figure 11: model already exhibited at several expositions
Figure 12: Plan of the pilot project
712 After having trained the craftsmen with this model a real roof structure made of steel for a single family house has been erected in Neuss, Germany,, see figures 12 and 13. Measurements on temperature behaviour and humidity control show no complications.
Figure 13: Realized house with steel rafters in Neuss, Germany
REFERENCES III
Moisio M. (1997), Steel House in a cold Climate, presented at ^97 Seoul International Seminar on LGSF Housing in Soeul, Korea
111
Nieminen J., Saari M., Salmi P., Tattari K. (1997), Low Energy Steel House, presented at Cold Climate HVAC ^97 in Reykjavik, Iceland
ly
Feldmann M., Heinemeyer C , Sedlacek G., Stangenberg H., Weynand K. (1998), Dachstuhl mit Sparren aus oberflachenveredeltem Stahl-Feinblech, research report P391 for Studiengesellschaft fiir Stahlanwendung, Diisseldorf, Germany
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
713
SUBSTITUTION OF TIMBER BY STEEL FOR ROOF STRUCTURES OF SINGLE- FAMILY HOMES M. Feldmann^ C. Heinemeyer' and G. Sedlacek* ^ Lehrstuhl fur Stahlbau, Rheinisch-Westfalische Technische Hochschule Aachen, D-52056 Aachen, Germany
ABSTRACT In Germany almost all roof structures of residential buildings are made of timber. In order to look for opportunities for a successful use of steel in housing a study was initiated to develop a steel roof structure for single family homes. The idea was to substitute the timber-rafters by modem and light weight cold-formed steel profiles and to develop a new steel roof section which fulfills the requirements for thermal and moisture insulation and maintains the classical techniques and sequences of the various trades involved in the building. In the course of the study a full-scale model of the steel rafter was assembled. With the experiences gainedfi'omthis model the execution of a roof for a single family home was planned and realized. These pilot projects showed substantial technological advantages as statical efficiency, good insulation properties, very easy handling and quick erection by the light weight and a high potential of prefabrication. These features together with the possibility to maintain the traditional way of laying the roof cladding on the new steel rafters attracted the interest of craftsmen due to their lightweight. The new solution is no obstacle for other trades.
KEYWORDS steel rafter, cold-formed steel profile, roof structure, roof constmction, roof cladding, erection sequence, thermal insulation, moisture insulation
INTRODUCTION Masonry, concrete and timber have been the dominant construction materials in housing in central Europe for a long time. The strong demands for energy savings, temperature- and moisture- protection, acoustic insulation and durability yielded into a high performance level in housing in Germany, that is reflected by various national codes, e.g DIN 4108 (1997). Solutions with timber and masonry were developed that allftilfillthe tight requirements of such codes. However steel solutions complying with these requirements so far were missing.
714
Figure 1: Substitution of timber by steel This has various reasons: In North America the housing standards are different. While in Germany housing is traditionally realized by brickwork in North American construction techniques using posts, rails and claddings are popular. So when talking of introducing steel in housing in our region we must realize that the use of steel in housing would not only mean the substitution of any material by steel, but also a change of the most popular way of construction in which all traditional trades are involved. There are various projects throughout Europe that were initiated to develop new integrated element systems for walls and floors with steel as structural material. These concepts aim at full and integral solutions considering all functional aspects of modem housing. However the project that is described here does not aim at such drastic solutions, it only looks for the substitution of timber beams by steel for roofing with the particular aim to keep all traditional trades in the works, seefigure1. The roof structure was found to be very convenient because the substitution of timber by steel can be easily justified by gain in erection time, static efficiency, energy savings and no great changes in the design works.
OBJECTIVE OF THE STUDY AND PROCEDURE The objective of the study was to develop a roof with steel rafters, moisture insulation, natural ventilation and thermal insulation. The cross-section for the rafters shall be adequate to fulfill modem requirements for insulation and sustainability. The roof cladding with tiles should be as for conventional roofs, the materials used should be commonly available. The connections of the steel rafters should be designed such that they can be executed by carpenters without problems. Within this study a full-scale model was produced as a prototype that is also a demonstration model for expositions etc. a roof for a single family house was designed and built.
715 REALISATION OF A NEW ROOF STRUCTURE MADE OF STEEL The main feature of the design was the substitution of the timber rafters by cold formed steel profiles. Two requirements had to be respected: a modem roof cross-section and no impairment for the craftsmen like the roofer.
»Y Y Y V Y -n
Roof cladding with tiles Lathing Rafter lathing Protection foil Ventilation layer '"T~^'^^~~ Additional rafter lathing •
xA
A
AWWYVVV
C-Profile Themnal insulation between rafters Themnal insulation underneath rafters Difftision-banier Inner cladding
Figure 2: composition of the roof To reduce the depth a cross-section with integral thermal insulation in between the rafters was chosen, see figure 2 and figure 3. Other arrangements with thermal insulation on or underneath the rafters are also possible. The cross-section was composed of: roof cladding with tiles (naturally ventilated) lathing rafter lathing protection foil additional rafter lathing that is fixed to the steel profile in advance and that is necessary if the roofer uses conventional nails to fix further lathing. If shot nails or screws are used this additional lathing is not necessary. ventilation layer (alternatively without ventilation layer but with moisture diffusible foil) 15 cm thermal insulation in between the rafters and 5cm thermal insulation underneath the lower flange of the profile to prevent temperature bridges and associated humidity problems. diffusion-barrier inner cladding (e.g. gipsum plates) In principle other sections with an additional timber shell (for extra requirements) and other claddings with a metal sheeting, slate, etc. are possible. For the cross-section with integrated insulation all usual details can be designed, see figure 4, figure 5 and figure 6. The depth of the steel profile results from the required insulation depth and is in the range of h = 120 mm to h = 150 mm. The spacing of the rafters is of about 0.60 cm corresponding to the width of insulation mats. Taking these conditions rafters made of steel up to 7.50 m length can be used. The thicknesses t of the profiles vary from t = 0.8 mm to t = 1.5 mm. The weight w of the steel rafters are in the range of w = 2.0 kg/m tow = 3.0 kg/m.
716 '^
Tile LatNiing_^
'\
>w
1
iti
Rafter / Lathing 77
^
'
•
^
/^^^V->^^ 1/ 1
Protection foil Additional rafteij lathing -C-Profile steel rafter
^~-
Figure 3: Roof components
./
.€! y^A^
^-^
Figure 2: Detail of an eaves
x
*V^/^)^NN, A^ X > ^
^^^
^^^CFigure 5: Detail of a ridge
Figure 6: Detail of a gable
In the study C-shaped profiles were chosen. Other geometries for the profiles for the steel-rafters would also be possible. Hereafter a list of suitable profiles is given, see figure 7: C-section hat-section with perforated upper flange or webs to allow for moisture diffusion rectangular hollow section I-section individual section that allows for joint detailing with gusset plates such as given in figure 7. hi general profiles with upper flanges supported by webs on both sides are to be prefered for reasons of a better lathing application.
I C-section
Hat-section
Rectangular Hollow section
I-section
Figure 7: Possible cross-sections
V
Individual section
717 TECHNICAL CONCLUSIONS Properties Unlike timber or concrete steel has no shrinkage or creep. Therefore there is no time dependent deformation, no twisting and no cracking. Tolerances can be tightened up so that geometry requirements from fixtures can be better fulfilled. Steel leads to a strong reduction of weight of the roof structure with economical benefits. By using modem producing processes steel is an environmental-friendly material. It has the ability to be recycled by 100 %. Statical aspects Due to the mechanical properties steel has a far better efficiency than timber and therefore the steel-rafters are light. The constructional depth of the profiles is controlled by insulation requirements. Therefore for usual spans from 4,00 m to 6,50 m the allowable limits of the material strength and deflection are rarely exploited. Stability problems do not occur due to sufficient bracing by the lathing. Insulation aspects The thermal insulation was performed with glass-wool. Finite Element calculations showed that k-values with k = 0,20 W/(m^K) ace. to DIN 4108 are possible. Figure 8, 9, and 10 show the modelling and results of the FE-calculations. For integrated insulation it makes sense to care for an additional thermal insulation underneath the bottom flange (about 5.0 cm) to prevent temperature bridges. Other researches III 111 have shown that a perforation of the web will reduce the temperature transition.
Figure 8: relevant components of the insulation Moisture insulation will be achieved by a diffusion barrier on the inner side of the thermal insulation. Care must be taken for sufficient ventilation as water condensation occurs inside the thermal insulation. This can be achieved by diffusible protection planes on the outer side, ventilation layers and perforation of steel members at locations of potential humidity barriers.
718 T = -10C !
Heat Transfer 25 W/mK
i j
ThiHtTiaf InsuMon 8 = 0,04 W/mK
1 Steer a«4SW/mK
,
!
I
Heat Transfer 7,69 W/mK Cladding 8 = 0,21 W/mK
T = 20C Figure 9: Heat conductivity of the components For family homes in Gemiany no particular measures for fire protection measures are necessary. However the application of a gipsum plate on the inner side of the roof produces a fire protection class of F 30, If corrosive atmosphere occurs at all, zinc coating prevents corrosion also at locations where nails or screws were driven.
Temperatyre DIsWbytion
imm: T» aro« m 7M wmm
Figure 10: Results of the Numerical Analysis Handling As the steel profiles have less than half the weight of comparable rafters made of timber the handling on site is rather easy. The erection of roof structures of single family houses can be done without lifting devices. The project showed that the roof structures made of steel were easy to build and they did not provide any complication for the roofer.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
719
VIBRATION PERFORMANCE TESTS FOR LIGHT-WEIGHT STEEL-JOIST FLOORS J. Kullaa^ and A. Talja^ ^ Helsinki University of Technology, Department of Mechanical Engineering, P.O. Box 4100, FIN-02015 TKK, Finland ^ VTT Building Technology, P.O. Box 18071, FIN-02044 VTT, Finland
ABSTRACT An experimental study of vibrations of light-weight steel-joist floors due to walking excitation was performed. Three different floor-types were tested, each with spans of 7.0, 7.8 and 8.8 m. The dynamic properties of the structure were determined experimentally. The walking tests were planned according to the preliminary on-site modal analysis defining the walking rate and the measurement point in order to get the maximum response due to walking. Surprisingly, the first mode of the structure was a sine wave in the lateral direction instead of a half-sine wave (bubble mode), which was not found. The result was verified in walking tests, resulting in higher deflections at the quarter-point than at the midpoint. The acceptance of vibrations due to walking was studied by subjective evaluation, which was based on body perceptions and on noise and the movements of various objects. The results of the calculation and testing methods were compared with the subjective opinions.
KEY WORDS Vibration, floors, walking, modal analysis, vibration criteria, natural frequency, damping
INTRODUCTION The acceptability of floors from the point of view of vibration depends on three main factors: floor characteristics, type of excitation acting on the floor, and the acceptable vibration limits (Rainer 1980). Floors in office or residential buildings are subject to dynamic loads induced by people walking on them. Floors with a fundamental natural frequency below 7 to 8 Hz have been observed to be prone to annoying walking vibrations (Bachmann et al. 1995). Walking rates range from 1.6 to 2.4 steps per second. If the forcing function due to walking is described in terms of Fourier components with frequencies that are multiples of the walking rate, it can be deduced that the components up to the third or fourth harmonic can be significant. Therefore, floors with the fundamental natural frequency of below 7.2 or 9.6 Hz may be susceptible to annoying vibration amplitudes. In this study the dynamic characteristics of the structure were determined with experimental modal analysis (Ewins 1989). Thereafter, the response to walking excitation was measured in order to capture
720
the maximum amplitudes due to walking. Subjective opinions were recorded and compared to different acceptability criteria. A more detailed description of the research can be found elsewhere (Talja 1997, Kullaa & Talja 1998, Talja & Kullaa 1998).
STRUCTURE The test series consisted of three different floor-types VPl, VP2 and VPS (Figure 1). The basic floor was type VPl, which was relatively light (83 kg/m^) and had a very low transverse stiffness. The joists were made from two cold-formed C-sections 250x71x63x21.5x2 placed back to back. The spacing of the beams was 400 mm. Floor-types VP2 and VP3 were assembled by changing the top layer of VPl. The mass of floor-type VP2 was 110 k g W and the ratio of longitudinal and transverse stiffness was about 1/400, which is roughly 10 times that of VPl. For floor-type VP3, the mass was 170 kg/m^ and the stiffness ratio was 1/60. The width of the floor was 8.3 m in all tests and the spans were 7.0, 7.8 and 8.8 m. The edges of the floors were simply supported and the floors were unfurnished.
9.y\AAAAAA/\AAA/\/\/\AAAAAA/: I T I
1 VPl ilimiUKiL 76f^
3^!/ \ | / \ /
iiinnniuin
30.8 mm 1 Two-layer gypsum board, type GL15 35 mm 2 Profile sheet RAN 35A/0,7 Zn, 2 screws in alternate 15 mm corrugations 250 mm 3 Plywood, glued tongue-and-groove joints, slefdriUing screws 0 6.3 cc 200 4 Load-bearing structure, C-profile C-250, sound insulation, mineralwool 50 mm 9 mm 25 mm 5 Wind barrier, gypsum board GTS 9 13 mm 6 Acoustic profile Z25 cc 400 7 Gypsum board - mass of the floor 110 kg/m^ - composite action between plywood and floor beam
\ T Q ^
VP2
iinnnnniuiiffffffi
I VP3 7"r
22 mm 1 Chipboard, glued tongue-and-groove joints 30 mm 2 Hard mineral wool OL-K 30 kN/m^ 3 Plywood, glued tongue-and-groove joints, self250 mm drilling screws 0 6.3 cc 200 4 Load-bearing structure, C-profile C-250, sound insulation mineralwool 50 mm 9 mm 5 Wind barrier, gypsum board GTS 9 25 mm 6 Acoustic profile Z25 cc 400 13 mm 7 Gypsum board - mass of the floor SSkg/m^ - composite action between plywood and floor beam
I
25...60 mm 1 Concrete K30-2 + mesh reinforcement #4 - 150 15 mm ^ ^^^^^ ^^^^^ ^ ^ ^ 35A/0,7 Zn, 2 additional screws 0 5.5 * 51 in altemate corrugations/floor beam, 250 nrni screw heads 15 mm uplifted 3 Plywood, glued tongue-and-groove joints, selfdriUing screws 0 6.3 cc 200 9 mm 4 Load-bearing structure, C-profile C-250, sound 25 mm insulation mineralwool 50 mm 13 mm 5 Wind barrier, gypsum board GTS 9 6 Acoustic profile Z25 cc 400 7 Gypsum board - mass of the floor ~ 170 kg/m^ - composite action between concrete, profile sheet and floor beams
Figure 1. Assembly of floor-types VPl, VP2 and VP3.
721 MODAL TESTING OF STRUCTURE Modal testing was performed on the floors using an impact hammer with a weight of 5 kg. The excitation and response were measured using a piezoelectric Kistler 9361A force transducer and five HBM B12/500 accelerometers attached to the top layer of the structure. The excitation (reference) point was determined so that no nodal point of the expected mode shapes coincided with the reference point. Fifteen measurement points (5x3) were chosen in order to identify the lowest 3 and 5 modes in the longitudinal and transverse directions, respectively. The frequency range of interest was 0 to 40 Hz. The signals were filtered with a 40 Hz low-pass filter, the sampling frequency being 400 Hz. Both the excitation and response were windowed with an exponential window. The measurement period was 10.5 s. In order to increase the frequency resolution, an adequate number of zeros was appended to the signal. The frequency resolution of 0.05 Hz was achieved. Every FRF was measured six times and averaged. The measurements were performed in a factory hall, which corresponded to almost ideal laboratory conditions. Therefore the coherence function was near unity. The modal parameters, natural frequencies, natural modes, damping ratios, and modal masses were determined with the modal analysis. The four lowest natural frequencies of all floors consisted of the half-sine in the joist direction, while in the transverse direction the modes varied from a full-sine to two and a half sine waves. The first two modes are shown in Figure 2. It was noticed that, surprisingly, the bubble mode did not exist. The bubble mode was found, however, in the VP3 floors with a concrete surface, but with an extremely high damping ratio. A high damping ratio for the lowest mode was also observed by Burstrand (1997). The reason for the non-existence of the bubble mode is not clear to the authors. Below 40 Hz, two longitudinal modes were found. The mesh in that direction was therefore sufficient. However, with the floors VPl, more than five natural modes in the transverse direction were found, implying that the mesh was not sufficient to distinguish all modes.
Figure 2. Two lowest natural modes. The modal parameters of the lowest modes are shown in Table 1 for different spans. The natural frequencies and modal masses increased from floor-type VPl to VP3. The frequencies decreased with an increasing span. The highest damping ratios were found in the VPl floors.
WALKING TESTS From the results of the modal analysis it was concluded that the position of the highest response amplitudes due to walking might not be the mid-point but rather the quarter-point in the transverse direction. In order to verify this, two walking routes were selected in the longitudinal direction, one along the centre-line and the other along the quarter-line. The test person, weighing 73 kg, walked the determined route in stockinged feet, at a rate controlled by a metronome. The dynamic deflections were measured with HBM B2 vibration transducers at the mid-point and quarter-point, respectively.
722 TABLE 1. LOWEST NATURAL FREQUENCIES/; AND CORRESPONDING DAMPING RATIOS ^/ AND MODAL MASSES MI FOR DIFFERENT FLOORS.
Floor-type VPl /
Mi
fi Hz
kg
1
fo" 1
U1 fo" 1
]! 1
fo" 1 2
[3
11.81 12.98
10.18 11.25
8.10 9.15 10.85
3.28 4.69
3.99 5.80
3.32 4.35 3.63
1210 850
1450 770
1400 940 980
Floor-type VP2 fi Hz span 7.0 m
Mi
kg
10.70 2.06 1490 1490 13.51 2.74 Span 8.8 m 1.25 2.01 1.26
H
^i
Mi 1
%
kg
11.97 13.38 15.90
1.10 1.84 2.45
1930 2000 2470
10.34 12.14 15.90
9.91 1.81 1.71
2110 1460 2970
8.12 9.91 14.00 19.84
11.93 1.31 1.65 1.34
2470 2700 2940
|
1650 12.09 1.70 14.78 2.33 1560 Span 7.8 m
8.76 11.78 16.10
Floor-type VP3 fi Hz
1630 2140 2180
|
|
3550 J
Moreover, to obtain the highest possible response, the walking rate was adjusted to the natural frequency of the structure. For some floors, two or three tests with different walking rates (1.6 to 2.4 1/s) were performed depending on the number of natural frequencies below 12 Hz (5* harmonic). The effects of the floor structure, span and walking rate on the maximum dynamic deflection are shown in Figure 3. It can be noticed that floor-type VPl gave the highest amplitudes, while the lowest were measured from VP3. With an increasing span the amplitude increased as expected. In addition, it can be seen that the response at the quarter-point was higher than that at the mid-point, which agrees with the results from the modal analysis. Moreover, the walking rate according to the lowest natural frequency gave a higher deflection at the quarter-point than the walking rate according to the second natural frequency. The opposite results were observed at the mid-point. The reason for this can be found by examining the corresponding natural mode shapes (Figure 2). In the lowest natural mode there is a node at the mid-point, while the maximum amplitude is found at the quarter-point. In the second mode the maximum occurs at the mid-point.
SUBJECTIVE PERCEPTIONS In addition to the measurements of dynamic properties, vibration magnitudes and human perceptions due to walking were also studied. Both the mid-point and the edge of the floor were considered. The walking person was a man weighting 73 kg, in stockinged feet, always passing the observation point from a distance of 0.6 m. The distance of the walking line from the edge was 1.2 m. The observations were: • body perception from a sitting position, • clinking of a coffee cup with spoon in cup and on saucer, • leaf movements of a 30-40 cm high pot plant (dracaena), • rippling of water in a glass bowl, and • chinking of a glass pane hung vertically from mirror hooks.
723
S
0.2
. I
VP1 7.0m f1
VP1 7.8m f1
VP1 7.8m f2
VP1 8.8m f1
VP1 8.8m f2
VP1 8.8m f3
iiiirmn VP2 7.0m
VP2 7.8m f1
VP2 8.8m
VP2 8.8m f2
VP3 7.0m fO
VP3 7.8m fO
VP3 7.8m f1
VP3 8.8m fO
VP3 8.8m f1
Floor type and walking rate
Figure 3. Maximum deflection amplitudes of different floors according to the position and the walking rate./o corresponds to the bubble mode frequency. The objects were set on a firm stand of height 1.2 m. Three persons gave their opinion of the observations. Additional feedback concerning body perceptions and the leaf movements of a pot plant in the middle of the floor was asked from people visiting the test site (6 opinions). Table 2 shows a summary of acceptability based on the criterion that the majority of the persons accepted the properties. The 50% criterion is quite optimistic but is justified by the fact that in a test situation people are probably more critical than in a residential one. The results indicate that the vibrations of objects at the supported edge parallel to the floor beams may be more critical than the body perception in the middle of the floor. TABLE 2. ACCEPTED FLOORS BASED ON BODY PERCEPTION ( B ) AND VIBRATION OF OBJECTS (A).
1 span 7.0 m 7.8 m 8.8 m
Floor-type VPl Middle Edge B A B A
Floor-type VP2 Middle Edge B A B A
Floor-type VP3 Middle Edge B A B A
V V V V
V
V V
V V
V V
V
COMPARISON OF MEASURED AND PREDICTED PROPERTIES Table 3 shows the results of the comparison between the predicted and measured properties. The properties were measured from the mid-point of the floors.
724 TABLE 3. CALCULATED AND MEASURED PROPERTIES OF TEST FLOORS FOR COMPARISON OF DIFFERENT DESIGN CRITERIA. SHADED ROWS SHOW THE FLOORS THAT WERE ACCEPTED BY THE MAJORITY. ^maxxalc
^,exp{mm)
Measured values
AISC
(mm) top 0.6 m pottoml ap^exn l o U 1 Gniax ^max Floor- / fo.exp surface offset |surface|(m/s ) factor 1 (m/s^) | (mm) | type (m) (Hz) 1.0 3.0 1.3 0.5 0.10 40 1.10 0.22 VPl 7.0 12 7.8 10 1.2 2.9 1.3 0.4 0.14 20 0.73 0.15 8.8 8 1.6 3.3 1.8 0.6 0.34 30 0.56 0.25 VP2 7.0 12 0.55 0.60 0.25 0.25 0.07 40 0.59 0.12 16 0.49 0.14 7.8 11 0.69 0.70 0.35 0.30 0.09 16 0.43 0.19 8.8 9 0.87 0.70 0.40 0.40 0.28 VP3 7.0 12 0.24 0.15 0,11 0.13 0.02 12 0.25 0.06 ^ o
1A
1A
A o
••/IS:
:a24^ 4.m
A tif 1 A A a
mm
Ohlsson (1988) sets the limit for fundamental frequency/o, static deflection bmax due to a 1 kN point load, and for a maximum velocity due to a 1 Ns impact load. The lowest frequency of the floor should be more than 8 Hz with the deflection less than 1.5 mm. The maximum velocity depends on the mass and damping of the floor. The method is widely used. It is also adopted in Eurocode for timber structures (ENV 1995-1-1, 1993) and with small modifications in the Australian standard (AS 3623, 1993). However, based on the subjective evaluation, Ohlsson's criterion is not satisfactory for the test floors. Onysko (1998) gives a criterion for static deflection only due to 1 kN, but the limit value depends on the span of the floor (Figure 4). The criterion agreed well with the subjective evaluation. According to the criterion, the allowed deflections due to 1 kN are 0.72, 0.68 and 0.64 mm for spans 7.0, 7.8 and 8.8 m, respectively.
Span (m) Figure 4. Onysko's deflection criterion. AISC/CISC criterion (1997) is based on the resonance between the fundamental frequency of the floor and harmonic multiples of walking frequency. The criterion is actually intended for heavy floors. The calculated peak acceleration ap, which strongly depends on the mass and damping of the floor, is compared to the limit value of 0.05 m/s^. To account for motion due to varying static deflection, an
725
additional criterion must be fulfilled: the maximum deflection under a concentrated load of 1 kN must be less than 1.0 mm, if the natural frequency is greater than 9 Hz. The AISC/CISC criterion agrees with the subjective evaluation, if lowest experimental natural frequency and measured damping ratio are used. Vibration measurements are generally considered in ISO 2631-1 (1985), ISO 2631-2 (1989) and ISO 2631-1.2 (1995): First the acceleration signal is filtered and weighted in one-third octave bands, and then the r.m.s. values of the weighted accelerations are computed. The influence of vibrations is assessed with an ISO factor, which is the ratio of the greatest weighted r.m.s. acceleration to the base value of 0.005 m/sl Samples including the maximum vibrations during a time period of 1.6 to 2 seconds were used. ISO factors of less than about 15 were usually accepted. However, the ISO factor does not correlate well with the subjective evaluation. Very often they indicate that longer spans are better than shorter spans, which disagrees with the subjective evaluations. The measured maximum dynamic displacement amplitude Umax agreed best with the subjective acceptability. The amplitudes of about 0.15 mm were accepted by the majority. The peak values of acceleration a^nox did not correlate so well with the subjective evaluation.
CONCLUSION Experimental modal analysis was used to identify the dynamic properties of the structure. These parameters can be used to evaluate and compare different structures and their susceptibility to vibrations. In this research, the results of the modal analysis were also used in the planning of the walking tests. The walking rate and measurement points were determined from the results of the modal analysis. Moreover, the modal analysis could explain why the deflections at the quarter-point were higher than those at the mid-point. The reason was that the lowest expected bubble mode did not exist, or that its damping ratio was very high. However, the reason for the non-existence of the bubble mode is not clear. All the floors could be classified as high-frequency floors, implying that no resonance response is expected. The walking rate was nevertheless seen to affect the maximum deflection. Comparisons of subjective opinions showed that the conditions proposed by Ohlsson (1988), which are given in many codes, may be unconservative for long-span steel-joist floors. The deflection criterion proposed by Onysko (1998) was more satisfactory. The AISC/CISC (1997) method, which depends strongly on mass and damping and which is actually not intended for high light-weight floors, also gave unexpectedly good results when the measured dynamic properties were used. However, because the evaluation of the acceptability of floor vibrations is a complex procedure, more consumer studies together with field measurements are needed. In a test situation people are more critical than in a residential one. In the case of light-weight floors, the effect of partitions and other non-structural components is stronger on floor properties than in the case of heavy floors. To understand better the effects of non-structural components, tests under different construction stages should be performed. The annoying vibrations of objects at the edges of the floors should also be considered.
ACKNOWLEDGEMENTS This article is based on the results of a research project that forms part of the Finnsteel Technology Programme and the VTT Steel Research Programme. The project was sponsored by the Technology Development Centre of Finland, Rautaruukki Oy, Gyproc Oy, Rannila Steel Oy and VTT Building Technology.
726 REFERENCES AISC/CISC. (1997). Steel design guide series 11. Floor vibrations due to human activity. American Institute of Steel Construction. 69 p. AS 3623. (1993;. Domestic metal framing. Standards Australia. ISBN 0-7262-831-6-9, 52 p. Bachmann, H. et al. (1995). Vibration problems in structures: Practical guidelines. 2"^ ed. Basel, Boston, Berlin. Birkhauser. 234 p. Burstrand, H. (1997). A study of human-induced vibrations in light weight floor structures. Swedish Institute of Steel Construction. Report 190:2. 25 p. + App. 12 p. ENV 1995-1-1. (1993). Eurocode 5: Design of timber structures. Part 1.1: General rules and rules for building. European Committee for Standardisation. 106 p. Ewins, D.J. (1989). Modal testing: Theory and practice. Taunton, Somerset, England. Research Studies Press Ltd. 269 p. ISO 2631-1. (1985). Evaluation of human exposure to whole-body vibration - Part 1: General requirements. International Standards Organisation. 17 p. ISO 2631-1.2. (1995). Mechanical vibration and shock - Evaluation of human exposure to whole-body vibration - Part 1: General requirements. Draft for voting. 33 p. ISO 2631-2. (1989). Evaluation of human exposure to whole-body vibration - Part 2: Continuous and shock-induced vibrations in buildings (1 to 80 Hz). International Standards Organisation. 18 p. Kullaa, J. & Talja, A. (1998). Vibration tests for light-weight steel joist floors — Dynamic properties and vibrations due to walking. Finnish R&D conference on steel structures (Terasrakenteiden tutkimus-jakehityspaivat), 25.-26.08.1998, Lappeenranta. Ohlsson, S. (1988). Springiness and human-induced floor vibrations. A design guide. Stockholm: Swedish Council for Building Research, 141 p. (Publication D12:1988.) ISBN 91-540-4901-6. Onysko, D. (1998). Development of design procedures for vibration controlled spans using engineered wood members (draft). Rainer, J.H. (1980). Dynamic tests on a steel-joist concrete-slab floor. Canadian Journal of Civil Engineering, Vol. 7, No. 2, 213-224. Talja, A. (1997). Criteria for floor vibrations of residences and offices. Finnish R&D conference on steel structures (Terasrakenteiden tutkimus-ja kehityspaivat), Oulu 16-17.1. 1997. Finnish Institute of Steel Construction. Talja, A. & Kullaa, J. (1998). Vibration tests for light-weight steel-joist floors — subjective perceptions of vibrations and comparisons with design criteria. Finnish R&D conference on steel structures (Terasrakenteiden tutkimus-ja kehityspaivat), 25.-26.08.1998, Lappeenranta.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
727
SIGNIFICANCE OF LOCAL BUCKLING FOR STEEL FRAMES UNDER FIRE CONDITIONS A. Y. Elghazouli and B. A. Izzuddin Department of Civil and Environmental Engineering, Imperial College London SW7 2BU, UK
ABSTRACT This paper deals with the behaviour of steel structures under fire conditions, with emphasis on the influence of local buckling of steel sections on the response of frame members and subassemblages. A computationally-efficient temperature-sensitive model for steel which accounts for local buckling is described. The model is implemented within an advanced frame analysis program which represents both geometric and material nonlinearities. Several examples are presented to illustrate the effects of local buckling on the response of steel members at elevated temperature as well as on the fire resistance of idealised structural configurations. It is shown that whereas in many cases local buckling may have an insignificant influence on the fire resistance of isolated members, particularly when non-slender crosssections are utilised, its effect should be accounted for when the interaction between the various structural members governs the overall fire resistance of the structure. KEYWORDS Fire conditions; elevated temperature; cross-section slendemess; local buckling; nonlinear response; frame analysis. INTRODUCTION Considerable attention has been recently directed towards investigating the performance of structural systems under fire conditions. Following close examination of the behaviour of large steel-framed structures subjected to major fires, it was observed that most of these buildings were significantly overdesigned and/or over-protected. Consequently, there is increasing awareness of the benefits of using more rational approaches which are based on true behaviour rather than on idealised representations of isolated elements. Due to the complex interactions which take place between various members in a frame, extensive redistribution of loads occurs during fire. These interactions are being examined through a number of experimental and analytical investigations in several countries, with a view to providing more rational design approaches. In order to undertake design studies for the purpose of providing improved code recommendations, the salient factors influencing the structural response under transient elevated temperatures, including load redistribution, should be adequately examined. For steel framed structures, accurate prediction of the performance under fire entails assessing the behaviour in both the elastic and inelastic ranges up to plastic collapse. One of the important phenomena affecting failure mechanisms is premature elastic, or inelastic, local buckling of the constituent plate components of a steel member. Although this effect may be represented numerically using detailed continuum finite elements, these models are computationally demanding. Therefore, their use in analytical studies on framed structures may be impractical and often unfeasible, particularly when extensive parametric smdies are undertaken on detailed structural configurations. In this paper, the influence of local buckling of the plate components of a steel section on the member as well as frame response under fire conditions is addressed. A computationally-efficient temperature-sensitive
728 model which accounts for local buckling of frame members is described. The model is based on a simplified representation of local buckling, yet captures the main features of the behaviour, and is particularly suitable for use within frame analysis programs. The model is implemented within the advanced nonlinear analysis program, ADAPTIC (Izzuddin, 1991), which represents both geometric and material nonlinearities and includes extensive facilities for static and dynamic analysis of three-dimensional steel, composite and reinforced concrete frames. Several examples are presented to illustrate the effects of local buckling on the response of steel members at elevated temperatures as well as on the fire resistance of idealised structural configurations. Specific situations in which the effects of local buckling are notably insignificant or relatively important are highlighted. NUMERICAL MODEL Elasto-Plastic Cubic Element In this study, use was made of the elasto-plasto cubic element (Izzuddin and Elnashai, 1993) within the nonlinear frame analysis program ADAPTIC (Izzuddin, 1991). The program was developed to provide an efficient tool for the nonlinear static and dynamic analysis of two and three dimensional framed structures.
Monitoring area
Figure 1: Monitoring areas for an I-section
Figure 2: Bilinear stress-strain relationship
The elasto-plastic formulation of the cubic element is derived in a local Eulerian system where six degrees of freedom are employed. The cubic formulation can model the spread of plasticity within the cross-section and along the length. The element response is assembled from contributions at two Gauss points where the cross-section is discretised into a number of monitoring areas, as shown in Figure 1. The formulation utilises a relationship between the direct material stresses and strains and allows various material models to be included. Thermal Model The capabilities of ADAPTIC have been extended to allow for analysis under elevated temperatures, by including a number of new elements and material models (Izzuddin et al, 1995; Izzuddin, 1997). Amongst the new facilities, a thermal model was developed and incorporated within the library of models available to the cubic formulation of the program. The thermal model accounts for the deterioration in the properties of steel at elevated temperature, with a kinematic bilinear model employed for the steady-state stress-strain relationship, as shown in Figure 2.
Figure 3: Reduction factors for parameters
Figure 4: Variation of thermal strain
729 The variation of the elastic modulus (E), yield stress (Cy) and strain-hardening modulus ([i) with temperature is assumed to follow a trilinear relationship. For each property, the reduction factors are obtained from a corresponding trilinear curve defined over the temperature domain, as illustrated in Figure 3. The thermal strain is also assumed to follow a trilinear relationship, as shown in Figure 4. The parameters of the various trilinear curves are user-defined, and may be determined on the basis of available information regarding the reduction factors for the elastic modulus and the effective yield stress, such as those suggested by Eurocode 3 (EC3, 1993). Non uniform temperature variations may be applied across the section and along the length of a member. The model has the advantage of computational efficiency, as well as the ability to represent strain reversals and non-monotonic temperature variation. This also allows for the effects of heating and cooling on the response of structural components to be readily modelled. Local Plate Buckling Local buckling might either reduce the ultimate strength of a steel section or diminish its rotational capacity in the inelastic range. This is largely dependent on the width-to-thickness ratios (b/t) of the constituent plate components of the section. The elastic critical stress of a plate segment may be determined from the slendemess, the restraint condition along the boundaries and the material properties. When the member cross-section is composed of various connected elements, an estimate of the critical stress may be determined by assuming for each plate element idealised support condition for the plate edges, depending on whether it is free or attached to another plate element. Also, when a plate element is relatively short in the direction of compressive stress, the behaviour of a unit plate width would resemble that of a column. The actual response is however relatively complex due to the influence of initial imperfections and residual stresses as well as interaction between the behaviour of various components within the section. Nevertheless, experimental studies on typical structural steel members indicate that, despite the large scatter observed in the results, the traditional plate buckling predictions may be used to determine the local buckling strength with reasonable accuracy. Fukomoto and Itoh (1984) established an extensive database of buckling tests on steel sections, and assessed the buckling strength of internal elements and outstands in both the elastic and inelastic range, depending on the plate slendemess parameter Xp given by:
b 112(1-v^)o.
where E, Oy and v are Young's modulus, yield stress and Poisson's ratio, respectively, and k is the buckling coefficient which is a function of the plate geometry and boundary conditions. Using appropriate values of k, i.e. 0.43 for outstands and 4.0 for internal elements, a good lower bound prediction of the nondimensional edge strain at buckling (ejEy) in both the elastic and inelastic range, may be represented by l/Xp\ such that: kTc'E Ey
(2)
12ay(l-v')(b/t)'
where £„ and Ey are the strains at the initiation of local buckling and the yield strain respectively. A simple approach is employed in this study to account for local buckling, which can be conveniently applied in frame analysis programs. The strain within the monitoring areas, to which the cross-section is divided as part of the analysis procedure, is continuously checked. The compressive strain is then compared against the critical strain given by Equation (2), but replacing the value of b or d with the distance (x^ or yj) from the monitoring point under consideration to the point of connection with another plate, as shown in Figure 5. In tension, the stress-strain relationship follows the bilinear relationship shown in Figure 2. When the buckling strain is exceeded at a specific monitoring area, the stress is reduced using an unloading relationship, as shown in Figure 6, for which:
e™ = e^ +
'^cr(i) i^crCpl)
£cr(i)J
where £„(i) and e„(pi) are the critical strains for the monitoring point and the plate, respectively.
(3)
730
vB
U >
^
^
Figure 6: Stress-strain in compression
Figure 5: Geometric parameters of an I-section
This relationship was calibrated with experimental results to provide an appropriate descending branch for the post-buckling stress of the overall plate component (Elghazouli, 1992) when more than 20 monitoring points are used across the length of each plate segment. The model also allows for strain reversals and recovery of stresses in tension. This approach has been found in previous studies (e.g. Ballio and Perotti, 1987; Elghazouli and Dowling, 1992) to give results which are in general agreement with experimental results on both steel and composite members. In order to account for the influence of local buckling at elevated temperature, this model was integrated within the thermal model described earlier. Consequently, local buckling would be considered according to the mechanical strain and stress at a given temperature. This implies the validity of the same plate buckling relationships at elevated temperatures based on the corresponding material properties, which seems to be in general agreement with the results of several researchers (e.g. Soares et al, 1996; Ranby, 1998). ILLUSTRATIVE EXAMPLES Moment-Curvature Response In order to illustrate the influence of the cross-section slendemess on the behaviour of steel members, Figure 7 (a) to (d) depict the moment, normalised to the yield moment at ambient temperature, versus mechanical curvature response for a steel I-section using the local buckling temperature-sensitive model described before. Such relationships are widely utilised by codes of practice as the basis for cross-section classification (e.g. EC3, 1993). The yield stress and elastic modulus at ambient temperature are assumed as 350 N/mm^ and 210 x 10^N/mm\ respectively. The parameters of the reduction factors for the trilinear curves for E and Gy are given in Table 1 and, for simplicity, the strain hardening factor is assumed as zero. For each case, and for comparison purposes, the analysis is carried out with and without account for local buckling. TABLE 1 PARAMETERS OF MATERIAL TRILINEAR CURVES 1 Property E
1
<JY
ti("C)
100 400
h 0.10 0.11
t.("C) 1000 800
i.ec) 1 1200 1 1200
Figure 7(a) shows the behaviour of a plastic section, for which the flange dimensions (bj x t,) are 172 x 14 mm and the web dimensions (d^x t^) are 332 x 20 mm. The moment versus mechanical curvature response is shown at ambient temperature as well as steady-state elevated temperatures of 300 and 600 °C. At ambient temperature, it is evident that the section reaches its plastic moment capacity which is gradually reduced by local buckling at high rotations. At 300 °C, the elastic modulus is reduced, and hence the rotation capacity is also affected. For a temperature of 600 °C, both E and Cy are lower, and the respective reduction in the moment due to local buckling is also indicated. In order to increase the slendemess of the same section, the flange thickness is reduced to 10 mm, leading to a compact section, for which the response is depicted in Figure 7(b). At ambient temperature, the plastic moment is reached but is not sustained for significant rotations. At elevated temperatures, the moment capacity is lower than the plastic moment mainly due to the reduction in the elastic modulus.
731 Similarly, in order to investigate the behaviour of semi-compact sections, the flange thickness of the same section is reduced to 8 nmi. As shown in Figure 7(c), at ambient temperature the moment capacity lies between the yield and plastic moments of the section, as expected. At elevated temperatures, the reduction in elastic modulus causes buckling to occur in the elastic range, and the moment capacity may falls below the estimated yield moment.
T
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
0
1
1
1
1
1
1
1
1
r-
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
Mechanical curvature (1/m)
Mechanical curvature (1/m)
(b) Compact section
(a) Plastic section
^^•••••««»»>rfMNMMMMPMM Amb.(exc.LB)
if^
^^^*^^^^^
^^
Amb.(incLB) 300{exc.LB) 300(inc.LB) 600(exc.LB) 600(inc.LB)
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
Mechanical curvature (1/m)
(c) Semi-compact section
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
Mechanical curvature (1/m)
(d) Slender section
Figure 7: Normalised moment versus mechanical curvature relationships Figure 7(d) shows the behaviour of a slender section. In this case, both the flange and web thicknesses are reduced to 4 nun. Typically, at ambient temperature, the moment capacity is lower than the yield moment of the cross-section. With the reduction in material properties at higher temperatures, local buckling is shown to occur at a relatively early stage, and the moment capacity is well below the yield moment in all cases.
732 Frame Members The influence of local buckling on the fire resistance of an isolated steel beam is assessed through a simple example in which a steel beam is subjected to increasing temperatures up to failure. The steel beam has a span of 6 m and is loaded with a uniformly distributed load of 30 kN/m. The flange and web dimensions are 172 X 8 mm and 332 x 7 mm, respectively, producing a nominally semi-compact section. The material properties are as reported earlier, and the parameters of the reduction factors are as shown in Table 1. The parameters of the thermal strain curve are given in Table 2. TABLE 2 PARAMETERS OF THERMAL STRAIN TRILINEAR CURVE
r 1
Eti 0.010
\CC) 756
t.m 850
et3
tsCC)
0.015
1200
1
The member is subjected to fire loading simulated by uniform temperature along the length and across the section of the beam, which is increasing gradually at the rate of 100 ^'C/min. The beam is modelled using 10 cubic elements, and the cross section is discretised using 100 monitoring areas in order to achieve high accuracy. The beam is restrained from rotation at both ends, but is considered with and without axial restraint. For comparison, the analysis was carried out excluding and including local buckling.
Temp. (deg.C)
Figure 8: Displacement of non-slender beam
Temp.(deg.C)
Figure 9: Displacement of slender beam
Figure 8 shows the vertical displacement at the centre of the beam as a function of the applied temperature. If local buckling is not considered, for the case of no axial restraint the beam fails after about 73 mins when the vertical displacements start to increase rapidly. In this case, when local buckling is considered, it is initiated at about 70 mins and the reduction in failure time is not noticeable. Because of the low indeterminacy of the system, failure is initiated when the yield moment is reached at the critical section. As expected, in this case the rotation capacity does not influence the behaviour significantly, and hence local buckling has almost no effect on the fire resistance of isolated flexural members with non-slender cross sections. For the case of axial restraint, the thermal expansion induces a compressive action in the beam at an early stage followed by increased vertical displacement before overall failure occurs. Local buckling is initiated after about 10 mins, and the displacement afterwards becomes relatively larger than that obtained from the analysis in which local buckling effects are ignored. In Figure 9, the vertical displacement of the beam is given for the same four cases described above, but with the flange section reduced to 4 mm in order to represent a slender section. The initial load is also reduced to 20 kN/m. For the axially unrestrained beam, local buckling is initiated after about 41 mins, leading to a reduction in the fire resistance by about 10%. Similarly, for the restrained beam, buckling is initiated after about 6 mins, and the displacement are relatively higher thereafter. The influence of local buckling becomes even more pronounced with further increase in cross-section slendemess. It should be noted
733 however that with appropriate design, a beam with slender cross-section would be initially designed for a reduced yield moment and hence this effect would implicitly be accounted for. Nevertheless, it would be important to consider the influence of local buckling when an analytical model is used to examine the behaviour, particularly when the interaction between several members within a frame is investigated. Structural Sub-Assemblages The sub-frame shown in Figure 10 is constructed to demonstrate some interactions which may occur between frame members under fire conditions. The material properties are the same as reported earlier, and the material factors are as shown in Tables 1 and 2. The column flange and web dimensions are 370 x 15 mm and 340 x 10 mm, respectively, which results in a non-slender section according to EC3 (1993). The beam has flange and web dimensions of 172 x 8 mm and 332 x 7 mm which may be classified as a semicompact section.
V
^
P^
^
6.0 m
Figure 10: Layout of idealised frame system A uniformly distributed load (w = 10 kN/m) is applied on the beam, and the column is also preloaded with 2.7 X 10^ kN. The top of the column is connected to an elastic vertical spring with a stiffness of 0.1x10^ kN/m representing approximately 13% of the axial stiffness of the column, which simulates possible restraint from other parts of the structure. The sub-frame is subjected to an increasing temperature at the rate of 100 °C/min for the beam and 50 °C/min for the column, assuming the provision of a degree of fire protection to the column.
Time (min) Time (min)
10
Figure 11: Axial force history in column
20
30
40
50
60
70
T
Figure 12: Vertical displacement at beam centre
734 Figures 11 and 12 depict the axial force in the column and the vertical displacement at the centre of the beam, respectively. The results are shown for the analysis undertaken excluding and including local buckling effects. As indicated in the results, local buckling occurs first in the column after about 28 minutes as a consequence of the increasing axial load due to the restraint to expansion. Thereafter, the axial load in the column is relatively reduced and hence the beam displacement increases accordingly. Overall failure occurs in the system after about 80 minutes compared to about 90 minutes when local buckling is not considered. It is evident that assuming that failure occurs at the initiation of local buckling would be considerably conservative. On the other hand, ignoring the influence of local buckling altogether would overestimate the fire resistance of the system. CONCLUDING REMARKS The influence of local buckling on the response of frame members is addressed in this paper. A thermalstructural model which accounts for local buckling is presented. The model maintains the computational efficiency of frame analysis programs, yet accounts for the salient features of the behaviour. A number of examples are presented to illustrate the effects of local buckling on the response of steel members at elevated temperatures as well as on the fire resistance of idealised structural configurations. For isolated members, local buckling does not influence the behaviour significantly unless very slender cross-sections are used or restraint from other parts of the structure is considered. For frame assemblages, local buckling may have a more pronounced effect, depending on the structural configuration as well as loading and restraint conditions, even when members with non-slender cross-sections are utilised. Whereas in many cases the onset of local buckling is not directly followed by failure, it may have a detrimental effect on the overall fire resistance of the structural system. Analytical approaches based on the former consideration or ignoring the latter may evidently lead to overconservative design or unsafe predictions, respectively. REFERENCES Ballio and Perotti, F. (1987). Cyclic Behaviour of Axially Loaded Members, Numerical Simulation and Experimental Verification, Journal of Constructional Steel Research, 7:1, 23-41. EC3 (1993). Eurocode 3. Design of Steel Structures, Commission of the European Communities, European Committee for Standardisation. Elghazouli, A. Y. (1992). Earthquake Resistance of Composite Beam-Columns, PhD thesis. Imperial College, University of London. Elghazouli, A. Y. and Dowling, P. J. (1992). Behaviour of Composite Members Subjected to Earthquake Loading, Proc. Tenth World Conf. on Earthquake Engineering, Madrid, 2621-2626. Fukumoto, Y. and Itoh, Y. (1984). Basic Compressive Strength of Steel Plates from Test Data, Proc. of Japan Society of Civil Eng., JSCE, No. 344/1-1, Structural Eng./Earthquake Eng., 129-139. Izzuddin, B. A. (1997). Quartic Formulation for Elastic Beam-Columns Subject to Thermal Effects, Journal of Engineering Mechanics, American Society of Civil Eng., 122:9, 861-871. Izzuddin, B. A., Song, L. and Elnashai, A. S. (1995). Adaptive Analysis of Steel Frames Subject to Fire Loading, Proc. Sixth Int. Conf on Computing in Civil and Building Engng, Berlin, 643-649. Izzuddin, B. A. (1991). Nonlinear Dynamic Analysis of Framed Structures, PhD Thesis, Imperial College, University of London. Izzuddin, B. A. and Elnashai, A. S. (1992). Adaptive Space Frame Analysis, Part II: Distributed Plasticity Approach, Proc. of the Institution of Civil Engineers, UK, Structures and Buildings, 99, 317-326. Ranby, A. (1998). Structural Fire Design of Thin Walled Steel Sections, Journal of Constructional Research, 46:1-3 Paper No. 176.
Steel
Soares, C. G., Gordo, J. M. and Teixeira, A. P. (1996). Elasto-Plastic Collapse of Plates Subjected to Heated Loads, Journal of Constructional Steel Research, 45: 2, 179-198.
Poster Session P5 COMPOSITE STRUCTURES
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
737
ELASTOPLASTIC LARGE DEFORMATION ANALYSIS OF CONCRETE-FILLED TUBULAR COLUMNS M.Shugyo^ and J.R Li^ ^Department of Structural Engineering, Nagasaki University, Bunkyo-Machi 1-14, Nagasaki 852-8521, Japan
ABSTRACT A numerical method for elastoplastic buckling analysis of concrete-filled tubular columns with hollow circular section is presented. The method is an advanced finite element method and the column is divided into short elements along the length. The elastoplastic tangent stiffness matrix in the method is constructed by using the tangent coeflftcient matrix obtained by numerical integration of the hardening moduli of the fibers about the member section to calculate the generalized plastic strain increments. The accuracy of the method is examined by comparing the results with available experimental ones.
KEYWORDS composite member, concrete-filled tubular column, elastoplastic analysis, buckling strength, large deformation
INTRODUCTION Concrete-filled steel tube is a typical composite member, that has high strength and high ductility because of the confinement effect of steel tube. Many investigations concerned with the mechanical behaviors of concrete-filled steel members have been done so far (Neogi et al., 1969 and Matsui et al., 1994) and their superiority has become clear fairly well. However, the studies are almost design criterion oriented ones using the experimental results of the beam-columns subjected to in-plane loadings. In the practical use of concrete-filled steel members it is very important to make clear the behaviors of beam-columns and frames under three dimensional loadings.
738 In this paper a numerical method for elastoplastic buckling and post buckling analysis of concretefilled tubular columns with hollow circular section is presented. The elastoplastic tangent stiffness matrix in the method is constructed by using the tangent coefficient matrix obtained by numerical integration of the hardening moduli of the fibers about the member section to calculate the generalized plastic strain increments. Since the method is a kind of FEM, it is also possible to analyze space frames of arbitrary configuration.
NUMERICAL METHOD The method is an advanced plastic hinge method based on the finite element method. The following assumptions are made concerning the plastic deformation in the yielded element: (1) A column deforms in a body and behaves according to the BernouUi-Euler hypothesis. (2) Only axial stress participates in yielding of a fiber. (3) Plastic deformation consists of only three components that correspond to axial force and biaxial bending moments. (4) No local buckling. (5) An actual generalized plastic strains in a short element (Figure 1) generally distribute nonlinearly (Figure 2(a)). It is supposed that generalized plastic strains distribute linearly with the values at the element nodes i and j (Figure 2(b)). (6) Incremental plastic deformations in the two 1/2 portions occur concentrically at the element nodes i and j respectively, where / is the length of the element.
Figure 1 : Member coordinate system and generalized stresses and strains plastic strain
Figure 2 : Assumption of generalized plastic strain distribution in an element
739 Geometrically
nonlinear
stiffness
matrix
Member coordinate system (x , y , z) are shown in Figure 1. From the assumption (1) we can obtain the following equation using the energy principle: dQ = K'dq'
(1)
in which K^ is the geomertrically nonlinear tangent stiffness matrix and Q and q^ are nodal force and nodal elastic displacement of an element, respectively. Both Q and q^ have 12 components. Plastic
tangent
coefficient
matrix for a cross
section
In the present method, plastic deformation increment is estimated utilizing tangent coefficient matrix for a cross section. The tangent coefficient matrix is obtained by numerical integration of tangent moduli of the fibers which compose the element. Figure 3 shows the model of stressstrain relationship of filled concrete used herein. The confinement effect of steel tube is taken into account as the perfect plastic and no collapse characteristics. The stress-strain relationship for steel is bilinear model.
F
^
1
J
1
/
o' • ^ cy
< 1 ^ Figure 3 : Stress-strain relationship of filled concrete
The components of generalized stress vector / and generalized strain vector S are shown in Figure 1. The components of / and S can be written as f = [fx ruy m,]
•\T
c.
; S=
r
[EQ
/
, -\T
(2)
where fx is an axial force, niy and ruz are bending moments, and the components of S are corresponding generalized strains, respectively. From the assumptions (2) to (4) the increments of generalized stresses are related to the fiber stress increments by dfx = /
dasdAs + / dacdAc
J As
J Ac
diTLy — I dasZdAg JAS
dm.
+
/
JA
darZdAr
- / dasydAs - / J As
(3)
dacydAc
J Ac
and the fiber strain increments are related to the increments of generalized strains by de = deo + zd(j)y — yd(j)z .
(4)
In Eqn. 3 subscript s and c denote the steel and the concrete, respectively. Using the present values of nodal forces and iterative procedure (Shugyo et al., 1995), we can obtain the following equations: d5\ = s'^df^ ; d6] = s)df^
(5)
740
where s • and s'j are plastic tangent coefficient matrices. The components of the tangent coefficient matrix are obtained by numerical integration. Now let us define plastic deformation increments dq^, dq^- as
rfqf=[d<,
0 0 0 del,
0 0 0 d^^ de%]'
(6)
which are the deformation increments due to the generalized plastic strain increments of an element. These plastic deformation increments can be obtained as described below. The generalized stresses at the member ends are obtained by the nodal forces with their coordinate transformation. Using these generalized stresses we can obtain the plastic tangent coefficient matrices s- and s'^ Representing the components of s[ by (sj^^)^, a new square matrix s^ of 6th order can be obtained as follows: 0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0
0 0 0
(7)
\^2^)i
Another matrix s^ which corresponds to s'j can be obtained similarly. In case of uniaxial bending, the plastic curvature increment distributes as shown in Figure 4 from assumption (5). Hence the plastic rotation increment at the member end i can be expressed as follows from assumption (6):
-del = [
;+o(rfC+'^0
Xo =
/ 3d<, + d4>l,
(8)
^-1
•T3
k-//2—IW/2Figure 4 : Assumption of plastic curvature distribution in an element The plastic deformation increments can be obtained by extending Eqn. 8 as dq\
I 3«r -s:
-s)
idQA \dQj
dQ, dQ,
(9)
Assuming that the total displacement increments dq are the sum of the elastic displacement dq^ and the plastic deformation dq^, we obtain dQ + R =[!-{- K'sP]~'K'dq
= K^dq
(10)
where R is the unbalanced force vector and / is the unit matrix. The numerical analysis can be carried out by Ramm's displacement control method using the elastoplastic tangent stiffness matrix K^. Coordinate transformation matrix of a member is updated and rigid body displacements are separated in each step by using rotation matrix(Crisfield, 1997).
741 NUMERICAL RESULTS A N D DISCUSSION Eccentrically
loaded concrete-filled
tubular
columns
In this section the accuracy of the present method is examined by comparing the results with available experimental ones. Figure 5 shows the model of eccentrically loaded column (Matsui
rL
i>f^'
tt
N,
Figure 5 : Eccentrically loaded concrete-filled steel pipe (Matsui et al., 1994) ( ^—
:z; 1200
Li •JL) = 4 K
^"^ 800 O
1600
0
^^ 3/^
f^
400
\
^
800
l\
O
400
\(X-\
5/^
0 '
-
0
1
8
L, •Jl\>-
K,
fV
\ 3n
5K
10
Lateral Displacement S (cm)
Lateral Displacement S (cm) 1600
/
^ ^ fe: 800
1200
O
(I J
1200
(
0
r
.^klD
L2
^
1200
K,
^^^
-A
400
^
T>K
3 AC
10
Lateral Displacement S (cm) 1600
^
^"^
800
O
400
Lk ID = 24 /c
1^
•-^r; ^
^
^ ^
3K
-*^ /
^ 17
1600
0
/
2 ^ 1200
Lateral Displacement S (cm)
^
^ !^
'Z^
O
<—^
TK 10
Lateral Displacement S (cm) Present analysis
^
1200 800 L 400
0
/i ; ;x'
AC
i4—««
:::^ ^ ^
A.
^ ^ ii--'
z^*/ D
=
3Af
5/
30
21ZZ
^ 10
Lateral Displacement S (cm) • Experiment(Matsui et al.)
Figure 6 : Load-lateral displacement relation at the mid span
742
et al., 1994). Sizes and mechanical properties of the column are as follows: outside diameter of the steel tube D=16.52cm, thickness ^=0.45cm, Young's modulus E'5=206.0GPa, yield stress crsy=353.0MPa, yield stress of filled concrete a^y = -34.70MPa, the stress-strain relation of the steel is bilinear and the strain hardening modulus after yielding Hs=E/lOO, and the yield strain of the filled concrete Scy = -0.0023 (see Figure 3). In the present analysis the columns were divided into 20 elements of equal length. For the column without eccentricity (e=0) sinusoidal distribution of initial displacement with the maximum value of Lfc/4000 was assumed and zero for e = K, 3K, and 5K, where /t=2.1cm. The comparison of both results of the experimental study by Matsui et al. and the present method are shown in Figure 6. Good agreement can be seen on the whole.
Three dimensional
frame
The present method is applicable to the analysis of frames which have concrete-filled steel tubular members provided that the finite element modeling does not conflict with the assumption (5). Figure 7 shows a space frame whose members are all the same steel tube of outside diameter D =4.82cm and thickness t =0.23cm. The material constants are the same as those of the above example. Each member divided into 6 elements by five nodes at the points of 1/10, 2/10, 1/2, 8/10 and 9/10 of the member length. Horizontal load-displacement curve under the constant vertical load A^ = O.SNy, where A^^ is the yield axial compressive load of the column, is shown in Figure 8. The broken line in the figure shows the result in the case that four columns are replaced by concrete-filled steel tube. The material constants of the filled concrete are also the same as those of the above example. The performance is improved remarkably.
L = 70.0cm Figure 7 : 3D frame subjected to eccentric horizontal load
r ^
^'° O
5 n
Steel ^t
i
Lateral Displacement u (cm) Figure 8 : Load-displacement relations
743 CONCLUSION A numerical method for elastoplastic buckling and post buckling analysis of concrete-filled tubular columns with hollow circular section and frames which have concrete-filled tubular members was presented. The accuracy of the method was confirmed by comparing the results with available experimental ones and it was shown that the present method can be used for the analysis of three dimensional frames which have concrete-filled tubular members.
References
Matsui C, Tsuda K, Ozaki I. and Ishibashi Y. (1994). Design Formula of Slender Concrete Filled Steel Circular Tubular Columns. Journal of Structural Engineering^ 40B, pp.403-410. Neogi P.K., Sen H.K. and Chapman J.C. (1969). Concrete-Filled Tubular Steel Columns under Eccentric Loading. The Structural Engineer^ 47:5, pp. 187-195. Shugyo M, Li J.P. and Oka N. (1995). Inelastic and Stability Analysis of Linearly Tapered Box Columns under Biaxial Bending and Torsion, Reports of Faculty of Engineering, Nagasaki University, 25:45, pp. 143-149. Crisfield M.A. (1997). Non-linear Finite Element Analysis of Solids and Structures, Vol.2 Advanced Topics, John Willy & Sons.
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
745
RESISTANCE OF COMPOSITE SECTION TO AXIAL LOADS AND BENDING: DESIGN AND ANALYSIS J. Brauns Department of Structural Engineering, Latvia University of Agriculture, 19 Academy St., Jelgava, LV-3001, LATVIA
ABSTRACT Composite columns and beams are a combination of concrete and steel columns combining the advantages of both types of materials. Eurocode 4 presents a simplified method for composite section design. For concrete-filled hollow section column, the plastic resistance of the cross-section is given as a sum of the components or taking into account the effect of confinement in the case of circular sections. In this study the stress state in composite columns is determined taking into account the relationships of the modulus of elasticity and Poisson's ratio on the stress level in the concrete. It is determined that the effect of confinement occurs at a high stress level when structural steel acts in tension and concrete works in compression. The stress state and load bearing capacity of section in bending is determined taking into account nonlinear dependence on position of neutral axis. The ultimate limit state of material is not attained for all the parts simultaneously. In order to improve the working conditions of a composite element and to prevent the possibility of a failure because of a small thickness of structural steel and fire, the appropriate strength classes of concrete and steel should be used.
KEYWORDS Composite section. Design method. Modulus of elasticity. Optimum design. Plastic resistance. Stress analysis. Ultimate moment
INTRODUCTION Composite columns and beams, formed as hollow steel sections filled with concrete have advantages in different architectural and structural solutions. The composite structure characterises with higher ductility than the concrete column and advantages of steel may be successfully used in connections. The concrete filling and reinforced bar increase rigidity and load-bearing capacity of the hollow steel section with no changing in external dimensions of column. The filling improves the fire resistance as
746 well. Composite column with a corresponding content of reinforcement can provide at least 90minutes of fire-resistance rating (R. Bergman, C. Matsui, C. Meinsma, & D. Dutta, 1995). In some countries (Australia, Canada) the design methods are very similar to the same in ENV 1994-11: Eurocode 4 (1992). In some cases these methods are not compatible with Eurocode 4 (EC4). The load-bearing capacity of composite column is obtained as a sum of the same values for the steel and concrete components and therefore no composite action is considered. According to EC4 to determine the resistance of a section against bending moment, a full plastic stress distribution in the section has been assumed. The internal bending moment resulting from the stresses is the resistance of the section against moment M pi. Rd. Reinforcement is included by adding the plastic bending capacity of the reinforcement alone. This leads to small deviation from the exact values. EC4 gives a simplified ultimate limit state design method for the composite structures, which is applicable for practical purposes. The method is based on consumption that ultimate strength of material is attained simultaneously in all parts of the section. Stress analysis of the composite column and beam is performed with purpose to obtain the maximal value of load-bearing capacity of the structure using components with the appropriate properties and taking into account the composite action.
ANALYTICAL MODEL FOR STRESS ANALYSIS The section presents a consideration on behaviour of the short concrete-filled steel column under axial loading. Disregarding the local effects at the ends, the stress state in the cross-section of the middle part of the column is analysed. A reinforced concrete core with radius R o is included in steel tube of thickness t (Figure 1). The reinforcement consists of symmetrically arranged longitudinal steel bars and circumferential reinforcing wire located at radius Rs from axis z. Reinforcement Structural steel
Concrete
Figure 1: Concrete-filled hollow circular column with reinforcement For radial displacements w and circumferential displacements u on the contact surface concretestructural steel (r = Ro) and concrete-reinforcement surface (r = Rs) the following equalities are valid: w
(1)
w^=w^; u^=u^
(2)
747 Here and below, the indices a, c and s refer to the steel tube, concrete and reinforcement. In the axial direction of the column, the cross-section area of the reinforcement is Az^ but in the circumferential direction - Ae^ placed with the given step. It is taken into account that the reinforcement is "spreaded" throughout the cross-sectional area, in result of that the anisotropic column is formed. The following relationships should be written assuming throughout the steel tube an in-plane membrane stressed state: (3)
oz Pc^-o
a
(4)
--— = const; Sft = (5) dz ^ r where pc - the contact pressure on the boundary surface steel tube-concrete, CTZ^,CTO^,SZ^ and se^ stresses and strains of the steel tube in the axial and circumferential direction. We assume the existing equality of stresses in the radial and circumferential direction (Or^ = GQ^) in concrete core. Taking into account the compatibility relationships (1) and (2), four equations can be written: 1
V^
CT^ —
1
— r ^ 7 ~—r<7fi = CTy — Gr + Go , ga z ga w gc z cc r p-c w '
(/) ^ ^
V"^
(Tr ~
v'^a
(O)
1
G^ —
V''
CJn ,
V^
Ga =
V'^n
1
-—^s 1 ^c ^zr ^ c ^z9 ^ c 7^7 =—rCTy — — T G r — CJft , pS z pc z pc r pc » ' l^y
E^0
L^y
J_>,
J^fJ
^e" =7:7^e --^^r 'T^^zES E' ' ^" ^0 "
/Qx (o) V /
W
were v ^ - is Poisson's ratio of the steel tube, vy^ - Poisson's ratio values of the reinforced concrete (i, j = r, z, 0 ; i 9t j). Eqs (6) - (9) have been written taking into account the transverse deformation. The equilibrium equations in the z and 0 directions are as follows
^^e+i^c^e+<^e=0,
A"a^
+A^GI
+ A | +q(A^
+ A ' ) = 0.
(10)
(11)
where A^ and A^ - the cross-sectional areas of the concrete and structural steel accordingly, E ^modulus of elasticity of the steel tube, Ez^ Er^ and Ee^- moduli of elasticity of the reinforced
748 anisotropic concrete core in the directions z, r and 0, Ez^ and Ee^ - moduli of elasticity of the reinforcement in directions z and 0, q - a uniformly distributed axial load. The load carrying capacity of composite elements in bending can be characterised by plastic deformation of structural steel in tension, by cracking of concrete in compression or by both. At the same time the stresses in reinforcement can be less or more of the yield limit of steel. The stress analysis of composite elements in bending has been performed with certain assumptions: 1) the hypothesis of flat sections has been used; 2) the concrete lying in the tension zone of the section is assumed to be cracked and is therefore neglected; 3) the conditions of deformation continuity of structural steel and concrete fulfil in the compression zone; 4) relationship for the stress distribution in the compression zone of concrete has been assumed in following form :a'f(z/x)"s
(12)
where cs^z - stresses in concrete at distance z from neutral axis; a^f - stresses in concrete in more compressed fiber; x - height of the compressed zone of concrete ; Uc - characteristic of the stress diagram form. The cross section of reinforced composite beam and the stress-strain distribution diagrams are shown in Figure 2.
Figure 2: Stress and strain distribution in composite beam On the basis of the equilibrium conditions of the internal forces and taking into account the hypothesis of flat sections, the position of neutral axis can be determined by using nonlinear equation: fcdb
(1 + E ^ / E " ) s ^
-2t(E^ + E%) + x[E^(-bt + 2 h t - 2 t ^ ) + E^(bt + 4ht + 2t^) + E ' A ' ] + (13) + E^(hbt + 2 h t ^ ) - E ^ ( h b t + 2h^t + 2 h t 2 ) - E ' A ' h o = 0 ,
where E^u and E^ - tangent and initial modulus of the concrete, E^ - modulus of constructional steel in tension, e^u - ultimate deformation of concrete in compression. The ultimate bending moment Mu was determined by using the condition of equality of the moments in section of the beam ^ 2r 1 Mu=bja'^f(z/x)"czdz-Ha^btx + - a^tx(x + t) + a^tt(h-x)(h + t - x ) + 0 ^ + a ' A ' ( h o - x ) + a%bt(h-x). where b and h - width and height of concrete section.
(14)
749 NUMERICAL SOLUTION AND ANALYSIS In order to perform the stress analysis we should solve the system (6) - (11) of six linear equations, which in matrix form is written as (12)
AX = B.
The vector components of the unknown quantities are stresses in the steel tube, concrete, longitudinal and spiral reinforcement, i.e.. X =
(13)
[c|,ag,c^<,§,.|,a^f.
The matrix of system A and vector of constants B is determined from the stress-strain relationships and equilibrium equations as follows: 1
v"
1
E^ v"
"E^
"E^.
1
v'.
v' ^ + v^ K E^ 1 v^
E" A^
"E"
K
E?" Eg
0
A'=
0
0
0
1
f c
E'.
IEJ
c "\ EQJ
0
0
0
0
A^z 1
0
0
Ro 0
0
1
v'ze
v'^
E'z
E?
0 1
n
(14)
~E|
h 0
0
0
B = [o, 0, - q ( A ^ + A'), 0,0, o]^.
Ae Rs 1
"E^ (15)
The following geometrical parameters are chosen for the analysis: diameter of circular hollow section d = 40.6 cm, thickness of the steel tube section t = 0.88 cm, reinforcement content p = 4 % of the concrete section, distance from the central axis to the reinforcement bar axis Rs = 15.5 cm, crosssectional area of the longitudinal reinforcement Az^ = 49 cm^ and secondary reinforcement Ae^ = 0.1 cm^ (per 1 cm of the column length). The modulus of elasticity for the reinforcement is E ^ = 200 000 MPa, for structural steel - E ^ = 210 000 MPa and for Poisson's ratio v ^ = 0.25. The initial tangent modulus and tangent modulus at the given stress level A.M. Neville (1981) have been used as the modulus of elasticity of the concrete. The Poisson's ratio of the concrete changes in accordance to the stress level, starting with v ^' = 0.14 in the initial stage. The modulus of elasticity of a reinforced anisotropic concrete core Ez^, Er^ and Ee^ as well as Poisson's ratio values are found on the basis of the reinforcement theory A.K. Malmeister, V.P. Tamuz, & G.A. Teters (1980) taking into account the content of the reinforcement p and mechanical properties of the constituents. The stress values in the material components and applied load N for two strength classes of concrete is given in Table 1. According to EC 4, for the concrete-filled circular hollow sections, the load-bearing capacity of the concrete is increased due to the prevention of transverse strain. This effect is shown in Figure 3. The transverse compression of the concrete (ar^) leads to three-dimensional effects, which
750
promotes increase of the column resistance. At the same time the circular tensile stresses (oe^) arise on the cylindrical surfaces reducing its normal stress capacity.
Figure 3: Stress distribution at the section of composite column Note that the axial stress in concrete is GZ^ = 0.75 fck when the Poisson's ratio of concrete v ^ increases and becomes higher than the same of the steel. In the region before fracture, the Poisson's ratio is approximately 0.33 A.M. Neville (1981). Circular tensile stresses GQ^ in the concrete core of the column change into compression stresses due to behaviour of steel tube and mentioned above features of concrete deformation process. TABLE 1 STRESS VALUES (MPa) IN CONSTITUENTS OF COMPOSITE COLUMN FOR RATIO d/t = 46
Strength class of concrete
Axial force N, kN
a/
C30/37
7261
-231
C20/25
4510
-122
a/
a/
a/
eye'
16.3
-30.1
-1.20
-235
72.0
9.5
-20.2
-0.68
-125
39.0
According to ENV 1994-1-1: Eurocode 4 (1992), the plastic resistance of the cross-section of a composite column with axial loading is given as the sum of the components: N pl.Rd = A % . + A % d + A / f , d ,
(16)
where A^ A^ and Az^ are the cross-sectional areas of the structural steel, concrete and reinforcement in the axial direction, and fyd, fed and fsd are design strength values of the materials mentioned above. In the case of steel grade Fe275 with characteristic yield strength fy = 275 MPa and relative slendemess ratio 0.15, the load-bearing capacity Npi.Rd = 7162 kN. Taking into account the effect of confinement for circular hollow sections, the load-bearing capacity Npi.Rd = 7935 kN. Here the analysis is performed
751 using the secant modulus of concrete according to the strength class of the concrete. According to our analysis (Table 1), the axial stresses in concrete are csi > fck, where fck is the characteristic strength of the concrete for the given strength class. The analysis of the load bearing capacity of the composite column is performed taking into account the design strength of steel and concrete as well as limit ratios of the circular hollow sections. It is shown in Figure 4 that the first load limiting factors are concrete design strength fed and d/t ratio. Using concrete of the strength class C35/45 and steel of grade Fe235 the load bearing capacity of the composite column increases by 18 %. In the case of a thin wall hollow section (d/t = 90) the steel economy is for 50 %. CT|
a;^(MPa)
p 30 fyd Fe275 _fcd_C35/45
200 V- 20
L_
1
fyd Fe235 4 ^y^^[^^. ted 030/37^;'^^' 1 1
•
100 1 ""^
,
50
ip
in
Cvl ,a>
CM 1
1 ,
77
90
d/t
Figure 4: Variation of stresses in structural steel and concrete in relationship of ratio d/t: 1,2- ^^ and <5i for the axial load value N = 4280 kN; 3,4 - cjz' and ^^ for N = 5058 kN. The result of the stress analysis or the composite beam is shown in Figure 5. By using nonlinear approach the position of neutral axis of reinforced concrete element depends on ultimate deformation of concrete in compression s'^u- Taking into account this peculiarity the stresses in structural steel a^ in tension zone are equal to the yield limit fy (steel grade Fe235) when s^u < 110"^. The ultimate bending moment Mu in this case is 40 % less than plastic moment Mpi. Rd determined according to EC4. The concrete filling leads to the increase of the load-bearing capacity that is much higher than that of steel elements and promotes the fire resistance as well. The concrete is contained within a circular rectangular steel profile and cannot split away, even if the ultimate strength of concrete is reached. Nevertheless, in order to prevent the possibility of a failure, especially, in the case of small thickness of the structural steel and fire, the appropriate strength classes of concrete and steel have to be used.
752 x/h
Mu(kNm)
1.0 r 200 r
0.5 h100
OL
0 eu-10''
Figure 5: Influence of ultimate concrete deformation in compression on predicted position of neutral axis, stresses in structural steel and ultimate bending moment: 1 - x/h, 2 - a^, 3 - Mu
CONCLUSIONS 1. On the basis of constitutive relationships for material components, the stress state in a composite column is determined, taking into account the dependence of the modulus of elasticity and Poisson's ratio on the stress level in the concrete. It is proved that the effect of confinement acts at a high stress level when the structural steel behave in tension and the concrete in compression. The main load limiting factors are concrete design strength fed and d/t ratio. In the case of concrete strength class C35/45 and steel grade Fe235 the load bearing capacity of the composite column increases by 18 %. In the case of a thin wall hollow section (d/t = 90) the steel economy is for 50 %. 2. By using nonlinear approach the position of neutral axis of composite beam depends on the ultimate deformation of concrete in compression 8*^u. When s^yi< \-\0'^ the ultimate bending moment Mu is 40% less than plastic moment Mpi Rd determined according to EC4 and the stresses in structural steel a^ in tension zone are equal to yield limit fy (steel grade Fe235). 3. In order to optimize the working conditions and cross section area of the composite structure as well as to prevent the possibility of a failure in the case of small thickness of structural steel and fire, the appropriate strength classes of concrete and steel have to be used.
REFERENCES Bergman, R., Matsui, C , Meinsma, C. & Dutta, D. (1995). Design Guide for Concrete Filled Hollow Section Columns under Static and Seismic Loading, Verlag TUV Rheinland GmbH, Koln, Germany Eurocode 4: (1992) Design of Composite Steel and Concrete Structures, Part 1.1: General Rules and Rules for Building ENV 1994-1-1
753 Eurocode 2: (1992) Design of Concrete Structures, Part 1.1: General Rules and Rules for Building, ENV 1992-1-1 Malmeister, A.K, Tamuz, V.P. & Teters G.A. (1980) Strength of Polymeric and Composite Materials (in Russian). Zinatne, Riga, Latvia Neville, A. M. (1981) Properties of Concrete, Pitman Publishing, London, UK
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
755
LIGHT-WEIGHT HOLLOW CONCRETE-FILLED STEEL TUBULAR MEMBERS IN BENDING Audronis Kazimieras Kvedaras Department of Steel and Timber Structures, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-2040 Vilnius, Lithuania
ABSTRACT This paper deals with the results of theoretical and experimental investigation for efficiency of lightweight hollow concrete-filled circular steel tubular members in bending. It is such type of composite members for that no regulations are included into any Code of Practice. There are shown reasons of technical and economical efficiency of such members usually connected with appearance of interaction between the external circular super-thin-walled steel shell and internal hollow concrete core. That interaction creates conditions to evaluate an increase in strength of both components of such members in bending. There are also presented the examples of practical use of those members for constructing of efficient structural systems. The paper describes such type of hollow composite members being very useful model to develop efficient lightweight structural members formed from a large scale of different building materials.
KEYWORDS Concrete-filled member, hollow composite section, stress behaviour, bending, stress interaction, design regulation, application, efficiency
INTRODUCTION One of the most effective ways for industrialization of construction and realization of the main structural materials (steel and concrete) is the widening of the use fields of structures produced by the means of centrifugal force too. The centrifuging process allows developing composite steel-concrete members with an effective hollow concrete core. For heavy loaded framework the combination of hollow concrete-filled steel circular sections with members including the concrete enhancement is already one of many examples of quite successful application of light-weight steel and composite steel-concrete structural systems where the total amount of used steel is rather less than of one for reinforcement in concrete structures used for the same purpose. The use of hollow composite steelconcrete members consisting of relatively thin-walled circular or rectangular steel tubes and spun or ordinary concrete is conceivable as means of economical improvement of the strength and ductility for structural members and their connections. Hollow composite members are now more widely used in
756 some countries as foundation piles, but according to our test data and development results a wide application of them in bridge piers as well as in columns, beam-columns and beams for buildings is looked forwards too. The comparison of results of calculations of resistance of short concrete-filled steel tubular elements in compression on the base of criteria of small and mediocre elastic-plastic strains with test data have showed a good agreement. It is usual to suppose an existing of the functional dependency expressing the magnitude of increase in resistance of composite member K^ upon the mechanical geometrical parameter, or steel contribution factor according to the Eurocode 4. There were calculated mean values of the factor Ke for the group of spun concrete-filled steel tubular members. These values were for the group of buckle members in uniaxial compression - Ke= 1.21; for the beam-columns - Ke= 1.27; for the beams - Ke= 1.32. For the stub columns of the same cross-section the mean value of this factor was only Ke = 1.17. That means more high efficiency of slender differently loaded hollow members against short ones and it is in some contradiction with limitation of EC-4 allowing to take into account the confinement effect only when slendemess ratio exceeds X > 0.5, or when the ratio between the length and outer diameter of a member exceeds 25. It is necessary to pay attention here that the greatest value of efficiency factor belongs to the hollow composite steel-concrete beams. Because of better geometrical properties the hollow composite steel-concrete beams are more efficient than those with solid concrete core. However, the problem of application of such hollow composite beams exists because no design recommendation is included into any Code of Practice including the EC-4. Therefore further this paper deals with presentation of methods for resistance calculations of hollow composite steel-concrete beams and comparison of the obtained theoretical results with experimental data own and taken from the references.
UNIAXIAL BENDING OF SOLID COMPOSITE BEAMS Usually uniaxial bending moment resistance of composite concrete filled circular steel tubular beam is calculated by using ideal plastic material models for concrete and steel. In Makelainen & Malaska (1997), formulae for circular concrete filled steel tubes with solid core are presented due to the Finnish design manual on Composite Structures. The design bending moment resistance Mu is defined by the plastic axial resistance Nps of the steel cross-section and a design eccentricity Cn which depends on the type of column cross-section: N,.=A,-f^,
(1)
M.=N^,-e„
(2)
The corresponding relationships are used to find the magnitude of this design eccentricity Cn
K,=—^:A-^
(3)
K.^jt'dllA'A^
(4)
2'(d = 2-K,'K^'S\n{K,) sin0 d^-dl
+ K,
(5) ,,,
757
'
3(2-0-sin0) d^=d-2-t
(8)
P = ^-^-L,lfy.
(9)
e„ = e.+ — - ( e . - e j . n
(10)
According to the Finnish practice, Makelainen & Malaska (1997), a centrally loaded and bent composite column is strong enough if the design moments are not exceeding the moment resistance in the main directions and the existing axial load is less than the axial resistance N^ = k^, • A^^ : M^<M^
(11)
M^<M^
(12)
N,
(13)
It is obvious that for the circular concrete filled steel tubes with solid core the conditions 11 and 12 only will be valid. In [2] as well as in [3] the design bending moment resistance Mu is defined as A ^ « = ^ p / • / . . • ( ! + 0.01.m)
(14)
Where \2 m = (lOO/PF^/)-((J-/)'-[(O.5-;r-0i)-cos0i+sin0,-1] +
+ O.25-/7-(t/-2O''[O.33-sin^0i-O.25-(2-e,-sin20i)-cos0i]}
(15)
Parameter m evaluates interaction effects that occur between the components of composite concrete filled beams. In design practice the numerical values of parameter m may be determined from the given in Codes charts due to values of p and ratio of external diameter and thickness of steel tube. The neutral axis is determined for each individual case and this is causing tedious and complicated calculations in Makelainen & Malaska (1997). More common case presents the method for determination of ultimate uniaxial bending moment according to Eurocode 4 (1996) presented in Makelainen & Malaska (1997). The similar expression is proposed in Kvedaras (1983) too. For determination of central angle which defines the height of compression zone of solid composite steelconcrete section the next expression is derived: (d = Arccos{[\-2'{t where
+ t^)ldf
'COs{Aj{t^'{d-2't-t^)
/^ = 0.5 • ( J - It) • (1 - V l ^ ) p = 02'a,JcT,,
+ 2't'(d-t))\}
(16) (17) (18)
758 k = CT,Ja,,
(19)
where ais and Oib are ultimate values of steel and concrete strengths obtained taking into account influence on them of confinement effects defined as for short composite columns, for instance, according to Kvedaras & Sapalas (1998).
UNIAXIAL BENDING OF HOLLOW COMPOSITE BEAMS It is a custom to think that the concrete filled steel tubes are most useful in those buildings where mainly compression forces are acting. However, the research data and building practice show those to being also efficient under other actions and types of cross-sections. The great efficiency of hollow composite beams evidently confirm the research data presented in Matzumoto, Fukuzawa & Endo (1976). It seems that limitation of EC-4 for resistance to local buckling to be assumed if d/t <90-£^ (wheres =^235//^ ) to be achieved may be extended to higher limit especially for hollow circular composite beams. The uniaxial bending resistance of hollow composite beams may be calculated according to Eqn.2. However, the plastic axial resistance Nps of the steel cross-section has to be expressed by Eqn.: ^p.=4-^s
(20)
where as is the ultimate stress in steel cross-section and its magnitude depends on the product of values of parameters K,=0.5'E,^AJE,-A,<1
(21)
and K^=fy,'W,,/W If the thickness of external steel tube t > tmin (where tmin = (d /1) ^fy^/E
(22) ) factor KA is taken equal to
unit in spite of real its calculated magnitude. To find the magnitude of ultimate stress in steel cross-section of hollow composite beam three conditions exist. 1. If product KA KW ^ fy the stress GS is to be find from Eqn.: (^s=(^ls'rMa
where
a,, = (4 / 3) • E. • (s, -f s^)
(23)
(24)
S=/./^.
(25)
^,=l-5-/,/^,,
(26)
€^=(or2-0.5'C7,)/E
(27)
759 (28) (29) (30)
plas"tic n e u t r a l
axis
Figure 1: Dimensions of cross-section of hollow concrete filled circular steel tubular beams and position of plastic neutral axis 2. If fy < K^'Kffr < fu , the stress Os is calculated due to Eqn. 23, ois - due to Eqn. 24 and 02 due to Eqn. 29, but instead of fy the product 0.5 (fy + fu) have here to be taken. 3. If product KA KW > fu the stress Os = fudDimensions of cross-section of hollow concrete filled circular steel tubular beam and position of plastic neutral axis are shown on Figure 1 Design eccentricity Cn is calculated according to Eqn. 10. The central angles fixing the position of plastic neutral axis 0, 61 and 02 are calculated so: 6 - due to Eqn. 16, 01 = Arc cos[(d • cos 0 ) /(J - 2 • 0]
(31)
02 = Arc cos[(d
(32)
•cos(d)l{d-2-t-2-tf,)]
In all three these cases the power k in Eqn. 16 have to be found so: (33) Distances Cb and Cg between the centres of compressed areas respectively of concrete core and of steel shell and gross cross-section are: e. =-
3 {d-2'tf
{d-2'tf'sm'(d,-[d-2'(t + tf,)f 'Sin'e^ •(2-01 - s i n 2 0 , ) - [ ^ - 2 ( r + r,)f •(2-02 -sin202)
(34)
760 2 ^5=
'
. 0+e. sin
3-(e + 0,)
d'-(d-2-tf
-'—r-^
2
-T
(35)
d^-{d-l'tf
Reduced thickness of hollow concrete core t^=().5'{d-2't)-^025'{d-2'tf-p't,{d-2't-t^)
(36)
where p = 0.24 oib / Ois • The verification of force equilibrium is carried out due to Eqn.: r ( J - / ) [ ; r - ( 0 - 0 , ) ] = (/?/8)[(c/-2-O'(20,-sin20i)-(^-2/-2/,)'(202-sin202)
(37)
Comparison carried out between the theoretical calculation and test results, own and presented at Matzumoto, Fukuzawa & Endo (1976) shows a good agreement.
CONCLUSION 1. Presented method for calculation of uniaxial bending resistance of hollow concrete filled circular steel tubular beams is in close connection with methods being used for such beams but with solid concrete core. 2. Good agreement between the theoretical and test results allows recommending an abovementioned method to use in design practice of efficient hollow concrete filled circular steel tubular beams.
References Pentti Makelainen & Mikko Maliska. (1997). Design Tubular Composite Columns According to the Finnish Code and Comparison with Eurocode 4. Concrete Filled Steel Tubes A Comparison of International Codes and Practices (Published by ASCCS), 39-58. BS 5400. Part 5. (1979). Steel, concrete and composite bridges. Part 5. Code of Practice for Design of Composite Bridges, British Standards Institution, London, UK Eurocode 4. (1996). Design of Steel and Concrete Structures, Part 1.1, general rules and rules for buildings, DD ENV 1994-1-1, British Standards Institution, London, UK MatzumotoY., Fukuzawa K. & Endo H. (1976). Manufacture and Behaviour of Hollow Composite Members. Final Report ofW^ Congress oflABSE (Tokyo, Sept 6-11. 1976), 389-394. Kvedaras A. (1983). Metal structures of concrete filled tubes. Editorial Publishing Council of the Lithuanian Ministry for Higher Education, Vilnius, Lithuania. (In Lithuanian) Kvedaras A.K. & Sapalas A. (1998). Research and practice of concrete-filled steel tubes in Lithuania. Journal of Constructional Steel Research, 49,197-212.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
761
AN "EXACT" FINITE ELEMENT MODEL FOR THE LINEAR ANALYSIS OF CONTINUOUS COMPOSITE BEAMS WITH FLEXIBLE SHEAR CONNECTIONS Ciro Paella, Vincenzo Consalvo , Emidio Nigro Department of Civil Engineering , University of Salerno Via Ponte Don Melillo - Fisciano (SA), ITALY
ABSTRACT In this paper a finite element model to analyse steel and concrete composite beams with flexible shearconnections is presented. The stiffness matrix of the element is deduced on the basis of the solution of the beam differential constitutive equation expressed in terms of curvature. The proposed mathematical approach permits to analyse structural problems with variable cross-sections and particular restraints utilising a beam discretization due to only structural and load discontinuity. KEYWORDS Steel-Concrete Composite Beam, Shear Connection, Matrix Analysis, Finite Element Method, Displacement Method, Elastic Behaviour INTRODUCTION The behaviour of steel and concrete composite beams is often influenced by the flexibility of the shear connections between the concrete slab and the steel profile. In order to use simple models, the shear connection flexibility can be neglected in the case of full connection design, according to Eurocode 4; in fact this hypothesis allows to consider the preservation of the plane sections (see Johnson (1994), Price & Anderson (1992), Eurocode 4 (1994)). Otherwise, when a partial interaction design is used, it is important to take into account the influence of the connector flexibility with reference both to the deflections and the moment redistribution in continuous beams. An extensive review of the main methods to analyse composite beams with incomplete interaction is reported by Leon (1996). These analysis methods, with reference to the mechanical model, may be classified in two relevant categories: methods for analyses in service conditions and methods for analyses at ultimate limit states. In the first case, the mathematical models generally assume linear constitutive laws for the component materials, the preservation of the plane cross-sections and the compatibility of the vertical displacements of the slab and the steel profile. In addition, if the load on the connectors does not exceed about half their ultimate strength, an elastic load-slip model may be assumed for the shear connection (Johnson (1994)). The elastic theory, due to Newmark & al. (1951), leads to a differential
762 equation of the second order in terms of curvature % or slip s , which has to be solved afresh for each type of loading. Cosenza & Mazzolani (1993) show the solutions of the differential problem for simply-supported beams subjected to relevant load conditions. In order to solve more general cases (continuous beams, various type of loading, variation of geometrical properties and mechanical non-linearities), approaches based respectively on finite difference method and finite element method have been implemented (see Aribert & Labib (1982), Amadio e Fragiacomo (1993)). Dezi & Tarantino (1993) propose a method for the viscoelastic analysis of composite continuous beams based on a step-by-step iterative finite difference approach obtained by the superposition of pseudo-elastic analyses. In the following an "exact" finite element model to analyse steel and concrete composite beams with flexible shear-connection is presented (Faella & al. (1997)). The stiffness matrix and the vector of equivalent concentrated forces of the element are deduced on the basis of the solution of the beam differential constitutive equation expressed in terms of curvature. This choice allows to reduce the approximation degree deriving from the beam discretization utilised in other approaches, as those based on the difference method or on finite element method with approximate shape functions. THE NEWMARK'S MATHEMATICAL MODEL FOR COMPOSITE BEAMS WITH ELASTIC SHEAR CONNECTION In the following it is briefly quoted the differential equation which governs the mathematical problem obtained assuming the hypotheses of the classical theory of composite beams with elastic shear connection (Newmark & al. (1951)). With reference to the notation of fig. 1, the slip s at concrete slab-steel profile interface holds: (1)
'*a,sup
ults: and, deriving with respect to z, it results: , _ du^
du^ d<^ (2) dz dz dz which expresses the variation of the slip s along z as a function of the strains e^ and (e^^-e^/^) concerning the centroids of the steel and concrete parts and of the curvature % . In (2) the shrinkage strain e^/, is introduced as positive value.
Figure 1: Notations for a steel-concrete composite section with flexible connection The equilibrium between the forces in the concrete slab and the steel profile (fig. 1) and the rotation equilibrium around the centroid of the concrete slab give the following equations: Fa=-Fc=F
(3)
763 M^+M^+F-d''
=M
(4)
being F the longitudinal shear force and M the bending moment. Introducing the dependence of M^ and M^ on the curvature, the relation (4) may be written as follows: M=xEcIc+XEaIa^Fd'=X'nabs+Fd*
(5)
where EI^i^^, expressed as the sum of EJ^ and EJ^, represents the flexural stiffness of the crosssection when the shear connection is absent. The same equation shows that the bending moment M is expressed by the sum of a part dependent on the curvature % and of a part dependent on the longitudinal shear force F. Equation (5) allows one to write the following relationships:
X = ^^if^
or F-^i^^^Pt.
E^abs
(6)
d
The assumed linear relationship between the longitudinal shear force per unit length F' and the interface slip s is expressed by the following: F' = ks (7) being k the constant stiffness of the shear connection. Using the compatibility equation (2), the equilibrium equations (6) and the constitutive relationship of the shear connection (7), the model leads to the following H-order differential equation in terms of curvature:
^hbs
where the constant terms a, EA ,d ^2^_k_EI_Ml_
.
£A* EI^j^,
'
E^full
E^abs
are expressed by
^^^.JE^AJJE.A,) E^A^+E^A^
^
^.2 ^EI f^
'
- EI gbs
^^^
EA""
and EIf^^ll represents the flexural stiffness of the beam with full interaction El full =EaIa+ E,I, + E^A^d^^
+ E,A,dl^
(10)
being dQ^ and dQ^. respectively the distances of the steel and concrete centroids from the centroid of the cross-section. THE PROPOSED "EXACT" FINITE ELEMENT FOR COMPOSITE BEAM WITH FLEXIBLE CONNECTION The differential equation (8) holds for beams of constant geometrical and mechanical properties; in addition, its analytic solution is possible only in the case of statically determinate structures and has to be made afresh for each type of loading. In fact, when the beam is statically indeterminate, the bending moment diagram M{z) is a priori unknown and the mathematical problem leads to a differential equation of FV-order in terms of curvature % or to a Vl-order differential equation in terms of displacement v , which results more difficult to solve. In order to obtain an easy matrix procedure able to solve more general cases (continuous beams, various type of loading, variation of geometrical and mechanical properties), it appears very effective to introduce a two-node one-dimensional finite element, characterised by six nodal displacement parameters: the rotation a^, the vertical displacement v,- and the slab-profile interface slip si for each node. The stiffness matrix and the vector of equivalent concentrated forces of the element, necessary for a displacement approach, are derived on the basis of the flexibility matrix and the vector of nodal displacements due to the external loads obtained utilising the solution of the differential equation (8).
764 Therefore, the name "exact" of the proposed finite element means that it is based on shape functions derived from the beam differential constitutive equation. The Flexibility Matrix of the simply-supported composite beam with flexible connection With reference to the simply-supported beam (fig. 2), the matrix relationship between the four nodal displacement (a/ , si , a,- , sp and the respective nodal forces (My , F^ , M; , F-), namely flexibility matrix, may be obtained solving the differential equation (8) with the four different boundary load conditions, in which alternatively one nodal force is equal to 1 and the others hold zero. Starting from the solution of the differential equation (8), expressed as a sum of a particular solution % j of the complete equation and of the general solution XQ of the associate homogeneous equation
x(0 = %o(0 + Xlt)
(11)
in the following two load condition cases will be explained in detail (M,- = 1 and F^ = 1); the remaining two cases will not be treated because of the symmetry of the structural problem.
Figure 2: Nodal forces and displacements of simply-supported composite beam with flexible connection a) First load condition (Mi = 1 , F/ = My = Fy = Oj An unitary concentrated moment is applied on the extreme / of the beam (fig. 3a). The particular solution Xi is: M(z}
z^ 1 ^ 1-EI full
Xl(0 = EI
(12)
full and hence the complete solution of the differential equation is: x(z)=Ci senh{az)+C2 cosh{az)-
1 EI full
1--
(13)
Making use of (6), the boundary conditions in terms of curvatures result: 5C(0)=
M{0)-F{0)d* EI abs
_
1
.
^^^yMiL)-FiL)d\^
EI abs
(14)
E^abs
and, solving with respect to the unknown constants Cj e C2 , finally it is obtained: C, =-
1 EI abs
EI abs - 1 \coth{aL) F7full
C,=-
1
EI abs
EI abs ~EIfull
(15)
765 The end rotations o^ij^. and a,-^f. are then easily deduced applying the Principle of the Virtual Works:
^iMi =jxt)-
L]]
0
1 + 3-
^EI full
L
^jMi
=\l{^)jdz
= -
1 + 6-
6EI full
EI full
EI full EI abs
( coth{aL) aL a^L^
-1
EI abs
(16)a
\ f -1
a L
QLLsen h{aL)
(16)b
The general expression of the interface slip s(z) holds, utilising (6) and (7): ^l^.>^^E'{z)^T{z)-x{z)EI^l,,
(17)
k k'd* which provides the end interface slips s^-^. and Sjj^.: EI 1 — abs EI full
1 ^iMi ~"
Lkd 1
^jMi —
1-
Lkd
^^abs
EI full
{aLcoth{aL)-\)
(18)a
)
1—
aL
seriih{aL)
(18)b
I
Mj=0
Mi=0
J
-3\
3.
1
Ga«
a) unitary concentrated bending moment b) unitary concentrated longitudinal shear force Figures 3a,b: Load conditions on extreme / b) Second load condition (Fj = 1 , M,- = My = F: = 0) In the case of concentrated longitudinal shear force applied on the extreme / of the beam (see fig. 3b), the bending moments are equal to zero over all the beam, because the longitudinal shear force is selfequilibrate (Mj = 0); therefore, the particular solution x^{z) vanishes: Xlf^) = 0
(19)
Making use of (6), the boundary conditions in terms of curvatures result: ^(0)-^(0)-^(0)-^*^ EI abs
d* El^ijg
xW=o
(20)
and, solving with respect to the unknown constants Cj e C2 , finally it is obtained:
c,=-
-coth{aL) EI abs
C, =EI abs
(21)
766 Due to the Betti's Theorem and the structural symmetry, the end rotation aj f. and aj p. are the same of the previously determinate interface slips Sij^. and s,y^.:
Making use of (17), the end interface slips Sj p. and Sj f. are the following: Sj f. = — 'Coth{aL) '
K sen h{aL)
(23)
The remaining two load conditions (M; = 1 and F; = 1) are not explicitly reported because the respective terms of the flexibility matrix may be derived by the previously described ones. In fact, the 4x4 reduced flexibility matrix D^ of the simply-supported composite beams with deformable connection, which represents the relationship between the force vector Q^ = (Mi, Fi, Mj, Fj) and the respective displacement vector 5^ = (a/, 5^, a,-, 5.), may be written as: a i,Fi Dr =
Drl
Dr2
Dr3
Dr4
^i,Fi
\^J,Mi
a i,M i
^i,Fi
^i,M i
^i,Fi
^J>Pi ^JMj
^j,Fj
(24)
where, due to the Betti's Theorem and the structural symmetry, for the 2x2 D^j minors it results: Dr4 = D rl
D'rlr ^ = Dr3 \ = D'r3
(25)
The Stiffness Matrix K of the composite beam with flexible connection By inverting the reduced flexibility matrix D^, the reduced 4JC4 stiffness matrix K^ of the simply-supported beam may be obtained. The 6x6 stiffness matrix K of the unrestrained beam (fig. 4), which represents the relationship between the nodal displacement vector 6 = (v,, a,, 5,, Vj, o^, Sj) and the nodal force vector Q = (Tj, Mj, Fj, Tj, Mj, Fj), may be then deduced on the basis of simple equilibrium conditions.
Mj
'T^
^H
Figure 4: Nodal forces and displacements of the composite beam with flexible connection (unrestrained beam) In fact, the terms which correspond to the forces and displacements of the reduced system are coincident (for example, /:2^2 = ^r(l,l)' ^2,3 = ^r(l,2) ' ^t^-)- The remaining terms, namely those of the first and fourth row and of the first and fourth column of the matrix K, may be expressed in the following way, making use of equilibrium and symmetry conditions:
767 (^2,2+^5,2) ^1.2 =^2,1 = - ^ 4 , 2 = - ^ 2 , 4 = - ^ 1 , 5 ="^5,1 =^4,5 =^5,4 =
L
^1,3 = ^3,1 = - ^ 4 , 3 = -^3,4 = -^1,6 = -^6,1 = ^4,6 = ^6,4 = -
kxx=-k
Al 4 = KA A — ^1,4
4,1
r/i^ v^c/or of the equivalent concentrated
(^2.3+^5,3)
(26)
(^2,1+^5,1)
nodal forces due to the external
loads
Once the stiffness matrix of the beam is known, in order to obtain the vector of the equivalent concentrated nodal forces it is necessary to determine the end displacements of the simply supported beam by solving the differential equation (8) in the presence of distributed external actions. With reference to uniformly distributed vertical load and shrinkage, the diagram of the bending moment along the beam is expressed by:
M(z) =
(27)
^Lz-z^)
The differential equation (8), utilising (27), becomes therefore: X -oc X = -
^ ( L Z - Z ^ )
El abs
kd'^Zsh
(28)
EI abs
2 EI full
whose complete solution is: x{z)=Ci
senh{az)+C2
cosh{az)-
EI full
z ' - L - z - ^
2 EI full
kd
-1
EI abs
a'-El
£ sh
(29)
abs
The boundary conditions may be written utilising (6) and posing % = 0 at the ends of the beam, being the nodal forces equal to zero. The constant C | and C2 hold, therefore: EI full
C, = a^
EI full
Co =-a"-
EI full
I
^Kbs
kd
cosh {aL ) - 1 ^
-1
'hull -1 EI abs
senh{aL)
I
a"^
e sh
cosh {aL ) - 1
EI abs
I
kd*Esh a^
senh ( a L )
J
(30)a
(30)b
EI abs
The end rotations are then determined applying the Principle of the Virtual Works on the simplysupported beam, whereas the end interface slips are derived utilising (17). It is obtained, therefore: ^oi,q
'^^oi,sh
^oi,q "^ ^oi,sh
(31)
^ro = ^oj,q
"*" ^oj,sh
^oj,q "•" ^oj,sh J being:
qL 01,q
"oj,q
2kd
EI full El abs
EL abs IEI full j
12
aH^
24 -+-
cosh (aL)
a'L'
f cosh (at)-
1-al
I
senh (at)
1 IJ
, \ — senh (aL) senh (aL)
(32)a,b
768 respectively the end rotations and slips of the beam due to the external loads and kd
Le.
1
2 ^oi,sh
^ oj ,sh
^sh a
fcosh{aL)-\^ y senh {aL) j aL
(cosh{aL)-\\ y senh (aL) J
(33)
(34)
respectively the end rotations and slips of the beam due to the shrinkage. The equivalent concentrated nodal forces may be deduced by the following compatibility equation system: Dr Qro + 8ro= »
(35)
which gives: Q,„ = - ( D r ) - i - 8 r o = -Kr-5ro
(36)
Therefore, the vector of the equivalent concentrated nodal forces Q„ of the unrestrained beam results:
CONCLUSIONS The proposed finite element model allows to analyse structural problems with variable cross-sections and particular restraints utilising a beam discretization due only to structural and load discontinuity and so reducing the approximation degree with respect to different approaches. Moreover, in the presented model the time-dependent analysis may be performed by using the simple algebraic methods; the concrete slab cracking may be also introduced by considering an appropriate value of the flexural stiffness of the slab over the cracked zone. Finally, the model may be easily developed in order to consider the non-linear behaviour of the shear connection. REFERENCES [1] JOHNSON R.P. (1994), Composite Structures of Steel and Concrete, Blackwell Scientific Publications, Oxford, United Kingdom [2] PRICE A.M., ANDERSON D. (1992), Composite Beams, in Constructional Steel Design, Elsevier Applied Science Publishers, par. 4.1 [3] EUROCODE 4 (1994), Design of Composite Steel and Concrete Structures: General Rules and Rules for Building, Pan 1.1 [4] LEON R.T., VIEST I.M. (1996), Theories of incomplete interaction in composite beams. Proceedings, Composite Construction in Steel and Concrete III, Irsee, Germany [5] NEWMARK N.M., SIESS C.P., VIEST LM. (1951), Tests and Analysis of Composite Beams with Incomplete Interaction, Proceedings, Society for Experimental Stress Analysis, Vol. 9, n. 1 [6] COSENZA E., MAZZOLANIS. (1993), Analisi in campo lineare di travi composte con connessioni deformabili: formule esatte e risoluzione alle differenze, 1° Workshop Italiano sulk Strutture Composte, Trento, Italy [7] ARIBERT J.M., LABIB A.G. (1982), Modele de calcul elasto-plastique de poutres mixtes a connection partielle. Construction Metallique, n. 4 [8] AMADIO C, FRAGIACOMO M. (1993), A finite element model for the study of creep and shrinkage effects in the composite beams with deformable shear connections, Costruzioni Metalliche, n. 4 [9] DEZI L., TARANTINO A.M. (1993), Viscoelastic Analysis of Composite Beams, 7° Workshop Italiano sulle Strutture Composte, Trento, Italy [10] FAELLA C, CONSALVO V., NIGRO E. (1997), Steel and Concrete Composite Beams with Deformable ShearConnections: an "Exact" Finite Element Model, XVI Congresso CTA, Ancona, Italy
Session AlO SPECIAL FEATURES IN MODELLING AND DESIGN
This Page Intentionally Left Blank
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
771
BEHAVIOUR OF A STRUCTURAL SHEET STEEL AT FIRE TEMPERATURES
Jyri Outinen and Pentti Makelainen
Laboratory of Steel Structures, Helsinki University of Technology, P.O. Box 2100, nN-02015 HUT, Finland
ABSTRACT
Extensive experimental research has been carried out during the years 1994-1998 in the Laboratory of Steel Structures at Helsinki University of Technology for investigating mechanical properties of various structural steel materials at elevated temperatures. One part of this research concerned on a structural steel grade S350GD+Z that is a commonly used material in thin-walled steel structures. In this research, the behaviour of the mechanical properties i.e. thermal elongation, modulus of elasticity and yield strength of steel S350GD+Z were examined at elevated temperatures using both transient state and steady state test methods.
KEYWORDS
High-temperature properties, mechanical properties, structural steel, transient-state, steady-state, tensile tests.
772 INTRODUCTION In this project the main aim was to study experimentally the behaviour of the mechanical properties of structural steel grade S350GD+Z. The tests were carried out using transient-state and steady-state test methods. The recorded test results for mechanical properties at elevated temperatures were then compared with those given in Eurocode 3: Part 1.2.
TEST METHODS
Transient-State Method
In transient-state tests, the test specimen was under a constant load and under a constant temperature rise. Temperature and strain were measured during the test. As a result, a temperature-strain curve was recorded during the test. Thermal elongation was subtracted from the total strain. The transient-state test method gives quite a realistic basis for predicting the material's behaviour under fire conditions. The transient-state tests (Outinen&Makelainen (1995)) were conducted with two identical tests at thirteen different stress levels. Heating rate in the transient state tests was 10°C min'. Temperature was measured accurately from the test specimen during the heating.
Steady-State Method
In the steady-state tests, the test specimen was heated up to a specific temperature. After that a tensile test was carried out. In the steady state tests, stress and strain values were first recorded and from the stress-strain curves the mechanical material properties could be determined. The steady state tests can be carried out either as strain- or as load-controlled. In the strain-controlled tests, the strain rate is kept constant and in the load-controlled tests the loading rate is kept constant.
TESTING FACILITIES
Test Pieces
The test specimen having rectangular cross-section was in accordance with the standard EN 10 002-5 (1992). The total length of the test specimens was 142mm and the original gauge length was 50mm.
773 Testing Device
The tensile testing machine and the extensometer used in the research projects is verified in accordance with the standard EN 10 002 (1992). The oven in which the test specimen is situated during the tests was heated using three separately controlled resistor elements. The air temperature in the oven was measured with three separate temperature-detecting elements. The steel temperature was measured accurately from the test specimen using a temperature-detecting element that was fastened to the specimen during heating.
TEST MATERIAL
The studied material was cold-rolled hot dip zinc coated structural steel
S350GD+Z (Z35)
manufactured by Rautaruukki Oyj. Test pieces were cut out from a cold-formed steel sheet with nominal thickness of 2mm, longitudinally to rolling direction. Steel material is in accordance with the requirements of the European standard SFS-EN 10 147.
MECHANICAL PROPERTIES AT ROOM TEMPERATURE
Five tensile tests longitudinally to rolling direction and four tests transversally to rolling direction were carried out at room temperature to determine the mechanical properties of the test material at room temperature. The results from the transient and steady state tests were compared with these results.
The tensile tests were carried out as stress rate-controlled. The stress rate of loading was 0.52 (N/mm^)/s which caused a rate of strain of 0.003 min'^ to the test specimen. The stress-strain curves were used to determine the tensile properties for each test specimen. The results are illustrated in Table 1. TABLE 1 MECHANICAL PROPERTIES OF THE TEST MATERIAL S350GD+Z AT ROOM TEMPERATURE. TEST PIECES LONGITUDINALLY AND TRANSVERSALLY TO ROLLING DIRECTION
Modulus of elasticity E 1 Yield stress Rpo.2 1 Ultimate stress Rm
Longitudinal 210 120 354.6 452.6
Transversal 209400 387.5
452.5
J
774 TRANSIENT-STATE TEST RESULTS Thermal elongation
Thermal elongation of the test material was determined with five tests at load level of 3N/mm^. Test specimen was heated with heating rate of 10°C/min until temperature was 750°C. Thermal elongation was measured during the heating process. The test results are illustrated in Figure 1. An analytical expression was fitted into the average test values by following equation:
Al/1 = 4.8x10'^ Ga^ + O.OOOOiea - 0.00026
(1)
where Al/1
is relative thermal elongation and
0a
is steel temperature.
1,2
100
200
300
400
Test1. Test 2. Testa. Test 4. Tests. -Average values
\
\
\
500
600
700
800
Temperature Og [°C]
Figure 1: Temperature dependence of thermal elongation of structural steel sheet S350GD+Z
Modulus of elasticity
Modulus of elasticity of structural steel sheet S350GD+Z was determined from the stress-strain curves which were converted from the transient state test results. The modulus of elasticity was determined as an initial slope of the stress-strain curves. Test results at temperatures 20°C - 700°C are compared in Table 2 with the values of mechanical properties given in Eurocode 3, in the European
775
Recommendations for the Fire Safety of Steel Structures (ECCS/TC3) and in the Finnish Code of Steel Structures (RakMK B7). TABLE 2 REDUCTION FACTOR FOR ELASTICITY MODULUS OF STRUCTURAL STEEL SHEET S 3 5 0 G D + Z AT TEMPERATURES 2 0 ° C - 7 0 0 ° C
Temperature
Test results
Eurocode3: 1.2
RakMK B7
ECCS/TC3
ET/210 120
ET/210 000
ET/210 000
ET/210 000
(N/mm^) 1 0.9 0,87 0.62 0.4 0.36 0.23 0.1
(N/mm^) 1.00 1.00 0.90 0.80 0.70 0.60 0.31 0.13
(N/mm^) 1.00 1.00 1.00 0.98 0.87 0.56 0.17 0.01
(N/mm^) 1 0.99 0.96 0.92 0.83 0.62 0.17
20 100 200 300 400 500 600
1
700
1
Yield Strength
Yield strength was determined for the test material S350GD+Z from the stress-strain curves based on the transient state test results. Test results for yield stress ao.2 (Rpo.2), based upon 0.2% nonproportional extension are shown in Figure 2. Test results are compared with the values for ao.2 according to Eurocode 3 and with the values according to the Finnish Code (RakMK B6/B7) . The reduction factors for effective yield strength fy given in Eurocode 3 are based upon 2.0% total strain. 400
1
350
1 S350GD+Z
1
Eurocode 3 : Part 1.2
300
RakMK B6/B7
250
*"*v^
200
^ N ^ S
150
N
^
100 50
200
300 400 500 Temperature 8. [°C]
600
700
800
Figure 2: Temperature dependence of yield stress ao.2 for steel S350 at temperatures 20°C-700°C
776 STEADY STATE TESTS
Steady state tensile tests were carried out for the test material S350GD+Z with three identical tests at each temperature of 300°C, 400°C, 500°C and 600°C. Tensile tests were carried out as stress ratecontrolled. The stress rate of loading was 0.52 (N/mmVs which caused a rate of strain of 0.003 min"^ in the test specimen. Furthermore, three steady state tests were carried out at temperature 400°C with stress rate of loading 5.0 (N/mm^)/s.
Stress-strain relationships
Stress-strain curves measured from the steady state tests at temperatures 300°C, 400°C, 500°C and 600°C are compared with transient state test curves in Figure 3.
400 350 CM
E
1 (0 (0
£
0)
^.^;:^^"^
300 n
250
,^'
,.^-'—
200
1
V
150 100 50
^^^^^.^^^ 1
- • •
•
^.y^
fi }C^
if
lur
1
0,2
0,4
0,6
0,8
Stea dy Stea dy Stea dy Stea dy 1
St. St. St. St.
300 400 500 600 1
1 1,2 Strain e [%]
Tr ans.st. 30 o°c —M—Tr ans. St. 4Co°c —•—Tr ans. St. 5Co°c —A—Tr ans. St. 6Co°c 1 1
1,4
1,6
1
1,8
1 y n 1
Figure 3: Measured steady state stress-strain curves for steel S350GD+Z at temperatures 300°C, 400°C, 500°C and 600°C compared with the measured transient state curves
777 The rate of loading in steady state tests has a substancial effect on test results. Measured steady state stress-strain curves with different loading rates at temperature 400°C are compared in Figure 4 with the transient state test results and stress-strain curves of Eurocode 3: Part 1.2.
400 350
,.^--'' -^
300 E 250
I
^ 200 i: 150
Loading rate = 5 N/mm2/s Loading rate = 0.52 N/mm2/s
0)
50
Transient state test
•'•'/
100
EC3:Part1.2
///
0 0,2
0,4
0,6
0,8 -, .1 „,, 1,2
Strain e [%]
1,4
1,6
1,8
Figure 4: Measured steady state stress-strain curves with different loading rates at temperature 400°C compared with transient state test results and stress-strain curves of Eurocode 3: Part 1.2
Mechanical properties of the test material based on steady state data
Mechanical properties of steel grade S350GD+Z determined with the steady state data are given in Table 3. TABLE 3 MECHANICAL PROPERTIES OF STEEL GRADE S350GD+Z AT TEMPERATURES 3 0 0 ° C , 4 0 0 ° C , 5 0 0 ° C AND 6 0 0 ° C BASED ON STEADY STATE DATA
Yield stress
Temperature (°C) 300
Yield stress Go.s (N/mm^) 325.5
Yield stress Gi.o (N/mm^) 362.4
02.0 ( N / m m ^ )
385.9
Ultimate stress fu (N/mm^) 451.0
400
279.8
306.6
335.4
389.1
500
182.1
200.9
214.9
262.8
600
104.9
116.4
136.3
156.5
j
778 CONCLUSIONS
In this research it became very clear that the EurocodeS: Part 1.2 should still be modified and a lot of experimental research should be carried out to have a proper basis on structural fire design of steel structures. The results from the steady state tests showed that the strain rate of the tests should be always in the limits of the high-temperature tensile testing standard. Otherwise the results are not comparable with the others. A suggestion to Eurocode 3: Part 1.2, ENV 1993-1-2, was made on the basis of this project concerning the strength and deformation properties of the studied steel grade S350GD+Z at elevated temperatures.
REFERENCES
European Committee for Standardisation (CENj (1993), Eurocode 3: Design of steel structures, Part 1.2 : Structural fire design, Brussels European Convention for Constructional Steelwork (ECCSj (1983), European Recommendations for the Fire Safety of Steel Structures, Elsevier Scientific Publishing Company, Amsterdam Standard EN 10 002 (1992): Metallic materials. Tensile testing. Parts 2-5: Brussels Outinen, J. & Makelainen, P.(1995j, Transient state tensile test results of structural steels S235, S355 and S350GD+Z at elevated temperatures, HUT/Dept. of Structural Engineering, Publication 129, Espoo Standard SFS-EN 10 147 (1992).- Continuously hot-dip zinc coated structural steel sheet and strip. Technical delivery conditions, (in Finnish), Helsinki Rakentamismaarayskokoelma, object B6 (1989), Ohutlevyrakenteet and B7 (1987), Terdsrakenteet (Finnish Codes of Building Regulations, Codes B6 and B7, Cold-Formed Steel Structures and Steel Structures), Helsinki
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
779
PERFORATED COLD FORMED STEEL C-SECTIONS SUBJECTED TO SHEAR (EXPERIMENTAL RESULTS) R.M. Schuster Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
ABSTRACT Presented in this paper are the experimental results of the tests carried out at the University of Waterloo of perforated cold formed steel C-sections subjected to shear. The objective of the overall research project was to establish an analytical method for calculating the shear resistance of perforated cold formed steel Csections subjected to constant shear. Presently in North America, no specific design rules on this topic are contained in either the Canadian Standard on the design of cold formed steel structural members, SI 36-94, nor in the American Iron and Steel Institute document, entitled. Specification for the design of cold-formed steel structural members -1996 edition. Both of these design documents do provide design rules for the shear resistance of solid web cold formed steel members, i.e., without web openings. A number of experimental tests were carried out in order to accomplish the objective of the overall research project. KEYWORDS Cold formed steel, shear, beams, perforations, holes, openings, C-sections, bending, flexure, testing INTRODUCTION Cold formed steel C-sections are used extensively in wind bearing curtain wall installations, load bearing wall stud assemblies and as floor joists. In general, these sections contain flat-punched perforations centred in the mid-depth of the web elements to accommodate bridging (bracing) channels, as well as, to serve as openings for electrical and service passage ways. In the North American industry of cold formed Csections, the depth of the perforations is either 20 mm ( 3/4"), 40 mm (11/2") or 65 mm (21/2"), v^th the most common being 40 mm (11/2"). The geometry of the perforations is commonly rectangular with the ends being either circular or diamond shaped. Totally circular perforations are also used, but are not as common as the rectangular type. With cold formed steel gaining rapid popularity in the residential construction industry in North America, it has become common practice for certain sub-trades to enlarge
780 existing perforations or to completely create new holes in order to accommodate the usual service requirements of a building. Often, these holes are exceedingly large and are not always placed in the proper position on the web of the section, as well as, along the length of the member. This is causing some concern regarding the structural integrity of the member as designed. Since bending is a common occurrence with these sections, shear is an important design criteria that must be considered by the designer. While the latest edition of the Canadian Standard on Cold Formed Steel Design, S136-94 [1] and the U.S. Specification, AISI - 1996 edition [2] do address the design of solid web members subjected to shear (without perforations), no specific design criteria for perforated web elements subjected to shear are given in either of these North American cold formed steel design documents. OBJECTIVE The overall objective of the research was to develop an analytical method for calculating the shear resistance of perforated cold formed steel C-sections. Contained in this paper are the experimental results only of the tests that have been carried out at the University of Waterloo. The primary objective of these tests was to establish what effect the perforations have on the shear resistance in comparison to the nominal shear resistance of solid web members calculated in accordance with either SI36 [1] or AISI [2]. One solid web specimen was also tested in order to substantiate the current design approach for such members in shear as specified by SI36 [1] and AISI [2]. The primary emphasis of the Waterloo experimental test program was to test a number of different perforation types and depths, using a number of different C-section depths and steel thicknesses. EXPERIMENTAL TESTING A total of 25 perforated beam specimens were tested, i.e., six of Type "B", five of Type "C" and 14 of Type "R". See Fig. 1 for specimen type. In addition, one solid web test was also carried out. Mechanical Properties of Steel Tensile coupon specimens were cut from web elements of the C-sections and machined according to ASTM A 370-88. Testing of these coupon specimens was carried out in the Materials Laboratory of the Mechanical Engineering Department at the University of Waterloo. Pertinent mechanical properties of these tests are summarised in Table 1. Description of Test Specimens All perforated and solid C-sections were shipped by the manufacturers to the Structures Laboratory of the University of Waterloo, where assembly of all the test specimens was carried out. Geometric cross sectional dimensions of the test specimens are given in Table 2. Since all of the specimens had web slendemess ratios larger than 91, stiffening channels were attached to the web elements at each reaction location and at the point of load application. This was done in an effort to prevent web crippling from occurring prior to shear failure. Each test specimen consisted of two C-sections connected together to form a box-beam as shown in Fig. 2. The length of each specimen varied, depending on the size of perforation. Aluminium angles (1 1/4x1 1^4 X 1/8 in.) were attached to the top and bottom of each specimen at approximately 305 mm (1ft) intervals with #12 self-drilling screws in order to prevent premature lateral-torsional buckling between the two C-sections.
781 Description of Test Set-up and Equipment Each specimen was tested as a simply supported beam in the same test frame, as shown in Figures 2 and 3. In order to subject the beam specimen to the largest shear force, the shortest possible specimen span was selected. One concentrated load was applied at the centre of the span, providing a constant shear region on each side of the applied load and a minimum moment at the point of load application. In order to prevent overall lateral-torsional buckling of the box beam assembly, lateral bracing was provided along the length of each specimen by using wooden blocks between two hot rolled channel bracing beams placed along the length of the specimen, one on each side. The test load was applied and monitored by means of an MTS Electro- Hydraulic Servo Control System. To obtain a continuous load-deflection read-out during each test, a Hewlett-Packard 7004BX-Y Recorder connected to a DC displacement transducer was used. Testing Prior to actual testing, each specimen was carefully positioned and aligned in the test frame, as shown in Fig. 3. A (DC) displacement transducer was positioned at midspan to document the load-deflection behaviour of each specimen. See Fig. 2 for schematic illustration and dimensional detail of the loading setup. Loading was applied at a constant rate until failure was experienced. EXPERIMENTAL RESULTS Specimen Designation The foUov^ng specimen designation was used throughout this paper: Example: SPECIMEN B200R-115 Example: SPECIMEN S90R B -Section type (see Fig. 1) S - Solid web 200 -Nominal depth of section (mm) 90 - Nominal depth of section (mm) R - Reinforced web R - Reinforced web 115 - Nominal depth of perforation (mm) Test Results In general, with all specimens, shear was the primary mode of failure. In some cases, local flange buckling was observed as well as web crippling (yielding) at the point of load application, but only after the clearly visible diagonal shear buckling lines had been formed. Fig. 4 shows a close up of specimen B200R-140 after failure and Fig. 5 shows all of the failed B200R specimens. The ultimate experimental test loads for each specimen are summarised in Table 3. CONCLUSIONS Presented in this paper are the experimental results only of the test program carried out at the University of Waterloo. No comparison with any analytical model(s) was made due to the length of paper restriction. ACKNOWLEDGMENTS The author wishes to thank Mr. Steve Fox of the Canadian Sheet Steel Building Institute and Mr. Roger Willoughby of the National Research Council's Industrial Research Assistance Programme for their most valuable help during the course of this phase of the project, as well as, for the financial support given by both the CSSBI and IRAP to have made this research project a reality. As well, the author wishes to thank Colin Rogers and Albert Celli, two former University of Waterloo students, who carried out the testing.
782 REFERENCES 1. 2.
SI36-94, "Cold Formed Steel Structural Members," Canadian Standards Association, Rexdale (Toronto), Ontario, Canada, December 1994. American Iron and Steel Institute: "Specification for the Design of Cold-Formed Steel Structural Members," 1996 Edition, Washington, D.C., June 1997.
NOTATIONS Fy Fu ky Pt V„ Vt
Yield Stress of Steel (MPa) Ultimate stress of steel (MPa) Shear plate buckling coefficient Ultimate total test load (kN) Calculated shear resistance per solid web C-section (kN) Ultimate test shear per C-section web (kN)
Table 1 Mechanical Properties Obtained from Coupon Tests t (mm)
(MPa)
(MPa)
Elongation (%)
1 All B200R
1.20
268
350
31.1
1
1 C200R-40
1.17
328
359
35.4
1
C200R-65
1.21
338
360
36.3
C200R-150
1.17
328
359
35.4
C150R-40
0.880
346
414
31.1
C90R-40
0.950
336
358
40.4
|S90R
0.820
284
350
32.2
1 All R90R
0.820
284
350
32.2
A11R150R
0.890
396
466
32.3
All R200R
1.18
254
311
37.0
SPECIMEN TYPE
Fu
Note: Values in table are average of three tensile coupon tests for each specimen type. Elongation based on 50 mm gauge length.
1
783 Table 2 Geometric Dimensions and Details of Test Specimens All dimensions in (mm)
1 SPECIMEN
t
D
B
d
dp
%
L
Le
a
h
1 B200R-40
1.20
203
41
12
38
102
762
140
381
196
78.9 1
B200R-65
1.20
203
41
12
64
114
787
140
394
196
66.2
B200R-90
1.20
203
41
12
89
156
870
140
435
196
53.5
B200R-115
1.20
203
41
12
114
165
889
140
445
196
40.8
B200R-140
1.20
203
41
12
140
191
940
140
470
196
28.1
B200R-150
1.20
203
41
12
152
219
997
140
499
196
21.7
1 C90R-40
0.950
92.1
41
12
38
117
540
76
270
86.4
24.2
C150R-40
0.880
152
41
12
38
117
540
76
270
147
54.3
C200R-40
1.17
203
41
12
38
117
794
140
397
196
78.9
C200R-65
1.21
203
41
12
64
117
794
140
397
196
66.1
C200R-150
1.17
203
41
12
152
222
1003
140
502
196
21.8
1 R90R-25
0.820
92.1
41
12
25
25
356
76
178
87.2
30.9
R90R-40
0.820
92.1
41
12
38
38
381
76
191
87.2
24.5
R90R-50
0.820
92.1
41
12
50
50
406
76
203
87.2
18.4
R150R-50
0.890
152
41
12
50
50
406
76
203
147
48.1
R150R-75
0.890
152
41
12
76
76
457
76
229
147
35.4
R150R-75
0.890
152
41
12
76
76
914
191
457
147
35.4
R150R-100
0.890
152
41
12
102
102
965
191
483
147
22.7
R150R-115
0.890
152
41
12
114
114
533
76
267
147
16.4
R200R-100
1.18
203
41
12
102
102
508
76
254
196
47.3
R200R-100
1.18
203
41
12
102
102
762
140
381
196
47.3
R200R-125
1.18
203
41
12
127
127
559
76
279
196
34.6
R200R-125
1.18
203
41
12
127
127
813
140
406
196
34.6
R200R-150
1.18
203
41
12
152
152
610
76
305
196
21.9
R200R-150
1.18
203
41
12
152
152
864
140
432
196
21.9 1
0.820
92.1
41
12
-
-
356
-
178
87.2
1 S90R
Note: See Figures 1 and 2 for description of terms. Also, inside bend radius, rj = 2t.
c
-
784 Table 3 Experimental Test Results and Comparisons
K
t (mm)
(kN)
(kN)
(MPa)
B200R-40
1.20
46.9
11.7
268
163
65.7
0.195
6.40
10.4*
B200R-65
1.20
44.9
11.2
268
163
55.1
0.324
6.33
10.3*
B200R-90
1.20
31.1
7.78
268
163
44.5
0.454
6.15
9.96*
B200R-115
1.20
24.4
6.10
268
163
34.0
0.584
6.12
9.90*
B200R-140
1.20
17.3
4.32
268
163
23.4
0.713
6.03
9.77*
B200R-150
1.20
12.6
3.15
268
163
18.1
0.778
5.96
9.65*
1 C90R-40
0.950
14.1
3.53
336
90.9
25.4
0.441
5.75
10.5*
C150R-40
0.880
29.6
7.40
346
167
61.7
0.260
6.53
5.55*
C200R-40
1.17
50.3
12.6
328
168
67.5
0.194
6.32
9.47*
C200R-65
1.21
49.9
12.5
338
162
54.6
0.324
6.31
10.5*
C200R-150
1.17
18.3
4.58
328
168
18.6
0.778
5.95
8.91*1
1 R90R-25
0.820
25.2
6.30
284
106
37.7
0.291
6.30
7.31*
R90R-40
0.820
20.2
5.05
284
106
29.9
0.437
6.18
7.17*
R90R-50
0.820
14.8
3.70
284
106
22,2
0.583
6.08
7.05*
R150R-50
0.890
42.3
10.6
396
165
54.1
0.345
7.44
6.54*
R150R-75
0.890
30.1
7.53
396
165
39.8
0.518
7.00
6.15*
R150R-75
0.890
24.8
6.20
396
165
39.8
0.518
5.75
5.06*
R150R-100
0.890
18.2
4.55
396
165
25.5
0.691
5.71
5.02*
R150R-115
0.890
13.5
3.38
396
165
18.4
0.777
6.56
5.77*
R200R-100
1.18
47.2
11.8
254
166
40.1
0.518
7.72
11.9*
R200R-100
1.18
40.6
10.2
254
166
40.1
0.518
6.40
9.84*
R200R-125
1.18
33.8
8.45
254
166
29.3
0.648
7.31
11.2*
R200R-125
1.18
29.8
7.45
254
166
29.3
0.648
6.27
9.64*
R200R-150
1.18
22.3
5.58
254
166
18.5
0.777
7.00
10.8*
R200R-150
1.18
20.5
5.13
254
166
18.5
0.777
6.17
9.48*
0.820
30.5
7.63
284
106
-
-
6.30
7.31* 1
SPECIMEN
S90R
H=h/t
C=c/t
dp/h
(kN)
Note: * Indicates elastic shear buckling based on Clause 6.4.5 of an unperforated web element[l].
785
V" 1
Specimen Type "B"
t
sa-
4-^r
Specimen Type "C"
, —
^
^
-
-
T Specimen Type "R"
Figure 1: Specimen I^pes
786
i
U—Lc-*^
--(E3J'-
m
4- D
K Stiffening channew^^^^^ JJl^LL n = 76.2 mm (3") 1^"^
h^^^^
71 C
D h
LJ
n=2t
i
J
T c
i t.- Lil
-B-H f*-B-150 mm (6 in.)— Section A-A Note: See Table 1 for specimen dimensions
SO.SiiMn
(2") •l2.7inm(l/2-)
t= 1.92 nmi (0.076")
Stiffening channel
Figure 2: Schematic of Typical Test Set-up and Specimen Cross-Section
787
Figure 3: Photograph of Test Frame Set-up with Typical Specimen in Position
Figure 4: Photograph of Failed Specunen B200R-140
788
Figure 5: Photograph of all FaUed B200R Specunens
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
'789
CARRYING CAPACITY OF GIRDERS WITH CORRUGATED WEBS H. Pasternak and P. Branka Department of Steel Structures, Brandenburg Technical University of Cottbus, PF 101344, 03013 Cottbus, Germany
ABSTRACT Girders with corrugated webs are an economic alternative to hot-rolled and plate girders. In comparison with conventional plate girders, they have the advantage to avoid the costly welding of web stiffeners which combined with automatic welding between the flanges and the web results in a very economic solution. The German guideline DASt-Ri 015 [1] is the only document which contains design rules for girders with trapezoidally corrugated webs. Certain knowledge about the behaviour and specific design rules for girders with sinusoidally corrugated webs are not available. For this reason, tests and finite elements calculations were carried out at the BTU to obtain the necessary informations about the behaviour of such girders. Tests on simply supported girders with sinusoidally corrugated webs and FE analysis showed, that the shear capacity of such girders estimated according to DASt is much lower than this from the experiment. Based on the tests results and FE analysis a new buckling curves and simple design procedures were developed.
KEYWORDS corrugated web, web buckling, web yielding, girders with thin webs
PRODUCTION AND APPLICATION Girders with a sinusoidally corrugated web (Figure 1) are produced automatically by the Zeman & Co. GmbH, Austria, using steel S235 and S355. The maximum height of the girders is 1500 mm, its minimal web thickness is 2 mm.
790
A-A
corrugated web
i
B-B
1
/\^-\..-N./>...-\..-3M/04^ J5L W
Figure 1: Girder with sinusoidally corrugated web TABLE 1 DIMENSIONS OF GIRDERS WITH SINUSOIDALLY CORRUGATED WEBS
b ti
Flange [mm] 200 - 430 10-30
h t hp
w
Web [mm] 500,625,750,1000,1250,1500 2,0 2,5 3,0 38 39 40 155
The production includes the following steps: the sheet (web) is removed from the coil, cut, profiled (Figure 2) and finally the flanges are welded to the web. The welding speed is 1 - 2,5 m/min.
Figure 2: Production line
791 Girders with a corrugated web may be used as rafters and columns of fi-ames or as crane runway beams. The behaviour of the corrugated web requires special constructional details, e.g. closed to the crane bracket additional web stiffeners seem to be necessary. In case of a rigid knee joint, a conventional (plane) web plate is used. The alternative with hinged beam-to-cohimn joints is shown in Figure 3.
Figure 3: Columns and frames under construction
RESEARCH Tests Three tests were carried out. The girder and the test arrangement are shown in Figure 4, the measured girder dimensions and some material properties are listed in Table 2. The web aspect ratio is a = a/h = 1. A-A
¥i
m + ^
Figure 4: Layout of the test girder
+t2
792 TABLE 2 DETAILS OF THE TEST GIRDER NO. 2
web details
flange details
h t E a b, b2 ti t2 fy [mm] [mm] [mm] [N/mm^] [N/mm^] [mm] [mm] [mm] [mm] 225 1470 1501 2,1 180000 250,2 250,3 12,4 12,5 The first two tests were conducted under monotonic loading, while the third girder was subjected to cyclic loading up to 480 kN. The number of cycles was equal to 500. After that the experiment was continued under monotonic loading. The applied load F and the central deflection v of the girder were monitored during all tests. The load deformation curves and the failure mode are given in Figures 5 and 6.
1000 /FEM
h
800 h
600
\ 1 Y TbstNo. 1 rest No. 12 \ 1
[ 400 200
Test N4. 3 \
//
\^M —
i
— -« 4
—• 6
8
10
Deflection V [mm] Figure 5: Load-deflection graph
Figure 6: The test girder No. 2 after failure The tests lead to three conclusions: local buckling does not appear, even for h/t = 750, the capacity under cyclic loading differs only 3 per cent from the value under monotonic loading, the buckling of the web occurs first after reaching the load capacity.
12
793 Finite elements calculations In addition, geometrically and physically non-linear finite elements calculations using ABAQUS were carried out [4]. The load deformation curve and the failure mode are given in Figures 5 and 7. From the stress distribution in Figure 8 follows that the web does not fail by buckling, but by yielding. Local buckling occurs only in case of extremely thin webs (not relevant in practice).
F = 814 kN, v = 4,5 mm
F = 790 kN, v = 5,1 mm
Figure 7: Deformation plot of the test girder no. 2 (magnified by 20) 225.0 210.1 195.1 180.2 165.3 150.3 135.4 120.5
105.5 90.62 75.68 60.75 45.82 30.89 15.96 1.024 I
Figure 8: Von Mises stresses (N/mm^) of girder no. 2
DESIGN PROPOSAL Figure 9 summaries the obtained results and gives a safe proposal for the reduction factor K^ g function of the relative slendemess Ip.
794 1,2 1,1
K,= 1.0
] 1,06-
_
0,9
^
0,8 •
\-
0,2 0,1
[
0,0 0,0
ik,
^ ^
^
FEM a = 3
V
Test
^ — p -
Nr As -f-
FEM a = 1
I 0,5 I 0,4 L Pi 0,3 \-
• • m
1
-
I 0,7 «S 0,6
(nm
^>t-vN>
j
1 / Xp»-5
- o i i 0,2
DASt-Ri015 1 . i . 1 0,4 0,6
V''
-
f
"•~T~"
— —
-D-
1
0,8
1,0
L....
i 1,2
•
1,4
1,(
1,8
Relative slenderness Xp [-]
Figure 9: Buckling curves The shear capacity of girders with sinusoidally corrugated webs may be obtained from Eqn. 1: VR,g,d = 0,58-K:,-fy,d-h-t
(1)
Tpi = ^ . ( D . - D 3 ) 0 , 2 5
(2)
with Kr-
and
1,0
Ip
f3Z
where Tpi critical buckling stress, Dx, Dy - stiffness of the orthotropic plate [2].
CONCLUSION Considering the favourable behaviour and the automatic production, girders with a thin, sinusoidally corrugated web are an economic alternative to hot-rolled and plate girders.
REFERENCES [1] [2] [3] [4]
DASt-Richtlinie 015: Trdger mit schlanken Stegen. Stahlbau-Verlagsgesellschaft, 1990. Easley J.T. (1975). Buckling Formulas for Corrugated Metal Shear Diaphragms. Journal of the Structural Division July 1975,1403-1417. Lindner J. and Aschinger R. (1988). Grenzschubtragfahigkeit von I-Tragem mit trapezformig profilierten Stegen. Stahlbau Vol. 57, 377-380. Pasternak H. and Branka P. (1998). Zum Tragverhalten von Wellstegtragem. Bauingenieur 73, 437-444.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
795
AN EXAMINATION OF WEB CRUSHING IN THIN-WALLED BEAMS. Rhodes J \ Nash D^ and Macdonald M^ ^ Department of Mechanical Engineering, University of Strathclyde Glasgow Gl IXJ, Scotland, UK ^ Department of Engineering, Glasgow Caledonian University Glasgow, Scotland, UK
ABSTRACT This paper briefly describes an initial investigation into the collapse behaviour of webs of thin-walled beams under concentrated loads. The general methods of analysis of this behaviour given in some current cold formed steel design specifications are outlined. Theoretical examinations of the buckling behaviour of channel sections under two flange loading are derived on the basis of a simple energy analysis and compared with the predictions of a finite element package. The development of a design procedure which makes use of the buckling analysis together with the Perry Robertson interaction formula to take account of material yielding, the effects of corner radii and imperfections is described and comparisons with experimental results are shown.
KEYWORDS Webs, crushing, failure, local buckling, thin-walled, beams, finite strip, finite element, design, specifications, flanges.
INTRODUCTION Web crushing at points of concentrated load or reaction in thin-walled beams is well known to be a significant problem and is of high importance in the field of cold-formed steel members, as such members are generally not stiffened against this type of loading. At points of concentrated loading , or at supports, severe lateral loading can result in a local buckling type failure in the web as illustrated in Figure 1. There are two types of loading and two different load locations which are of prime interest. The different types of loading are (a) Loading through one flange only and (b) Loading through two opposite flanges at more or less coincident locations on the beam. The two different load locations which are often studied separately are loads applied near an end of a beam (sometimes termed exterior loading) and load applied within the span of a beam (termed interior loading). In general loading applied near the end of a beam is more severe than the same load applied in the interior.
796
Section A-A Figure 1. Web crushing at a support point Web crushing in cold formed steel beams has been studied extensively in the United States, (e.g. (1)(4)) and in Canada, for example (5), (6). Through these researches design specifications dealing with the web crushing, or web crippling, capacity of beams under various conditions have been set up, based mainly on an experimental background. The design expressions used in most cold formed steel specifications are empirical in nature, as there is a large number of variables involved in the resistance to web crushing. There have been many attempts also to apply theoretical analysis to these problems, and the buckling behaviour of webs in the presence of localised loads. Of note among these is Khan and Walker (7). In recent years also the plastic mechanism approach has been used by different authors, e.g. Bakker and Stark (8), Setiyoni (9), Young and Hancock (10), to provide alternative formulations to the solution of web crushing problems. However, in cold-formed steel design the specifications are at present empirical. The AISI Cold-Formed Steel Specification (11), for example, has equations dealing with a wide variety of conditions. The corresponding British Specification (12) has its web crushing rules completely based on those of the AISI specification, and thus uses substantially similar equations. Two main drawbacks to the empirical nature of the design rules are as follows:- (i) the rules are strictly confined to the range for which they have been proven, and (ii) it is often difficult to ascertain the engineering reasoning behind the different parts of the rather complex equations. It is perhaps because of this second point that in the final stages of the development of the new Eurocode dealing with cold formed steel member (13) the UK/AISI rules have been adopted, but applied to different conditions than those for which these are used in the AISI or UK specifications. In the Eurocode dealing with hot rolled steelwork (14), however, the behaviour of webs under localised transverse loads is treated as a buckling problem, incorporating an effective width of compressed web together with the European column curves. While the extension of such an approach to deal with members of general shape is perhaps a difficult task, the use of powerful analysis methods such as the finite element method provides at least one basis of approach to the derivation of more theoretically based design rules for cold formed sections. A good start for the development of such rules is to ascertain how the rules setup compare with the existing empirical design rules, as these are known to have reasonable accuracy within their range of application.
797 EXISTING DESIGN RULES In this paper we shall consider only interior load, i.e. loading applied within the span of a beam. We shall further confine the considerations to two flange loading, as shown in Figure 2 in the case of a channel section beam. For two opposite loads or reactions as shown the British cold formed steel code gives the following expression, derived from the AISI code with safety factors omitted:P^ = t^k (122-22k)(l06-.06
r / 0 ( 0 . 7 + 0.3(0/90)'){4800 - l4{D/t)}x{l
tftft
rffff
+ 0.0013(A^/0}
(1)
Load applied at end of radius End elevation
Figure 2. Two opposite lateral loads or reactions acting on web of a channel section. In the above expression t is the material thickness, k is a measure of the material yield strength, i.e. k= py/228 where py is the material design strength in N/mm^, r is the corner radius between web and adjacent elements and 9 is the angle between the plane of the web and the bearing surface. The rules indicate that the web crushing capacity is dependant upon material thickness, the yield strength, D/t, N/t and r/t ratios and the angle of the web from the vertical. The dependence on D/t and N/t does not appear, at first glance, to be very significant, but as these ratios can be quite large the first glance may be rather deceptive. The rules have been validated within certain limits, for example in the British standard it is stated that D/t should not be greater than 200 and r/t not greater than 6. Because of the empirical nature of the rules, stepping outside the validated region leads into the unknown
THEORETICAL INVESTIGATION OF TWO-LOAD CASE To gain some insight into web crushing behaviour in beams under two flange loading two theoretical approaches were initially considered. Firstly finite element analyses of a channel section beam, loaded as shown in Figure 2, was carried out using the finite element package ANSYS. In the package the bifurcation buckling analysis was employed and the buckling loads obtained were used for comparison with the results of a rather crude and simple energy buckling analysis of channel type sections under web crushing conditions. Thereafter the energy approach was used in a web crushing analysis with column curve "c" of the Eurocode. This curve was used since the Eurocode itself (13) suggests curve "c" together with an effective width of web. In the finite element analysis beams of unit (1 mm) thickness were considered and depths of 50, 100, 150 and 200 mm were examined. For the bulk of the channels investigated a set geometry of flange=web/2, lip=flange/4 was employed, but a specific investigation of lip width was also developed. In the case of plain channels, i.e. with no lips, the deflected forms at buckling were as shown in Figures 3. This figure shows a three-dimensional view of the section, illustrating the localisation of the buckling (close to the load application) and the symmetry of deformations.
798
Figure 3. Three dimensional view of plain channel section under web buckling For lipped channels, as shown in Figure 4, the symmetry of deformations is again evident, but in this case the buckled form shows that although there is localisation of deformations of the buckle in the loaded web, the lip deflections are not localised, and the deformations take place over a much larger length of beam at the lips than in the web.
Figure 4. Three dimensional view of lipped channel section under web buckling An energy analysis was set up to provide a relatively simple, if rather crude, model of the behaviour of a channel section under two flange loading which could be examined without the need to continually consult the finite element package. For the channel section shown in Figure 5 the analysis assumes deflected forms as follows:Web:-
w = A | s m ^ + C s m — r ^ I cos
Flanges:-
. = A ^ ( l + 3 C ) ^-k\^
2\
Lips:-
, = ^i^(l+3C)(l-A:)cos^ — d a
(2)
, KX
(3)
(4)
799
Figure 5. Co-ordinate system for channel section. Substitution of these expressions into the energy equations for plates, performing the necessary integrations and applying the Principle of Minimum Potential Energy yields an expression for the magnitude of the stress applied at the loading points to cause buckling. The magnitudes of a, C, and K can be adjusted either as part of the analytical process, or numerically, to obtain the least value of the applied stress, and applied load. Per, to cause elastic web buckling. The dimension a which gave the minimum buckling load was in almost all conditions just over twice the web depth plus the load length. The buckling loads so derived are purely approximations to the elastic buckling load, and do not necessarily describe the web crushing behaviour, which is influenced by buckling, yielding, imperfections etc. To use the buckling loads derived as an aid to finding the web crushing load the combination of the derived elastic buckling load with the Perry-Robertson approach as used in ECS:Part 1.1 would seem to be a potentially useful method. It could be expected that the nature of web buckling is such that there is little postbuckling reserve capacity, and so the Perry-Robertson approach is apt. Thus, having obtained Per and Ps the web crushing capacity can be derived as follows:(5)
Pc=X^Ps but
;^ < 1
where
X=
(6)
with
(p = 0.5[l + r ] + I ' ]
(7)
~^ = {PslPj'
(8)
In ref. (10) it was established that for very compact channels with small web depth to thickness ratios plastic mechanism analysis gave good results using the plastic moment capacity of a length of web obtained using an assumed load dispersion angle. This suggests that for such channels the Perry factor r| should be taken as equal to 4(r+t/2) where r is the inside bend radius, and that in evaluation of the squash load, Ps, the full buckle width a corresponding to Per should be used. These were incorporated into the analysis, and the Perry factor and buckle width were obtained using these considerations for all beams examined. The web buckling loads Per obtained on the basis of Equations 2-3 agreed very well with the Finite element results for plain channel sections, being generally of the order of 5%-10% greater than the finite element results. In the case of lipped channels the effects of lip bending are dealt with using equation 4. It was found that, mainly because of the changes in wavelength from web to flange free edge caused by the lip, the inaccuracy of the simple approach increased with lip width. To take this into account, and give a fairly
800 good agreement with the finite element analysis, an effective width of lip was used in the above equations. The effective lip width used was as follows:bi =\Ob,l{bi
+10)
(9)
The use of this effective lip width produced very good correlation between simple analysis and the finite element results for the buckling load.
COMPARISON OF CALCULATED CAPACITIES WITH DESIGN CODE RULES AND WITH EXPERIMENTAL RESULTS The theoretical analysis suggests that there can be substantial differences in the buckling loads calculated for plain and lipped channels, whereas design codes tend to suggest that these can be covered by the same formula. There are available test results on channel sections, and here we can examine two investigations into plain channel sections. Some previous experiments available to the authors on channels under two flange interior loads are those of Ref (9), although the experiments here on this topic are limited to a set of 18 tests on plain channels, with material thickness of 1.11 mm, flange widths nominally 30 mm and web depths of 70 mm and 100 mm nominal magnitudes. The average material yield strength for the specimens tested was 315 N/mm^. Comparisons of these test results with the predictions of BS 5950: Part 5 and the energy method described previously are shown in Figures 6 and 7.
^
o
«»o | 4 O •a
Calculated —
BS5950:Pt5
o Experiment
1
20
30
40
50
Bearing Width N mm
Figure 6. Comparison of predicted capacity with Experiments of Ref (9) - 70mm deep specimens In Figure 6, which deals with tests on 70mm deep specimens the energy approach gives conservative results in comparison with experiment while BS 5950:Part 5 gives overestimates of the experimental capacity.
801
4 h
—Calculated
^
-—BS5950:Pt5
2 Y
•
Experiment
•
30
20
40
50
Bearing Width N mm
Figure 7. Comparison of predicted capacity with Experiments of Ref (9) -100mm specimens In Figure 7, while the BS 5950: Part 5 curve still overestimates the experimental results, the energy approach again underestimates the experimental results. Ref. (10) also reports a number of tests on plain channels, mainly of compact dimensions, under two flange interior loading, and gives full details of cross sections and properties. A comparison of these results with the values calculated using the method described here, and those obtained using the British Code, BS 5950:Part 5 are shown in Figure 8. In this figure the calculated results and the experimental
1.4 r 1.2 I-
o
1 liJ X ^ 0.8 UJ
datum
0.6 0.4 |-
1
^xx
o
Pexp/Pcalc
X
Pexp/Pcode
0.2 I0 40
80
D/t
Figure 8. Comparison of Experimental results from Ref 10 with calculated and code values results are in fairly good agreement over the range of D/t ratios from 20 to 65, whereas the experimental results become very substantially less than the British code predictions as the D/t ratio reduces. It must be emphasised here that this is not simply a criticism of the British Code. These rules were originally taken from the AISI code, and are now also incorporated in EC3:Part 1.3. Thus all of these codes are non-conservative in the area discussed.
802 SUMMARY AND CONCLUSIONS The use of buckling analysis with appropriate consideration of plastic behaviour would seem to give a reasonable assessment of the web buckling of thin-walled cold formed channel sections under two flange interior loading as shown here. There is no obvious reason why this approach cannot be extended to produce a general method of web crushing analysis, suitable for all sections and loading conditions. For the particular loading condition examined here current design specifications seem to be rather nonconservative.
REFERENCES 1. Winter G. and Pian R. H. B.(1945). Crushing strength of thin steel webs. Cornell Bulletin 35. Pt. 1 2. Zetlin L. Elastic instability of flat plates subject to partial edge loads. Journal of the Structural division, ASCE, Vol. 81. 3. Hetrakul N. and Yu W. W. (1979) Cold-formed steel I beams subjected to combined bending and web crippling. Thin-Walled Structures. Eds J Rhodes and A. C. Walker, Granada Publishing. 4. Yu W. W. (1981) Web crippling and combined web crippHng and bending of steel decks. Civil engineering study 81-2, University of Missouri-Rolla. 5. Wing B. A. and Schuster R. M. (1981) Web crippling and the interaction of bending and web crippling of unreinforced multi-web cold formed steel sections. University of Waterloo, Canada. 6. B A Wing and R M Schuster. (1986) Web crippling of decks subjected to two-flange loading. Proc. Eighth Int. Conf on Cold-formed Steel Structures, St. Louis, USA, 371-402 7. M Z Khan and A C Walker. (1972) Buckling of plates subject to localised edge loading. The Structural Engineer, Vol. 50, No. 6. 225-232 8. Bakker M. C. M and Stark J. W. B. (1994) Theoretical and experimental research on web crippling of cold formed flexural steel members. Thin-Walled Structures Vol. 18, No. 4 9. Setiyono H. (1994) Web crippling of cold formed plain channel steel section beams. Ph. D. Thesis, University of Strathclyde. 10 American Iron and Steel Institute. (1996) Specification for the design of cold formed steel structural members. 11 British Standards Institution (1987). BS 5950: British Standard for Structural use of steelwork in building: Part 5. Code ofpractice for design of cold formed sections 12 CEN ENV 1993-1-3: 1996. Eurocode 3: Design of Steel Structures. Part 1.3: General Rules. Supplementary rules for cold formed thin gauge members and sheeting 13 CEN ENV 1993-1-1: 1992. Eurocode 3: Design of Steel Structures. Part 1.1 General rules and rules for buildings. 14 British Standards Institution (1985). BS 5950: British Standard for Structural use of steelwork in building: Part 1. Code of practice for design in simple and continuous construction: hot rolled sections.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
803
EXPERIMENTAL INVESTIGATIONS AND MODELLING OF STEEL GRIDS Piotr Konderla^ and Jakub Marcinowski^ ^ Institute of Civil Engineering, Wroclaw University of Technology, 50-370 Wroclavs^, Wybrzeze Wyspianskiego 27
ABSTRACT Flooring grids made of steel rods of rectangular cross sections welded together are analysed experimentally and theoretically in the paper. The grids of a square pattern, hexagonal pattern and restrained hexagonal pattern are considered. The details of experimental analysis and its results are presented in the paper. Three load cases were taken into account: knife loading along midspan, the concentrated load on a 200x200 mm punch applied at the centre and the concentrated load on a 200x200 mm punch applied at the midspan of a one free edge. On the basis of the experimental results the theoretical, discrete model of the grid was created. To reduce the static model of a structure to the linear discrete model, the method of homogenisation was applied. The analysis was confined to the physically and geometrically linear range. As the result of the procedure the comparatively easy but very accurate manner of Idad carrying capacity assessment of such kind of grids was proposed.
KEYWORDS Flooring grids, catwalks, static analysis, experimental investigations, homogenisation, discrete theoretical model.
INTRODUCTION Steel flooring grids being a part of typical catwalks are subject of interest in the paper. Grids are made of steel rods of rectangular cross sections welded together creating characteristic mesh pattern. The most frequently used mesh patterns are: square pattern (grid Kl), hexagonal pattern (grid K2) and restrained hexagonal pattern (grid K3). The typical grid is supported along two opposite edges and two other are free. Li spite of the apparent simplicity of this structural member it is not easy to estimate its load carrying capacity for various load cases by a simple formula. The main objective of this paper is creating the discrete theoretical models versatile enough to tackle various load cases and various bars arrangement accurate enough to fiilfil requirements of engineers.
804 Discrete calculation models presented in the paper were created as a result of experimental analysis and an attempt of estimation of load carrying capacity of a grid in the simplest by accurate way. The main idea of the adopted approach lies in the stiffiiess homogenisation in both direction of the grid plane (Sokolowski (1957), K^czkowski (1980)). The analysis was confmed to the physically and geometrically linear range. Details of the calculation models were presented step by step. Results obtained by means of the proposed models coincide quite well with experimental results. Some examples of comparisons are inserted. Theoretical models are general enough to take into account other cases of loading. A comparatively simply form of coefficients of governing equations makes possible optimisation of grids as far as mesh patterns are concemed. The details of the test stand and experimental analysis are presented in the paper as well. A large number of steel grids of various mesh patterns were examined experimentally. Grids of 1x1 m made of rods of various cross sectional dimensions, supported along two opposite edges were considered. Three load cases were taken into account: knife loading along midspan, the concentrated load on a 200x200 mm punch applied at the centre and the concentrated load on a 200x200 mm punch applied at the midspan of a one free edge. Three grids of every kind were investigated. All grids were taken from the manufacturer and have been chosen by random way.
EXPERIMENTAL INVESTIGATIONS The scheme of the test stand presenting its main components was shown in Figure 1. The stand is composed of a rigid frame (1) attached to the steel base (2). Two beams of I-section (3) stabilised by four bars (4) fastened with screws to webs of I-beams were placed on the base. On these beams grids being examined in experiments were supported. The force was applied by means of hydraulic jack (5) one end of which was attached to the horizontal beam of the frame whereas the other was put on the top of a force transducer (6). The force transducer in turn was placed on the rigid square plate (7) resting on the examined grid (8). hi the first case of loading this plate was placed in the edge as shown in Figure 2a. hi the second case of loading the plate was placed exactly at the centre. Li the third case of loading the load was applied to the grid via the beam resting in the middle of the grid span.
Figure 1: Scheme of the test stand
805
Figure 2: Load cases and mesh patterns of grids considered As far as a measurement, apparatus is concerned besides the force transducer also inductive displacement, transducers (9) fastened to the base of the stand were used. In both devices change of voltage after appropriate amplification was registered by analogue measurement, instrument, and then the same signal was converted into digital form by analogue-digital card installed in the computer. The computer program prepared deliberately for the purpose of this experiment made possible automatic registration of data and their further elaboration in form of graphic plots and data files. The smgle experiment, was carried out. as follows. Hand operated hydraulic pump was put. in motion for a moment. The piston of a hydraulic jack started to move exerting a pressure on the grid via the force transducer and the square plate. The accompanying force and the displacement, being the result, of the force action were registered just by pressing a key on the computer keyboard. Then the hydraulic pump was again put in motion for a moment and the resulting force and displacement were registered once again. In this step-by-step maimer the load process was continued till the load-displacement, characteristic observed on the comput.er monitor started to curve significantly. It. was the signal that, the physically nonlinear range was reached. From this very moment, the return way started. The pressure in hydraulic pump was gradually reduced by means of the valve in hydraulic pump. The return way was also accomplished in the step-by-step manner. The experiment ended at the moment, of a total reduction of the jack exertion. The course of experiment, was the same in every case of loading and in every case of the analysed grid.
PHYSICAL MODELS OF FLOORING GRIDS Beam Models The single beam model of flooring grids was constructed with adoption of the following assumptions: a) the flooring grid of rectangular projection is simply supported (hinge like mode of supporting) on two opposite edges, b) the flooring grid is made of repeatable members equally distributed along x axis, c) thQ members are made of straight, or bent, bars of rectangular cross section dxh, d) the loading is symmetrical with respect to x axis, e) the problem is physically and geometrically linear. The homogenisation of the stif&iess of flooring grid was made according to the above assumptions. The bending stiffiiess Ely of a single grid element, and the equivalent, stif&iess EIx of couplers were determined. The beam of stif&iess EIx resting on spring supports of stif&iess ky (Figure 3) was assumed as the model of flooring grid. The stif&iess coefficient, ky is equal to the point, load applied to a single member of a grid causing the unit, deflection of a member.
806 Physical model of the grid Kl The grid is composed of straight flat steels connected laterally by coupling rods (both create square mesh pattern). It is being assumed that every single bar of a grid works independently. The influence of coupling by lateral bars is neglected (Figure 4a). Parameters of the model are:
£;/,=0; EIy=EIo;
ky =
ABEIo'
(1)
EI,
'_yA.±±±±±±± Figure 3: Beam model of the flooring grid 1^
•l^
dh' Elr^ = 12 single bar
where
is the bending stif&iess of a
»l
VH-
'
Figure 4: Repeatable elements of theflooringgrids Physical model of the gridK2 The set of two flat steels shown in Figure 4b is repeatable element of the grid. The equivalent stiffiiess was obtained by comparing the deflections of a section of a segment of actual beam with the deflection of a prismatic beam of Ely stif&iess. Both members were loaded by couple of moments applied at the ends as it was shown in Figure 5a. From the static analysis we obtain
_Jo_ ^EI,
b+
2 cos a(Y sin^ a + cos^ a)
w.
JL BEI^
(2)
807 M
M=l
s' M=
M
M=\
h—"+"—H M 1 / ' ~ > _ _ _ _ £ ~ N M=l
Figure 5: Equivalent stiffiiesses of grid elements
where y =
GKo _ EIQ
2d^ (l + v)h^
Comparing deflections of both systems and taking into account the following relationships: c = / c t g a , 6 = (ZQ -fctga)/2, 0 = 2/'/Zo, one obtains
EIy=2EIo
sin a(y sin^ a + cos^ a) Y sin^ a(sin a - S cos a) + sin a cos^ a + d(l - cos^ a ) '
(3)
and ^ -
48M. J3
(4)
The equivalent stiffiiess of the single coupler was obtained in analogous way (Figure 5b) Ell - ^h sina(YCOs^ a + sin^ a)
(5)
and similarly the stiffiiess of the beam of this model _2Li EI^ L . , 2 • 2 \ EI, = = — sm a(Y cos a + sm a). " 3 2Zo 3Zo
(6)
808 Physical model of the gridKS In the case of this grid the one straight flat steel and two bent flat steels as m the grid K2 create repeatable member of the grid (Figure 4c). The stiffiiess of the model of this grid was determined in analogous way as in the case of the grid K2 and hence
EIy=EIo
1+
2 sin a(y sin a + cos^ a ) Y sin^ a(sin a - 0 cos a ) + sin a cos^ a + ^(1 - cos^ a )
(7)
(8)
S/_ = —siaa(ycos^ a + sin^ a).
(9)
Plate model of the grid The grid was treated as the orthotropic plate. Plate stiffiiesses were taken as average stiffiiesses of the grid in x and;; directions. If the plate stif&iess matrix is denoted by D
0 'D„ D^ D : Dy. D„ 0 , 0 0 ^G^\
(10)
and then D "
= ^ ; D = ^ , 2Zo' "^ 2f '
D^
= Dy^ = 0
(11)
The stiffiiess Gxy was obtained as a result of the analysis of the static scheme shown in Figure 5c. The numerical solution of the orthotropic plate was obtained by means of our own computer program in which FEM was exploited (Zienkiewicz (1989)).
VERIFICATION OF PHYSICAL MODELS The physical model prepared allows us to find relationship between a particular loading of the grid and a corresponding deflection within elastic range of deformations. In order to verify the correctness of models, grid displacements obtained as a result of application of these models were compared with results obtained in experiments. In calculations it was assumed that three load cases were taken into account: - the load uniformly distributed along midspan line ("knife loading" along midspan), - the uniformly distributed load on a 200x200 mm punch at the centre, - the uniformly distributed load on a 200x200 mm punch at the midspan of a free edge. During the experimental investigations displacements were measured at 15 measurement points for :r = {0, ± 250, ± 500} and y = {0,± 250} . For the grid of every kind and for every case of loading
809 experimental investigations were performed for the series of three grids. The results presented are average values from every series. At the same points displacements were calculated by means of both theoretical models: the beam model and the plate model. In Figure 6 displacements of the grid center as a function of the load P for the „knife" loading (cf Figure 2c) were shown. Displacements of the midspan section {y = 0) of the grid K3 for the punch loading at the center (cf. Figure 2b) was shown in Figure 7, whereas displacements of the same section of this grid evoked by the punch applied at the free end (cf Figure 2a) was shown in Figure 8.
jQ
^• y<'
experiment beam model plate model O X A
/j = 25 /2 = 30 /2 = 40
w [mm] 1
2
3
4
Figure 6: Displacement of the central point
Figure 7: Deflection of the midspan line
5
no
beam model plate model O h = 25 X h = 30 A /2 = 40
-500
-250
0
250
500
Figure 8: Deflection of the midspan line
CONCLUSIONS It follows from the inserted graphical comparison of the experimental results and results obtained from the physical models, that these last possess bigger stifBiess in comparison to real structures. The source of this discrepancy lies in the lack of ideal joints of particular grid members. These joints were made in form of point welding of adjacent flat steels. The geometrical imperfections of actual grids are the other cause of the observed discrepancies. All grids were warped in smaller or greater level. The prepared models together with stif&iesses determined in form of analytical functions make possible the direct exertion analysis of a grid and allow us to perform the optimisation of the structure as well.
REFERENCES K^czkowski Z. (1980). Plates. Static calculations, Arkady, Warszawa, Poland, (m Polish) Sokolowski M. (1957). Calculations of the elasticity constants for orthotropic plates. Arch.Int Lqd, 3, no 4, 457-485. (in Polish) Zienkiewicz O.C. and Taylor R.L. (1989). The Finite Element Method, McGraw-Hill Book Company, London, UK
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
811
ULTIMATE STRENGTH AND BEHAVIOR OF WELDED CURVED ARCH BRIDGES T. YAMAO'\ T. AKASE'^ and H. HARADA'^ ^Department of Civil Engineering and Architecture, Kumamoto University, Kumamoto, 860-8555, Japan ^Maeda Co. Ltd., Kyushu Branch 2-4-8, Watanabe-tori, Chuohku, Fukuoka, 810-0004 ^Asia Planning Co. Ltd. 7-15-27-101, Toroku, Kumamoto, 862
ABSTRACT This paper presents the results of ultimate strength analysis of deck-type bridge arches which are composed of curved pair ribs with lateral members. A parametric study was carried out by using the elasto-plastic finite displacement method on various theoretical models by changing the ratio of rise-tospan and the distance between right and left hinged arch springings. Ultimate strength of deck-type bridge arches which are composed of curved pair ribs subjected to vertical and horizontal loads are higher than that of ordinary deck-type bridge arches which are composed of parallel twin ribs. It is found that the effects of curved rise-to-span ratios of arch ribs on the ultimate strength are large. KEYWORDS Curved arch bridge, curved pair ribs, ultimate strength, strut, vertical and horizontal load, FEM INTRODUCTION Ordinary arch bridges are composed of parallel twin ribs braced with lateral members of truss or transverse beams. A serious problem for long-spanned arch bridges is tendency for failure due to inelastic lateral instability. Many studies have reported on inelastic lateral stability of arch bridges, but few studies have been made on the ultimate strength of curved pair ribs braced with lateral members. According to our analysis, it is found that pair curved structures subjected to axial compressive loads resist higher buckling loads in comparison with usual two straight columns tied with struts, Yamao and etc.(1995). New pairs of curved arches tied with struts as sliovm in Figure 1 are considered to possibly have higher inelastic lateral buckling load limits. This paper proposes the results of ultimate strength analysis of deck-type bridge arches which are composed of curved pair ribs with lateral members as shown Figure 1. A parametric study was carried out by using the elasto-plastic finite displacement method on various theoretical models by changing riseto-span ratios of the arch ribs, the span length, and the distance between right and left hinged arch springings. Ultimate strength of deck-type bridge arches which are composed of curved pair ribs subjected to vertical or horizontal loads are higher than that of ordinary deck-type bridge arches which are composed of parallel twin ribs. It can be recognized that the effects of curved rise-to-span ratios of arch ribs on the ultimate strength and out-of-plane bending moment about the vertical axis of curved pair ribs are large.
812 Vertical load
^ ^
Horizontal load
Figure 1 Pair of curved arches tied with struts SCOPE OF PARAMETRIC STUDY Analytical Models The analytical models for arch bridges studied herein are shown in Figures 2,3. Model-A and Model-B are thought to be representative of actual deck-type arch bridges. Model-A, in which three columns are hinged to the arch ribs at both ends, is a span length (l=36m) as shown in Figure 2. Model-B, with 15 columns with a same end conditions, is also a span length (1=126m). These models are built of two curved arch ribs tied with struts which have hinged ends, and only Model-B has a pair of diagonal bracings above the crown in order to restrict the torsional deformation. The coordinate axes and the cross sectional dimensions of the arch ribs are shown in Figures 2 and 3, where a represents the distance between left and right arch springings. The analytical models are assumed to have no residual stresses and initial crookedness modes are described by sinusoidal half-waves with amplitude WQ =L/1000 in the z-direction. The value L/1000 is specified in the Japanese Specifications for Highway Bridges (JSHB). The configuration of the curved arch ribs is a parabola and the struts have the same cross sectional area i4/(=0.2/4, A\ cross sectional area of an arch rib) for arch structures. The uniform load distributed along the full span length of the arch, p, is assumed to be the dead and live load condition. It is converted to 17 or 21 equivalent concentrated loads for each arch rib and applied to nodal points along the deck-girder. Geometrical and structural properties of the models used in the numerical analysis are summarized in Table 1. The slendemess ratios, Zy, of an arch rib is defined as, Ay=l/ry , in which A> denotes the radius of gyration of the cross section about the vertical axis. Arch rise-to-span ratio, fyi, of the analytical model is determined by the conditions of the actual arch bridge. Curved rise-to-span ratios, fj\, of the arch rib as shown in Figure 2(b) depend on the distance a and are taken to be 0.0,md 0.072 or 0.033 equal to the width of the deck plate at the midspan. That is. Type Al(fi/1=0.072) or Type Bl (fb/l=0.033) are the models with curved arch ribs(a=0). Type A2 and B2 TABLE 1 PROPERTIES OF ANALYTICAL MODELS
Items Young's modulus E (kN/mm^) Yield stress ay (N/mm^) Span length 1 (m) Bridge length L(m) Slendeness ratio X y=l/ry Archrise-to-spanratio f/l 1 Curvedrise-to-spanratio fj\ 1 Distance of between both arch springings a(m) 1 Cross sectional area of rib A(cm^)
Model-A 206 314 36 63 110 0.167 0.0,0.072 0,520 593
Model-B 206 353 126 134 276, 360 0.258 0.0, 0.033 0, 840 540, 698
813
(a) Global figure
r
^ 1, L85cm 7B.3cm
a fb
-jl
L85cin
(b) Plane
figure (c) Cross section of Type A Figure 2 Analytical Model-A
r
(a) Global figure M
UOcm
1*
1 P
10cm
•Z
Y
1^
n
90cm
i
1 •*• 10cm
leotm
-J
^-
lOcm
185cm
J L14an
(b) Cross section of Type B1 ,B2 (c) Cross section of Type B3 Figure 3 Analytical Model-B
814
10 (a)Model-A
v(cm) (b)Model-B Figure 4 Vertical load vs. in-plane displacement
20 v(cm)
(fb/l=0) are the arch models with parallel ribs. Type B3 is built of arch ribs with 1.5 Elz(E: Young's modulus, Iz: moment inertia of a cross-section about strong axis) . The analytical models were chosen to investigate the effect of the curved rises y& of the arch rib upon the ultimate strength and elasto-plastic behavior. Computation Methods A finite element method developed for large spatial deflections and elasto-plastic behavior of thinwalled fi'ames and arches with closed cross sections was used to determine the ultimate strength of arch bridges, Komatsu & Sakimoto(1976). The method was formulated under the following assumptions and idealizations: a)closed cross-section, b)elastic-perfectly plastic material, c)uniform torsion, d)yield criterion of von Mises, e)plastic stress-strain relation of Pmdtl-Reuss, f)no change of position of the shear center after yielding, g)Bemoulli-Navier's hypothesis in bending, h)small strains, and i)no local buckling and no cross-sectional distortion. With this method, the behavior of the system may be traced from the beginning of loading until the maximum load capacity as determined by lateral deflection is reached. As the load levels are increased, the output of the computer program traces the growth of vertical, lateral and rotational movement as well as the growth of yielded regions within the system. Since these deformations correspond to nonlinear pre-buckling deformations, the analysis is more sophisticated and realistic than expected for the bifiircation solution. For numerical computations, curved arch ribs are divided into 12 or 15 straight member elements and each intersection point of the members is treated as a nodal point. The cross sections of arch rib elements and other member elements are layered into 20 and 16 segments respectively to analyze the development of the plastic zone on the cross section. RESULTS AND DISCUSSIONS Figure 4 shows the relations between the vertical loads (p/py, py : vertical load causing the yield stress at the springings) and the in-plane displacement (v) of Models A and B. The in-plane displacements (v) are measured at the arch crown and uniform load pais the dead load. It is apparent that the in-plane behavior of both Models are almost the same until the maximum load is obtained, except for Type B3 which has a larger flexural rigidity. Figure 5 shows the relations between the horizontal loads (p/pw, pw: horizontal load causing the yield stress at the springings) and the out-ofplane displacements (w) of both Models. In this case, the horizontal load pw is a live load that is increased after the dead load pd is applied in the in-plane direction. It is can be seen that the maximum load of Types Aland Bl, with pairs of curved arch ribs, is larger than that of Types A2 and B2, with parallel arch ribs and such curved arch bridges experience smaller out-of-plane dis-
815
400
v,w(cm)
600 v,w(cm
(b) Model-B (a) Model-A Figure 5 Horizontal load vs. displacement
a(kgf^cn?)
a(kg£^cn?)
700
1300 -
^TypeBl
1200 -
/ '^'^\ \
1100 -\ 1000
D
/?'
\\ "'^\ ^ V
/!
V.
^TypeB2
p
900 800
^
Jo-£ll
1/2 (a) Model-A (b) Model-B Figure 6 Axial stress distributions of the archrib(p/pw=1.0) M(kgf' cm) [XHfl
1/2 (a) Model-A
M(kgf • cm) [xio"]
1/2 (b) Model-B
Figure 7 Out-of-plane bending moment distributions of the archrib(p/pw=l.0)
1
816 placements than that of the usual parallel type. The axial stress distribution and the out-of-plane bending moment of Models A and B subjected to a horizontal load(p/pw=1.0) are given along the span length as shown in Figures 6 and 7. The axial stresses of pairs of curved arch ribs(Types Al, Bl) are slightly larger than that of the usual parallel arch type(Types A2, B2). However, the out-of-plane bending moment near the springings of pairs of curved arch ribs decreased more than that of the usual parallel arch type. The effects of the curved rise-to-span ratio on the out-of-plane bending moment are clearly recognized. The out-of-plane deflection modes of Model-B are shown in Figure 8 for conditions just prior to collapse (p/pw=1.3). From this figure, the effects of curved ribs on the restraint of out-of-plane displacement can be seen. Figure 9 shows the yielding zones of Model-B due to tension, compression and both stresses, with a horizontal load(p/pw=1.3). It is seen that there are more yielding zones in the arch ribs of Type B2 in comparison with that of Type Bl, and yielding zones of Type B2 are concentrated in the arch crown. This is due to the out-of-plane displacement of Type B2 being larger than that of Type Bl. Moreover, it was found that the diagonal bracing member above the arch crown was effective in preventing the torsinonal deformation of arch ribs subjected to a horizontal load. Next, the effects of arch rise-to-span ratio on the ultimate strength and behavior of an arch bridge is considered. Figure 10 shows the relations between the vertical loads (p/py, py : vertical load causing the yield stress at the springings) and the in-plane displacements (v) of Model-B versus changes in the arch rise-to-span ratio. For Type Bl, with a small arch rise-to-span ratio, an increase in the maximum load is not observed. However, it is can be seen that the maximum load of Type B3, with the same arch rise-to-span ratio and larger in-plane flexural rigidity, approaches the maximum load of Type Bl, with a large arch rise-to-span ratio. As can be seen from these figures, the effects of arch rise-to-span ratio on the ultimate strength can be recognized.
(a)TypeBl
(b)TypeB2 Figure 8 Out-of-plane deflection modes of Model-B
817 C Yielding due to tension # Yielding due to compression ^ Yielding due to tension and compression
Load direction (a) TypeBl
(b)TypeB2 Figure 9 Yielding zones of Model-B
TypeBl(4/l=0.15)
10
20
^cm)
Figure 10 Vertical load vs. in-plane displacement CONCLUSIONS Ultimate strength of deck-type bridge arches which are composed of curved pair arch ribs tied with struts under vertical and horizontal load conditions were studied theoretically. A parametric study was carried out using the elasto-plastic finite displacement method on various theoretical models by changing rise-to-span ratios of the arch ribs, the distance between right and left hinged ends of the arch ribs. From this study the following conclusions can be drawn: l)lJltimate strength of deck-type bridge arches which are composed of curved pair ribs subjected to vertical and horizontal loads increased compared to that of ordinary deck-type bridge arches which are composed of parallel twin ribs.
818 2) The effects of curved ribs on the restraint of out-of-plane displacement can be seen. 3) It is found that the effects of curved rise-to-span ratios of arch ribs( fe/l) on the ultimate strength of deck-type bridge arches which are composed of curved pair ribs are large. ACKNOWLEDGMENT We wish to thank Mr. H. Kayashima, a student at Kumamoto University for his help in carrying out the calculations. REFERENCES Fukumoto, Y.(1987). Guidelines for Stability Design of Steel Structures. Japan Society of Civil Engineers, (in Japanese) Japan Road Association(1994). Specificationfor Highway Bridges. Komatsu, T. and Sakimoto, T.(1976). Nonlinear Analysis of Spatial Frames consisting of Members with Closed Cross-Section. Proc. ofJSCE, No. 252,143-157. Sakimoto, T.(1991). Structural Mechanics. Morikita. (in Japanese) Timoshenko, S.P. and Gere, J.M.(1961). Theory of Elastic Stability. McGraw-Hill, 2nd edn., New York, 46-59. Yamao, T. Ishihara, Y. and Hirai, 1.(1995). Buckling Strength and Behavior of Two Plane Curved Members Tied with Struts. Journal of Structural Engineering Vol. 41A, 229-234. (in Japanese)
Session B3 RESPONSE TO DYNAMIC AND ALTERNATING LOADS
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Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
821
BEHAVIOR OF STEEL PIERS SUBJECTED TO VEHICLE COLLISION IMPACT Y.Itoh\T.Ohno'and C.Liu' ' Center for Integrated Research in Science and Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan 'Department of Civil Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan
ABSTRACT In this paper, the behavior of steel piers subjected to the vehicle collision impact is studied using a dynamic FEM analysis method. An original FEM model of a truck has been developed for collision analyses of steel piers having rectangular and circular sections. The results of collision analyses using the truck and car models are compared with respect to the deformation responses of vehicles and piers. In addition, the effect of filling the base of the pier with concrete is investigated on the impact energy absorption characteristics of steel piers.
KEYWORDS Concrete-filled pier, Dynamic analysis. Energy absorption, FEM, Steel pier, Vehicle collision impact
INTRODUCTION The dynamic collision impact of vehicles is significant during the design of the transportation infrastructures such as roads, bridges, and guard fences. Due to the difficulty of remedying the damaged highway bridge piers under the traffics, it is very important to avoid a large deformation of the piers due to the collision impact of vehicles within the service life of piers. As a matter of fact, the massive concrete piers are not likely to be damaged from the vehicle collision impact due to their large rigidity. In the case of the collision impact between the steel piers and vehicles, the local displacement of steel piers frequently happens so that the repair is required to recover the pier structure. Therefore, the partially concrete-filled steel piers are presented to improve the steel piers against the natural and artificial disasters such as earthquakes and vehicle collision impacts (Suzuki et al. 1996). With the improvement of the road network and the vehicle capacities, and the increase of the allowable maximum weight of trucks from 20 tf to 25 tf from November 1994 in Japan, a research effort is needed to recognize the capacities of the steel piers to the collision impact of vehicles from
822 both the function and safety viewpoints. It should be noticed that the types of vehicles such as cars and heavy trucks are highly related to the collision impact on the bridge piers. Several research approaches have been carried out such as the impact simulation between vehicles and roadside safety hardware (Wekezer et al. 1993) and a finite element computer simulation for the vehicle impact with a roadside crash cushion (Miller and Carney 1997). Because of the huge consumption of time and cost, it is impossible in the field to measure the collision performances of the bridge piers on many occasions. In this research, by taking the advantages of both computer software and hardware, the collision impact process between the vehicles and the bridge piers is simulated inside the laboratory based on the presented numerical calculation models for both vehicles and bridge piers. A nonlinear, dynamic, three-dimensional finite-element software LS-DYNA3D is capable of simulating the vehicle impact into the steel bridge piers on an enhanced Workstation (Hallquist 1991). The remainder of this paper is organized as follows. The next section presents three FEM analytical models for the steel bridge piers, the autocars and trucks. Then, in the two following sections, these models are applied for studying the collision impacts in two cases between the autocars and the steel piers, and between the trucks and the steel piers. The last section summarizes the findings of this research.
DEVELOPMENT OF ANALYTICAL MODELS Pier Model In this research, the models of both pipe and box steel bridge piers are prepared according to the standard diagrams used for designing the existing urban highway bridges in Nagoya as shown in Figs. 1 and 2, respectively. For studying the efficiency of the filled concrete at the lower part in the steel pier, both steel piers and partially concrete-filled piers are formulated and analyzed using the abovementioned LS-DYNA3D software. The boundary conditions are considered as a free end and a fixed end at the top and bottom sides, respectively. For simplicity, the steel pipe pier, the partially concretefilled pipe pier, the steel box pier, and the partially concrete-filled box pier are abbreviated to be SP, CP, SB, and CB, respectively. In addition, the symbols CM and TM represent the autocar model and the truck model, respectively.
Figure 1: Pipe-section pier model
Figure 2: Box-section pier model
The steel is assumed to be an isotropic elasto-plastic material following the von-Mises yielding
823 condition. The strain hardening is taken into consideration in the stress-strain relationship. The yield stresses of SM570 used for the pipe pier and SM490 used for the box pier are 450 MPa and 352 MPa, respectively (Specifications 1996). The Young's modulus and the Possoin's ratio of steel are 206 GPa and 0.3, respectively. The concrete filled in the steel pier is assumed as a general elasto-plastic material. This means that the concrete will be in the general elasto-plastic condition while the concrete in the compressive side reaches the yield point and only the cut-off stress is available once the tensile stress increases to the allowable tensile strength. The concrete compressive and tensile strengths are 23.52 MPa and 2.20 MPa, respectively. The Young's modulus and the Possoin's ratio of concrete are 24.36 GPa and 1/6, respectively. Autocar Model Various FEM models of autocars have been developed in the manufacturing companies and research institutes for detecting the performances of their products. These models are not easy to find in the published literatures. The car model used in this research is modified according to the Ford Taurus Car Model developed by the American National Crash Analysis Center. The FEM model is shown in Fig. 3 with the values in three dimensions. The numbers of nodes and elements in the FEM model are 28353 and 26729, respectively. All materials such as the steel, bumper, glass, gum and others are assumed to be isotropic elasto-plastic following the von-Mises yielding condition. However, the specific characters of each material are taken into consideration in its stress-strain relationship.
Figure 3: Autocar model
Figure 4: Truck model
Truck Model The collision accidents between trucks and bridge piers are serious due to the massive volume and weight of a truck. However, there are few research publications on the development of models used for analyzing the collision impact (Zaouk et al. 1996). The trucks whose weight is 25 tf is studied by modeling the truck frame, engine, driving room, cargo, tiers and so on. Fig. 4 shows the FEM model of the ladder-type truck frame whose two side members are of channel sections so that some facilities such as the fuel tanks and pipelines can be attached inside the side members. The thickness of the side member is 8 mm, and the yield stress is 295 MPa. The general elasto-plastic stress-strain relationship is adopted. The solid element with the same shape and volume is modeled for the engine and the transmission, and their weights are adjusted according to the practical vehicles. The tiers, wheels, and gears of a truck influence its behaviors during the collision impact significantly. The connection of the tier and the wheel is assumed to be a rotational joint so that the movement of the wheel can be simulated. The driving room and other small portions are also modeled for the purpose of the numerical calculation. In this model, the numbers of nodes and elements are 7224 and 6327, respectively. The Young's modulus of steel is 206 GPa. The Possoin's ratio of steel is 0.30. The shear modulus is 15.85 MPa. Furthermore, the effect of the strain velocity rate is checked on the deformation of the bridge piers. It
824
is assumed that the yield stress increases following the strain velocity rate as described in Itoh et al (1996). Fig. 5 shows the deformations of a pier with time in two cases with and without the strain velocity rate. In this calculation, the collision speed is 80 km/h, and the pier is a steel pipe pier. According to the deformation tracks as shown in this figure, considering the strain rate can reduce the maximum response deformation, the residual displacement and the response period about 15%, 25% and 20%, respectively. At about 0.12 second after the collision impact, the displacement increases rapidly within a very short time. In the following sections of this paper, the increasing of the yield stress due to the strain velocity rate is not considered for the purpose of safety.
With strain rate
0
0.05
0.1
0.15
0.2
0.25
TIME(sec)
Figure 5: Effect of strain velocity rate
Figure 6: Deformation of a collided car
IMPACT ANALYSIS OF CAR AND HIGHWAY PIER In the following calculation of this section, the vehicle is assumed to impact a pipe pier with a speed of 100 km/h. To investigate the most dangerous effect, the impact angle is 90^, which means the movement direction crosses the vertical center line of the bridge pier. Fig. 6 shows the deformed shape of a car after the collision. The time-deformation curve of a steel pier without filled concrete is shown in Fig. 7. It can be noted that the maximum displacement of the bridge pier is only 1.8 mm and the residual displacement is almost 0. This represents that most kinetic energy is transferred to the internal energy of the car with its plastic displacement, and the impact nearly affects the bridge pier after a very short time of 0.05 second. This result implies that it is unnecessary to analyze the case of the concrete-filled bridge pier. Fig. 8 shows the time-acceleration curve at the rear end of the vehicle. It is obvious that this is a multi-cycle curve with a maximum value of around 90 g. The symbol g is the gravity acceleration. The impact lasts about 0.1 second. lUU
^
|\
80
w
z2T 60
"
(T 1 A (1 \n f\
H
2 W -1
<
0 0.05
0.1
0.15
TIME(sec)
Figure 7: Time-deformation relationship
J1
'^O
w u U ^„ 20
1
/ Jy
\vn \\
' \\l
^
\
1
NJ
|i ~ 1 n 1
yi 1
K
K ,.w\
, , , , , , , . 0.05
. ^^^"^^,
0.1
TIME(sec)
Figure 8: Time-acceleration relationship
0.15
825 Further study is carried out for comparing the calculation results and the experimental results on a vehicle impacting an ideal rigid wall (Emori 1993). The acceleration-deformation relationships from the calculation and experiment are compared in Fig. 9. The dotted line shows the plastic spring constant of vehicles from the experiment. This constant is about 41g/m for all types of vehicles with various speeds. It can be found the increase trend of the calculation value is closely near to this experimental value. Fig. 10 shows the changes of the kinetic energy of a vehicle, the internal energy of the vehicle, and the internal energy of the pier after the collision impact. About 95% of the kinetic energy is shifted to the internal energy of the car. This is why the deformation of the vehicle is very large as shown in Fig. 6 after the collision impact.
E S 4
> 2 O
500 1000 DEFORMATION(mm)
1500
U.(15
0.15
0.1
TIME(sec)
Figure 9: Acceleration-deformation relationship
Figure 10: Energy absorption
IMPACT ANALYSIS OF TRUCK AND HIGHWAY PIER Impact Analysis of Steel Pipe Pier Fig. 11 shows the time-deformation relationships in four cases that have different types of piers (with or without the filled concrete) or speeds (50 km/h or 100 km/h) as represented in the legend of the figure. It is obvious that the filled concrete is able to reduce the residual deformation at the speed of 100 km/h or 50 km/h. Furthermore, the ratio of the residual deformation versus the maximum deformation in the case with the filled concrete is about 10% greater than the case without the filled concrete for either speed. The changes of the acceleration at the end of the truck with time in these four cases are shown in Fig. 12. SP-TM-lOO SP-TM-050 CP-TM-lOO CP-TM-05()
^ 200 \
^
1
;'
2 130 I -SP-TM-100 SP-TM-050 -CP-TM-100 -CP-TM-050
^ g 100 I1 ^ U ^^ I U 30 I
j 1 1
<
0.1 0.15 TIME(sec)
Figure 11: Time-deformation relationship
.1
••••..
,
,
1
'•^ 1 . . .
0.05
/
1 f 1 0.05
, 1
,
•
0.1 TIME(.sec)
,
,
,
,
0.15
Figure 12: Time-acceleration relationship
1
0.2
826 The peaks of accelerations are 48g, 84g, 117g, and 240g respectively in cases of SP-TM-050, SP-TM100, CP-TM-050, and CP-TM-100. Their impact periods are 47 msec, 51 msec, 19 msec, and 24 msec, respectively. In cases with the filled concrete, the peak accelerations of the piers increase to 2.5 times of the values without the filled concrete. However, the impact periods decrease to about 40% of the values without the filled concrete. This implies that a truck colliding with the concrete-filled piers is damaged seriously due to the large rigidity of the bridge piers. The effects of the impact degree on the deformations of trucks and piers are studied by means of the changes of deformations with the acceleration at the end of a truck. Fig. 13 shows the maximum deformations of piers in the horizontal axis and the acceleration of the end of the truck in the vertical axis, respectively. Fig. 14 shows the relationship between the relative deformation of the front of the truck due to the collision stroke and this acceleration. It can be inferred that the recovery deformation of the pier is obvious and the deformation of the truck is irreversible from these figures. 250 r ^200
SP-TM-lOO SP-TM-050 CP-TM-100 CP-TM-050
;
;;200
o a 150
SP-TM-lOO SP-TM-050 CP-TM-100 CP-TM-050
o
h u u
<
'i^\
u u
<
100
1
1
200
300
1
/ .
1
400
Q Jg^^r-^a-j^
500 1000 1500 DEFORMATION(nini)
DEFORMATION(mm) Figure 13: Acceleration-deformation relationship
Figure 14: Acceleration-deformation relationship
The energy shift is studied for the two cases of SP-TM-050 and CP-TM-050. In the case of piers without the filled concrete as shown in Fig. 15, the kinetic energy of the pier is always zero. The kinetic energy of a moving truck rapidly shifts to the internal energy of both the truck and the pier after the collision impact. In the case with the filled concrete, the filled concrete is able to absorb a rather large portion of the kinetic energy of the truck, which is greater than the internal energy of the pier as shown in Fig. 16. Further, more kinetic energy of the truck is transferred to the internal energy of the truck compared to Fig. 15, and at about 0.14 second much kinetic energy of the truck shifts the internal energy suddenly. 2.5 r
Truck Kinetic Truck Internal Pier Kinetic Pier Internal Concrete Internal
Truck Kinetic Truck Internal Pier Kinetic Pier Internal
0.1
0.15
0.25
TIME(i;ec) Figure 15: Energy absorption for SP-TM-050
0.05
0.1
0.15
TlME(.sec) Figure 16: Energy absorption for CP-TM-050
827 Impact Analysis of Steel Box Pier The impact analysis is further carried out for a truck with the speed of 50 km/h and a box pier with or without the filled concrete. Fig. 17 shows the deformations of piers with time. The maximum deformations of SB-TM-050 and CB-TM-050 are 115 mm and 43 mm, respectively. Their residual deformations are 92 mm and 36 mm, respectively. The effect of the filled concrete is rather large. Similar results to the pipe piers can be found on the time-acceleration relationship as shown in Fig. 18. The filled concrete increases the peak acceleration from 39g to llOg, and decreases the impact lasting time from 50 msec to 24 msec.
12.5
-SD-'J'M-050 CB-TM-050
OJJ
z ino o H
7.5 S W
50
LJ
U 25 <
0.25
Figure 17: Time-deformation relationship
Figure 18: Time-acceleration relationship
The effects of the impact degree on the deformations of trucks and box piers are also studied by means of the changes of deformations with the acceleration at the end of a truck. Fig. 19 shows the maximum deformations of box piers in the horizontal axis and the acceleration of the end of the truck in the vertical axis, respectively. Fig. 20 shows the relationship between the relative deformation of the front of the truck due to the collision stroke and its acceleration. 150 SB-TM-050 CB-TM-050
125
-SB-TM-050 • CD-TM-()50
CO
o i"o H
S 75 • tu •J
S 50 u •
25
50
75
100
DEFORMATION(mm)
Figure 19: Acceleration-deformation relationship
<
500
1000
1500
2000
DEFORMATlON(mm)
Figure 20: Acceleration-deformation relationship
Further study is carried out for recognizing the energy shift. In the case of without the filled concrete as shown in Fig. 21, the kinetic energy of a truck rapidly shifts to the internal energy of both the truck and the pier after the collision impact. In the case with the filled concrete, the filled concrete is able to absorb a rather large portion of the kinetic energy of the truck as shown in Fig. 22. Further, more kinetic energy of the truck is shifted to the internal energy of the truck compared to the case without the filled concrete. These results are similar to the pipe piers as mentioned in the above sub-section.
828 2.5 Truck Kinetic Truck Imernal Pier Kinetic Pier Internal
0.1
Truck Kinetic Truck Internal Pier Kinetic Pier Internal Concrete Internal
0.15
TIME(sec)
Figure 21: Energy absorption for SB-TM-050
Figure 22: Energy absorption for CB-TM-050
CONCLUSIONS This research focuses on the impact analysis between the vehicles and the bridge steel piers using the presented FEM models. The main conclusions can be stated as follows: (1) It is made possible to simulate the collision impact process and to determine the maximum deformation, residual deformation and the energy shift by using LS-DYNA3D, a nonlinear dynamic analysis software of structures in three dimensions. (2) It is found that the calculation results of the maximum deformation, residual deformation and the energy shift are highly related to the relative rigidity between the vehicle and the pier. (3) The filled concrete is very significant for both the pipe and box steel piers against the collision impact of a heavy truck.
REFERENCES Emori I. (1993). Vehicle Accident Engineering. Technology Publishing House, Tokyo (in Japanese). Hallquist J. (1991). LS-DYNA3D Theoretical Manual, Livermore Software Technology Corporation, LSTC Report 1018, University of California. Itoh Y., Sasada T., and Ohno T. (1996). Nonlinear Impact Behavior Considering Strain Rate. Steel Construction Engineering, 3:11, 47-58 (in Japanese). Miller R and Carney J. (1997). Computer Simulations of Roadside Crash Cushion Impacts. Journal of Transportation Engineering, ASCE, 123:5, 370-376. Specifications for Highway Bridges. (1996). Japan Road Association, Tokyo (in Japanese). Suzuki T., Usami T., Itoh Y, and Teshima K. (1996). Behavior of Concrete-Filled Steel Slender Columns through Pseudo-Dynamic Tests. Journal of Structural Mechanics/Earthquake Engineering, JSCE, 537/1-35, 77-88 (in Japanese). Wekezer J., Oskard M., Logan R. and Zywicz E. (1993). Vehicle Impact Simulation. Journal of Transportation Engineering, ASCE, 119:4, 598-617. Zaouk A. , Bedewi N. , Kan C. and Schinke H. (1996). Evaluation of a Multi-purpose Pick-up Truck Model Using Full Scale Crush Data with Application to Highway Barrier Impacts, 29th International Symposium on Automotive Technology & Automation, Florence, Italy.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
829
STABILITY ANALYSIS OF BRIDGE PIERS SUBJECTED TO CYCLIC LOADING E. Yamaguchi\ Y. Goto^ K. Abe\ M. Hayashi^ and Y. Kubo^ ^Department of Civil Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, JAPAN Department of Civil Engineering, Nagoya Institute of Technology, Nagoya 466-8555, JAPAN
ABSTRACT Since the 1995 Hyogo-ken Nanbu Earthquake, much research has been conducted to improve the seismic performance of bridge piers in Japan, and the seismic design specifications for highway bridges have been revised. Since the revised specifications require nonlinear dynamic analysis for the seismic design of steel bridge piers, the knowledge of the mechanical characteristics of bridge piers under cyclic loading is essential. However, such mechanical characteristics are not yet well understood for many types of bridge piers. In the present study, we conduct the finite element stability analysis of two steel bridge piers to help establish the numerical method to evaluate the mechanical characteristics of bridge piers. The bridge piers are modeled by shell elements and beam elements so as to reproduce local buckling. Three different elastic-plastic constitutive models are employed for the analysis of each bridge pier: the isotropic-hardening model, the kinematic-hardening model and the three-surface model. The present numerical study then concludes that the three-surface model is clearly superior to the others, but that the difference between constitutive models is not so crucial when the out-of-plane displacement is pronounced.
KEYWORDS Steel bridge pier, seismic design, cyclic loading, plasticity model, three-surface model, finite element analysis, hysteresis loop, local buckling.
INTRODUCTION Extensive damage of steel bridge piers was observed in the aftermath of the 1995 Hyogo-ken Nanbu Earthquake, Japan. Since then, much research effort has been made to improve behaviors of steel bridge piers during a severe earthquake (Kensetsu 1996). Since the damage was found attributable to cyclic horizontal load due to the earthquake coupled with the weight of a superstructure, steel bridge pier specimens have been tested under the combination of constant vertical load and cyclic horizontal
830
p
>L
oo
mmMM^.
unit: mm
800 (a) Pipe-sectional steel bridge pier
(b) Stiffened box-sectional steel bridge pier
Figure 1: Bridge pier specimens load. From such experiment, one can evaluate the mechanical characteristics of a bridge pier, which are to be used for nonlinear dynamic analysis required in the revised seismic design specifications for highway bridges (Japan 1996). The experimental data are undoubtedly valuable. Nevertheless, it is not practical to depend solely on experiment for evaluating the mechanical characteristics of bridge piers since experiments are costly: the number of experiments is inevitably limited. Therefore, the numerical simulation of steel bridge piers by the finite element method has been also explored (Structural 1997). However, the accurate and reliable numerical procedure has not been well established yet. Against the background of the above information, in the present study we carry out the stability analysis of two steel bridge piers subjected to constant vertical load and cyclic horizontal load. The study employs the finite element method with shell and beam elements, and simulates the local buckling and elastic-plastic deformation. Particular attention is paid to the comparison of three plasticity models: the isotropic-hardening model, the kinematic-hardening model and the three-surface model. The experimental data of the two bridge piers are available (Public 1997; Yoshizaki et al. 1997), and the numerical results in the present study are compared with them. Based on this comparison, the effectiveness of the numerical approach is discussed.
EXPERIMENT We briefly describe the experiment of the two bridge piers which we simulate numerically in this study. Figure 1 illustrates the bridge pier specimens. Both of them are basically cantilevers fixed at the bottom and free at the top. The plate thickness in Figure 1 (a) is 9 mm, while in Figure 1 (b) the plate thicknesses of a flange and a web are both 7 mm, and the thickness of a stiffener is 8 mm. The comers of the cross section in Figure 1 (b) are rounded with a radius of 142 mm. This type of bridge pier (Figure 1 (b)) is often employed in a densely populated district in Japan from an aesthetic point of
831
^600
a
-
^600
s
^ 400
^ 400,
<> 200 0
*^ 200 1
»
J
0.1
0.2
0.3
0.4
00
(a) Pipe-sectional steel bridge pier
1
0.1
1
0.2
1
0.3
0.4
oP
(b) Stiffened box-sectional steel bridge pier
Figure 2: Uniaxial stress-strain relationships 10,
NO. OF CYCLES
Figure 3: Predetermined displacement path view. The uniaxial material tests were performed: the stress-strain relationships in terms of true stress c and logarithmic plastic strain e^ are shown in Figure 2. In each experiment, the axial compressive force P is 15% of the nominal squash load of the cross section Py, and the horizontal load H is applied to trace the predetermined path of the horizontal displacement 5 at the loading point (Figure 3). 5y is the displacement at the initial yielding due to the horizontal force and given by
8y =
Hy(hi+h2r 3EI
(1)
where hi = specimen height; h2 = distance between the top of the specimen and the loading point; and EI = bending rigidity, while Hy is defined by
«'=h-5j;f h2
(2)
832
(a) Pipe-sectional steel bridge pier
(b) Stiffened box-sectional steel bridge pier
Figure 4: Finite element meshes where Cy = nominal yield stress; A = cross-sectional area; and Z = section modulus.
MODELING FOR ANALYSIS Due to symmetry, only a half of each bridge pier is analyzed. The finite element meshes are illustrated in Figure 4. For the pipe-sectional steel bridge pier, 416 shell elements and 4 beam elements are employed, while 2966 shell elements and 33 beam elements are utilized for the other. The lower portion of each bridge pier is modeled with finer mesh since local buckling is expected to take place. The material behavior is assumed to be described by the plasticity theory of von Mises type. We consider three plasticity models in this study: the isotropic-hardening model, the kinematic-hardening model and the three-surface model They are different fi*om each other in terms of a hardening rule. Except the third model, the plasticity models employed herein are conventional and the explanation can be found in many books (e.g. Chen 1994). The three-surface model proposed by Goto et al. (1998) is an extension of the two-surface model (Dafalias and Popov 1976) and can reproduce the nonlinear features of structural steel such as the yield plateau and the shrinkage of elastic region. The model consists of three surfaces: the bounding surface, the yield surface and the discontinuous surface. The yield surface changes its size as it moves within the region defined by the bounding surface, while the discontinuous surface lying between the bounding and yield surfaces controls the hardening modulus. Multi-surface plasticity models usually require a number of material parameters, and so does the three-surface model. However, by performing various structural analyses and comparing the numerical results with experimental data, the values of some material parameters have been fixed, so that in practice all the material parameters can
833 1
EXPERIMENT ISOTROPIC
EXPERIMENT KINEMATIC ^lyJiLuTiP' -
''**^/nt\
5 0
"^<^^^lih
X
„
-12
flTcriTT*''*
5 0
^'^Mlli'h
. 1
0 5/5,
12
(b) Kinematic-hardening model
(a) Isotropic-hardening model 2 EXPERIMENT THREE-SURF.
X X
0 5/5,
-12
12
—
1
—
^^'^Mk -2
-12
0 5/5,
12
(c) Three-surface model Figure 5: Horizontal load-displacement curves (pipe-sectional steel bridge pier) be obtained only from the result of a uniaxial material test. In the present study, the parameters are determined by the experimental data shown in Figure 2
NUMERICAL RESULTS The numerical results of the pipe-sectional steel bridge pier together with the experimental data (Public 1997) are presented in Figure 5. They are in the form of hysteresis loop of the horizontal loaddisplacement relationship at the loading point. In the case of the isotropic-hardening model, the point of ultimate strength in the hysteresis loops is located in a position quite different from that of the experiment. The kinematic-hardening model tends to overestimate the structural stiffness in the reversed loading paths, which can be attributed to the constant elastic region of this plasticity model. The three-surface model produces the hysteresis loops that are very similar to the experimental result even in the region far beyond the point of ultimate strength. In general, however, it may be stated that all the numerical results due to the three constitutive models are reasonably good with the discrepancy of the ultimate strength from the experimental value being less than 5%. Figure 6 summarizes the numerical results of the stiffened box-sectional steel bridge pier with round corners together with the experimental data (Yoshizaki et al. 1997). Because of the presence of
834 -EXPERIMENT -KINEMATIC
5 0
-2 -12
(a) Isotropic-hardening model
12
(b) Kinematic-hardening model
2
12
(c) Three-surface model Figure 6: Horizontal load-displacement curves (stiffened box-sectional steel bridge pier) stiffeners, the reduction of the post-peak strength is less sharp than that of the pipe-sectional steel bridge pier. All the numerical results have more or less captured this phenomenon. The characteristics of each plasticity model described above are observed also in this analysis. However, the discrepancy from the experimental result is larger herein, although the three-surface model still yields a reasonably accurate result with respect to the shape of hysteresis loop and the strength as well: the discrepancy of the ultimate strength is only about 7% whereas the kinematic-hardening model underestimates the strength by 11%. Figure 7 shows the deformed configurations at the point of maximum horizontal displacement. Local buckling is clearly observed near the bottom end in both bridge piers. The out-of-plane displacement in the pipe-sectional steel bridge pier has taken place in a narrow portion and is quite pronounced, while it is less localized in the box-sectional steel bridge pier. These buckling modes compare very well with the experimental observations. The difference in terms of a buckling mode can explain the difference between the two bridge piers regarding the strength reduction in the post-peak region. And it may be realized that a constitutive model plays an important role when the out-of-plane displacement is small, while it is not so crucial when the geometrical change due to local buckling is pronounced.
835
(a) Pipe-sectional steel bridge pier
(b) Stiffened box-sectional steel bridge pier
Figure 7: Deformed configurations CONCLUDING REMARKS Using the three plasticity models, we have conducted the finite element stability analysis of the two bridge piers under cyclic horizontal loading. In this numerical study, we observe that different constitutive models lead to different structural responses. And we may conclude that the three-surface plasticity model is clearly superior to the others. Nevertheless, we can expect reasonable accuracy with the conventional plasticity models of the isotropic-hardening model and the kinematic-hardening model if the geometrical change due to local buckling is pronounced. These conclusions imply that the finite element analysis conducted with some care can take the place of experiment for evaluating the mechanical characteristics of steel bridge piers under seismic loading.
REFERENCES Chen, W.F. (1994). Constitutive Equations for Engineering Materials, Vol.2, Elsevier, Amsterdam, Netherlands. Dafalias, Y.E. and Popov, E.P. (1976). Plastic Internal Variables Formalism of Cyclic Plasticity. J. Applied Mechanics, 43, 645-651. Goto, Y, Wang, Q., Takahashi, N. and Obata, M. (1998). Three Surface Cyclic Plasticity Model for FEM Analysis of Steel Bridge Piers Subjected to Seismic Loading. /. Structural Mechanics and Earthquake Engineering, JSCE, 591/1-43,189-206. Japan Road Association (1996). Design Specifications of Highway Bridges: Part V Seismic Design, Maruzen, Tokyo, Japan. Kensetsu Tosho (1996). Bridge and Foundation Engineering, 30:8, Tokyo, Japan. Public Works Research Institute et al. (1997). Technical Report of Joint Research on Seismic Design
836 for Highway Bridge Piers, PWRI, Ministry of Construction, Japan. Structural Engineering Committee (1997). Proceedings of Nonlinear Numerical Analysis and Seismic Design of Steel Bridge Piers, JSCE, Tokyo, Japan. Yoshizaki, N., Murayama, T., Yasunami, H., Natori, T. and Tuji, H. (1997). An Experimental Study on the Cyclic Elasto-plastic Behavior of Steel Box Column with Round Comers. Proceedings of Nonlinear Numerical Analysis and Seismic Design of Steel Bridge Piers, JSCE, 339-346.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
837
FATIGUE STRENGTH OF LONGITUDINAL JOINTS FOR ALL STEEL SANDWICH PANELS
P. Kujala and K. Kotisalo Helsinki University of Technology, Ship Laboratory PL 4100, 02015 TKK, Finland
ABSTRACT Applications of all metal sandwich panels on cruising ships are studied in this paper. The main topic discussed is the laser welding of these panels and the effect of production process on the fatigue strength of these panel and especially on the fatigue strength of the joints used. A number of test panels are produced for tension fatigue testing of the joints. The initiation of the fatigue cracks is described and the effect of production on the location of crack intiation is discussed. The main concern is the accuracy of the corrugation of the core plate used. The corrugation is "hand made" causing some problems for the welding process. The planned applications of the all-steel sandwich panels on a crusing ship are discussed. At present the technology is available to install these panels as longitudinal bulkheads on a crusing ship. The application of these panels as upper deck structures need some further development. KEYWORDS Sandwich panels, laser welding, fatigue, ship structures INTRODUCTION The demand for faster, bigger and lighter ships has increased the need for more efficient structures. Sandwich panels form one type of efficient structure enabling the application of steel, aluminium or composites in the construction. The present interest in steel sandwich structures has been awakened by the developments in laser welding technology enabling efficient production of these panels. The US Navy has studied the applications of laser welded corrugated core steel sandwich panels (LASCOR) since 1987 (Marsico et al., 1993). So far they have built a number of applications on Navy ships using stainless steel as a material with plate thicknesses varying from 0.6 to 1.6 mm. These applications include bulkheads and decks in accommodation areas, hatch covers, deck edge elevator doors, and hangar bay division doors on conventional navy ships. The maximum weight
838 savings on these applications have been reported to be 30 to 50 %. Meyer Werft shipyard in Germany has studied since 1993 applications of laser welded steel sandwich panels onboard cruising ships and also installed a number of panels as bulkheads and staircase landings for passenger ships (Roland, 1996) The studies related to the all steel sandwich panels at HUT/Ship Laboratory were initiated in 1987 (Tuhkuri, 1991, Tuhkuri, 1993) by research to apply all steel sandwich panels as shell structures of an icebreaker. The application to the shell of an icebreaker was found to be problematic due to a high demand for the local strength of structures under ice loading. Later, the studies related to the applications of all steel sandwich panels as deck and bulkhead structures of a cruising ship were initiated (Kujala et al., 1995). The studies included development of design methods, weight optimisation, ultimate strength testing under hydrostatic loading, fire and noise testing of the panels. Thereafter the studies continued by developing design approach to analyse ultimate strength of sandwich panels under e.g. hydrostatic and wheel loading (Kujala, 1998a). In addition, development of composite coatings for precurved sandwich panels were studied (Kujala, 1998b). The precurved panels were planned to be used on a railway cargo wagon with wheel loading as the main design criteria. One important topic for application of all steel sandwich panels onboard ships is the design of joints to attach the panels together and to the surrounding structure. This problems has been studied by conducting extensive fatigue testing for various types of joints specially designed for this purpose. The studies include e.g. fatigue testing of the panels used as longitudinal bulkheads (Kujala and Salminen, 1997) and as deck panels (Kujala et al., 1998, Kujala and Salminen, 1998). As a result of these studies, the first all steel sandwich test panels used as longitudinal and transverse bulkheads were installed on a cruising ship in 1997 under construction by Kvaemer Masa Yards, Helsinki New Shipyard. In this paper, the fatigue testing of the deck sandwich panels are described in more detail. Finally, the obtained S-N curves are used to evaluate the fatigue life for the tested panels when installed on a cruising ship navigating on the northern Atlantic and the Caribbean Sea. DESIGN OF THE DECK PANELS The deck panels for cruising ship are designed assuming a maximum longitudinal tensile stress of 170 MPa and a compressive stress of 70 MPa (Kotisalo, 1998). In addition a 5 kPa water pressure on deck is included as loading on the panel. The panel is assumed to be supported by webframes with a spacing of 2900 mm. Three types of material are used: Racold 320 cold formed steel, Polarit 725 and 757 stainless. The measured yield stress for the Racold is 405 MPa, for Polarit 725 the measured yield stress is 335 MPa and for Polarit 757 the measured yield stress is 312 MPa. The weight optimised height of the sandwich panel is 60 mm and plate thicknesses are 1 mm. The weight of the panels is 33 kg/m^, which is about 60 % of the weight for a typical deck structure with conventional stiffened plating. For deck structures the most critical joint is the longitudinal joint between the panels. Figure 1 illustrates the joint studied. C-profiles are attached at the end of the sandwich panel and these profiles are then welded by conventional welding on the webfi*ame.The plate thickness of the Cprofile is 3 mm. This joint is critical because the attachment of the corrugated core to C-profile is practically difficult causing high stress concentrations on the surface plate when the longitudinal stresses fi"om the core run over the joint.
839 Face sheet Core Face sheet
Web frame
Figure 1. Illustration of the fatigue tested longitudinal joint.
PRODUCTION OF THE PANELS The panels were laser welded by Laserplus Oy in Riihimaki. The welding was done with a 2 kW fast axial flow C02, which was integrated to a Trumpf TLC105 5-axis laser processing workstation. The system consists of the laser, the standard zoom system (which keeps the raw beam size constant through the workstation), 4 flat mirrors and the focusing lens. One of the mirrors is XIA mirror ensuring a circular polarized beam. The beam mode was TEMOl. Size of the raw beam is about 20 mm in diameter at the focussing optics. The focusing was performed with a standard ZnSe planoconvex focusing lens with focal length of 127 vam. The theoretical diameter of the focal point was about 0.15 nmi. The welding head, nozzle, lens holder and lens protection cross-jet arrangement were designed and manufactured by Laserplus. The laser power used was 1.3 kW on the top of the workpiece and welding speed 2 m/min. Shielding gas used was argon with feed rate of 15 1/min. The gas was introduced with a separate gas nozzle, hole diameter 10 mm. Shielding gas was fed from the leading edge of the weld pool in an angle of 35° degrees. Focal point position was on the top of the top sheet. The parts were placed in a fixturing frame and fixed such that the position of the parts was constant during the welding. The real fixturing, which pressed the parts together removing the air gap between the sheets, was carried out with separate fixture system moving with the laser beam so that the joint is fixtured around the welding point with four balls. The pressure around the welding point was created with a spring. The focal point and weld pool located in the middle of the four pressing points (Kujala and Salminen, 1998). FATIGUE TESTING OF THE LONGITUDINAL JOINTS Test arrangements The tests aimed to simulate the fluctuating longitudinal stresses on the deck panels. The length of the tests panels were 700 mm and breadth 230 mm, see Figure 2. The panel breadth included two longitudinal corrugations. The studied joint was located at midspan. The force was produced with a hydraulic cylinder and it was measured with a force gauge installed between the test piece and the
840
hydraulic cylinder. The force was introduced to the structure through the riveted and clued plates on the surfaces of the test pieces.
Figure 2. Illustration of the test piece used to study the tensile fatigue of deck panels.
Tests results The tests included altogether 3 longitudinal ultimate strength and 15 fatigue tests for the C-profile joints. This means that 1 ultimate load and 5 fatigue tests were made with each material used. Results of the ultimate strength tests are presented in Table 1. The ultimate strength tests shows that the strength of the laser welded joints is adequate. TABLE 1. RESULTS OF THE ULTIMATE STRENGTH TESTING
Test Racold320(RCVl) Polarit725(P725Vl) Polarit757(P757Vl)
Ultimate load (kN) 205 227 191
Nominal stress (MPa) 427 472 398
During the fatigue testing, fatigue cracks were typically observed to initiate at the end of the weld between the surface and the core plate. This is the location where the core plate ends near the transverse weld of the C-profile and the surface plate, see Figure 3. In some tests fatigue cracks initiated at the transverse weld between the C-profile and the surface plate. In all these cases the penetration of the laser weld was insufficient. Unsatisfactory penetration of the weld was a consequence of the too wide air gap between the C-profile and surface plate. This is again caused by the inaccurate height of the corrugated core. Figure 4 summarises the fatigue test results. The fatigue strength of a joint is usually described by the S-N curves NS"" =C
where A^ is the number of cycles to failure under a constant stress range S, m and C are fatigue strength exponent and fatigue strength coefficient, respectively. No significant difference in fatigue
841 behaviour between studied materials was observed. Based on results, given in Figure 4, the S-N curve parameters for the studied joint can be estimated as: m=3.07 and C=6.25 10*^. Figure 4 shows also the S-N curve for fatigue class 45 as a reference class. The FAT class 45 is the class for a transverse butt weld on plate welded from one side only without backing bad.
Figure 3. Typical location of the fatigue crack on the longitudinal deck joints.
X P725CV
1000.00 -
+ P757CV • RCV — 45
"eo* Q.
— C-prof
s
OT 100.00 -
"^--x^g^::::^
a 10.00 1-E+04
1.E+05
1.E+06
N
Figure 4. Obtained fatigue test results for the longitudinal joints. The fatigue class obtained for the studied joints is fairly low. The main reason for this is the problematic laser welding of the joints due to the inaccuracies in the panel dimensions. In addition the lack of attachment of the corrugated core to the C-profile due to the manufacturing difficulties lowers the fatigue class of the joint. Kotisalo (1998) has also studied by finite element calculations the case when the corrugated core is attached to the C-profile by welding. Kotisalo founded out that the stress level can then be about 15 % smaller than with tested joints. Based on that 15 % decrease of the stresses on the most critical area, can S-N curve parameters be estimated as: m=3.07 and C=15-10*^
842 APPLICATION OF THE TESTED PANELS ONBOARD PASSENGER CRUISE SHIP In the following, fatigue strength of metal sandwich panels are studied by replacing normal deck structures by steel sandwich panels. A passenger cruise ship built in the 80's for the Caribbian Sea is used as an example ship. The main dimensions of the ship areilength over all, LQA = 230.90 m, breadth, B=29.20 m, draught, T=7.80 m, height to 4th deck (main), D4=19.4 m and height to 9th deck, D9=32.95 m. An integrated program system has been developed for spectral fatigue analysis of ship structures (Kukkanen, 1996). The procedure is based on linear theory and only low frequency loads are taken into account. The hull girder is assumed to behave as a rigid beam when evaluating the loads. The evaluation of fatigue life is based on standard well known approaches to determine long-term loading for ships and the application of Miner's rule. Long-term prediction of responses are derived by weighting the cumulative distribution function of the short-term response peaks by the occurrence probabilities of different operational and environmental conditions during the assumed 20 year lifetime period of the ship (Kujala et al., 1998). Two different combinations of sea areas are considered in the investigations: the Caribbean Sea (70 %) and Atlantic (30 %), and the Caribbean Sea only (100 %). Figure 5 shows a typical vertical bending stress distribution on the cross-section. Calculated fatigue damage and fatigue life are given in Table 2 for the assumed ship operating areas in the Caribbean Sea and Atlantic, and in the Atlantic. ROYAL PHINCSSS. MIDSHIP. MODEL 10, RB3UC5D WIHDOWS . 27.11.96 THO NON>UNEAR STRESS DISTRIBUTION MAXIMUM- ^4112^02
-.2oE>42
0
3ae*oz
9th deck
4th deck
Figure 5. Calculated bending stress distribution at midship cross-section (Kujala et al., 1998).
843 TABLE 2. CALCULATED FATIGUE LIFE FOR THE ALL STEEL SANDWICH DECK PANELS.
Deck 4. deck 5. deck 6. deck 8. deck
Caribbean Sea and Atlantic Fatigue life Tf [years] 85.7 92.6 17.0 5.0
Caribbean Sea Fatigue life Tf [years] 127.1 199.4 33.6 8.4
As can be seen from Table 2, the calculated fatigue life of the all steel sandwich panel is adequate only on the decks 4 and 5 when North-Atlantic and Caribbean Sea is used as the studied sea areas. If the 15% decrease of stress level due to the welding of the core plates to the C-profiles is taken into acoount, then the deck panels can be applied as high as the 7. deck with a fatigue life longer than 20 years on the North Atlantic and Caribbean Sea as the studied sea areas (Kotisalo,1998). This indicates the existing possibilities for longitudinal joints after ftirther development. The better accuracy of the corrugated core height can remarkably increase as such the fatigue life for the joints. Also, longitudinal joints can be ftirther developed to increase the fatigue life. CONCLUSIONS Applications of all steel sandwich panels on ship, trains and cranes structures have been extensively studied in Finland in recent years. Under considerations have been panels with height varying from 50 to 150 mm and with plate thickness varying from 0.5 mm to 3 mm. Several experiments and numerical stability analyses have been conducted to evaluate the ultimate and fatigue strength of these panels. The results are used to develop sound design basis for these structures. Laser welding offers a flexible fast welding method for corrugated panels. The moving fixture system makes it easy to change flexibly the welded pattern on the part. The successftil welding, with reasonable trough put time and acceptable production costs, requires careful pre-work and wellorganized flow of work during manufacturing. An all-steel sandwich panel welded by laser is a new solution to obtain light and efficient ship structures. It has high potential for weight savings, but also require new approaches e.g. for detail design of the joints for attachment of these panels to each other and to surrounding structures. The fatigue tests summarised in this paper indicate that the panels can, at present, be used as lower deck structures on a cruising ship. Further development is, however, needed before these panels can be applied e.g. on the upper deck panels of a cruising ship. One of the main problems is accurate production of the corrugated core. Also the longitudinal joints between the panels need some ftirther research and development work. ACKNOWLEDGEMENTS The work presented in this paper is based on the research conducted in a national research project. The projects are financed by the Technology Development Center (TEKES) and the Finnish companies: Finnish Board of Navigation, KCI Konecranes International, Kvaemer Masa Yards,
844 Outokumpu, Rautaruukki, VR Engineering. These financing supports are here gratefully acknowledged.
REFERENCES Kotisalo, K., 1998. Fatigue strength of the longitudinal joints of the corrugated core steel sandwich panels. Diploma thesis. Helsinki University of Technology. Faculty of Mechanical Engineering.(In Finnish). Kujala, P., Metsa, A., Nallikari, M., 1995. All metal sandwich panels for ship applications. Shipyard 2000, Spin-off project. Helsinki University of Technology, Ship Laboratory, Report M-196. 65 p. Kujala, P., Salminen A., 1997. Fatigue strength testing of laser welded all steel sandwich panels for ships. 6th NOLAMP Conference. Lulea, Sweden. August 27-29. Kujala, P., 1998a. Ultimate strength analysis of all steel sandwich panels. Journal of Structural Mechanics, Vol. 31, No 1-2, 1998, Pp. 32-45. Kujala, P., 1998b. Strength testing under wheel loading for precurved all steel sandwich panels with composite coatings. 4th International Sandwich Constructions Congress, Stockholm, 9-12 June. Kujala, P., Kukkanen, T. and Kotisalo, K., 1998. Fatigue of all steel sandwich panels. Applications on bulkhead and decks on cruising ship. The 7th International Symposium on practical Design of Ships and Mobile Units. PRADS'98. Hague, 20-25 September. Kujala P. and Salminen, A., 1998.Laser production and fatigue strength of all steel sandwich panels. 17th International Congress on Applications of Lasers and Electro-Optics. ICALEO'98, November 16-19, Orlando, Florida. Kukkanen, T. 1996. Spectral fatigue analysis for ship structures. Uncertainties in fatigue actions. Licentiate's Thesis, Helsinki University of Technology. Faculty of Mechanical Engineering. 101 p. Marsico et al. , 1993. Laser welding of lightweight structural steel panels. Proceedings of the Laser Materials Processing Conference. ICALEO'93. Orlando, 1993. Roland, F., 1996. Trends, problems and experience with laser welding in shipbuilding. IIW Shipbuilding Seminar, Odense, April 17-19. Tuhkuri, J., 1991. Sandwich structures for ships under ice loading. Licenciate thesis. Helsinki University of Technology, Faculty of Mechanical Engineering. Tuhkuri, J., 1993. Laboratory tests of ship structures under ice loading. Vols 1-3. Helsinki University of Technology, Ship Laboratory, Report M-166.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
845
LOCAL AND OVERALL INTERACTION BUCKLING OF STEEL COLUMNS UNDER CYCLIC LOADING T. Usami and H. B. Ge Department of Civil Engineering, Nagoya University, Nagoya 464-8603, JAPAN
ABSTRACT This paper deals with the local and overall interaction buckHng of steel columns subjected to cyclic lateral loads combined either with a concentric axial force or an eccentric axial force. Steel columns analyzed are of pipe-sections. The radius-thickness ratio and slenderness ratio are taken as main parameters. In the analysis of eccentrically loaded columns the effect of eccentricity of axial force is also investigated. To account for material nonlinearity, a modified two-surface model is employed to trace accurately the inelastic cyclic behavior of steel. As a result, some design formulas are proposed to predict the interaction buckling strength and ductility of steel columns under cyclic loading. KEYWORDS CycHc loading, Ductihty, Elasto-plastic analysis, Local buckling, Overall buckling. Pipe-section, Strength. INTRODUCTION It is well known that local buckling may interact with overall buckling in steel structures such as columns and beam-columns composed of thin-walled sections. In the past decades, the local and overall interaction buckfing of steel columns has been extensively investigated. However, most of the investigations were carried out under monotonic loading conditions. Recently, cyclic behavior of steel columns is becoming a hot topic in the field of seismic design. Especially, the available ductility of the columns is an important factor. For such a purpose, a great number of experimental investigations on the cyclic behavior of steel columns have been conducted several years ago (e.g., Usami et al. 1992). On the other hand, efforts in predicting numerically the cyclic behavior of steel structures were also made by many researchers. One of the achievements is the development of a material model named the modified two-surface model (e.g., Shen et al. 1995). This plasticity model can be used to predict the cychc behavior of steel structures with good accuracy (Mamaghani et al. 1996; Gao et al. 1998). The present paper is concerned with a numerical investigation into the interaction buckfing of steel columns under cyclic loading. The steel columns analyzed are of uniform pipe-sections. Both the
846
(a) Centrally loaded column
(b) Eccentrically loaded column
(<^) ^^°^^ Section
Figure 1: Analytical model
centrally loaded columns and eccentrically loaded columns are considered. T h e radius-thickness ratio parameter and slenderness ratio parameter are taken as main parameters. In the analysis of eccentrically loaded columns the effect of eccentricity of axial force is also investiagted. To account for material nonlinearity, the modified two-surface model (2SM) is employed to trace accurately t h e inelastic cyclic behavior of steel. As a result, some design formulas are proposed to predict t h e interaction buckling strength and ductility of steel columns under cyclic loading. NUMERICAL METHOD For thin-walled steel columns of uniform pipe-sections subjected to a constant axial force and cyclic lateral loads, local buckling always occurs near the base of the columns. Therefore, beam-column elements are employed for the upper part of the column, while shell elements which can consider t h e local buckling are employed for the lower part of the column, as shown in Figure 1. T h e b e a m element type employed is a 2-node open section beam (B310SH) with hybrid formulation, while the shell element type is a four-node doubly curved shell element (S4R) which has only one sample point but five layers are assumed across the thickness (ABAQUS 1995). In the analysis, only half of t h e column is modelled because of the symmetry of both geometry and loading. In addition, a stiff plate with infinite bending stiffness is assumed in the interface between t h e beam-column elements and shell elements. T h e modified Newton iteration technique coupled with the displacement control method is used in t h e analysis. Details of the solution procedure can be found in the ABAQUS theory manual (1995). T h e displacement convergence criterion is adopted in the analysis and the convergence tolerance is taken as 10~^. The initial geometrical deflections and residual stresses are not taken into consideration. T h e radius-thickness ratio parameter, i?^, and the column slenderness ratio parameter. A, are defined as follows:
R, = v^3(l - u^:
E 2i
(1)
847
A =
2h
1
Rt=0.115 X=0.26
(2)
in which (Ty = yield stress; E = Young's modulus; v = Poisson's ratio; D = diameter of the column; and t = thickness; h = column height; and r = radius of gyration of cross section.
CENTRALLY LOADED COLUMNS In this section, numerical results of centrally loaded columns reported in a previous paper (Gao et al. 1998) are briefly summarized here.
^
0
t
f ^
r t
\
fI 1 i / 'V''^'-^
^'^^r?^
/ / / J 1 f / /ji^^^
r ^^i^ii
*'"4^~y--j^-yT^
t
lu-i
1
Experiment |
•_
-10
\
H
^**^' L-i
j
fm-t'-^'.i'' jv::- _
10
5/5^
Figure 2: Comparison between analysis and test . . „.
,,. _ .,. —, —,— _^.. _,
E x p e r i m e n t a l Verification As an example, a comparison between the present analytical result and test result (Nishikawa et al. 1996) is first presented. T h e tested specimen is of A = 0.26 and Rt = 0.115. T h e measured strain hardening modulus, Est, is E/iO, and the strain at the onset of strain hardening, Sst-, is 14 times of the yield strain, €y. During the test, the column is subjected to a constant axial load of P/Py = 0.12 and cyclic horizontal displacement at the tip. Here, Py denotes the squash load.
A 0.5
1
1
0.02
0.04
0.06
Figure 3: Strength of centrally loaded columns 8
A
6
The curves of nondimensionalized horizontal load, H/Hy, versus horizontal displacement, S/6y^ from both the experiment and 2SM analysis are shown in Figure 2. The notations, Hy and 6y, represent respectively the yield load and yield displacement considering the effect of axial load (Gao et al. 1998). From the figure, it can be observed that t h e shape of the hysteresis loops from the present analysis is in good agreement with the experimental result. This indicates that by accurate description of material behavior, the developed F E M formulation based on the 2SM can accurately predict the cychc behavior of such columns. As reported in the previous paper, however, traditional plasticity models such as the von-Mises yield criterion incorporated with either isotropic hardening or kinematic rules can not give a satisfactory prediction to the actual behavior.
Computed Proposed
Computed Proposed
4 k
2 h 0.04
0.08
KXA
0.02
Computed Proposed
0.04
Figure 4: Ductility of centrally loaded columns
Interaction Strength and Ductility To obtain design formulas of the interaction strength and ductihty of t h e pipe-section columns, a total of 16 columns are analyzed. The ranges of the parameters are summarized here: (l) Rt = 0.031-0.115 {D/t = 2 9 - 1 2 2 ) ; (2) A = 0.25-0.50; and (3) P/Py = 0.10-0.30. T h e material properties are: cXy = 235.4 MPa, E = 206.0 GPa, i/ = 0.3, E/Est = 40.0, and Sst/Sy = 10.0. Moreover, all the columns are loaded with one cycle of horizontal displacement reversal at each displacement a m p h t u d e (i.e., ±6y^ i 2 ^ y , • • •).
\
848 T h e computed interaction strengths, HmaxIHy^ are plotted against a multiplication of the parameters Rt and A in Figure 3. The proposed formula is as follows: Hrr,
0.02
H,,
(3)
+ 1.10
As can be seen from Figure 3, for a constant Rt, the ultimate strength increases with the decrease in A. Likewise, for a constant A, the strength is improved as Rt decreases. On the other hand, two equations are given to obtain the available ductility of the columns. In t h e present study, the ductility is defined with Sj^JSy and Sg^/Sy. Here, 6^ is the displacement corresponding to the maximum horizontal load, and 695 is the displacement at the point where the strength is 95% of the ultimate strength after the peak load. T h e two formulas are plotted in Figure 4 and expressed as 1
(4)
3(Rt\°~ 0.24 8.
(5)
(1 + PiPyfin^imt
W i t h t h e above formulas, the available strength and ductility of an existing column can be easily evaluated. Inversely, for a given ductility demand, the value of Rt and A can be determined when either of t h e m is determined. ECCENTRICALLY LOADED COLUMNS In this section, discussion is Hmited to eccentrically loaded columns subjected to a constant axial force and in-plane cyclic lateral loads. For an eccentrically loaded column, the axial force P is applied eccentrically at a distance e from the centroidal axis of the colunm. Thus, the eccentricity ratio, e//i, is also considered as a structural parameter. The ranges of all the parameters are : (1) Rt = 0.05 ^ 0.10; (2) A = 0.25, 0.30; (3) PjPy = 0.15; and (4) ejh = 0.0 ~ 0.5. In this study, 12 columns with different values of the above parameters are analyzed. T h e analytical model, as shown in Figure 1(b) in which the action of t h e axial force applied eccentrically is simulated by an end moment MQ = P - e, is adopted bucause the axial force is small. Before we proceed to the discussion of the interaction buckling behavior of t h e eccentrically loaded column under cyclic lateral loading, it would be pertinent to investigate t h e load-deformation relation between the two types of columns in the elastic range. C o r r e l a t i o n in t h e Elastic R a n g e He
Figure 5 shows the loading condition and deflected shape of the centrally and eccentrically loaded columns. In t h e figure. He and He are horizontal loads, and 8c and 8^ are horizontal displacements. Note that the subscripts c and e denote respectively the central and eccentric cases. Moreover, the displacement 8^ is measured from the original undeformed centroidal axis of t h e column. According to the elastic theory neglecting shear deforamtion, 8c and 8e are calculated as
'^fe^
-^— '^I^TT/
Figure 5: Two types of columns
849 Rt=0.075 A,=0.30
-10
Rt=:0.115 A,=0.26
r
-J
1 ~t
7, 1
'
.**
1 ..y^fifi\M^ ^ ^ ^ r * ft h i.d^
I
I Pl-«^
Pw 1- 5
-10
,
-j Pl-e2
1
0
1
5
10
5/5y 5/8y Figure 6: Comparisons of hysteretic curves between two types of columns
^c
=
(6)
ZEI
Se=So +
(7)
3EI
where I = inertial moment of cross section; SQ = initial horizontal displacement resulted from the eccentric vertical load, and given by ^0 =
(8)
2EI
From Eqs. (6) and (7), we have Hf, — Hr.
3Mo 1
2h SQ
[{6, - «,) - So]
(9)
Here, Sg — 8c stands for the horizontal displacement difference. It is clear that for a given displacement difference between the eccentrically and centrally loaded columns, there exists a definite relation between the corresponding horizontal loads. For example, when (5^e — <^c = 0 we have He — He = —3Mo/(2h), This relation indicates that in the elastic range, for the same horizontal displacements of these two columns, the difference in the horizontal loads keeps a constant value of 3Mo/{2h). Likewise, if 6e — 6c = 6o/3 is assumed, Eq. (9) gives He — He = —Mo/h. It should be noted that all the above derivations are based on the elastic theory without considering shear deforamtion. However, as is well known that large plastic deformation will be induced during a severe earthquake event. Therefore, to find out a relation between He and He that is vaHd in both the elastic and plastic ranges, an extensive elastoplastic large displacement finite-element analysis is needed.
850 C o m p a r i s o n of H y s t e r e t i c C u r v e s b e t w e e n C e n t r a l l y and E c c e n t r i c a l l y L o a d e d C o l u m n s T h e computed hysteretic curves of four eccentrically loaded columns are illustrated in Figure 6, together with those of corresponding centrally loaded columns. In each plot of Figure 6, the dashed curve is the result of the centrally loaded column while t h e real curve stands for the result of the eccentrically loaded column. It is observed t h a t the load-carrying capacity in the eccentric side is greatly reduced due to the presence of the eccentric force. In contrast, the maximum strength in the opposite side is much larger than that of the centrally loaded column. Figure 7 shows t h e results of the maximum strength difference between the centrally and eccentrically loaded columns. T h e specimen number is taken as the abscissa, while the maximum strength difference \Hc,max—He,max\ normalized by Mo/h is adopted as the ordinate. Here Hc,max and He,max stand for the maximum strengths of the centrally and eccentrically loaded columns, respectively. For simplicity, Hc,max cind He,max refer to the absolute values of the maximum strength at both the eccentric and opposite sides, respectively. It must be noted that the reference value of Mo/h is determined by trial and error. Actually, five different values that correspond to five different displacement differences (i.e., 6e — 6c = 0, 6o/3, SQ/2, 2 ^ O / 3 , and ^o) were tested. As a result, it is found t h a t t h e case of 6e — Sc = SQ/S is an ideal reference because the maximum strength difference nondimensionalized with Mo/h is almost around the unity for each column, as shown in Figure 7. In other words, the relation between Hc,max and He,max can be approximately written as follows: Mo
He.
Here, symbol "-|-" is for the case in the eccentric side; and symbol "-" denotes the case in the opposite side.
(10) I • I ' I '
' I ' I '
^ 1 2
I
^^H^^^6^&&i>no^±^
To find out a relation between He and He at an arbitrary loading state, comparisons of the n 0.8 h strength differences are shown in Figure 8. As I 0.6 in t h e case of the peak point, the normalized Bccentric Side horizontal displacement, 6e/Sy, is taken as the O Opposite Side 0.4 abscissa, and the value of \Hc — He\/[Mo/h) 2 4 6 8 10 12 14 16 18 20 is taken as t h e ordinate. Here, He represents Specimen Number the horizontal load at displacement 8c^ while He denotes the horizontal load at displacement 8e. Figure 7: Difference in maximum strength Comparisons of the strength differences indicate t h a t : (1) In the elastic range, the strength difference is approximately around the unity. This fact coincides well with t h e previous theoretic derivation. (2) In the elastoplastic range, the strength difference is also close to the unity in each different displacement level. Therefore, it can be concluded t h a t in the case of Se — Sc = So/3, there exists a definite strength correlation between the eccentrically and centrally loaded columns as follows: Se — 6c = So/3
Hc-He
= ±
Mo
(11) (12)
Figure 9 shows comparisons of computed hysteretic curves of the eccentrically loaded columns and those obtained using both the above correlation and computed results of corresponding centrally loaded columns. T h e real curve stands for the analytical result (designated by F E M in the figure), while the dashed curve refers to the predicted hysteretic curve (designated by P R E D in
851 1.6 1 Pl-el T - —I—r • 5e-5c=5o/3 1.4 1.2 e 1 0.8 f ••* \ X 0.6 0.4 Rt=0.115 0.2 A,=0.26 • 0 8 10 - 1 0 - 8 - 6 - 4 - 2 -10-8 -6 - 4 - 2 0 2 4 6
«••
1 _..•• 1 •• • • • • <
1
5c/5y
(a)
(b)
1.6 —I 1 1 1 1 1 P8-el T— 1.4 1.2 / 1 ^ ^ A 1 :§ • ' •< 0.8 0.6 0.4 Rt=0.075 0.2 A,=0.30 ...1 1 .,.. 1 1 1 0 - 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 10
1 ••••
(c)
1 P8-e2
r •••.
T~
1
2
4
6
8 10
1
1
1
1
1
A A
• ' • <
Rt=0.075 A,=0.30 -10-8 - 6 - 4 - 2
(d)
^*^^y
0
^A
0
!_.,.. 1
1
i
2
6
8 10
4
^A
Figure 8: Strength differences at different displacement level the figure). It is observed that the two curves at each plot show a good agreement with each other at both the elastic and inelastic ranges, which implies that the proposed correlation for the centrally and eccentrically loaded columns is quite convincing. Accordingly, the hysteretic curve of an eccentrically loaded column can be conveniently obtained from that of the centrally loaded column. Interaction Strength and Ductility In the previous section, the strength and ductility formulas of the centrally loaded columns are available as given by Eqs.(3) to (5). Thus, the interaction strength and ductility formulas for eccentrically loaded columns can be written as He.
_
0.02
Hy
+ 1.10T
1 3{Rt <^e,95
Sy
AO-5)o.«
0.24
l±
P e Hy h
(14)
3($„
,
(1 + P / P J 2 / 3 ^1/3 R^
(13)
±A
(15)
3^^
where Se,m = displacement at the maximum strength point for the eccentrically loaded column; <^e,95 = displacement at the point corresponding to 95% of the ultimate strength after peak load for the eccentrically loaded column; symbol "±" = positive value taken in the eccentric side and negative value employed in the opposite side; and symbol "=F" = meaning which is reverse to "±". Based on the theoretic derivation and numerical analysis, it can be concluded that Eqs. (13) to (15) can be used to determine the interaction strength and ductihty of the eccentrically loaded columns in seismic design.
852
2 1
r Rt=0.075 ~T A,=0.30
1
k
X
-1
\i^;^^Y/////^
F(a) ,
-3
1 ^ '.
I
_,__^ - 1
0
-10
2 1
5
JJ^^\ H
-2
10
8/5y
f Rt^O.115 A,=0.26
1
'
1 Pl~el
-2
-T
'
I
w^IL L
'^T^Tr}ff.
F(b) , I _; _ 1
-10
-5
—J PRED]
0
-10
10
5/5y -|
'
1 Pl-e2
A,=0.26
LLw.^ ^
F(c) , LA^;—1_
5!Hi^
1
1-
r
1 P8-e2 1 1
9T^^^?^Tj'^f~~H
1 Rt=0.115
0 -1
Rt=0.075 ^=0.30
1
^A^Sa^-|
P^ 0
5
P8-el rj
H r PREDI
1
0
5/8,
S
F(d) , l„._i
10 - 1 0
1
-5
1
^ff^
-j
/f^ M^r^"^
1
— — — PREDI 1
0
8/5y
5
10
Figure 9: Comparisons of hysteretic curves between analysis and prediction CONCLUSIONS This paper presented the numerical results of both the centrally and eccentrically loaded steel pipe-sectional columns under cychc loading. The modified two-surface model was used as the constitutive law of steel. Comparing of the analytical results with experimental results of a tested specimen showed that the plasticity model can be employed to predict the cyclic behavior of steel structures. Discussion was also made to the correlation of the hysteretic characteretics between the centrally and eccentrically loaded columns. As a result, a definite relation between the two types of columns was found and this finding is expected to be very useful in the effective use of available data obtained from the centrally loaded columns. REFERENCES Usami, T., Mizutani, S., Aoki, T., and Itoh, Y. (1992). "Steel and concrete-filled steel compression members under cyclic loading." Stability and ductility of steel structures under cyclic loading, Y. Fukumoto, and G. C. Lee, eds., CRC Press, Boca Raton, Fla., 123-138. Shen, C , Mamaghani, I. H. P., Mizuno, E., and Usami, T. (1995). Cyclic behavior of structural steels. II: Theory. J. Engrg. Mech., ASCE, 121:11, 1165-1172. Mamaghani, I. H. P., Usami, T., and Mizuno, E. (1996). "Inelastic large deflection analysis of structural steel members under cychc loading." Engrg. Struct., 18(9), 659-668. Gao, S. B., Usami, T. and Ge, H. B. (1998). Ductility evaluation of steel bridge piers with pipe-sections. J. Engrg. Mech., ASCE, 124:3, 260-267. ABAQUS/Standard User's Manual. (1995). Hibbitt, Karlson &; Sorenson, Inc., Ver. 5.5, Vol. I &;II. Nishikawa, K., Yamamoto, S., Natori, T., Terao, K., Yasunami, H., and Terada, M. (1996). "An experimental study on improvement of seismic performance of existing steel bridge piers." J. of Struct. Engrg., JSCE, 42A, 975-986 (in Japanese).
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) ©1999 Elsevier Science Ltd. All rights reserved.
853
STEEL SHEAR PANELS FOR ANTI-SEISMIC ELEMENTS
M.Yamada Dept., Architecture, Facul. Engrg., Kansai University, Iwakura, Chuzaiji-cho, No. 156, (606-0021) Sakyo-ku, Kyoto, Japan
ABSTRACT In order to make clear the resisting mechanism of steel shear panels as anti-seismic element and to establish their calculation method for anti seismic design, tests were carried out on steel shear panels with various plate thicknesses. Their resisting mechanism is the diagonal tension field formed after the buckling of conjugate diagonal compression field. Deformation processes are computed as the superposed resistances of surrounding frame and builted in steel shear panel. Steel shear panel is assumed to be a diagonal tension member with an equivalent effective width of 1/3 of span and height of panel under smaller story sway angle R<0.005, and 1/2 of span and height of panel under larger story sway angle R>0.005. Calculated results are compared with tested results and coincide very well until ultimate states.
KEY WORDS Anti-seismic element. Buckled wave. Cyclic shear, Effective width. Seismic desgin. Steel shear panel, Sway angle. Tension field. Ultimate state.
854 INTRODUCTION Steel shear panels have been applied as anti-seismic element in steel buildings in Japan,e.g.the head office of Obayashi Corporation, in Osaka, Japan, Yamada (1980), the headquarter of the Japan Federation of Economic Organization, in Tokyo, Japan, and the New Kobe City Hall, in Kobe, Japan, Yamada (1996). Especially the New Kobe City Hall had shown very good resisting behavior under the last Kobe Earthquake, 17. Jan. 1995, Yamada (1996). However there are yet only few researches, Yamada(1992), Caccese et al. (1993), Elgaaly et al. (1993), Sugii and Yamada(1996), on their resisting behavior under cyclic loading until fracture experimentally as well as analytically. In order to make clear the resisting mechanism under cyclic shear loading and to establish a calculating method of the resistance and deformation of steel shear panel, a series of tests were carried out on steel shear panels with various steel panel thicknesses.
TESTS Test Pieces Tests were carried out on 1/10 scale models with various steel panel thicknesses, surrounded by rectangular composite rigid frame made of 2 steel channels of 20 X 40 X 20mm, covered by reinforced concrete with 2- (J) 3mm reinforcements such as shown in Figure 1. Mechanical properties of steel panels are indicated in Table 1.
Tm
2-0 3,
2 - [ 20 X 40 X 20 t=2.1mm ^-*lloi« ^ > i l i ^
^
Mm
Stimip
%
R=<5 /H
^
\^^
^r"
^
yr
• 4 < ] - ^ ^
-l6Ch 30 -H h-
480 L=540
H60k -H K 30-
ts=0.4,0.6,0.8, 1.0, 1.201111 :
Figure.l : Test piece TABLE 1 MECHANICAL PROPERTIES OF STEEL PANELS Test Pice iSWT04CYC SWT06CYC SWT08CYC SWTIOCYC SWT12CYC
Plate Thicknes ts(mm) 0.4 0.6 0.8 1.0 1.2
Yield Point a y (MPa) 238 218 229 230 234
Tensile Strength O max (MPa) 335 340 343 344 354
Elongation i S max (%) 42 33 40 46 37
855 Testing Methods Cyclic horizontal force Q was loaded at the top of specimen and horizontal sway 6
was measured
by dial gages such as illustrated in Figure 1. Horizontal sway angle R were indicated as R= d /H. Cyclic loading were carried out by incremental sway angle amplitudes of
R= ± 0.001, ± 0.003,
± 0.005, ± 0.007, ± 0.010, ± 0.030, ± 0.050. Test Process At the sway angle of R=0.003, the first buckling wave was formed along the diagonal direction and the diagonal tension field was formed along this buckling wave. Then at the following inverse loading after the vanishing of the first buckling wave, inverse buckling wave was formed along the conjugate direction at the sway angles of R=-0.001 and 0.003, and the inverse diagonal tension field was formed along this conjugate buckling wave. With the increase of load ampletudes, buckling waves of shear panel were reciprocally formed and vanished at the sway angles of about 0.001 ^ 0.003. Figure 2 shows a typical horizontal force (Q)-sway angle (R) relationship of test piece with a steel panel thickness of 1.2mm and the formation processes of buckling wave^. Figure 3 shows the buckling waves which correspond to the small triangles T indicated in the hysteresis loops in Figure 2. Q(kN)
P
1
(5 (mm) (rad) 0
100
0.001 0.002 0.003 0.004 0.005 ( R = ± l / 1 0 0 0 ~ ± 5/1000) •
1"
-^^
h
(a)
Q(kN) "h
50
-3.00
-2.00^
>/<1.00
Cb)
/ y^ ^ y 1 . 0 0 ^ ^ ^ 2.00
0
6 (mm) 3.00
^ tf
^^V^ -50
1
-100 • -0.010
-0.005
()
( R = ±7/1000~±
0.005
Figure 2 : Hysteresis loops of ts=1.2mm
10/1000) R(rad) 0.010
J '^"'^"•'-^.1 (b)
IXI (c)
O:^
1
h
(d)
^
(e)
Figure 3 : Formation of buclding waves at each loading stage corresponding to • marks in Figure 2
856
m [
-100 t-^-0.012 -0.008
> -20 -40 -60 -80 -100 ' -0.012
0 0.004 R(rad) (a)
0.008 0.012
100 80 60 40 20 0 -20 -40 -60 -100 -0.012
-0.008 -0.004
0 0.004 R(rad) (c)
0.008
^
R ^
-0.012
-0.008
-0.004
0 R(rad) (e)
0.004
0.008
Figure 4-1: Tested Results (R<0.001)
• • •
j
-0.004
•;SWTD4(7yc
i
0 0.004 R(rad) (a)
0.008 0.012
1SWT06CY-C -0.004
0 R(rad) (b)
0.004
0.008
0.012
0 R(rad) (c)
0.004
0.008
0.012
Aim ^ H—
-0.008
100 80 : I D SBR 60 40 20
-0.004
@ FY
r
0
i::l-£'^-^''
-20 -40 -60
0.012
•
ASHP ^FY
-0.008
iQSB^ JQSBR
-0.012
•
. . ' . . . ' ' . -0.008
100 , 80 IUSBV 60 IQSBR 40 20 0 -20 -40 -60 -100 -0.012
I
'-'—!-
\
. SWT04CYC -0.004
ASB6
1
^swriDCvc 1
-0.008
-0.004
0 R(rad) (e)
0.004
0.008
0.012
0.004
0.008
0.012
Figure 4-2: Caluculated Results (Be=l/3L)
Figure 4 : Hysteresis loops of steel panel shear wall R ^ 0.010
857 120 80
f ; ]
120
l-IU J5BY ' A SW 80 '"^^ SBH , ^ FY
I ;
40
I 0 -40 -80
40
I \^ fwA t
•
-
. >
C^'J^"-^^ I ° ^-40
SWT|04CrC
*SWt04GYCJ
-0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 R(rad)
-0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 R(rad)
(0
(0
120 80 40 ^
0 -40 -80 SWT06CYC -120 -0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 R(rad) (g) 120 ( SBR i) FY 80
-0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 R(rad) (g)
E
^ 40 I 0 -40 SWTOgCYC -120 t
-0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 R(rad) (h)
-0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 R(rad) (h) 120.
' A SM^ SBR ^ n r d feBY
• rfl
,.7-«--® -0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 R(rad)
(0
(0
120 I
LlQ feBY LID
§
0 -40 -80
-0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 O.I R(rad)
(J)
Figure 5-1: Tested Results (R>0.001)
SWTTOCYC
-0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 R(rad)
SBR
@ FY
[.
, 9'-: ;
\ \
. j . Q
\ >
: l-.'^
;
,
,
, 4
^
.
.
. ^
;
jSWTucYc j
-120 t -0.1 -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 R(rad)
0)
Figure 5-2: Caluculated Results (Be=l/2L)
Figure 5 : Hysteresis loops of steel panel shear wall R ^ 0.010
858 Test Results Tested results of horizontal load (Q)-horizontal sway angle (R) relationships are illustrated in Figures 4-1 (a)-(e) and in Figures 5-1 (a)-(e). Figure 4-1 shows the tested hysteresis loops of smaller sway angle amplitudes R ^ 0.010 and Figure 5-1 shows the tested hysteresis loops of larger sway angle amplitudes R ^ 0.01 until fracture. Final fracture occurred through the fatigue break down of steel shear panel by the repeated reciprocal buckling.
CALCULATION Resisting Mechanism Resisting mechanism of a steel shear panel is idealized as a diagonal tension field with an effective width of initially 1/3 and finally 1/2 of span and height of shear panel such as shown in Figure 6.
M "y • /^^7Est=0.03E
•'" h'^l y -a^
/
" E=2.0 X 10« (kg/cm*)
[^v
/ / -
11
"y 5 -M„
Figure 6 : Tension field model of shear panel
Figure 7 : Stress-Strain,
Figure 8 : Moment-curvature
relationships of steel
relationship of surrounding
panel for calculation
frame for calculation
Composite Frame Steel Panel Total Resistance R(rad)
FY SR SBY SBO
Yielding of Surrounding Composite Frame Begining of Resistance by Bracing Steel Panel Yielding of Bracing Steel Panel Vanishing of Resistance by Bracing Steel Panel R
Figure 9: Superpose of Each Resisting Elements
859 Assumptions Stress-strain relationship of steel panel is assumed to be a tri-linear process such as illustrated in Figure 7. Moment-curvature relationship of surrounding composite frame is assumed to be a bi-linear process such as illustrated in Figure 8. Calculation of Load(Q)-Sway angle(R) Relationships Load(Q)-Sway angle(R) relationships are calculated by the superposition of resistances of the diagonal tensile steel bracing band with effective width Be and that of the surrounding composite frame under the same displacements such as shown in Figure 9.
Calculated Results Calculated results are shown in Figures 4-2, 5-2 with critical points of the yielding of surrounding frame FY, the initiation of resistance of steel bracing SBR, the yielding of steel bracing SBY, and the vanishing of resistance of steel bracing SBO (see in Figure 9).
DISCUSSIONS Shear resistace snd sway angle relationships of steel panel shear walls were tested with various steel panel thicknesses and compared with calculated results. With the increase of steel panel thicknesses, the hysteresis loop incresed its rigidity and resistance and became stable. On the contrary the hysteresis loops of thinner steel panel, shew slipping and hard spring type loops. Reversal of buckling waves brought some unstable slipping of hysteresis loops under lower resistance zones. Ultimate resistances were reached at the sway angle of about R=0.03 unrelated with panel thicknesses. Numbers of buckling waves in steel panel were many (6-7) at thinner plate thickness (ts=0.4mm) and few (2-3) at thicker plate thickness (ts=1.2mm). The hysteresis loops of the tested results shown in Figures 4-1,5-1 coinside very well with the calculated results shown in Figures 4-2,5-2, the coinsidence between tested and calculated hysteresis loops is very well. The corresponding critical points in calculated hystersis loop indicate and clarify the resisting mechanism of shear panel very well.
860 CONCLUSION Tests of steel panel shear walls under incremental cyclic shear were carried out on verious panel thicknesses until fracutre. Steel shear panels formed diagonal tension field after the formation of buckling wave in the conjugate compressive direction. Effective width of this diagonal tension field may be assumed to be as 1/3 of span and height of panel at the smaller sway angle amplitudes R ^ 0.005, and 1/2 of span and height of panel at the larger sway angle amplitudes R ^ 0 . 0 5 . By this assumption the calculated results of elasto-plastic hysteresis loops coincide very well with tested results until fracture.
Acknowledgements This research was carried out by the grant in aid (C) of Ministry of Education, Japan(1994) and Colaboratory Research in Kansai Univ.(1994). The auther would like to express this hearty thanks to them and the kind supply of materials to the Nippon Steel Corporation. Thanks are dedicated to Mr. and Mrs. Sugi-i, and Miss. Fukuda for their cooperation on this research too.
References Caccese V. Elgaaly M. Chen R.(1993). Experimental Study of Thin Steel-Plate Shear Walls under Cyclic Lo2idJ.Struc.Engrg.ASCE 119:2,573-587. Elgaaly M. Caccese V. Du C.(1993). Postbuckling Behavior of Steel-Plate Shear Walls unde Cyclic Loads. J.Struc.Engrg.ASCE,ll9i2 588-605. Sugii K. Yamada M. (1996). Steel Panel Shear Walls, With and Without Concrete Covering, Proc.ll World Conf. Earthquake Engineering (WCEE), Acapulco, Mexco. Yamada M. (1980). Bauen in erdbebengefahrdeten Gebieten — Beispielhfte Losungen. Deutsche Bau zeitung 80:11, 24-34. Yamada M. (1992). Steel Panel Encased R.C. Composite Shear Walls, Composite Construction in Steel and Concrete II , Proc, Engrg. Foundation Conf. Stru, Div. ASCE, Potosi, Missouri. 899-912. Yamada M. (1996). Das Hanshin Awaji-Erdbeben, Japan 1995 —Schaden an Hochbauten, Bauingenieur, 71:2 73-80.
Light-Weight Steel and Aluminium Structures P. Makelainen and P. Hassinen (Editors) © 1999 Elsevier Science Ltd. All rights reserved.
861
RESULTS FROM LOW CYCLE FATIGUE TESTING Anders Salwen and Tom Thoyra Steel Structures group, Department of Structural Engineering, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
ABSTRACT In this paper the results from three different types of tests performed will be discussed. All of the construction details are part of KB5/Fast Bridge 48. This is a new type of rapid assembled military bridge constructed by Karlskronavarvet. The performed tests all concern the low cycle fatigue behaviour in extra high strength steel, EHS, and ultra high strength steel, UHS. Since these steels are relatively new products there are no design codes available for these steels and therefore design through testing must be performed. Since these applications are from real structures there are no corresponding detail classes in the codes. The results from these tests shows that the current design codes for fatigue can be used and the codes will give fatigue strength well on the safe side. KEYWORDS High strength steel. Low cycle fatigue, Welded joints, Fatigue test. Buttering, Box column, Truss bridge. INTRODUKTION An application of the EHS and UHS steel is in the KB5/Fast bridge 48 system. The KB5/ Fast bridge 48 is a truss structure bridge made of Weldox 1100 and Weldox 700. The truss bridge has been developed and manufactured by Karlskronavarvet for the Swedish Defence Material Administration. The bridge is 48 m long and can be assemble by a seven men crew in less then 90 minutes. The bridge is designed to carry loads up to Military Load Class 70, MLC 70 is about 65 metric tones. The Department of Structural Engineering at the Royal Institute of Technology has been involved in the design and testing of many of the details in the KB5 bridge mainly because no material data of the Weldox 700/1100 is possible to find in any codes. In this paper some of these details will be discussed.
862 FATIGUE BENDING TEST OF BOTTOM TRUSS MEMBER JOINTS OF KB5 The first test, Edlund & Hoglund (1994), studied the fatigue strength of welded connection between the bottom truss member and the diagonals, refer Figure 1.
studied area
Stiffeners
FIGURE 1: ELEVATION OF A SECTION OF KB 5 / FAST BRIDGE 48
The fatigue strength of the upper edge of the bottom truss member was determined by applying a transversal load to a part of the truss member that had been turned upside down, refer Figure 2.
ii no
^
^ ^
"ZX
41^
T
, b ^
3500
^
a
—••
FIGURE 2: PART OF BOTTOM TRUSS MEMBER, TURNED UPSIDE DOWN, USED IN THE BENDING TESTS
Two beams were tested. For the first beam three different load ranges were used and for the second the load range was kept constant. The maximum force is based on the theoretical ultimate load according to the effective cross section regarding buckling. The beams dimensions, forces and number of cycles is given in Table 1. TABLE 1 DIMENSIONS, FORCES, STRESSES AND LOAD CYCLES TO FAILURE FOR THE TWO BEAMS
Beam 1
a b h [mm] [mm] [mm] 161 172 283
284
160
166
Thickness [mm] 5.2
5.2
Force [kN] 10,4- 132 30,3- 180 10,5- 189 10,4- 190
Stress [MPa]
cycles
388 479 573 616
1 - 3000 3001 - 12000 12001 -14496 1 - 7544
To compare the results, the fatigue classes were calculated. The C class is identify by the characteristic fatigue strength at 2 million cycles, Eqn. 1 according to Boverket (1994).
863
/..
^2-10'^^ (1)
When two or more different load range had been applied the Palmgren-Miner sum was used to determine the C class, Eqn. 2. C-
n,a, 210^
K
3 \, «,C7;
210^
(2)
For beam one and two the C classes were calculated to 93 MPa and 96 MPa. FATIGUE TENSILE TEST OF BOTTOM TRUSS MEMBER JOINTS Another design detail that was tested was the welded connection between the diagonal truss member and the upper part of the bottom chord, Edlund & Hoglund (1994). The diagonal connects to the bottom chord in a sharp angel. This makes the welding of the connection between the diagonals and the chord on the acute side almost impossible. The solution was to use the buttering technique. A three mm high weld deposit was attached to the upper side of the bottom chord and the connection plate was then welded to the bottom chord from the obtuse angel side, refer Figure 3. In the buttering weld a consumable with lover strength then in the real weld was used in order to even out the stresses in the joint.
FIGURE 3: USAGE OF BUTTERING TECHNIQUE
FIGURE 4: TESTED COUPLING DESIGN TYPES FOR T H E H D S B
864 Two types of test specimen was used, se Figure 4. Five test specimens of each type were tested. The results from the testing is given in Table 2 and Table 3. TABLE 2: DATA FOR THE FIVE TEST SPECIMEN FROM FATIGUE TENSILE TEST TYPEI
Specimen no.
Min Load [kN]
Max Load [kN]
1 2 3 4 5
0,9 1,1 32,1 32,1 32,1
139 139 170 173 173
Average crosssection [mm^] 258,8 259,1 259,1 262,9 261,9
Min Stress [MPa]
Max Stress [MPa]
Stress Range [MPa]
3 4 124 122 123
537 537 656 658 657
534 533 532 536 534
No. of cycles before failure >3850 2872 5263 4205 4001
TABLE 3: DATA FOR THE HVE TEST SPECIMEN FROM FATIGUE TENSILE TEST TYPE2
Specim en no.
Min Load [kN]
Max Load [kN]
1 2 3 4 5
40,8 41,3 41,4 41,8 41,9
216 215 217 217 216
Average crosssection [mm^] 601,2 596,9 602,2 602,5 601,3
Min Stress [MPa]
Max Stress [MPa]
Stress Range [MPa]
67,9 68,6 68,8 69,3 69,7
359 360 360 360 359
291 291 291 291 289
No. of cycles before failure 44949 62013 65145 32769 64557
When testing the two first specimen of type 1 both tensile and bending deformation occurred due to the low load and the asymmetry of the specimen. When the applied load was increased for the reaming tests, including those of test type 2, only minor bending occurred. Using statistical evaluating a calculation of the detail class C, according to Boverket (1994), gives a value of C « 60 MPa for test type 1 and C ~ 61 MPa for test type 2. In Boverket (1994), C is given for a number of details but not any are like the ones we have tested. The main purpose of these tests was to determine a corresponding value for the detail class C.
FATIGUE TESTING ON BOX COLUMNS MADE OF EXTRA HIGH STRENGTH STEEL The high strength steels Weldox 700 and 1100 are quenched and tempered with following annealing. At the production the annealing temperature has been 500^ C but today the annealing temperature is 200^ C. At this test, Salwen (1997), six box columns were simply supported and exposed to four point bending. Reinforcement plates were welded on the tensile flange between the loads. These plates were either welded over the corner or on the plane surface of the beams. The purpose of this investigation was to determine weather there were any differences in fatigue life for the two types of reinforcement plates and if the annealing temperature had influence of it. The experimental arrangement and the cross section of the two types, refer Figure 5. During each test the strain distribution in the vicinity of the reinforcement was measured with strain gauges. The cross section and the strain gauges location for the tested beams is shown in Figure 6.
865
5-H
Beam 1 & 5 low tempered 2 high tempered
160 Beam 3 & 6 low tempered 4 high tempered
Type 1
Type 2
V
^ F
J
Hydralic jackl HEA160
HEA160 Reinforcement plaqe
Stiffeners
225
Deflection meters
750 1000 1250
2000
FIGURE 5: EXPERIMENTAL ARRANGEMENT AND CROSS SECTION OF THE TESTED COLUMNS
^13
weld
I
13-15 I
6
5 225
4
weld
12_J^
10-12
FIGURE 6: BEAM DIMENSION AND LOCATIONS OF STRAIN GAUGES.
For the first four beams two different load ranges were used. The first load range was between 10-140 kN and the second was between 10-190 kN. The second load range was applied after 12133 cycles for the four first test and from the beginning for the two last ones. The stress distribution for beam no. 1 after 12133 load cycles at a load range of 180 kN and the stress distribution along the centre line of beam no 5 at the same load range, but after 100 load cycles, is shown Figure 7. To compare the results, the fatigue class C were calculated according to Eqn. 1 and 2. Table 4 shows the load range cycles and the calculated fatigue class C.
866
FIGURE 7: STRESS DISTIBUTION FOR BEAM NO. 1 (LEFT) AND BEAM NO. 5 (RIGHT)
TABLE 4: BEAM TYPE, LOAD AND LOAD CYCLES TO FAILURE
Nr. 1 2 3 4 5 6
Load rkNI Type 1, low temp. 10-140 for « < 12133 1, high temp 10-140 for « < 12133 2, low temp. 10-140 for n < 12133 2, high temp 10-140 for n < 12133 1, low temp. 2, low temp.
Load rkN1 Cycles to failure 10-190 for « > 12133 18231 10-190 for « > 12133 15294 10-190 for n > 12133 16177 10-190 for n > 12133 13846 10-190for all« 6517 10-190for all w 8330
The design class is based on constant amplitude tests calculated from the mean line for a large number of tests minus 2 standard deviations. The design C class for reinforcement plates is between 45 - 50 MPa for steel with an yield strength up to 700 MPa according to Boverket (1994). CONCLUSIONS The results from the tests reported in this paper is summarised in Figure 8. These testes were all performed mainly because the details are not represented in the current Swedish design codes. However using suitable detail class in the design codes will give a result on the safe side. The results from the bending tests, in Figure 9 referred to as Beam 1 & 2 and Box 1 - 6, and the tensile tests, in Figure 9 referred to as Test type 1 and Test type2, indicates that a high static strength does not increase the fatigue strength of welded structures except for low cycles where yielding is an upper limit for the maximum stress.
867
Load cycles
1'E+05
FIGURE 8: S N - DIAGRAM OVER THE RESULTS FROM THE TESTS REPOERTED IN THIS PAPER.
REFERENCES Boverket (1994), BSK 94 Boverketshandbok om stalkonstruktioner (Swedish regulations for steel structures, in Swedish), Boverket S. Edlund and T Hoglund (1994), Tests of bottom truss member joints of KB5/Fast bridge 48, Technical report 5, Steel structures, Department of structural Engineering, Royal Institute of Technology, Teknikringen 78, SE-10044 Stockholm Sweden Federal Republic of Germany, United kingdom and United States of America. (1984), Trilateral design and test code for military bridges and gap crossing equipment A. Salwen (1997), Provning av utmattningshallfastheten hos hoghdllfastladbalk med forstdrkningspldtar (in Swedish), Technical report 10, Steel structures. Department of structural Engineering, Royal Institute of Technology, Teknikringen 78, SE-10044 Stockholm Sweden
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KEYWORD INDEX calculation 257 catwalks 803 CEN/TC 250/SC 3/PT 4 399 CEN/TC 265 399 channel column 45 channel sections 171 clamped 335 classification of cross sections 503 cUnch 577 codification 475 cold-formed 107, 115, 141, 179, 317 member(s) 131, 293 steel 3, 123, 569, 593, 779 steel design 89 steel profile 713 steel wall frames 37 collapse 99 columns 3, 155, 441, 627 combined members 69 composite(s) 317, 465 construction 423 member 737 section 745 compressed bent 415 compression 123, 647 computer 115 aided design 359 program 249 concentrated plasticity 655 concrete-filled member 755 concrete-filled pier 821 concrete-filled tubular column 737 connection(s) 449, 569, 577 consistent approximation 241 construction 367 continuous 189 corner bolt system 465 corrosion 703 corrugated 115 sheet 79 web 789 coupled instabilities 155 crack arrest 223 crack detection 215 creep 197, 679 crest-fixed trapezoidal steel cladding systems 609 critical force 415
advanced analysis 19 air tightness 689 allowable stress 399 aluminium 69, 433, 441, 449, 487, 561, 679 alloy beams 637 alloys 457, 655 box sections 515 extrusions 465 structures 475, 647 amplification factor 61 analyse 327 analysis 283 Annex J 335 Annex L 335 anti-seismic element 853 application 755 astron 335 austenitic stainless steel 233 axial load 171 bars 647 base 335 beam(s) 3, 449, 779, 795 columns 171 element 249 beam to column joint 293 beam-column 141, 179 bearing design 601 bending 171, 755, 779 bending moment resistance 503 bifurcation analysis 27 blind rivets 495 bloating 555 bolted connection(s) 539, 601 bolts 569 box column 861 break-away column 205 bridges 487 British Standards 69 buckled wave 853 buckling 3, 19,79, 115, 171, 679 length 61 mode 265 buckling of studs 37 buckling strength 737 building(s) 335, 423 building industry 561 buttering 861 869
870 cross-section 487 slendemess 727 cross-sectional classification 637 crushing failure 795 C-section(s) 115, 779 curtain walls 433 curved arch bridge 811 curved pair ribs 811 cyclic behaviour 539 cyclic loading 283, 293, 577, 829, 845, 853 cyclic shear 853 cylindrical shells 233 cylindrical storage tank 367 damping 719 databases 377 deformation 257 degradation laws 539 design 3, 327, 441, 475, 487, 627, 689, 795 analysis 69 codes 457 formula 609 method 523, 745 principles 351 regulation 755 strength 45, 569 development 351 discrete joints 415 discrete theoretical model 803 displacement 415 method 761 distortional buckling 27, 45, 89, 123, 131 double bottom 367 double shell 367 ductihty 449, 655, 845 demand 457 durabihty 703 dynamic 211 analysis 53, 821 earthquake 351, 377 EC3 335 eccentricity 171 ECCS column curves 441 economy 351 edge stiffener 45 effective section 131 effective thickness 515 effective width(s) 155, 273, 853 efficiency 755 elastic and elasto-plastic range 163
Keyword Index elastic behaviour 761 elastic foundation 107 elastic stabiHty 61 elasto/plastic 69 elastoplastic analysis 737 elasto-plastic analysis 845 elevated temperature(s) 233, 727 energy 689 absorbing 205 absorption 53, 821 dissipation capacity 539 method 107, 273 environment 689 erection sequence 713 Eurocode 3 Part 1.4. 627 Eurocode 9 433, 441, 475, 515 experiment(s) 257, 301 experimental 317 analysis 647 investigation(s) 45, 803 extrusion 487 fabrication 327 failure 171 failure mode 601 fatigue 215, 223, 837 test 861 FEM 53, 811, 821 fine grained micro-alloyed steel 367 finite element 795 analyses 465 analysis 69, 301, 585, 829 method(s) 141, 155, 761 modelling 317, 609 finite rotations 241 finite strip 115, 795 finite strip buckling analysis 27 finite strip method 423 fire 561 fire conditions 727 fire protection 547 fire resistance 523 fire resistivity limit 561 fire tests 523 fireproof 561 fire-resistance coating 555 fixed 335 fixed-ended 45 fixed-ended columns 27 flanges 795 flexural buckling 27, 441, 627
Keyword Index flexural wrinkling 301 flexural-torsional buckling 27, 131 flexure 779 flooring grids 803 floors 719 foam joints 301 folded shell model 433 fracture mechanics 215 frame 335, 343 frame analysis 727 frames 19 full scale wall frame tests 37 generalised beam theory 3 geometrical imperfection 249 geometrically exact 241 girders with thin webs 789 girt 351 global buckling 265 glulam structures 327 gross section 131 guard fence 53 HAZ 449 heat and mass transfer 689 heat conduction 547 heat transfer 703 heavy truck 53 high strength steel 45, 861 high-temperature material data 679 high-temperature properties 771 holes 779 hollow composite section 755 homogenisation 803 hybrid design 465 hydrated micas 555 hysteresis loop 293, 829 impact test 205 imperfection sensitivity 89 imperfections 155 indoor climate 689 infra red measurement 585 initial geometrical imperfections 211 instabihty phenomenon 265 interaction 171, 189 buckling 141 intermediate support 189 internal negative pressure 399 Internet 377 inverse methods 547, 679 inverse techniques 215
871 Java 377 joint(s) 577, 655 connection 617 flexibility 343 modelling 539 rotation 343 laminated structures 415 large deformation 737 large scale tests 399 laser welding 837 lateral torsional buckling of beams 503 leaning columns 61 light gauge steel structures 689 light steel framing 317 light weight members 351 lighting column 205 light-weight steel 593 limit state 257 limit state design 189 load capacity 69 load carrying capacity 155 load combination 359 load effect interaction 359 load test 293 load-bearing structures 523 load-carrying capacity 99 local buckling 27, 89, 131, 141, 265, 293, 423, 515, 637, 727, 795, 829, 845 local dimpling and pull-over failures 609 low cycle fatigue 861 LRFD vs. ASD 385 material 189 matrix analysis 761 mechanical fasteners 495 mechanical properties 771 member resistance 61 metal structures 179 micas 555 minimum weight design 385 modal analysis 719 modelling 189 moderate rotation(s) 241, 249 modulus of elasticity 745 moisture 689, 703 insulation 713 monopanel 415 mounting 327 multimedia 377 multiple column curves 627 multi-spring element model 293
872 natural frequency 719 nonlinear 19, 249 analysis 241 response 727 stability 163 nonuniform girders 385 numerical analysis 637 openings 779 optimum design 745 orthotropic material 99 orthotropy 163 overall buckling 845 partial interaction 317 partial safety factor 399 path following algorithm 249 perforated section 131 perforation(s) 123, 779 pipe-section 845 plane frames 61 plasterboard lining 37 plastic design 457 plastic hinge 655 plastic hinges 19, 457 plastic resistance 745 plasticity 257 plasticity model 829 plate girders 385 plates with loaded edges clamped 273 post buckhng deformabihty 293 post-buckling behaviour 211 postcritical behaviour 155 press-join 577 production cost reduction 465 program 115 pull-through 495 purlin(s) 115, 351 railway bridges 223 Raleigh-Ritz 107 Ramperg-Osgood law 433 rectangular box-columns 273 rectangular hollow section 293 relaxation 197 residual stresses 141, 211 resistance 189 retrofitting 223 rivet 577 riveted beams 223 road side safety 205 roof cladding 713
Keyword Index roof construction 713 roof structure 713 roof truss 593 ROSETTE-joint 585, 593 rotation capacity 457 rotational capacity 637 safety 351 sandwich panel(s) 189, 197, 301, 837 screw 577 sectional behavior 283 seismic design 829, 853 self drilling screws 495 self tapping screws 495 serviceabihty 189 shape 487 imperfections 257 shear 79, 495, 779 shear buckhng load 241 shear connection 317, 761 shear lag 569 shear strength 577, 585 sheet steel 601 shell 257 shell buckling 233, 399 shell structure 265 ship structures 837 shipbuilding 561 softening 449 space frame(s) 19, 617 space structures 647 space truss 617 spatial structures and stabihty 19 specifications 795 square hollow section 141 stability 211 stability design 233 staggered holes 569 stainless steel 503, 523, 627 standard rheological model 197 static analysis 803 steady-state 771 steel 69, 107, 351 steel and timber 495 steel beam 79 steel bridge pier 829 steel bridges 215, 223 steel construction 423, 555 steel elements 327 steel frame 293 steel members 283 steel pier 821
Keyword Index steel rafter 713 steel shear panel 853 steel sheet joining 593 steel slender web 211 steel structure(s) 179, 335, 547, 577, 617, 703 steel-concrete composite beam 761 stiffened steel plate 385 stop-hole 223 strain hardening 503, 637, 655 strength 449, 845 stress analysis 745 stress behaviour 755 stress interaction 755 structural analysis 359 structural collapse 617 structural damage 215 structural experiment 377 structural stabiHty 179, 617 structural steel 771 structures 487 strut 811 stub column tests 515 supporting sheet 415 supports from Aluminium 495 sway angle 853 sway frame 359 tangent rigidities 27 tank design 399 tensile tests 771 tension 495 field 853 member 569 test 123 strength 45 testing 779 technique 131 tests 561, 647 thermal bridges 689 thermal conductivity 547 thermal insulation 689, 713 thermal performance 703 thermovision technique 585 thin walled cross section 343
873 thin walled structure(s) 79, 495 thin-walled 107, 115, 179,795 thin-walled member 265 thin-walled structures 99, 155, 163 three-surface model 829 tilting and pull-out 495 traffic accident 205 tram structures 465 transfer matrix method 265 transient-state 771 truss bridge 861 truss testing 593 two-surface plasticity model 283 ultimate 189 moment 745 state 853 strength 273, 811 uniformly varying load 273 unprotected structures 523 vehicle coUision impact 53, 821 vermiculite 555 vertical and horizontal load 811 vibration 211, 719 criteria 719 viscoelastic adhesives 465 viscoplasticity 679 walking 719 wall stud 123 warping 249 web(s) 79, 795 web buckhng 789 web crippling 107 web yielding 789 weld 449 welded joints 861 welded sections 423 welding 367 wind uplift 609 work-hardening 433 z-section 115
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AUTHOR INDEX Author
Author
Page
Abe, K. Akase, T. Akesson, B. Ala-Outinen, T. Al-Emrani, M. Alghamdi, S.A.
829 811 223 523 223 385
Ge, H.B. Gedeonov, P.P. Gedeonova, T.P. Gon9alves, R.M. Goto, Y. Guggenberger, W.
Bakker, M.C.M. Baldassino, N. Baniotopoulos, C.C. Baptista, A. Baroudi, D. Belica, A. Bohm, S. Boon, B. Boruszak, A. Branka, P. Brauns, J. Brekelmans, J.W.P.M. Bueno, F.G.F.
107 131 433 61 547 335 407 671 343 789 745 317 179
Hancock, G.J. Harada, H. Hassinen, P. Hautala, K.T. Hayashi, M. Heinemeyer, C. Hopperstad, O.S. Hornung, U. Hudramovych, V.S.
Camotim, D. Chen, H. Chen, Y. Chernoivan, N.V. Chistyakov, A.M. Co§kun, H. Consalvo, V. Couchman, G.H. Crocetti, R.
61 19 293 415 415 351 761 317 223
da Silva, R.M. Davies, J.M. De Matteis, G. de Paula, F.A. Deus, E.P. Dunne, B.D. Edlund, B. El-Boghdadi, M.H. Elghazouli, A.Y. Paella, C. Fakury, R.H. Feldmann, M. Fick, K.F. Freitas, A.M.S.
Itoh, Y. Izzuddin, B.A. Johannesson, G. Kaitila, 0 . Kedziora, S. Kerstens, J.G.M. Kesti, J. Kolakowski, Z. Kolari, K. Koltsakis, E. Komann, S. Konderla, P. Konovalov, P.N. Kotelko, M. Kotisalo, K. Kouhia, R. Kowal-Michalska, K. Kubo, Y. Kujala, P. Kullaa, J. Kurzawa, Z. Kurzyca, J. Kvedara, A.K.
617 3 637, 655 617 215 69 223 385 727 539, 663, 761 617 713 495 179
Laakso, K. LaBoube, R.A. Landolfo, R. Langseth, M. 875
Page 845 555 555 617, 647 829 241, 249 45, 115, 131, 141,601 811 189 233 829 713 515, 637 399 257 53, 377, 821 727 689 593 163 107 123, 585 163 577 433 585 803 415 99 837 679 163 829 837 719 343 197 755 205 569 515, 637 451, 515, 637
Author Index
876 Larsen, P.K. Lebedev, A.A. Leutenegger, S. Li, J.P. Liew, J.Y.R. Lindner, J. Liu, C. Lu, W. Macdonald, M. Mahendran, M. Makelainen, P. Malite, M. Mamaghani, LH.P. Mandara, A. Marcinowski, J. Matusiak, M. Mazzolani, F.M. McAndrew, D. Mennink, J. Miyake, Y. Moen, L.A. Mori, M. Mossakovsky, V.L Murkowski, W. Murzewski, J. Muzeau, J.P. Myllymaki, J.S.
449 257 465 737 19 155 53,,821 585 69,,795 37, 30L,609 123, 585, 593,,771 647 283 45L,655 803 449 451, 475, 637, 655,,663 301 487 265 637 53 257 343 359 61 547.,679
Narayanan, R. Nash, D. Nieminen, J. Nigro, E.
273 795 703 761
Ohga, M. Ohno, T. Oiger, K. Orlowska, K. Orlowsky, Y. Outinen, J.
265 821 327 561 561 771
Panagiotopoulos, P.D. Papangelis, J.P. Pasternak, H. Pedreschi, R.F. Peil, U. Pekoz, T. Piluso, V. Preftitsi, F. Rapp, P. Rasmussen, K.J.R. Rass, F.V.
433 115 585 ,789 309 215 89 515, 539 ,663 433 197 27, 441 ,627 415
Ravinger, J. Rhodes, J. Rizzano, G. Rogers, C.A. Rondal, J. Rusch, A. Rzeszut, K.
211 69, 17L, 795 539.,663 601 44L,627 155 343
Saal, H. Sales, J.J. Salonvaara, M. Salwen, A. Salzgeber, G. Schafer, B.W. Schmidt, H. Schneider, W. Schuster, R.M. Sedlacek, G. Shanmugam, N.E. Shen, Z.Y. Shigematsu, T. Shnal, T. Shugyo, M. Silvestre, N. Snijder, H.H. Soetens, F. Stangenberg, H. Starlinger, A. Sully, R.M. Szostak, W.
399 647 703 861 249 89 233 407 779 503 ,713 27^,273 293 265 561 737 61 107 487 503 465 141 197
Talja, A. Tang, R.B. Tang, Y. Tasarek, J. Taylor, G.T. Telue, Y. Thiele, R. Thoyra, T. Toma, A.W.
531 , 719 609 293 79 69 37 407 861 317
Usami, T. Uy, B.
845 423
Valtonen, J. Van den Brande, E.L.M.G. Venturini, W.S. Vojvodic Tuma, J.
205 317 215 367
Wang, G.Y. Wazaki, H. Weijs, H. Wright, H.D.
293 377 671 423
Author Index Yamada, M. Yamaguchi, E. Yamao, T.
877 853 829 811
Young, B. Yu, W.W.
27, 45 569
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