Flexural Behaviour and Design of Coldformed Steel Beams with Rectangular Hollow Flanges

Transcription

1 Flexural Behaviour and Design of Coldformed Steel Beams with Rectangular Hollow Flanges By Somadasa Wanniarachchi School of Urban Development A THESIS SUBMITTED TO THE SCHOOL OF URBAN DEVELOPMENT QUEENSLAND UNIVERSITY OF TECHNOLOGY IN PARTIAL FULFILLMENT OF REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY December 005

2 Acknowledgement I wish to convey my appreciation and wholehearted sense of gratitude to my principal supervisor Professor Mahen Mahendran for his enthusiastic and expert guidance, valuable suggestions, constructive criticism, friendly discussions, and persistent supervision during my research study. I am indebted to him for his constant encouragement and meticulous efforts in correcting faults and suggesting improvements. I also want to express my sincere thanks to associate supervisor Dr. Thishan Jayasinghe for his valuable suggestions, advice and assistance towards achieving the research objectives. I would like to thank the Department of Education, Science and Training (DEST) for providing an International Postgraduate Research Scholarship (IPRS) to conduct this research project, Queensland University of Technology (QUT) for providing financial support and materials for experiments, QUT structural laboratory and workshop staff for their assistance with experiments, QUT computing services for the facilities and assistance with finite element analyses, as well as my fellow post-graduate students for their positive suggestions and help throughout this research project. I gratefully acknowledge the provision of study leave by University of Ruhuna in Sri Lanka to undertake postgraduate studies in overseas. Finally, I have deeply appreciated the continuing patience and sacrifices of my wife and daughter whose love and support has been a constant source of encouragement and guidance to me. Moreover, I gratefully acknowledge my family members in Sri Lanka for their patience and encouragement during my study. Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges iii

3 Keywords Flexural behavior, hollow flange beams, rectangular hollow flange beams, cold-formed steel beams, distortional buckling, lateral tortional buckling, buckling tests, section moment capacity, finite element analysis. Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges iv

4 Abstract Until recently, the hot-rolled steel members have been recognized as the most popular and widely used steel group, but in recent times, the use of cold-formed high strength steel members has rapidly increased. However, the structural behavior of light gauge high strength cold-formed steel members characterized by various buckling modes is not yet fully understood. The current cold-formed steel sections such as C- and Z-sections are commonly used because of their simple forming procedures and easy connections, but they suffer from certain buckling modes. It is therefore important that these buckling modes are either delayed or eliminated to increase the ultimate capacity of these members. This research is therefore aimed at developing a new cold-formed steel beam with two torsionally rigid rectangular hollow flanges and a slender web formed using intermittent screw fastening to enhance the flexural capacity while maintaining a minimum fabrication cost. This thesis describes a detailed investigation into the structural behavior of this new Rectangular Hollow Flange Beam (RHFB), subjected to flexural action The first phase of this research included experimental investigations using thirty full scale lateral buckling tests and twenty two section moment capacity tests using specially designed test rigs to simulate the required loading and support conditions. A detailed description of the experimental methods, RHFB failure modes including local, lateral distortional and lateral torsional buckling modes, and moment capacity results is presented. A comparison of experimental results with the predictions from the current design rules and other design methods is also given. The second phase of this research involved a methodical and comprehensive investigation aimed at widening the scope of finite element analysis to investigate the buckling and ultimate failure behaviours of RHFBs subjected to flexural actions. Accurate finite element models simulating the physical conditions of both lateral buckling and section moment capacity tests were developed. Comparison of experimental and finite element analysis results showed that the buckling and ultimate failure behaviour of RHFBs can be simulated well using appropriate finite element models. Finite element models simulating ideal simply supported boundary Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges v

5 conditions and a uniform moment loading were also developed in order to use in a detailed parametric study. The parametric study results were used to review the current design rules and to develop new design formulae for RHFBs subjected to local, lateral distortional and lateral torsional buckling effects. Finite element analysis results indicate that the discontinuity due to screw fastening has a noticeable influence only for members in the intermediate slenderness region. Investigations into different combinations of thicknesses in the flange and web indicate that increasing the flange thickness is more effective than web thickness in enhancing the flexural capacity of RHFBs. The current steel design standards, AS 4100 (1998) and AS/NZS 4600 (1996) are found sufficient to predict the section moment capacity of RHFBs. However, the results indicate that the AS/NZS 4600 is more accurate for slender sections whereas AS 4100 is more accurate for compact sections. The finite element analysis results further indicate that the current design rules given in AS/NZS 4600 is adequate in predicting the member moment capacity of RHFBs subject to lateral torsional buckling effects. However, they were inadequate in predicting the capacities of RHFBs subject to lateral distortional buckling effects. This thesis has therefore developed a new design formula to predict the lateral distortional buckling strength of RHFBs. Overall, this thesis has demonstrated that the innovative RHFB sections can perform well as economically and structurally efficient flexural members. Structural engineers and designers should make use of the new design rules and the validated existing design rules to design the most optimum RHFB sections depending on the type of applications. Intermittent screw fastening method has also been shown to be structurally adequate that also minimises the fabrication cost. Product manufacturers and builders should be able to make use of this in their applications. Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges vi

6 Publications Publications in Preparation: 1. Wanniarachchi, KS. and Mahendran, M. (005) Section Moment Capacities of Rectangular Hollow Flange Beams.. Wanniarachchi, KS. and Mahendran, M. (005) Experimental Investigation of Member Buckling Behaviour of Rectangular Hollow Flange Beams 3. Wanniarachchi, KS. and Mahendran, M. (005) Finite Element Modeling of Rectangular Hollow Flange Beams 4. Wanniarachchi, KS. and Mahendran, M. (005) Development of Design Models for Local Buckling of Rectangular Hollow Flange Beams 5. Wanniarachchi, KS. and Mahendran, M. (005) Lateral Distortional Buckling Design of Rectangular Hollow Flange Beams Target Journals: Journal of Constructional Steel Research, Thin-Walled Structures, Structural Engineering and Mechanics, American Society of Civil Engineers J. of Structural Engineering Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges vii

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15 ( 7!!*;*!! ( 7!! ( ( -*! (. 7 4!- B 6 ( ' 7 4!- C 6 ( & 7 4!- D 6 ( 3!7 ;!!)!* ( <!! ( % #8 "!!;"66! ( )!*!!!,*!!/+,.!!/+,. -,17/+,. (!"!/!!+6!.. 9"0!!*!!!/+,. '!)!/+,. & ;!!,1. 3,1. < ;!. % 7!!-!!9E.!/+,!!. -*!. 7 - B 6. (!;!!-!,1!/+, C 6. ' ;!,18 "!*;!0. & 7 - D ;!!. <!5>!!76*9. 0. %!5>!!76*0 0!!*!$%%3* (< (% ( ( (( (. (3 (< (% ((....<...(.'.<....(.&.3.%.(.((.(%.. Flexural Behaviour and Design of Cold-formed Steel Beam with Rectangular Hollow Flanges xvi

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17 ' 7 4!- /+, 6*5E!!!8"*1 B 6 ' 7 4!- /+, 6*!*1)!*!!8"*1 C 6 ' ( 7 4!- /+, 6*!*1;*!!8"*1 D 6 '. 7 ;!!)!*!) /+, ' ' ;!"1!7/+,!! '( '( '( '(. '(' ' & ;!"1!7/+,9!! '(' ' 3 )!*!!59!*6!5!,1 7!! &!!!!/+, & Local Buckling of the Hollow Flange s Top Plate Element & ( Local Buckling of the Hollow Flange Flange s Web Element &3 &. Local Buckling of RHFB s Web Element & ' Graphical Comparison of the Effect of Geometric Imperfection and Residual Stress on Moment Capacities of RHFBs & & Deformation Shape of RHFB for Different Fastening Arrangements '(< &( &. &% &. &' & 3 Stress Contours of RHFB for Different Fastening && Arrangements & < Close-up View at a Screw Location Under Flexural Loading &3 & %! Comparison of Moment Capacities of Slender RHFBs with Different Fastening Arrangement & %" Comparison of Moment Capacities of Compact RHFBs with Different Fastening Arrangement &! Moment Capacity Curves for Compact RHFB Sections from FEA & " Moment Capacity Curves for Slender RHFB Sections from FEA & & && &3 & Comparison of Moments Capacities with AS 4100 Predictions &(( & Comparison of Moment Capacities with Pi and Trahair s (1997) Predictions &(. Flexural Behaviour and Design of Cold-formed Steel Beam with Rectangular Hollow Flanges xviii

18 & ( Comparison of Moment capacities with AS/NZS 4600 Predictions &. Comparison of Moment Capacities with Avery et al. (000) Predictions & ' Comparison of Moment Capacities with Mahaarachchi and Mahendran (005) Predictions & & Comparisons of Moment Capacities Predicted by Equation6.16 with FEA Results & 3 Comparison of Moment Capacities Predicted by Equations 6.16 and 6.17 and FEA Results & < Comparison of Moment Capacities Predicted by Equation 6.17 and FEA results and other Existing Design Rules & % Comparison of Predicted Moment Capacities and FEA Results for G300 and G550 Steel Slender RFHBs &(% &. &.( &.& &.3 &.3 &.% Flexural Behaviour and Design of Cold-formed Steel Beam with Rectangular Hollow Flanges xix

19 !" #$ %&'( ) +, - *!& '(./ * +) %.5 $46 + ))+ +,&&'78$,4 + +!%$! $,77, +0 ++! $%&7'& +0 : +! $ %&$;&' ' $! $ %& + * +* 7 + &' 7!&': 0 7$!, &, 7< $ 0.5(: 0+ 4 & 00! $7%&$<& ;&' 7'& 0 &'.#' &!&':3, 4 $ &, %: &'.#' &$.5!&!& ' &'.#' &$.57!& ' + &'.!& &.53,, 3( 9 Comparison of FEA Results with the Predictions from Current Steel Design Standards for G300 Steel 9 Comparison of FEA Results with the Predictions from Current Steel Design Standards for G550 Steel 9 Comparison of FEA Results with the Predictions from Current Steel Design Standards for G300 Steel * * 9 9 9* Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xx

20 9 Comparison of FEA Results with the Predictions from Current Steel Design Standards for G550 Steel 9+ Comparison of FEA Results with the Predictions from Current Steel Design Standards for G300 Steel 9+ Comparison of FEA Results with the Predictions from Current Steel Design Standards for G550 Steel 90 Comparison of Residual Stress Effects on RHFB Moment Capacities 90 Comparison of Initial Geometric Imperfection Effects on RHFB Moment Capacities 90 Comparison of Combined Effects of Residual Stress and Initial Geometric Imperfection on RHFB Moment Capacities 9 Comparison of Moment Capacity Results of RHFB-080t f -080t w - 150h w -G300 (slender) for Different Fastening Arrangements 9 Comparison of Moment Capacity Results of RHFB-10t f -10t w - 150h w -G300 (slender) for Different Fastening Arrangements 9 Comparison of Moment Capacity Results of RHFB-300t f -300t w - 150h w -G300 (compact) for Different Fastening Arrangements 9$ Comparison of Moment Capacity Results of RHFB-500t f -500t w - 150h w -G300 (compact) for Different Fastening Arrangements 99 Member Moment Capacities of Compact RHFB Sections from FEA 99 Member Moment Capacities of Slender RHFB Sections from FEA 9* Comparison of FEA and AS 4100 Section Moment Capacities for Compact RHFB Sections 9* Comparison of FEA and AS 4100 Section Moment Capacities for Slender RHFB Sections 9 Comparison of FEA and AS/NZS 4600 Section Moment Capacities for Compact RHFB Sections 9 Comparison of FEA and AS/NZS 4600 Section Moment Capacities for Slender RHFB Sections 9 Comparison of Predicted Moment Capacities using New Design Formula with FEA Results for G300 Steel Slender RHFBs 9 Comparison of Predicted Moment Capacities using New Design Formula with FEA Results for G550 Steel Slender RHFBs ) ) * 9) 9 Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxi

21 Notation Abbreviations AISI AS 4100 = American Iron and Steel Institute = Australian Standard for the Design of Steel Structures AS/NZS 4600 = Australian Standard for the Design of Cold-formed Steel Structures ASD BHP BMT BSI C3D8 CHS COV CSA FEA HFB LRFD LSB MPC PDT PTM QUAD4 RHFB RHS S4 S4R4 SHS SPC TCT TRIA3 UC = allowable stress design = BlueScope Steel Products = based metal thickness = British Standards Institution = eight node linear brick element = circular hollow section = covariance = Canadian Standard Association = finite element analysis = hollow flange beam = load and resistance factor design specification = light steel beam = multipoint constraint = potentiometric displacement transducers = Palmer Tube Mills = quadrilateral shell element = rectangular hollow flange beam = rectangular hollow section = quadrilateral general purpose shell element with four nodes and six degrees of freedom per node = quadrilateral thin shell element with four nodes, reduced integration, and five degrees of freedom per node = square hollow section = single point constraint = total coated thickness = triangular shell element = universal column Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxii

22 Symbols ν λ ρ ρ nom ρ true δ δ l β o λ d λ e α m,c b λ s α s α sd λ sp λ sy ε m ε n ε p(ln) φ A B b b e b f C p d D E f c, σ c f cr F m f max = poisons ratio = half-wavelength at distortional buckling, or non-dimensional slenderness = effective width factor = measured longitudinal stress (tensile coupon tests) = true longitudinal stress (modified stress) = global imperfection = local imperfection = target reliability index = non-dimensional member slenderness for distortional buckling = plate element slenderness = moment modification factor = section slenderness = slenderness reduction factor for lateral tortional buckling = slenderness reduction factor for lateral distortional buckling = section plastic slenderness = section yield slenderness = measured longitudinal strain = measured longitudinal strain (tensile coupon test) = true longitudinal strain (modified strain) = capacity reduction factor = cross section area = element overall width (HFB) = element flat width (general) = effective width of flat plate = flange width (RHFB) = correction factor = web depth (HFB) = overall depth (HFB) = young s modulus of elasticity = critical stress = elastic critical stress for local buckling = mean of the fabrication factor = maximum edge stress Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxiii

23 f od f ol f u f y,σ y f * G h f h l h w I I x I y I w I y J J e k k φ L L e, l e L la m M M * M b M m M c M o M od M s M TH M u M y n P = distortional buckling stress = local buckling stress = tensile strength = material yield stress = applied stress =shear modulus = flange height (RHFB) = lip height (RHFB) = web height (RHFB) = second moment of area = second moment of area about major principal axis = second moment of area about minor principal axis = warping constant = moment of inertia about minor axis = polar moment of inertia = effective polar moment of inertia = local buckling coefficient = rotational spring stiffness = member length = effective length = initial lever arm length = degree of freedom = applied moment = design bending moment = nominal member moment capacity = mean of the material factor = critical moment =flexural torsional buckling moment resistance = distortional buckling moment resistance = nominal section moment capacity = Thin-wall buckling moment = ultimate moment capacity = yield moment = number of tests = applied jack load Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxiv

24 P m P y R o t t f t w V F V m V p V Q w = mean value of the tested to predicted load ratio = squash load = corner radius (HFB) = steel thickness = thickness (RHFB) = thickness of web (RHFB) = covariance of the fabrication factor = covariance of the material factor = Covariance of the tested to predicted load ratio = covariance of load effect = plate width Z, Z x or Z f = full section modulus about major axis Z e = effective section modulus Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxv

25 The work contained in this thesis has not been previously submitted for a degree or diploma at any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made. Signature:.. Date:.. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxvi

26 CHAPTER 1 Introduction 1.1 General In steel structures, two primary structural steel member types are used: hot-rolled steel members and cold-formed steel members. The hot-rolled steel members are formed at elevated temperatures whereas the cold-formed steel members are formed at room temperatures. Until recently, the hot-rolled steel members have been recognised as the most popular and widely used steel group, but since then the use of cold-formed high strength steel structural members has rapidly increased. However, the structural behaviour of these light gauge high strength steel members characterised by various buckling modes such as local buckling, distortional buckling, and flexural-torsional buckling is not yet fully understood. Open coldformed steel sections such as C-, Z-, hat and rack sections are relatively common because of their simple forming procedures and easy connections, but they suffer from certain buckling modes due to their mono-symmetric or point symmetric nature, high plate slenderness, eccentricity of shear centre to centroids and low torsional rigidity. It is therefore important that these buckling modes are either delayed or eliminated to increase the ultimate capacity of cold-formed steel members. This study is aimed at developing an innovative cold-formed steel beam with two torsionally rigid rectangular hollow flanges and a slender web formed using intermittent screw fastening to enhance the flexural capacity at minimum fabrication cost. The new cold-formed steel beam introduced in this research is referred to as Rectangular Hollow Flange Beam (RHFB) to differentiate from the conventional hollow flange beams (HFB) containing triangular flanges. This study therefore involves investigations into the flexural behaviour of RHFBs comprising various steel grades, steel thicknesses, section sizes and screw spacings to fully understand the primary buckling and ultimate failure characteristics, and to derive suitable design rules for the new RHFB flexural members. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-1

27 This chapter discusses the significance and importance of this research under the headings of: conventional cold-formed steel section types, the development of HFBs, research needs for RHFBs, objectives and scope of the research program, and method of investigation. 1. Conventional Cold-formed Steel Section Types The use of cold-formed steel structures is increasing rapidly around the world. The main use of cold-formed steel members is found in the construction of residential and other low rise buildings such as commercial, industrial and institutional buildings. Figure 1.1 illustrates some of the commonly used cold-formed steel section types in the above applications. They include channel (C-) sections, Z-sections, angles (L-), hat sections, I- sections and tubular sections such as rectangular hollow sections (RHS) and square hollow sections (SHS). Figure 1.1: Commonly used cold-formed steel sections (From Yu, 000) These sections are commonly in used, but they are more susceptible to structural instabilities due to their geometrical shapes. The characteristics due to monosymmetric or point-symmetric nature of these sections are not normally encountered in doubly symmetric sections such as I- sections or tubular sections (i.e. RHS, SHS, CHS). Therefore, combining the advantages of hot-rolled I-sections (better stability) and conventional cold-formed sections such as C- and Z- sections (high strength to Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-

28 weight ratio) can produce improved cold-formed steel sections that can be made using modern technologies available in the cold-formed steel industry. Complex structural shapes may now be formed in two or more parts and then assembled into a single shape. This may have the advantage of combining different material qualities and thicknesses into a single component. However, the use of higher strength steels is inevitably accompanied by the reduction in thickness of the section and it may result in more slender sections which could be structurally unstable. Structural behaviour of the commonly used cold-formed steel sections (see Figure 1.1) has been well researched in the past. However, only limited research has been undertaken to investigate the structural performance of other cold-formed steel member types. Therefore, there is an urgent need in cold-formed steel industry to look beyond the conventional cold-formed steel sections and generate more structurally efficient cold-formed steel sections in an economical manner. One of the typical examples for an advanced cold-formed steel section produced by using modern cold-formed steel technology is the hollow flange beam (HFB), which includes two closed triangular hollow flanges and a web connected using electric resistance welding method. The HFB was first developed by Palmer Tube Mills Pty. Ltd. in the early 1990s. Section 1.3 discusses the development of HFBs in detail. 1.3 Development of Hollow Flange Beams (HFB) Figure 1. Closed-cell Section Types Investigated by O Connor et al. (1965) The history of HFB can be traced back to 1965 when O connor et al. (1965) first showed that the inclusion of various closed cells to I- section beams (see Figure 1.) improved their buckling behaviour significantly. They found that this improvement Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-3

29 of buckling behaviour was mainly due to increase in torsional rigidity. This led the researchers to focus on cold-formed steel sections with torsionally rigid flanges, which can delay or eliminate structural instability problems effectively. The socalled HFB beam is one such cold-formed steel section with torsionally rigid flanges. During the early 1990s, Palmer Tube Mills Pty. Ltd. mass produced cold-formed, high strength steel beam sections with two closed triangular hollow flanges (see Figure 1.3). This is a structurally efficient steel section made from a single strip of high strength steel using an automated fabrication process of cold-forming and electric resistance welding. Although the electric resistance welding method used by Palmer Tube Mills is adequate, it makes the manufacturing process somewhat complicated and expensive. This was one of the reasons for the discontinuation of the triangular HFB production in Further, it was capable of producing only one group of HFB with 90 mm wide triangular flanges. The use of other flange shapes (i.e. rectangular or square or other geometry) and widths (60 mm to 50 mm) could considerably improve the structural efficiency of HFBs while eliminating or delaying many undesirable buckling modes. (a) Isometric view (b) Sectional view Figure1.3 Geometric Shape and Sectional Parameters of HFB (From Dempsey, 1990) Consequently, Zhao and Mahendran (001) at Queensland University of Technology initiated a research program to investigate the structural behaviour and design of such hollow flange beam sections as compression members. Their study used Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-4

30 rectangular hollow flanges and various manufacturing methods such as spot welding, self-pierced riveting and screw fastening to form Rectangular Hollow Flange Beam (RHFB) sections from a single steel strip (see Figure 1.4). Their study has identified that the type of fastening and spacing does not affect the member compression capacity significantly. However, the structural behaviour and design of RHFB as flexural members will be different and therefore further investigations are needed to identify their failure modes and develop suitable design rules for RHFB as flexural members. Therefore this research is into the flexural behaviour and design of coldformed steel beams with rectangular hollow flanges (RHFB) made of separately formed flanges and web connected by simple screw fastening. Section 1.4 describes the necessity of further research on RHFB flexural members. b f h f t f h w t w h l h f (a) Sectional view (b) Isometric view Figure 1.4: Geometry of a Typical RHFB 1.4 Research Needs of RHFBs Past research has identified that the flexural capacity of Palmer Tube Mill s triangular hollow flange beams is reduced drastically due to the lateral distortional buckling failure compared to the conventional hot-rolled I- sections. This is mainly due to the presence of slender web and torsionally rigid flanges. Due to the unique fabrication method and their lateral distortional buckling behaviour, the HFB is not completely compliant with the Australian Standards for the design of hot-rolled (AS Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-5

31 4100, 1998) and cold-formed (AS/NZS 4600, 1996) steel structures. Therefore, further research is necessary to improve the existing HFB with the elimination or delay of undesirable lateral-distortional buckling failure and recommend suitable design rules. The proposed RHFB in this study considered existing shortcomings in the conventional HFB and addresses them carefully to give better structural performance than conventional HFB at a lower production cost. In the proposed RHFB, screw fastening was introduced as an alternative manufacturing method to minimize the production cost, whereas an innovative method of joining the web and flanges separately was considered to give the designers a large range of very efficient RHFB beams with varying combinations of web and flange thicknesses (see Figure 1.4). The use of thicker web will considerably increase the lateral distortional buckling capacity of rectangular HFB flexural members. The section geometry of HFB considered in this study was confined to rectangular hollow flanges, since they provide better connection capability than conventional triangular hollow flanges. Furthermore, manufacturing of the former is also much easier than the latter with the proposed fabrication methods in this research program. However, sufficient research data is not available to conclude all the above presumptions on this new beam type and need to be investigated. The design rules for RHFB must also be formulated as currently available design rules proved inappropriate for the similar sections in the past. This research is therefore aimed at finding appropriate solutions for the following unanswered questions (problem definition): 1. Is it feasible to produce the innovative RHFB sections shown in Figure 1.4, which will be structurally efficient and economically sound as flexural members?. What are the effects of intermittent screw fastening method on the flexural member performance (including buckling and ultimate strength) compared to the continuous welding method? i.e. which method can be recommended? Does the increased discontinuity in web-flange connection reduce the flexural capacity noticeably? Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-6

32 3. Do the various combinations of web and flange thicknesses improve the flexural capacity of RHFB eliminating or delaying any undesirable buckling failures? 4. Are the current design rules applicable to the new RHFB or is there a need to develop new design rules? 1.5 Objectives Overall Objective To investigate the fundamental buckling and ultimate strength behaviour of a group of innovative cold-formed steel beam sections with rectangular hollow flanges (RHFB) made by assembling two separately formed flanges and a web using screw fastening method, and to develop appropriate design methods for the said RHFB flexural members Specific Objectives 1. Investigation of flexural behaviour and ultimate section moment capacities of innovative RHFBs using a series of short span beam tests, and the comparison of experimental ultimate section moment capacities with the predictions from the current design rules.. Investigation of flexural behaviour and ultimate member moment capacities of innovative RHFBs using a series of lateral buckling tests, and the comparison of experimental ultimate member moment capacities with the predictions from the current design rules. 3. Development of accurate finite element models for the innovative RHFBs subjected to flexural actions, and validation using experimental results (from objectives 1 and ), and use them to investigate the local, lateral distortional and flexural (lateral) torsional buckling modes of failures. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-7

33 4. Investigation of the effects of relevant parameters (section geometry, material properties and fabrication methods) on the local, lateral distortional and flexural (lateral) torsional buckling capacities of the innovative RHFB flexural members using a series of parametric studies, and determination of the applicability of current design rules in AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996) or develop alternative design rules. 1.6 Research Methodology In the first phase of this study, independent reading and literature review as outlined in Chapter was undertaken to develop the required knowledge in this research field. Following the literature review, laboratory experiments were carried out to understand the local, lateral distortional and flexural torsional buckling modes of failures of RHFBs, and also to develop the required data base for finite element model validation. The laboratory experiments included a series of full-scale tests of section and member capacity using this new RHFB sections shown in Figure 1.4. The tests were conducted on a group of innovative RHFB with the various combinations of geometric parameters, member lengths and steel grades. Sectional dimensions and member lengths of all specimens were selected in such a way that each specimen failed under certain pre-determined buckling modes. Following the laboratory experiments, analytical investigations on RHFBs were conducted using finite element models in order to fully understand the local, lateral distortional and lateral torsional buckling failure modes. Avery et al. s (000) and Yuan (005) finite element models were reviewed, and modified to include the new features associated with the innovative RHFB sections shown in Figure 1.4. The latest developments in finite element modelling and the many features of finite element software ABAQUS were also introduced in the finite element modelling. Finite element models were developed separately for experimental and ideal boundary conditions for the purpose of model validation and parametric studies, respectively. The capability of the developed finite element models to simulate the local, lateral distortional, and flexural torsional buckling behaviour was validated by using the test results obtained from the laboratory experiments. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-8

34 Further analyses were carried out to investigate the effect of relevant parameters (i.e. geometrical dimensions, material properties, member lengths and manufacturing methods) on local, lateral distortional and flexural torsional buckling capacities. Therefore a series of parametric studies were conducted to obtain an extensive behavioural data base. These results were then used to develop appropriate behavioural models for the new RHFBs and also determine the applicability of current design rules within the AS 4100 and AS 4600 provisions or develop alternative design rules. 1.7 Thesis Layout The detailed investigations of flexural behaviour of innovative RHFBs using an extensive series of experimental studies, finite element analyses and development of improved design rules for the design of new RHFBs for flexural action are presented in this thesis as seven different chapters. The contents of each chapter are described briefly next. Chapter 1: This chapter presents a brief introduction about this research project including the areas of conventional cold-formed steel section types and their various buckling failure modes, development of hollow flange beams and their advantages, research needs and problem definition, overall and specific objectives and the research methodology adopted in this study. Chapter : A summary of current literature relating to various aspects of coldformed steel flexural members, independent reading and critical analyses of previous findings are presented in this chapter. The broad areas included in this chapter are special characteristics and design considerations of cold-formed steel members, common cold-formed steel sections, their applications, advantages, disadvantages and different buckling failure modes. This chapter also includes details about the development of new beam types and different design procedures for cold-formed steel members. Experimental and analytical investigations conducted by previous researchers are also described evaluating their findings and method of testing. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-9

35 Chapter 3: This chapter describes the experimental investigations on material properties and section moment capacities of RHFBs. All the experimental results are presented, compared with predictions from the current design methods and appropriate recommendations made for RHFBs. Chapter 4: This chapter presents the details of laboratory experiments of lateral buckling tests on RHFBs to investigate the lateral distortional and lateral torsional buckling behaviour. All the experimental results are presented and the current design rules are reviewed. Comparisons of experimental results with the current design rules are also presented. Chapter 5: This chapter presents the details about the development of finite element models to simulate laboratory tests described in Chapters 3 and 4. The procedures of simulating loads, boundary conditions, material properties, initial conditions and validation of the developed finite element models are presented using experimental results. Chapter 6: The detailed finite element analyses to investigate the effect of various parameters on the flexural behaviour of RHFBs are discussed and a wide range of data base was obtained to develop suitable design rules for the new RHFBs. The new design rule was developed and compared with the current design rules to check the applicability. Chapter 7: In this chapter, a summary of the most significant findings of this research is presented with the recommendations for further research. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 1-10

36 CHAPTER Literature Review.1 General Due to increasing interest among researchers, a large number of publications dealing with the cold-formed steel structural members are in existence. However, the socalled hollow flange beams (HFB), which were developed in early 1990s, are not well researched until recently and therefore their publications are limited. This chapter aims to provide a brief review of previous research investigations on the cold-formed steel beams with special attention to the HFBs.. Special Characteristics and Design Considerations of Coldformed Steel Members Unlike conventional hot-rolled steel members, there are certain unique characteristics related to cold-formed steel members, particularly due to their forming process and the use of thinner material. Some of these special characteristics and design issues are discussed in the following sections...1 Methods of Forming In general, two manufacturing methods are used to produce various shapes of coldformed steel sections (see Figures.1 and.), and they are cold roll-forming and press brake operations Cold Roll-forming The cold roll-forming process consists of feeding a continuous steel strip through a series of opposing rolls (see Figure.1a) to deform the steel plastically to form the desired shapes. The process involved in cold-forming a Z- section is illustrated in Figure.1b. A simple section may be produced by as few as six pairs of rolls but a complex section may require as many as 15 sets of rolls (Yu, 000). This method is Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -1

37 usually used to produce cold-formed steel sections where a large quantity of a given shape is required. (a) Cold Roll-forming Machines (Yu, 000) (b) Roll-forming Sequence for a Z- Section (Hancock, 1998) Figure.1: Cold Roll-Forming Processes However, a significant limitation of this method is the time taken to change rolls for different size sections. Consequently, adjustable rolls are often used which allow a rapid change to a different section width or depth. From a structural point of view, roll-forming may produce a different set of residual stresses in the section and hence the section strength may be different in case where buckling and yielding interact...1. Press Braking The equipment used in the press brake operation essentially consists of a moving top beam and a stationary bottom bed that produce one complete fold at a time along the full length of the section (see Figure.). This method is normally used for low volume production where a variety of shapes are required and the roll-forming tooling costs can not be justified. However, this method has a limitation that it is difficult to produce continuous lengths exceeding approximately 5 metres (Hancock, 1998)... Common Section Profiles and Their Applications Cold-formed steel shapes can broadly be classified into two groups: individual structural frame members, and panels and decks. The former includes sections shapes Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -

38 Figure. Press Braking (Karren, 1967) such as I, L, C and Z, which are commonly used in engineering practices of coldformed steel construction. However with the improvement of industrial coldforming processes, more complex section types are possible (see Figure.3) and offer competitive solutions to achieve structural weight reduction and high strength. There are wide range of applications for these section types: typical Z or C sections are used as purlins and bracings in roof and wall systems in residential, commercial and industrial buildings, C- or tubular sections are normally used as shelf beam and upright frames in steel racks, and circular, square or rectangular hollow sections are used for structural members such as chords and webs in plane and space trusses. The panels and decks are used mostly for roof decks, floor decks, wall panels, sliding materials and bridge forms (Yu, 000). Figure.3: Various Shapes of Cold-formed Steel Sections (Yu, 000) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -3

39 ..3 Effect of Cold-forming When steel shapes are cold-formed by either press-braking or cold-rolled-forming, there is a change in mechanical properties of the material due to cold working of the metal. Because the material properties undoubtedly play an important role in the performance of structural members, it is important they are included in the design of cold-formed sections. Macdonald et al. (1997) described that the yield strength, and to a lesser extent the ultimate strength, are increased and ductility is reduced as a result of this cold working, particularly in the bends of the section. Consequently, the material properties of a formed section may be markedly different from those of the virgin sheet material from which it is formed. The tests conducted by Karren and Winter (1967) illustrated the variation of mechanical properties from the parent material at the specific locations in a channel section (see Figure.4). Figure.4: Effect of Cold-work on Mechanical Properties in a Channel Section (Karren and Winter, 1967) Hancock (1998) stated that the research investigations by, Karren (1967) and Chajes (1963) on the influence of cold working in steel Winter (1968) indicated that the changes of mechanical properties due to cold work are caused mainly by strain hardening and strain ageing as illustrated in Figure.5. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -4

40 Figure.5: Effect of Strain Hardening and Strain Aging on Stress-strain Characteristics (Chajes et al., 1963) Current cold-formed steel design standards: AS/NZS 4600 (SA, 1996), Specification for the Design of Cold-formed Steel Structural Members (AISI,1996), BS5950-Part 5 (BSI, 1998) and EC3 (ENV, 1996) make use of this yield strength increase and give many design recommendations including methods on how to compute the increase in yield strength gained from cold working and procedures for full-section test. A comparison of the Specification for the Design of Cold-formed Steel Structural Members (AISI, 1996) and AS/NZS 4600 (SA, 1996) equations to calculate the enhanced yield strength of cold-formed sections shows that they are almost the same with the exception that Specification for the Design of Cold-formed Steel Structural Members (AISI, 1996) equations use a weighted average method to approximate the full cross-section tensile yield strength, while AS/NZS 4600 (SA, 1996) equations allow the calculation of enhanced corner yield strength. In the case of Euroode 3 and BS 5950-Part 5 equations, they are almost identical with the exception that for Eurocode 3, the limiting values of increased average yield strength gained from cold-forming allows for greater cold-formed section yield strength to be considered in design. Some of the existing cold-formed steel design standards and their design aspects are discussed in the next section. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -5

41 ..4 Cold-formed Steel Design Standards Specifications and standards for the design of cold-formed steel structural members are available in many countries. The design clauses for cold-formed steel structural members were first introduced with the preparation of the American Iron and Steel Institute Specifications in 1946, using the research work on cold-formed members of Professor George Winter at Cornell University (AISI, 1946). The British Steel Standard, BS 449 (BSI, 1959) was modified in 1961 to include the design of coldformed members by the inclusion of Addendum No. 1 (1961) based on the work of Professor A.H. Chilver (BSI, 1961). In Australia, the Australian Standard for the design of cold-formed steel structural members, AS 1538 (SAA, 1974) was first published in It was based mainly on the 1968 edition of the American Specifications but with some modifications to the beam and column design curves to keep them aligned with the Australian Steel Structures Code ASCA (SAA, 1968). In Australia, a significant revision of the 1974 edition of AS1538 was produced using the 1980 and 1986 editions of the American Iron and Steel Institute Specification (AISI, 1980 and 1986) in However, they were all in an allowable (permissible) stress format (ASD). The American Iron and Steel Institute produced a limit state version of their 1986 specification in 1991, called the Load and Resistance Factor Design Specification (LRFD) (AISI, 1991). In 1990, Standards Australia published the limit state design standard for steel structures called AS 4100 based on the load factor and capacity factor approach similar to that used for LRFD in the USA. In 1993, Standards Australia and Standards New Zealand commenced work on a limit states design standard for cold-formed steel structures to suit both countries (SA, 1996). The new standard called AS/NZS 4600 is based mainly on the latest AISI specifications (AISI, 1996). In the UK, BS5950, Part5 is the principal source of guidance for the design of cold-formed structural steel work (BSI, 1998). Other international standards for cold-formed steel structures which are in a limit state format include, the Eurocode3 EC3 (ENV, 1996) and the Canadian Standard CAN/CSA S (CSA, 1994). The corresponding design approach of steel structural design standards, AS 4100 and AS/NZS 4600, will be discussed later in this chapter. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -6

42 ..5 Special Design Criteria A set of unique problems pertaining to cold-formed steel design has evolved mainly due to the thinner materials and cold-forming process used in the production of coldformed sections. Hence, unlike the usually thicker conventional hot-rolled steel members, the design of cold-formed steel members must be given special considerations during the design phase of such members. A brief summary of such considerations is listed next Local Buckling and Post Buckling Strength Individual elements forming cold-formed steel members are usually thin with respect to their width. Therefore, they are likely to buckle at a lower stress than yield point when they are subjected to compression, bending, shear or bearing forces. However, unlike one-dimensional structural elements such as columns, stiffened compression elements will not collapse when the buckling stress is reached, but they often continue to carry increasing loads by means of redistribution of stresses (Winter, 1970). The ability of these locally buckled elements to carry further load, known as post buckling strength, is allowed in the design to achieve an economic solution. Figures.6(a) and (b) illustrate two cases of local buckling of thin-walled box and plate girders. The applied sagging bending moment induces longitudinal compressive stresses in the top flange plate, causing local buckling in the top flange. Detailed descriptions of local buckling effects on the behaviour of cold-formed steel members are presented in Section.3.1. (a) Box Girder (b) Plate Girder Figure.6: Local Buckling of Compression Flanges (SCI, 1998) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -7

43 ..5. Torsional Rigidity Many of the steel shapes produced by cold-forming are monosymmetric open sections with their shear centre eccentric from their centroid as illustrated in Figure.7. The eccentricity of loads from the shear centre axis will generally produce considerable torsional deformation in the thin-walled beams as a result of flexuraltorsional buckling (see Fig..7). The torsional rigidity of an open section is proportional to t 3 (t, is material thickness) so that the cold-formed steel sections consisting of thin elements are relatively weak against torsion. Hence torsional stiffness of cold-formed steel members is an important criterion in the design of coldformed steel sections to achieve an economic solution. Figure.7: Torsional Deformations in Eccentrically Loaded Channel Beam (Hancock, 1998)..5.3 Distortional Buckling Thin-walled flexural or compression members composed of high-strength steel and/or slender elements in the section, which are braced against lateral or flexuraltorsional buckling, may undergo a mode of buckling commonly called distortional buckling (Hancock, 1997). The previous research studies (Ellifritt et al., 199, Kavanagh and Ellifritt, 1993 and 1994) have shown that a discretely braced beam, not attached to deck and sheeting, may fail either by lateral-torsional buckling between braces, or by distortional buckling at or near the braced point. Two modes of distortional buckling are specified in the cold-formed steel design standard, AS/NZS 4600 (SA, 1996). The first one is flange distortional buckling, Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -8

44 which involves rotation of a flange and lip about the flange/web junction of a C- section or Z-section and the second one is lateral-distortional buckling, which involves transverse bending of vertical web (see Figures.8 (a) and (b)). Flange distortional buckling is most likely to occur in the open thin-walled sections such as C- and Z- sections while lateral distortional buckling is the most likely in beams, such as hollow flange beams, where the high torsional rigidity of the tubular compression flange prevents it from twisting during lateral displacement (Pi and Trahair, 1997). The distortional buckling concept is first introduced into AS/NZS 4600 in its 1996 version (SA, 1998b). Section.3.. of this chapter gives a comprehensive review of distortional buckling. (a) Distortional Buckling (b) Lateral Distortional Buckling Figure.8: Buckling of a Channel Section and a Hollow Flange Beam (SA, 1998b)..5.4 Connection Types The generally used connection types in the cold-formed steel construction include; welds, bolts, screws, rivets and other special devices such as clinching, nailing and structural adhesives (see Figure.9). (a) Clinching (b) Screw Fastener (c) Bolt Fastener Figure.9: Generally Used Cold-formed Steel Fasteners Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -9

45 Due to the comparative low thickness of the material, connection technology plays an important role in the development of structures using cold-formed steel members. Although the above mentioned conventional methods of connections are available and used in cold-formed steel constructions, they are practically less appropriate for thin-walled member connections in terms of cost, quality and construction efficiency (Lennon et al., 1999). The self-piercing riveting introduced commercially by HENROB is a recently discovered connection type with many advantages compared with other conventional methods used in cold-formed steel connections (Voelkner, 000, see Figure.10). Therefore, the choice of connection type is an important decision in cold-formed steel manufacturing, because it affects the combinations of cost, quality and construction efficiency of the whole project. Figure.10: Cross section of a Self-piercing Rivet (Voelkner, 000).3 Flexural Behaviour of HFBs The behaviour of flexural members is governed by several parameters including their geometric shape and section properties, loading pattern, material properties, support conditions etc. Unlike hot-rolled heavy steel sections, structural behaviour of coldformed beam sections such as HFBs is mostly characterised by their high strength thinner elements composed in the section. In the design of cold-formed steel flexural members, the moment resisting capacity and stiffness of the beam are the most important criteria. The moment resisting capacity of flexural members is limited by various buckling modes including local, lateral distortional and flexural-torsional, particularly when the section is fabricated from thin material. A brief review of flexural behaviour and design aspects of cold-formed steel beams, especially the HFBs, is presented in the following sections. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -10

46 .3.1 Buckling Behaviour of HFBs Subjected to Bending It has been found that the buckling behaviour of triangular HFB sections is different to that of conventional hot-rolled I sections and the cold-formed open sections such as C- and Z- sections (PTM, 1990). A series of finite strip analyses for the case of uniform moment for C-, Z-sections and triangular HFBs revealed that the buckling stress corresponding to local buckling modes in triangular HFB sections are much greater than the yield stress (PTM, 1993). However, the yield stress is only slightly higher than local buckling stress in the case of C- and Z- sections. Thus, the local buckling is only a minor issue for HFBs in bending, whereas it tends to dominate in the cold-formed open sections (Heldt and Mahendran, 199). However, research has identified that the behaviour of triangular HFBs is significantly influenced by the lateral-distortional buckling mode of failure (Dempsey, 1990, 1991). Unlike the commonly observed lateral-torsional buckling (flexural-torsional buckling) of steel beams, the lateral-distortional buckling of triangular HFBs is characterised by simultaneous lateral deflection, twist and cross-sectional change due to web distortion (Avery and Mahendran, 1997). The graphs in Figure.11 represent the buckling stress versus buckle halfwavelengths for the two sections subjected to pure bending about their major principle axes so that their top flanges are in compression while their bottom flanges are in tension as in a conventional beam. The buckling stress is the value of the stress in the compression flange farthest away from the bending axis when the section undergoes elastic buckling. Figure.11 clearly demonstrates that at short half-wavelengths (50 mm-500 mm), the changed buckling mode from local buckling in the unattached flange element in HBS changes to local buckling of the top flange at a higher stress in HBS1. At long half-wavelengths ( m 10 m), the increased torsional rigidity of the flanges has increased the buckling stress in execs of 100 percent for lengths greater than 5 m. The mode of buckling at a half-wavelength of 5 m for the open sections (HBS) is a conventional flexural-torsional buckle. The flexural-torsional buckling mode of the open HBS section involves longitudinal displacements of the cross-section (called warping displacements) such that the longitudinal displacements at the free edges of- Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -11

47 Figure.11: Buckling Stress versus Half-wavelength (Hancock, 1998) the strips are different from the longitudinal displacements of the web at the points where the free edges abut the web. The mode of buckling at 5 m for the section with closed flanges (HBS1) shows a new type of buckling mode not previously described for sections of this type. It involves lateral bending of the two flanges, one more than other, with the flanges substantially untwisted as a result of their increased torsional rigidity. The web is distorted as a result of the relative movement of the flanges. The mode is called lateral distortional buckling. It has substantially increased buckling stress value over that of the flexural-torsional buckling of the open section HBS. Hence it is not valid to compute the flexural-torsional buckling capacity of HBS1 using conventional buckling formulae as this would produce erroneous results as shown by the dashed line in Figure.11 (Hancock, 1998). Therefore, it can be concluded that the lateral distortional buckling mode of failure is the most significant criterion for the closed triangular HFBs, however, local buckling is also a possible mode of failure Local Buckling The individual plate components forming cold-formed steel sections are normally thin compared with their width. This may instigate local buckling of plate elements in cold-formed sections before yield stress is reached. Local buckling in plate elements involves flexural displacements, with the line junctions between plate elements remaining straight (see Figure.7). The local buckling failure in thin- Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -1

48 walled sections can occur under compression, bending or shear loading. Previous researchers (Bleich, 195: Allen and Bulson, 1980 and Troitsky, 1976) have extensively investigated and summarised the elastic critical stress for local buckling. The elastic critical stress for local buckling ( f cr ) of a plate element is determined using Bryan s (1891) differential Equation (.1) based on small deflection theory; Et 3 4 w x 1(1 ν ) x x y y x w 4 w w + = f t (.1) Bryan s differential equation has been developed based on a rectangular plate of width w, length a and thickness t, with in plane stress f x acting on the plate as shown in Figure.1. The solution of Bryan s differential equation for the elastic critical local buckling stress (f cr ) is given by; f cr kπ E = 1(1 ν ) t w (.) Figure.1: Rectangular Plate Subjected to Compression Stress (Hancock, 1998) The elastic critical local buckling stress (f cr ) is a function of the elastic material properties (E,ν), plate slenderness ratio w/t, and the restraint conditions along the longitudinal boundaries represented by the value k, where k, E and ν are called as plate local buckling coefficient, elastic modulus and the Poisson s ratio, respectively. For example, a steel plate with simply supported edges on all four sides and subjected to uniform compression will buckle at a half-wavelength equal to the plate Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -13

49 width (w) with a plate buckling coefficient (k) of 4.0. A plate element is defined as slender if the elastic critical local buckling stress (f cr ) calculated using Equation. is less than the material yield stress (f y ). A slender section will buckle locally before the squash load (P y ) or the yield moment (M y ) is reached. If the elastic critical buckling stress (f cr ) exceeds the yield stress f y, the compression element will buckle in the inelastic range (Yu, 000). Equation. can be used for the local buckling of plates subjected to bending and shear (Trahair and Bradford, 1988). The buckling of disjointed flat rectangular plates under bending with or without longitudinal loads has been investigated by many researchers; (Yu, 000). The bending buckling coefficient, k for long plates was found to be 3.9 for simple supports and 41.8 for fixed supports by Timoshenko (1959). The relationships between the buckling coefficient, k and the aspect ratio a/h (where a and h represent length and height of the web, respectively) was presented by Gerard and Becker (1957) as shown in Figure.13. Bulson (1969) also showed the influence of bending stress ratio f c /f t on buckling coefficient k, when a simply supported plate is subjected to a compressive bending stress higher than the tensile bending stress. A summary of local buckling coefficients, k with corresponding halfwavelengths of the local buckles for a long rectangular plate subjected to different types of stress (compression, shear, or bending) and boundary conditions (simply supported, fixed, or free edge) is given in Table.1. Figure.13 Bending Buckling Coefficient k Vs Aspect Ratio, a/h (Gerard and Becker, 1957) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -14

50 Table.1: Local Buckling Coefficient (Hancock, 1998) Note: L=Plate length, b=plate width Although local buckling occurs at a stress level lower than the yield stress of steel, it does not necessarily represent the collapse of the members. In the case of considerably low (b/t) ratios, failure is governed by post-buckling strength which is generally much higher than local buckling strength. For example, a plate subjected to uniform compressive strain between rigid frictionless platens will deform after buckling, and will redistribute the longitudinal membrane stresses from uniform compression to those shown in Figure.14. Although the stiffness reduced to 40.8% of the initial linear elastic value for a square stiffened element and to 44.4% for a square unstiffened element, the plate element will continue to carry load (Bulson, 1970). The theoretical analysis of post buckling and failure of plates is extremely difficult, and generally requires a sophisticated computer analysis to achieve an accurate solution (Hancock, 1998). Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -15

51 Figure.14: Redistribution of Stress after Post buckling of Uniformly Compressed Plate Element (Hancock, 1998) The buckling behaviour of triangular HFB sections was investigated by Dempsey (1990) using a finite strip buckling analysis program BFINST6. His buckling analysis has shown that the buckling coefficients (k) are generally equal to or greater than 4.0 for flange element and the web element, thus verifying that the flange and web elements are adequately stiffened. Figure.15 shows the buckling stresses over a wide range of half-wavelengths. Local buckling occurs in the top compression flange at a half-wavelength of approximately the flat width of the compression element (Point A). Both of the flange return and the compression portion of the web do not experience local buckling because the stresses are lower and are not uniform and their flat width to thickness ratio (b/t) is much smaller. Figure.15: Different Buckling Modes and Buckling Stresses for a Triangular HFB Subjected to Bending (Dempsey, 1990) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -16

52 By rearranging Equation. for elastic critical stress for local buckling of a plate element in compression, and substituting the buckling stress at point A from the finite strip buckling analysis, the buckling coefficient can be calculated for the top flange (Dempsey, 1990). Table. shows the values of k calculated using Equation.3 for a series of triangular HFBs. For all except the thicker 300 mm deep sections, k 4.0, which more than satisfies the assumed support conditions for the edges of the flange compression element. For those cases where, k < 4.0, local buckling does not occur because the flat width to thickness ratio is sufficiently high. The ratio of the radius to thickness of the flange bends also appears to affect the calculated value of k. The smaller the ratio (within the range shown), the smaller the buckling coefficient. This would indicate that for a given material thickness, sharp corners do not provide as much stiffness to the edge of the flange as wider corners (Dempsey, 1990). Table. Local Buckling Coefficients of Flange for Major Axis Bending (Dempsey, 1990) Designation B t R b/t f ol k 300HFB HFB HFB HFB HFB HFB HFB HFB Note: B, t, and R - geometric parameters of triangular HFBs (see Figure 1.3) f ol - local buckling stress in MPa and k plate buckling coefficient.3.1. Distortional Buckling Distortional buckling is a mode of failure where a section changes its cross-sectional shape under compressive stress. It may occur in thin sections in compression or bending at stresses significantly below the yield stress, especially for high strength steels (Hancock and Rogers, 1998). The wavelength of distortional buckling is Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -17

53 generally intermediate between that of local buckling and global buckling which places it firmly in the practical range of member lengths (Davies and Jiang, 1998). Past investigations have revealed two distinctive distortional buckling modes that commonly observed in cold-formed steel members namely flange distortional buckling and lateral-distortional buckling (see Section..5.3). The flange distortional buckling of cold-formed C- and Z-section steel members has been extensively investigated: Lau and Hancock (1987) presented distortional buckling formulae for channel columns, Hancock (1997) provided a design method for distortional buckling of C-section flexural members, Lau and Hancock (1990) studied inelastic buckling of channel columns in the distortional mode, Jiang and Davies (1997) derived design approaches for distortional buckling of channel sections, Hancock et al. (1994) provided design strength curves for thin-walled C- sections undergoing distortional buckling, Rogers and Shuster (1997) investigated the distortional buckling of cold-formed steel C-sections in bending, and Teng et al. (003) studied distortional buckling of channel beam-columns. The formulae to predict the elastic distortional buckling stress (f od ) of thin-walled channel section columns with a range of section geometries were presented by Lau and Hancock (1987). They were derived based on an approximate model of the distortional buckling as shown in Figure.16a. The distortional buckling formulae for sections in compression were later modified by Hancock (1997) to allow them to apply to distortional buckling in flexure based on a revised distortional buckling model as shown in Figure.16b. (a) Compression (Lau and Hancock, 1987) (b) Flexure (Hancock, 1997) Figure.16: Analytical Models for Distortional Buckling in Compression and Flexure Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -18

54 The rotational spring stiffness, k is given in Lau and Hancock (1987) as: 3 Et 1.11f ' ( ) od b w λ k = φ 1 (.4) b 0.06λ w Et bw + λ where E is the modulus of elasticity, t is the thickness, b w is the width of the web and f od is the compressive stress in the web at distortional buckling, computed assuming k is zero. In Equation.4, is the half-wavelength of the distortional buckling, and is given by: 0.5 I 4.8 xf b f bw λ = (.5) 3 t in which, I xf and b f are moment of inertia and width of the compression flange, respectively. When the web of the C-section is subjected to compression as shown in Figure.17a, it is treated as a simply supported beam in flexure (see Figure.18a). The rotational stiffness at the end would then be EI/L as a result of the equal and opposite end moments. When the web of the C-section is subjected to flexure as shown in Figure.17b, it is treated as a beam simply supported at one end and built in at the other (see Figure.18b). The rotational stiffness at the free end would then be 4EI/L. Hence it can be concluded that the change in restraint to bottom flange from Figure.17a to Figure.17b will approximately double the torsional restraint stiffness k (Hancock, 1997). (a) (b) Figure.17: Buckling of a C- section Under Compression Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -19

55 (a) Figure.18: Symmetric and Asymmetric Restrained Bending (Hancock, 1997) (b) On this basis, Equations.4 and.5, which were developed for the compression members by Lau and Hancock (1987), were later revised by Hancock (1997) for flexural members as: 3 4 = Et 1.11 f ' ( ) od b w λd k φ 1 (-6) b w λd Et λd bw λd bw 0.5 I 4.8 xf b f bw λ = d (-7) 3 t Clause of the cold-formed steel design standard, AS/NZS 4600 provides design methods for flexural members subjected to distortional buckling for two cases (SA, 1996): (a) Distortional buckling involving rotation of a flange and lip about the flange/web junction of a channel or Z-section, and (b) Distortional buckling involving transverse bending of a vertical web with lateral displacement of the compression flange. The elastic distortional buckling stress f od is calculated using equations provided in Appendix D of the AS/NZS These formulae are based on Hancock s (1997) distortional buckling formulae for C-section flexural members, and so that they may Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -0

56 only cover the Part a of Clause AS/NZS 4600 has recommended the equations given in Appendix D to be used in calculating the distortional buckling stress (f od ) in the case of lateral distortional buckling (Part b of Clause ) which is likely to occur in beams, such as HFBs. However, these equations provided in Appendix D of the AS/NZS 4600 have been primarily developed for open C-sections by Hancock (1997) and hence the use of the same equations to calculate distortional buckling stresses for other section geometries subjected to lateral distortional buckling mode is debatable. This position is supported by the statement from Avery et. al. (000), who state that the lateral distortional buckling is not encompassed by the design formulae contained in either the Australian Steel Structures (AS 4100) or Cold-formed Steel Structures (AS 1538), which was later revised to the current version of Cold-formed Steel Structures, AS/NZS The elastic lateral distortional buckling of triangular HFBs has been investigated to some extent by past researchers: Dempsey (1990) analysed the elastic lateral distortional buckling of simply supported triangular HFBs in uniform bending using a finite strip method incorporated in the computer program Thin-wall (Hancock and Papangelis, 1994), Heldt and Mahendran (199) conducted investigations of lateral distortional buckling of triangular HFBs using both buckling analysis and experiments, Mahendran and Doan (1999) carried out lateral distortional buckling tests of triangular HFBs, Avery and Mahendran (1997) investigated the use of web stiffeners to eliminate the lateral distortional buckling of triangular HFBs and Pi and Trahair (1997) have developed a nonlinear inelastic method to analyse the lateral distortional behaviour of triangular HFBs. Mahendran and Doan (1999) indicated that research has identified the flexural capacity of triangular HFB is limited under certain restraint, span and loading conditions by the lateral distortional buckling mode of failure (see Figure.8). They have also indicated that the cross-sectional distortion causes significant strength reductions, and is particularly severe in short to medium spans. Dempsey (1990) demonstrated the change of buckling modes at different halfwavelengths (see Figure.15). At long wavelengths, the buckling curve is similar to, but lower than the flexural-torsional buckling curve which is shown as a dashed line Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -1

57 in Figure.15. The HFB is actually buckling in a distortional mode since the member cross-section does not maintain its original shape. The distortion occurs as double curvature of the web as the compression flange displaces laterally while the tension flange remains in its original position. As the half-wavelength is increased even further, the tension flange also displaces laterally so that distortion reduces until the buckling mode is almost totally lateral buckling, and the distortional buckling curve approaches the flexural-torsional buckling curve. Several factors influence the reduction of buckling stress due to distortion, and it appears that a relationship between the reduction in buckling stress and the member geometry has not yet been established, even though this behaviour has been recorded for thin-web I-beams (Bradford and Trahair, 198). Generally the behaviour seems to be a function of the torsional rigidity of the compression flange, the slenderness of the web, and the unrestrained length of the beam (Dempsey, 1990). Past research (Avery and Mahendran, 1997: Bradford and Trahair, 198) has also demonstrated that the provision of web stiffeners and batten plates enhance the lateral distortional buckling strength, as they act to prevent distortion by coupling the rotational degrees of freedom of the top and bottom flanges. Pi and Trahair (1997) stated that the survey of research information on triangular HFBs indicated that there is no simple formulation for predicting the effect of lateral distortional buckling on the lateral buckling of HFBs. On this basis, they attempted to find a simple but sufficiently accurate closed form solution for the effects of web distortion on the elastic lateral buckling of simply supported triangular HFBs in uniform bending. They also attempted to develop an advanced theoretical method of predicting the effects of stress-strain curve, residual stresses, and geometrical imperfections on the strengths of HFBs that fail by lateral-distortional buckling. The equation for flexural-torsional buckling moment resistance M o was modified by Pi and Trahair (1997) by introducing an effective torsional rigidity GJ e in place of the nominal torsional rigidity GJ to calculate the lateral distortional buckling moment resistance M od. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -

58 The elastic flexural-torsional buckling moment resistance without web distortion is given as: M o π EI = y w L π EI GJ + L (.8) in which, EI y = minor axis flexural rigidity; EI w = warping rigidity; and L = span length. This formula is revised to include the effect of web distortion as: M od = π EI y π EI GJ + e L L w (.9) The approximate lateral distortional buckling moment resistance, M od is obtained by using the approximate effective torsional rigidity GJ e given by; GJ e = ( GJ ) GJ F F + 3 Et L 0.91 π d 3 Et L 0.91 π d c c (.10) in which, GJ F = torsional rigidity of a hollow flange about its own axis; E = Young s modulus of elasticity, d c = clear web depth and t = web thickness. M yz, M yzd Flexural torsional and lateral distortional Buckling Moment, M TW Thin-wall values Figure.19 Lateral Distortional Buckling Moments (Pi and Trahair, 1997) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -3

59 The elastic lateral distortional buckling moments, predicted by Thin-wall (M TW ) and obtained from Equation.9 (M od ) were compared with flexural torsional buckling moment (M o ) for two triangular HFB sections as shown in Figure.19. It can be seen that the approximate values M od are in close agreement with the accurate Thinwall values M TW, and also these lateral-distortional buckling values are significantly lower than the flexural-torsional buckling moments M o..4. Design Procedures for HFBs Current specifications for the design of flexural steel members are based on semiempirical equations, used to estimate the ultimate section and member capacities. The capacity of a flexural member in a steel structure is determined using the appropriate specification equations and compared with the member forces corresponding to the ultimate applied loads, typically obtained from a simple elastic analysis. Effects of local buckling are accounted for by using the effective section concept. The current AS4100 and AS4600 specifications for the design of members subjected to flexural loading with or without full lateral restraint are presented in Sections.4.1 and.4.. However, shear, bearing, flange curling and web crippling are not considered in this study as they are outside the scope of this research Design procedures of AS 4100 The nominal section moment capacity (M s ) is defined in Clause 5..1 (SA, 1998) as follows: M s = Z e f y (.11) The effective section modulus (Z e ) shall be either plastic section modulus or reduced section modulus to allow for flexural local buckling. The effective section modulus is specified in Clauses 5..3, 5..4, or 5..5 (SA, 1998) as follows: λ Z = Lesser of S or 1.5Z (.1) s λ sp e Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -4

60 λ sp λ sy λs < λs λsy Z e = Z + ( Z Z ) c λsy λsp (.13) λ s > λ sy λsy Z e = Z λs (.14) where S and Z are the plastic and elastic section modulii, respectively. Z e is the effective section modulus as specified in Clause 5..3 (SA, 1998), which is either S or 1.5Z. The section slenderness ( s ) is taken as the value of the plate element slenderness ( e ) for the element of the cross-section which has the greatest value of ( e / ey ). The plate element slenderness ( e ) is defined in Clause 5.. (SA, 1998) as a function of the element clear width (b), thickness (t), and yield stress (f y ): b f y λ e = (.15) t E The section plasticity and yield slenderness limits ( sp, sy ) are taken as the values of the element slenderness limits ( ep, ey ) given in Table 5.. (SA, 1998) for the element of the cross section which has the greatest value of e / ey. The limiting slenderness ratios were established from lower bound fits to the experimental local buckling resistance of plate elements in uniform compression and flexure. The nominal member moment capacity (M b ) of members with full lateral restraint is specified in Clause 5..1 (SA, 1998) as equal to the nominal section moment capacity of the critical section. The critical section is defined in Clause (SA, 1998) as the cross-section which has the largest value of the ratio of the design bending moment (M * ) to the nominal section capacity in bending (M s ). The nominal member moment capacity (M b ) of a beam without full lateral restraint has been specified in Clause (SA, 1998) as: M b = α α M M (.16) m s s s Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -5

61 where the moment modification factor ( m ) shall be determined from one of the methods described in Clause (SA, 1998). For uniform bending moment distribution m =1.0. The slenderness reduction factor ( s ) is defined in Clause (SA, 1998) as: M s M s α s = (.17) M o M o where, reference buckling moment (M o ) is defined in Clause (SA, 1998) as: M o π EI Le GJ π EI + Le = y w (.18) Therefore, the member capacity of a beam subjected to a uniform bending moment can be rewritten as: M b M M s = o M M s o M s M s (.19) Avery et al. (000) pointed out that Equation.19 is based on the lower bounds of the test results for I-section beams, and therefore its suitability in the design of HFB beams requires further investigations. Bradford (199) states that the relationship between distortional buckling strain, yielding and elastic distortional buckling is the same as that between the lateral buckling strength, yielding and elastic lateral buckling. This implies that if the reference buckling moment (M o ) in Equation.19 is replaced with the elastic distortional buckling moment (M od ), the AS 4100 procedure shall be suitable for the hollow flange beams. This approach was investigated by Pi and Trahair (1997), in which they used Equations.9 and.10 to determine M od for use with AS 4100 procedure. Their research showed that Equation.19 has to be slightly modified to that given by the following equation in order to improve the accuracy in predicting the flexural member capacity of triangular HFBs. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -6

62 M b M M s = o + M M s o M s M s (.0).4.. Design procedures of AS/NZS 4600 The nominal section moment capacity (M s ) is specified in Clause 3.3. (SA, 1996) in a similar fashion to AS 4100 as follows: M = Z f (.1) s e y However, unlike AS 4100, the effective section modulus (Z e ) is based on the initiation of yielding in the extreme compression fibres. Therefore AS 4600 does not allow for the inelastic reserve capacity of the section. The effects of local buckling are accounted for by using reduced (effective) width (b e ) of non-compact elements in compression for the calculation of the effective section modulus. In the effective width approach, the non-uniform stress distribution over the entire width of plate element (b) due to redistribution of stress after buckling is replaced by a uniformly distributed stress equal to the edge stress (f max ) acting over a fictitious effective width (b e ) as shown in Figure.0. Figure.0: Stress Distribution in Stiffened Compression Element (SA, 1998b) The effective width concept was first introduced by von Karman et al. (193) and since then extensive investigations on light-gauge, cold-formed steel sections have been carried out. The following equations to calculate effective width (b e ) was developed by Winter (1946) for a stiffened compression element simply supported at both longitudinal edges. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -7

63 b e = ρ b (.) where, is the effective width factor defined in Clause..1. (SA, 1996) as: 0. 1 λ ρ = 1.0 (.3) λ where is the slenderness ratio and is calculated as: 1.05 b f max λ = (.4) k t E In Equation.4, k is the plate buckling coefficient and depends on edge boundary conditions and type of stress (see Section.3..1). For nominal section moment capacity (M s ) calculations, f max is assumed equal to yield stress, f y, for the extreme flange element The nominal member moment capacity (M b ) is specified in Clause as the lesser of the values calculated in accordance with members subjected to lateral buckling or distortional buckling. Therefore, unlike AS 4100, AS 4600 does provide equations specifically intended for the design of members subjected to distortional buckling in Clause (SA, 1996) as follows: M c M b = Z e (.5) Z f For hollow flange beams, it is appropriate to determine the effective section modulus Z e at a stress level M c /Z f, where M c is the critical moment as defined in Equations.6 and.7. For λ < d λ d M c = M y 1 4 (.6) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -8

64 For λ d 1 M c M y λd = (.7) The non-dimensional member slenderness ( d ) is given by: M y λ d = (.8) M od Avery et al. (000) studied the flexural capacity of triangular HFBs and pointed out that the member capacity predicted by AS 4600 is, on average, more accurate and precise than the AS 4100 predictions. Their study further indicated that AS 4600 equations overestimate the capacity of hollow flange beam sections for intermediate spans, and therefore the detrimental effects of web distortion are not accurately accounted for. Hence, the AS 4600 equations cannot be safely used for the design of hollow flange beam members subjected to uniform bending Trahair s Design Procedures A modified design procedure for triangular HFB flexural members based on Trahair s (1997) equations was proposed by Avery et al. (000) as a more accurate and reliable alternative to the AS 4600 design methods. Trahair (1997) equations for flexural member capacity are given next. M b a b = b + n 1+ cλd M s M s ; b M o M (.9) The non-dimensional member slenderness ( d ) is given by; M s λ d = (.30) M od The coefficients (a, b, c, and n) for a range of hollow flange beam sections were found by using the least square method with the total error defined as the square of the difference between the normalized analytical capacity (i.e. M u /M s ) and the normalized design capacity (i.e. M b /M s ). The results indicated an unacceptable Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -9

65 maximum unconservative error of more than 10 percent. Therefore the coefficients were established by Avery et al. (000) for separate group of sections with the same thickness as given in Table.3. The member capacity predicted by Avery et al. (000) using the modified Trahair equations was found to be significantly more accurate and precise than the AS 4100 predictions. Table.3: Coefficients for Trahair s (1997) Member Capacity Equation Coefficient t=3.8 t=3.3 t=.8 t=.3 Overall a b c n Finite Element Analysis Finite element analysis (FEA) of cold-formed steel structures plays an increasingly important role in engineering practice, as it is relatively inexpensive and time efficient compared with physical experiments, especially when a parametric study of cross-section geometries is involved. Furthermore, it is difficult to investigate the effects of geometric imperfections and residual stresses of structural members experimentally. Therefore, FEA is more economical than physical experiments, provided the finite element model is accurate and the results could be validated with sufficient experimental results. The finite element analysis process involves three major phases; 1. Pre-processing The purpose of pre-processing is to develop an appropriate finite element mesh, assign suitable material properties, and apply boundary conditions in the form of restraints and loads.. Solution While the pre-processing and post-processing phases of the finite element method are interactive and time-consuming for the analyst, the solution is usually a batch process, and is demanding of computer resources. The governing equations are assembled into a matrix form and are solved numerically. The assembly process depends not only on the type of analysis (e.g. static or dynamic), Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -30

66 but also on the model s element types and properties, material properties and boundary conditions. 3. Post-processing After a finite element model has been prepared and checked, boundary conditions have been applied, and the model has been solved, it is time to investigate the results of the analysis. This activity is known as the postprocessing phase of the finite element model. Post-processing begins with a thorough check for problems that may have occurred during the solution stage. Most solvers provide a log file, which should be searched for warning or error messages, and which will also provide a quantitative measure of how well behaved the numerical procedures were during solution. Finite element modelling requires care to guarantee good results. Bakker and Pekoz (003) gave an overview of possible errors, which might occur during linear and non-linear finite element analysis. Table.4 shows a summary of errors that can occur during finite element modelling. Table.4 Overview of Possible Errors During FEA (Bakker and Pekoz, 003) Reality Idealization error Mechanical model Input error Discretization error: equilibrium approximated Geometry error: Geometry approximated Shortcoming in element formulation Program bugs Finite element model Solution error Convergence error Program bugs Nodal displacements Program bugs Derive results according to finite element model PREPROCESSING SOLUTION Rendering error: Postprocessor inter/extrapolates differently than finite element model: Integration points nodes contour plots Program bugs Results according to postprocessor POSTPROCESSING Interpretation error: postprocessor shows something else than is expected, for instance averaged instead of unaveraged stresses Interpretation of results Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -31

67 .5.1 Analysis Types Non-linear analysis The load capacity of steel members of moderately high slenderness is not easy to determine because of its dependence on a large number of parameters related to geometric and material properties. The non-linear static analysis is therefore used to determine the complete load-displacement behaviour of structures. In non-linear analysis, the load is applied incrementally, with the stiffness calculated at each step. The non-linear behaviour of structures occurs as a result of these material and geometric non-linearities. Material non-linearity A linear static analysis assumes that the material is within the elastic limit, and that it follows a simple linear stress-strain curve. The problems where this is not the case include those exhibiting plasticity and creep of the material. For such problems an idealised stress-strain curve must be supplied to the finite element program, and is usually approximated in a bilinear or multilinear way, depending on the particular material, as illustrated in Figure.1. Geometric non-linearity A large-displacement analysis is required when the structural displacements become so large that the original stiffness matrix no longer adequately represents the structure. In such cases, the structure stiffness matrix needs to be adjusted accordingly. There are two ways in which this can be achieved. The first approximate method assumes that the size of the individual element is constant, so that reorientation of the elemental stiffness matrices due to the elements rotation and/or translation is all that is required. The second method is more accurate, and recalculates the stiffness matrices of the elements after adjusting the nodal coordinates with the calculated displacements. It is quite conceivable that largedisplacement problems can themselves experience stress-stiffening effects, in which case the geometry stiffness matrix must also be included in the large-displacement solution. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -3

68 Figure.1: Material Non-linearity.5.1. Buckling Analysis The buckling analysis is used to predict the buckling loads and the corresponding buckling shapes. The buckling load is generally used as a parameter in determining the post-buckling strength of members. The buckling shape is used for the description of the geometric imperfections when the maximum amplitude of the imperfection is known but its distribution is not known. Superimposing of multiple buckling shapes may be used as the initial geometric imperfection in post-buckling analysis. The post-buckling analysis is needed to investigate the load-deflection behaviour. Pekoz et al. (003) pointed out that several approaches are possible depending on the selected algorithm and how the boundary conditions are applied. When the loads can be applied by means of prescribed displacement, increment method (where proportional displacements are applied) is used. In other cases, the modified Risk method (where proportional loads are applied) is used in order to be able to pass limit points. Both approaches are effective in obtaining non-linear static equilibrium states during the unstable phase of the response. In both cases initial geometric imperfections must also be introduced to obtain some response in the buckling mode before the critical load is reached. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -33

69 .5. Initial Conditions.5..1 Geometric Imperfections Geometrical imperfections refer to the deviation of a member from perfect geometry (see Figure.). Imperfection of a member includes bending, warping and twisting as well as local deviations. Local deviations are characterized by dents and regular undulations in the plate. Schafer and Pekoz (1998b) cited that previous researchers have measured geometric imperfections of cold-formed steel members such as C-, Z- and RHS sections. However, existing imperfection data provides only a limited characterisation of imperfections. For plate thickness less than 3 mm, Schafer and Pekoz (1998b) provided simple rules for so-called Type 1 imperfections for width/thickness w/t < 00, and Type imperfections for w/t < 100 (see Figure.1). For Type 1 imperfections an approximate expression d w was given as a simple linear regression based on the plate width. They also gave an alternative rule based on an exponential curve fit to the thickness. d te (d 1 and t in mm) t 1 6 For Type imperfections the maximum deviation is approximately equal to the plate thickness: d t (a) Type 1 (b) Type Figure.: Definition of Geometric Imperfections (Schafer and Pekoz, 1998b) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -34

70 When precise data of the distribution of geometric imperfection is not available, three approaches have been used (Pekoz et al., 003). One is to use an imperfection based on superimposing multiple buckling modes and controlling their magnitudes. The magnitude of imperfection can be controlled by using existing statistical imperfection data where the maximum values are provided. On the other hand, if imperfection measurements are conducted, the imperfection spectrum generated from the imperfection measurements may be used to approximate the imperfection magnitude corresponding to a particular eigenmode. Another method is to use a stochastic process to generate signals randomly for the imperfection geometric shape. However, a large number of measurement data is also required to have a reasonable stochastic model. An initial geometric imperfection shape introduced by superimposing the eigenmodes for local and distortional buckling is shown in Figure.3. (a) Local buckling (b) Distortional buckling (c) Imperfection Figure.3: Geometric Imperfection (Pekoz et al., 003) Schafer and Pekoz (1998b) suggested that at least two fundamentally different eigenmode shapes should be summed for the imperfection distribution in a limited study. Numerical modelling of triangular HFBs by Avery et al. (000) used nominal global imperfection magnitude of L/1000 based on the AS 4100 recommendation of tolerance for compression members. The magnitudes of the local flange and web imperfections were conservatively taken as the manufacture s fabrication tolerance (PTM, 1993) as shown in Figure.4. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -35

71 Figure.4: Local Imperfections (Avery et al., 000).5.. Residual Stresses Generally, residual stress includes two types: flexural (or bending) and membrane. In cold-formed members residual stresses are dominated by a flexural, or through thickness variation (Schafer and Pekoz, 1998b). This variation of residual stresses leads to early yielding on the faces of cold-formed steel plates. Adequate computational modelling of residual stresses is troublesome for analyst, and the inclusion of residual stresses (at the integration points of the model for instance) may be complicated. Furthermore, selecting an appropriate magnitude is made difficult by a lack of data. As a result, residual stresses are often excluded altogether, or the stress-strain behaviour of the material is modified to approximate the effect of residual stresses. Residual stresses are idealized as a summation of flexural and membrane stresses (see Figure.5a). However, Schafer and Pekoz (1998b) state that this idealization is a pragmatic rather than scientific choice. The average bending residual stresses for a cold-formed C-section as a percentage of yield stresses are shown in Figure.5b. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -36

72 (a) Definition of Flexural and Membrane Residual Stresses (b) Average Flexural Residual Stress as % f y Figure.5: Membrane and Flexural Residual Stresses (Schafer and Pekoz, 1998b) Doan and Mahendran (1996) suggested a residual stress model for triangular HFBs based on measured residual stresses (see Figure.6). The same residual stress model was used by Avery et. al. (000), to model geometric imperfections of triangular HFBs considered in their finite element modelling. However, significant modifications will be needed if the same model is considered in this research study involving rectangular HFBs. Figure.6: Bending Residual Stress Distribution for Outside Fibre, Expressed as a Percentage of the Yield Stress (f y ) with Tension Positive (Doan and Mahendran, 1996).5.3 Finite Element Analysis of HFBs The structural behaviour and failure mechanism of cold-formed steel beams with hollow flanges have been studied by past researchers (Avery et al., 000, Pi and Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -37

73 Trahair, 1997) using finite element analyses. Therefore, it is now apparent that the structural behaviour of hollow flange beams can be predicted by finite element analysis. Since analytical studies are considerably less expensive than testing, and as it opens up the possibility of extensive parametric studies, finite element modelling plays an important role in the investigation of flexural behaviour, aiming at developing appropriate design rules for rectangular HFBs. Applications of FEA to triangular HFBs in investigating their structural behaviour are discussed next. Flexural capacity of triangular HFBs was investigated recently by Avery et al. (000) using finite element analyses. From their investigations, they discovered that the elastic lateral-distortional buckling moment and the ultimate capacities of triangular HFBs can be accurately predicted from their finite element analyses and therefore used them in the development of design curves and suitable design procedures. The study involved two models, namely experimental model and ideal model. The experimental model was developed to match the actual test members, and the ideal model was developed incorporating ideal constraints and nominal imperfections to generate member capacity curves (see Figures.7(a) and (b)). The ABAQUS S4R5 shell elements were employed in the models and the results showed that this element type provides sufficient degrees of freedom and hence can explicitly model the local buckling deformations and spread of plasticity effects. The R3D4 rigid body elements were also used to model the pinned end conditions. The loads and boundary conditions, as used by Zhao et al. (1995) in the study of lateral-buckling of cold-formed RHS beams, were used in these models to provide idealized simply supported boundary conditions and a uniform applied bending moment. The ideal support boundary conditions used in the models were: vertical and lateral translational restraint, twist restraint, freedom to rotate about the major and minor axes, and no warping restraint. The lateral tortional buckling formula used in AS 4100 was also derived based on the same conditions (SA, 1998). However, they have not been able to eliminate the warping restraints due to the overhang in the experimental models. The models incorporated all the significant effects that might influence the ultimate capacity of triangular HFB beams, including material inelasticity, local buckling, member instability, web distortion, residual stresses and initial geometric imperfections. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -38

74 Initial geometric imperfections used in this model were based on the values of fabrication tolerances specified in AS 4100 (1998) and triangular HFB design manual (PTM, 1993). Residual stresses were modelled using Doan and Mahendran s (1996) residual stress model which was based on the measured residual stresses (see Sections.5..1 and.5..). However, this investigation was limited to triangular HFBs, fabricated from a single steel strip using electric resistance welding. Therefore, the results of this study are not applicable to other geometric shapes and manufacturing methods. (a) Experimental HFB Model (b) Ideal HFB Model Figure.7: Finite Element Models of HFBs (Avery et al., 000) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -39

75 Avery and Mahendran (1997) studied the lateral-distortional buckling of hollow flange beams with web stiffeners using finite element analysis. The finite-element analysis program, MSC/NASTRAN was used for this study, with the aid of MSC/XL as the pre-processor to generate model and MSC/XL and AVS as postprocessor for the visualization of results. The quadrilateral shell elements (QUAD4) in the NASTRAN library are flat, with four nodes and six degrees of freedom per node, and were used in this finite element modelling. Triangular shell elements (TRIA3) were used to model the stiffeners, since trapezoidal shape of stiffeners forced unacceptable distortions of QUAD4 elements. The mesh detail of the model is shown in Figure.8. Figure.8: Finite Element Mesh (a) HFB Model (b) Stiffener Mesh (c) Web Distortion of Unstiffener Mesh (Avery and Mahendran, 1997) Only half of the beam was modelled by making use of the symmetry of geometry and loading conditions about the centre plane of the span, so that the size of the model and hence solution time and computational effort are reduced. The support conditions used in this model were similar to those used by Avery et al. (000) as described earlier. This ideal model was used in the parametric studies using elastic buckling analysis. In this study, a modified model was also developed to represent actual experimental set-up, and was referred to as the experimental model, and was used in the comparison with experimental results. It was found that there is Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -40

76 negligible difference between the ideal and experimental models and so that this indicates that the warping restraint provided by the cantilever is an insignificant factor in the experimental model. In the nonlinear ultimate strength analysis of the experimental model, including geometric imperfection and material nonlinearities, an initial imperfection was assumed as recommended by Salmi and Talja (199). It consists of linear variation in lateral displacement for all the nodes on the crosssection, varying from zero at the support to a maximum value of two wall thicknesses at midspan. Some other studies involving finite element analysis of cold-formed steel beams included a finite element study by Wilkinson and Hancock (1999) to predict the rotation capacity of RHS beams. ABAQUS Version 5.7 (HKS, 1998) was used in this investigation. A quarter of the experimental RHS beam was modelled due to its geometric symmetry. The S4R5 shell element was used to model the beam while the C3D8 brick element was used to model the loading plates. The welding joint between the loading plate and the RHS beam was modelled using C3D6 elements. The model details are as shown in Figure.9. Three types of material properties for the flanges, web and corners were used and the nonlinear analyses included bending residual stresses. Figure.9: Physical Model and Typical Finite Element Mesh (Wilkinson and Hancock, 1999) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -41

77 .6 Experimental Investigation Experimental methods are the base and a necessity for scientific research even though they are very time-consuming and expensive. The mathematical formulae can only be used to predict the capacities of idealized structures where a number of assumptions have been made. Experimental results can be used to verify the numerical models that can then be used to expand the results to enable a full understanding of the structural behaviour and the development of design rules. Some of the experimental investigations of cold-formed steel beams are discussed next. Zhao et al. (1995) conducted a series of lateral buckling tests of cold-formed RHS beams to improve existing design rules for RHS beams. The section size used in the testing program was 75 mm 5 mm.5 mm. Spans were varied from 000 mm to 7000 mm in order to produce a substantial range of slenderness ratios. The loading system included a gravity load through the centroid of the section and the support system was designed to ensure that simple support end conditions were achieved. The layout of test setup is shown schematically in Figure.30a. The support system used in this study (see Figure.30b) was similar to that used by Trahair (1969) in his elastic lateral buckling tests of aluminium I-beams and later by Papangelis (1987) in his flexural-torsional buckling tests of arches. (a) Test arrangement (b) Support system Figure.30: Lateral Buckling Test of RHF Beams (Zhao et al., 1995) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -4

78 The test beams were simply supported both in-plane and out-of-plane. The in-plane vertical deflections were prevented by the supporting tracks but the in-plane rotations were not restrained, ie. the beam was free to rotate about the horizontal axis (x 1 x 1 ). The out-of-plane deflections and twist rotations were prevented by the prismatic spigots but minor axis rotations were not restrained, ie. the beam was free to rotate about the vertical axis (y 1 -y 1 ) (see Figure.30). However, warping displacements were not prevented except by the adjacent cantilever lengths. The restraint to warping provided by the cantilever lengths can be considered minimal because significant warping does not occur in tubular sections, compared with I-sections. Unlike RHS beams, rectangular HFB considered in this research program are open steel beams and hence they are expected to induce significant amount of warping displacements compared with RHS beams. Therefore, warping effect needs to be accounted for if the same test arrangement is used for rectangular HFB testing. The loading system included gravity loads being applied by suspending lead blocks on a platform which is supported by hangers. However, the gravity loading system can be replaced by a power control loading system to ensure identical bending moments at the ends of the span. Zhao et al. (1995) cited that the loading system used in their study was similar to those used by Cherry (1960) and Hancock (1975), where the vertical load applied acted through the centroid of the section and no restraints were applied against out-of-plane movement at the loading point. Mahendran and Doan (1999) conducted an investigation into the lateral-distortional buckling behaviour of hollow flange beams with triangular flanges. A purpose-built test rig was used in this study to obtain the bending capacity of hollow flange beams under uniform bending moment. The test rig included a support system and a loading system, which were attached to an external frame consisting of a main girder and four columns as shown in the schematic diagram in Figure.31. The support system was designed to ensure that the test beam had simply supported end conditions, whereas the loading system was designed in such a way that no restraints were induced as the beam deformed during loading. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -43

79 Figure.31: Schematic Diagram of Test Rig Including Support System (Mahendran and Doan, 1999) Two vertical loads were applied at the end of two overhangs to produce a uniform bending moment within the span of the specimen. The simply supported end conditions of the span were simulated in a similar way to that of Zhao et al. (1995) used for the rectangular hollow sections (RHS) but were modified to suit the triangular HFBs. However, warping restraints induced by overhang of the beam could not be eliminated in this system. The same support system can also be applied to the innovative rectangular HFB beams considered in this research program with minor modifications. However warping restraint effect need to be eliminated to obtain ideal pinned end conditions. The loading system included two hydraulic jacks instead of gravity loading system used by Zhao et al. (1995). They were operated under load control to ensure that the same load was applied and hence identical bending moments were provided at the ends of the single span. The loading system was designed such that there was no restraint in lateral and longitudinal directions from the jacks to the overhang at the loading positions. The load was applied through the shear centre of the cross section to eliminate the load height effects. Mahendran and Avery (1996) conducted buckling experiments to investigate the effects of type, thickness, location and number of web stiffeners on the lateral buckling behaviour of triangular HFBs. The results of these experiments were also Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -44

80 used to validate the finite element model developed by Avery et al. (000). The tests included ten 6 m long triangular HFB specimens, which were loaded to failure under a constant bending moment within a span of 4.5 m as illustrated in Figure m 4.5 m 0.55 m Figure.3: Schematic Diagram for Lateral Buckling Tests of HFBs (Mahendran and Avery, 1996) The experimental set-up used in this study included specially designed loading device and a support system. The support system provided restraint to vertical and lateral translation at the supports, and prevented from twisting about the longitudinal axis of the member, while being free to rotate about the major and minor axes. The support system included two mild steel plates placed between the HFB and each roller support. The plates were separated by a stainless steel sheet attached to the top plate and a Teflon layer connected to the bottom plate. A steel pin fixed to the top plate fitted into a hole in the bottom plate. The plate could therefore rotate freely on the low friction Teflon/stainless steel interface, but prevented relative translation by the pin. A Rectangular Hollow Section web stiffener was used to prevent twist at the support, and connected the HFB specimen to the top plate, allowing rotation about the minor axis without lateral defection. Two load-controlled hydraulic jacks, located on the overhangs were used to apply the loads the web stiffeners at the support prevented any local bearing failure of the bottom flange. Although the support conditions assumed in this experimental program were pinned, they can be neither fixed nor pinned, but partially restrained due to friction forces induced between different components within the support system. The bottom plate of the support was placed and clamped to the roller support which could have restrained the major axis rotation to some extent. Similarly the minor axis rotation could have been restrained to some extent by the friction forces due to the top and bottom plate rotation. Twisting of the beam sections at the supports were prevented Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -45

81 using stiffeners attached to the web at the supports but it might not have been possible as stiffeners themselves are free to move and rotate. However, the simplicity of this support system is a big advantage compared with other complicated support systems used by Mahendran and Doan (1999) and Zhao et al. (1995) as described earlier. Some other experimental research on the lateral buckling strength of cold-formed steel beams included Pi and Trahair (1998a, b) who investigated lateral buckling capacities of cold-formed lipped Z- and C- section beams to find improvements for the future design code formulations. Although the support system was designed to achieve simple support end conditions in these tests, they were different and complicated than the above mentioned loading and support systems due to different geometric configurations of these section types (see Figure.33). A gravity loading system was used for beam loading. This system applied the vertical load in the loading drum. A low friction bearing system was used to maintain vertical line of action and hence lateral buckling restraint effect was eliminated. The lengths and load heights were selected so that the tests would supply experimental data in the intermediate slenderness range, for which inelastic buckling was expected to control the transition from the section capacity (for low slenderness C- and Z-s) to the elastic buckling capacity (for high slenderness C- and Z-s). These test arrangements are not suitable for the buckling tests of rectangular HFBs considered in this research program since they were not designed for the doubly symmetric sections (only for C- and Z-). (a) Test Arrangement for C- and Z- sections (b) Support System for C-section Figure.33: Lateral Buckling Tests of Cold-formed C- and Z- section Beams (Put et al., 1999) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -46

82 (c) Support System for Z- section Figure.33: Lateral Buckling Tests of Cold-formed C- and Z- section Beams (Put et al., 1999).7 Summary of Literature Review Findings An extensive literature review as described in this chapter has enabled the accumulation of the required knowledge in the following topics: types of coldformed steel sections used for flexural members, effects of cold-forming, special design criteria for cold-formed steel design, failure modes of cold-formed steel beams, current cold-formed steel design standards and procedures, finite element analysis and experimental investigations of cold-formed steel beams. The main focus of all the above topics was based on the HFBs as flexural members. A summary of the literature review is presented next. Typically used cold-formed steel sections for flexural members, such as C-, Z- and hat sections, are found to be more susceptible to structural instabilities due to their profile geometry. However, the characteristics due to monosymmetric nature of the C- sections and the point symmetry nature of Z-sections are not normally encountered in doubly symmetric sections such as I-sections and tubular sections (i.e. RHS, CHS, SHS). Therefore, the recently invented cold-formed steel section known as HFB, comprising two triangular hollow flanges is considered to be structurally more efficient than conventional sections such as C- and Z- sections. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -47

83 Cold working during the formation of cold-formed steel sections affect the mechanical properties of the formed sections due to strain hardening and strain ageing. The resulting changes in material properties must be included in the design of cold-formed steel members to achieve structurally efficient members in an economical manner. Current cold-formed steel design standards (see Section..4) allows for this effect by introducing average yield strength (f ya ) for coldformed sections. Local buckling and post-buckling strength of cold-formed steel members subjected to compression or flexural actions play an important role in the design of cold-formed steel structures. The inclusion of these buckling effects in coldformed steel design is important to achieve more structurally efficient coldformed structures in an economical manner (see Section..5.1). Torsional rigidity is also an important criterion in the design of cold-formed steel members, since torsional rigidity of an open section is proportional to the cubic power of thickness (t 3 ), resulting in low torsional rigidity. However, hollow sections such as RHS, CHS and SHS have high torsional rigidity because of their geometry. The so-called HFBs comprising two triangular hollow flanges connected by a web also have a high torsional rigidity and therefore their lateral torsional buckling capacity is expected to be higher. The distortional buckling is one of the most important buckling failure modes for the practical cold-formed steel beams (see Sections..5.3 and.3.3.). However, accurate design rules are not available in the current cold-formed steel design standards to deal with distortional buckling. The Australian cold-formed steel design standard AS/NZS 4600 has included improved design methods for distortional buckling, however they were based on the C- and Z- sections and hence their applications for other section types such as HFBs will need significant modifications. The design approach proposed by Pi and Trahair (1997) to calculate the elastic distortional buckling moment capacity of HFBs is only valid for HFBs Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -48

84 comprising triangular hollow flanges and hence the application of the same equation for other section types such as rectangular HFB may need appropriate modifications. Previous researchers have recommended that the application of AS 4100 design formulae for triangular HFBs needs modifications (see Section.4.1). The member moment capacity equation provided in AS 4100 has been based on lateral tortional buckling and the lower bound of the test results for I- section beams. However, it has been indicated that if the reference buckling moment for lateral tortional buckling (M o ) given in AS 4100 is replaced by elastic lateral distortional buckling moment (M od ), AS 4100 design procedures shall be suitable for triangular HFBs. This needs to be investigated for rectangular HFBs. The member moment capacity of triangular HFBs calculated using the AS/NZS 4600 approach is more accurate than the AS 4100 approach. However, it was found that the AS/NZS 4600 equations overestimate the capacity of triangular HFBs for intermediate spans, and therefore, the detrimental effects of web distortion are not accurately accounted for. Therefore the AS/NZS 4600 equations cannot be safely used in the design of triangular HFB members subjected to uniform bending. Previous researchers have used finite element analyses to investigate flexural behaviour of triangular HFBs. They have shown that the structural behaviour of HFBs can be predicted by finite element analysis if it is used accurately to model the beam under investigations with inclusion of appropriate geometric imperfections, residual stresses, material characteristics, loading and boundary conditions. Experimental researches have also been carried out by previous researchers to investigate the flexural behaviour of triangular HFBs, and sometimes to validate finite element models. This literature review showed that the uniform bending moment distribution within a selected span is the common practice for buckling tests, since these conditions allow comparing experimental and theoretical results Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -49

85 accurately. Two explicit methods have been used by previous researchers to generate uniform moment conditions over a span of the beam. In the first method, two equal overhang loads are applied at an equal distance outside the supports to generate a uniform bending moment between the supports. In the second method, two equal loads at an equal distance from the supports but within the span are applied to generate uniform bending moment between the loading positions. However, it is clear and understandable from the literature that the former method is more common among researchers than the latter since the former method allows the simulation of a uniform bending moment within the entire span, and hence the boundary conditions in analytical models can be set-up easily. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges -50

86 CHAPTER 3 Experimental Studies of Material Properties and Section Moment Capacities of RHFB 3.1 General This chapter describes the experimental studies of material properties and section moment capacities of Rectangular Hollow Flange Beams (RHFB) based on a series of laboratory experiments. Sixteen tensile coupon tests including all the steel grades and thicknesses were conducted using specimens taken from the same batch of steel sheets that were used in the section and member capacity tests. The main objective of the tensile test program was to obtain accurate stress-strain relationships for the three steels with steel grades G300, G500 and G550 and different thicknesses that were needed in the section capacity calculations and the numerical modelling of RHFBs. Twenty two section capacity tests of RHFBs on short and fully laterally restrained flexural members were conducted under simply supported end conditions, and the test results were compared with the predictions from the current design rules in the Australian steel design standards AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996) to verify their applicability to RHFBs. 3. Material Property Tests 3..1 Material Description The sheet metal manufactured by BlueScope Steel Limited in Australia was purchased from the Smorgon Steel Sheet Metal Suppliers in Victoria, Australia to fabricate the test specimens for section and member capacity tests of RHFBs. Therefore tensile coupon tests were also conducted using the same sheet steels to obtain the relevant material properties. Three steel grades, G300, G500 and G550, Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-1

87 were chosen and their nominal yield strengths (f yn ) are 300 MPa, 500 MPa and 550 MPa, respectively, and the nominal tensile strengths (f un ) are 340 MPa, 500 MPa and 550 MPa, respectively. These steel sheets were manufactured to comply with the Australian Standard Steel Sheet and Strip-Hot-dip zinc-coated or aluminium/zinc-coated AS 1397 (SA, 001). The milling process during the production phase of the steel sheets causes the grain structure of cold reduced steels to elongate in the rolling direction, which results in an increase in strength and a decrease in ductility (BHP, 199). The effects of cold working are cumulative, i.e. grain distortion increases with further cold working, however, it is possible to change the distorted grain structure and control the steel properties through heat treatment. BHP (199) reported that various types of heat treatment exist and are used for different steel products. G300 sheet steels are fully recrystallised, i.e. the grain structure is returned to its original state although some preferred grain orientation remains whereas G500 and G550 sheet steels are stress relief annealed, i.e. recrystallisation does not occur. The high yield stress and ultimate strength values of G500 and G550 sheet steels are obtained by means of an alloying process, i.e. high strength low alloy steels. The typical chemical compositions of steels from the three steel grades G300, G500 and G550 and different thicknesses are shown in Table 3.1. Table 3.1: Chemical Composition of Steel Used in the Tests (BHP, 199) Steel Grade Nominal Chemical composition (%) thickness (mm) C P Mn Si S Ni Cr Mo Cu Al Ti Nb G G G Note: Additional data is presented in Appendix 3A Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-

88 All steels used in this test program were cold reduced to the required thickness, with an aluminium/zinc alloy (zincalume AZ), or zinc coating (Galvanized Z). Wills (198) and Wills and Lake (1988) have reported that the behaviour of coated G300 sheet steels is dependent on the composite action that occurs between the zinc or aluminium/zinc coating and the base metal. However, specific references which detail the influence of metallic coating on G550 sheet steels are not available although it can be assumed that a composite action occurs, as found for G300 sheet steels. However, it is assumed that the contribution of metallic coating to the structural strength of RHFBs in terms of section and member capacities is insignificant and therefore the base metal thickness (BMT) is used in place of total coated thickness (TCT). The BMT for each steel grade and thickness was determined using acid itching method. For this purpose, three steel strips, 5 mm 100 mm were cut from each steel grade and thickness giving a total of 4 specimens. The TCT of each specimen was measured before they were immersed in the hydrochloric acid to wash off the metallic coating (see Figure 3.1 (a)). The specimens were taken out after approximately 30 minutes in the hydrochloric acid and were washed in pure water before the BMT was measured (see Figure 3.1 (b)). The details of applied metallic coating types, the measured TCT and BMT and the calculated coating thicknesses for different steel grades and thicknesses are listed in Table 3.. (a) Specimens in hydrochloric acid Figure 3.1: Base Metal Thickness Measurement Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-3

89 (b) Specimens after coating wash-off Figure 3.1: Base Metal Thickness Measurement Table 3.: Metallic Coating Details and Measured Thicknesses of Steel Sheets Steel Grade G300 G550 Nominal BMT (mm) Coating type Measured (mm) TCT BMT Calculated CT (mm) 0.55 Zincalume (AZ150) Zincalume (AZ150) Zincalume (AZ150) Galvanized (Z75) Zincalume (AZ150) Galvanized (Z350) Zincalume (AZ150) G Galvanized (Z350) Note: Coating Thickness, CT = TCT BMT 3.. Test Specimens and Test Set-up Sixteen tensile test specimens including two specimens from each steel grade and thickness were taken from the same steel batch that was used in the section and member capacity tests. This allowed the determination of an accurate stress-strain relationship for each steel grade and thickness used in the tests that can be used in the section and member capacity calculations of RHFBs. The material properties of cold reduced steels have been shown to be anisotropic (Wu et al., 1995, Dhalla and Winter, 1971). Hence all the tensile test specimens were cut in the longitudinal direction with respect to the rolling direction of steel sheets, as it was the same longitudinal direction along which the test beams used for section and member Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-4

90 capacities were made. Specimen size and shape are important variables which can affect its behaviour. Accurate and consistent fabrication procedures were used for all specimens included in this test program to ensure that test specimens were of near identical size and shape. Various standards exist which specify the requirements for the testing of tensile specimens. Tensile specimens for this test program were prepared in accordance with the Australian Standard Methods for Tensile Testing of Metals AS 1391 (SA, 1991). A typical tensile test specimen used in this test program is shown in Figure 3. (a) whereas Figure 3. (b) shows some of the strain gauged tensile test specimens. The thickness and width of all the test specimens were measured at three different locations within the constant gauge length. The average cross-sectional dimensions are presented in Table 3.3. R0 13 mm 5 mm 40 mm 80 mm 40 mm 00 mm (a) Nominal Dimensions of a Typical Tensile Test Specimen (b) Strain Gauged Tensile Test Specimens Figure 3.: Tensile Test Specimens Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-5

91 Table 3.3: Measured Dimensions of Tensile Test Specimens No Steel grade Nominal Measured Thickness (mm) Thickness (mm) TCT BMT Width B (mm) G G G The tensile test set-up is shown in Figure 3.3. All the tests were carried out using a 300 kn capacity Shimadzu testing machine. All the operations were performed automatically after the tensile specimen was mounted in the machine. The load was monitored using the Labtech Realtime Visionpro software, while the test data were logged using the Labtech Notebook data acquisition software. The specimens were loaded as specified in AS 1391 (SA, 1991). It specified that the elastic strain rate can be at any convenient rate up to approximately one half of the force value corresponding to the expected or specified yield point, and beyond this force, the test (i.e. plastic strain) shall be carried out within a strain rate range of s -1 to s -1 and aimed at a target value of s -1. A mm strain gauge installed at the mid-height of the test specimens (see Figure 3.(b)) was used to measure the strains during the tests. The stress and strain measurements were used to derive the stress-strain relationship and thereby determine the modulus of elasticity (E). However, the strain gauges alone were not sufficient to capture the entire range of elongation of the test specimens. Therefore an extensometer was used to obtain the stress-strain curve until the specimen failed. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-6

92 Specimen mounting handle Specimen Top jig Grip jig 50 mm extensometer Strain gauge Bottom jig Figure 3.3: Tensile Test Set-up 3..3 Test Procedure The top and bottom jigs of Shimudzu test machine were aligned with its vertical axis. The bottom jig was further adjusted to ensure that the grips of jigs were oriented in the same direction. One end of the strain gauged tensile specimen was installed inside the top grip ensuring that the vertical axis of the specimen and the machine coincided. The top jig was then moved down carefully to install the bottom end of the specimen in the bottom grips without twisting or bending. Rogers and Hancock (1996) stated that Yates (1993), and Maladakis and Ayoub (1994) experienced problems with specimen twisting and bending while the grips were tightened, i.e. the top end of the specimen rotated with respect to the bottom end. It was necessary to centre the test specimens in the grips and plumb the coupon with respect to vertical axis using a small levelling instrument. This eliminated the possibility of load eccentricity and flexure of the test specimen during testing. All of these procedures were required to ensure that the applied loading was concentric during testing. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-7

93 A 50 mm extensometer was attached to the central portion of the constant gauge length after the tensile specimen was installed and aligned with the vertical axis of testing machine (see Figure 3.3). The tests were undertaken using a cross-head speed of about mms -1 that gave a target strain rate of about s -1. The applied tension load and extensometer and strain gauge readings were recorded through a data acquisition system attached to a personal computer, and were used to plot the stress-strain graphs and hence calculate the basic material properties for each test specimen as described in the following section Tensile Test Results and Discussion The stress versus strain graphs which describe the general behaviour of eight selected tensile coupon test specimens from the initial elastic portion of the stress-strain curve to failure are presented in Figures 3.4 (a) and (b) for the steel grades of G300, and G500 and G550, respectively. Appendix 3B presents the stress versus strain graphs for all other tensile tests. All the strain values were calculated using the displacement readings obtained by the extensometer, divided by the original gauge length of 50 mm. The stress values were calculated using the tensile load data output divided by the initial cross-sectional area based on BMT Stress (MPa) G mm G mm G mm G mm % Strain (a) Stress-Strain Curves for G300 Steels Figure 3.4: Typical Tensile Stress versus Strain Curves for Different Steel Grades and Thicknesses Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-8

94 Stress (MPa) G550_0.55 mm G550_0.75 mm G550_0.95 mm G500_1.15 mm % Strain (b) Stress-Strain Curves for G500 and G550 Steels Figure 3.4: Typical Tensile Stress versus Strain Curves for Different Steel Grades and Thicknesses Table 3.4: Tensile Test Results Test No Grade Nominal f y (MPa) f u (MPa) E (GPa) Thickness (mm) Msd. Ave. Msd. Ave. Msd. Ave G G G Note: Msd. Measured Ave Average f u / f y Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-9

95 Stress (Mpa) (MPa) Slope = E =15GPa Tensile strength fu=380 MPa Fracture point Lower yield point, fy = 30 MPa, e = % Strain (a) 1. mm G300 steel Slope = E = 17 GPa Tensile strength, fu =639 MPa Stress Stress (MPa) (Mpa) % offset yield, fy = 610 MPa Fracture point % Strain (b) 0.95 mm G550 steel Figure 3.5: Illustration of Basic Material Properties Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-10

96 All G550 sheet steels tested during this tensile test program yielded gradually with minimum strain hardening (see Figure 3.4 (b)) whereas G300 sheet steels displayed a sharp yield point, followed by a yield elongation plateau and then a strain hardening region (see Figure 3.4 (a)). The material properties, i.e. yield stress (f y ), ultimate tensile strength (f u ), and Young s modulus (E), for all types of sheet steels used in this tensile test program were obtained based on AS 1391 (SA, 1991) recommendations. Figures 3.5 (a) and (b) illustrate the basic material properties for grades G300 and G500/G550 steels, respectively. As the yielding was gradual for G500/G550 sheet steels, their yield stresses were calculated using the 0.% proof stress method, whereas the yield stresses of G300 steels were directly read from the graphs at the sharp yield points. The yield stress and ultimate tensile strength values calculated using the base metal thickness (BMT) for all the steel types were significantly above the minimum specified values except for 1.9 mm G300 steel (see Table 3.4). The presence of higher yield stresses has been previously documented (Rogers and Hancock, 1996) and is a result of the sheet steel forming process. The stress-strain curves were linear only for small strains. The Young s modulus of elasticity (E) was calculated based on the average slope of the stress-strain curve over the initial elastic region. An important parameter in the ductility requirements for plastic behaviour of a material is the ratio f u /f y, which was also calculated for each test specimen and is given in Table 3.4. The G500 and G550 steels exhibit a consistent ultimate strength to yield stress ratio (f u /f y ) closer to 1.0 (1.01 to 1.06). This clearly indicates the lack of strain hardening in these high strength steels. During the tensile tests it was observed that the G300 steel had greater ductility, whereas the G500 and G550 steels demonstrated reduced ductility. Failure was also sudden in the latter. Figures 3.6 (a) and (b) show the typical G300 and G500/550 steel specimens after failure. A more ductile fracture with cross-section necking can be seen in Figure 3.6 (a) for G300 steel, whereas Figure 3.6 (b) demonstrates sudden fracture behaviour for G500 and G550 steels. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-11

97 Fracture point Fracture point (a) G300 Steel (b) G500 and G550 Steels Figure 3.6: Tensile Specimens after Failure The tensile test results are summarised in Table 3.4. The comparison of these tensile test results with the results provided by the steel manufacturers (see Appendix 3A) shows a good agreement. The average yield stress values presented in Table 3.4 are used in the calculation of section capacities while the Young s modulus of elasticity values were used in the stiffness calculation for member capacities of RHFBs using the current design rules given in the design standards AS/NZS 4600 and AS Clause of AS/NZS 4600 (SA, 005) recommends that the structural steel shall comply with one of the following standards: AS 1163 (SA, 1991), AS 1397 (SA, 001), AS/NZS 1594 (SA, 00), AS/NZS 1595 (SA, 1998) and AS/NZS 3678 (SA, 1996), as appropriate. Previous investigations (CASE, 00) at the University of Sydney on certain crosssections have shown that they do not comply with the above standards due to the manufacturing process used. For those situations where Clause is not satisfied, AS/NZS 4600 allows the use of other steels, the properties and suitability of which are in accordance with Clause of AS/NZS 4600 (SA, 005). According to Clause (b) of AS/NZS 4600 (SA, 005), G550 steels with thickness less than 0.9 mm, the yield stress (f y ) and the tensile strength (f u ) used in design are taken as 90% of the corresponding specified values or 495 MPa, whichever is the lesser, and for steel less than 0.6 mm in thickness, the tensile yield stress (f y ) and the tensile strength (f u ) used in design are taken as 75% of the corresponding specified values or 410 MPa whichever is the lesser. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-1

98 3.3 Section Capacity Tests Test Specimens Since the flexural behaviour of RHFB sections has not been investigated yet, it is important that the key parameters are chosen carefully in the design of this test program. A number of sections with different key parameters (i.e. section geometry, material thickness and yield stress) were selected in the test program. A schematic diagram of an RHFB cross-section is shown in Figure 3.7. There was a total of section capacity tests in this investigation. b f t f h f Screw spacing along the beam s t w h w h l h f Figure 3.7: RHFB Cross-section All the test specimens were 1130 mm long and were fabricated by assembling two separately formed rectangular hollow flanges to a single web plate using Hi Tek self drilling screws of size mm at 50 mm and 100 mm spacings. The rectangular hollow flanges of sizes 50 mm 5 mm with 15 mm lips were formed by using the press-braking method. Three steel grades of G300, G500 and G550 were used with nominal thicknesses of 0.55, 0.80, 1.0 and 1.90, 1.15, and 0.55, 0.75 and 0.95 mm, respectively. The test specimens were labelled so that the specimens variable parameters: flange and web thicknesses, web height, specimen length, steel grade and screw spacing could be identified from the label, as illustrated in Figure 3.8 for a typical RHFB specimen. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-13

99 Rectangular Hollow Flange Beam Screw spacing = 50 mm RHFB 10t f 055t w 150h w G300 50s Flange Thickness = 1.0 mm Web Thickness = 0.55 mm Steel grade = G300 Web Depth = 150 mm Figure 3.8: Specimen Labelling The specimen label consists of the beam type (i.e. RHFB) followed by a series of numbers and scripts. For example, the label RHFB-10t f -055t w -150h w -G300-50s defines a rectangular hollow flange beam (RHFB) specimen made of 1. mm flange thickness (10t f ), 0.55 mm web thickness (055t w ), 150 mm web height (150h w ), using grade G300 steel with screws at 50 mm spacing. Flange width and height, lip height and specimen length were not included in the labelling since they were the same (50, 5, 15 and 1130 mm) for all the test specimens. With a constant overhang of 30 mm on each end, the specimen length of 1130 mm gave a span of 1070 mm in all the tests. Geometric imperfections and overall section dimensions were measured for each test specimen, from which centreline dimensions of specimen cross-section were calculated. Table 3.5 presents the values of measured imperfections and calculated centreline dimensions from the measured values for section moment capacity test specimens. A typical test specimen is shown in Figure 3.9. Electrical strain gauges and displacement transducers were installed on the specimens at appropriate locations before testing. (a) Overall View (b) Close-up View of Cross-section Figure 3.9: Typical Specimen Used in Section Capacity Tests Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-14

100 50 mm screw spacing 100 mm screw spacing (c) Close-up View of Screw Fasteners Figure 3.9: Typical Specimen Used in Section Capacity Tests Table 3.5: Measured Cross-section Dimensions and Imperfections Beam RHFB Designation b f (mm) h f (mm) Flange h l (mm) Measured dimensions t f (mm) h w (mm) Web t w (mm) Maximum global Imperfection δ (mm) 1 RHFB-10t f -055t w -100h w -G300-50s RHFB-10t f -055t w -100h w -G s RHFB-080t f -080t w -150h w -G300-50s RHFB-080t f -080t w -150h w -G s RHFB-10t f -10t w -150h w -G300-50s RHFB-10t f -10t w -150h w -G s RHFB-080t f -190t w -150h w -G300-50s RHFB-080t f -190t w -150h w -G s RHFB-10t f -055t w -150h w -G300-50s RHFB-10t f -055t w -150h w -G s RHFB-075t f -075t w -100h w -G550-50s RHFB-075t f -075t w -100h w -G s RHFB-075t f -075t w -150h w -G550-50s RHFB-075t f -075t w -150h w -G s RHFB-115t f -115t w -150h w -G500-50s RHFB-115t f -115t w -150h w -G s RHFB-075t f -115t w -150h w -G550-50s RHFB-075t f -115t w -150h w -G s RHFB-115t f -075t w -150h w -G500-50s RHFB-115t f -075t w -150h w -G s RHFB-095t f -055t w -150h w -G550-50s RHFB-095t f -055t w -150h w -G s Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-15

101 3.3. Section Properties Section properties of test specimens based on the measured dimensions of RHFBs were calculated using an Excel spreadsheet program (see Appendix 3C). This spreadsheet was first used to calculate the basic section properties such as I xx, I yy and Z x and the results were compared with corresponding results from the well known buckling analysis program Thin-wall (Papangelis, 1994). A close agreement of results demonstrated the accuracy of the spreadsheet. Thin-wall was used to obtain other section properties, I w and J, which were not calculated by this spreadsheet. The section property results are presented in Table 3.6. The same spreadsheet program was also used to obtain the effective section properties of RHFBs based on the effective width concept in accordance with the steel design standards AS 4100 and AS/NZS 4600 (see Appendix 3C). The effective section properties were used to calculate the section capacity of RHFBs using the design rules specified in AS 4100 and AS/NZS 4600, and the results are discussed in Section Geometric Imperfections The magnitudes of member imperfections were measured for each test specimen using a Wild T05 theodolite and a new equipment shown in Figure 3.10, which was specially designed and fabricated to measure geometric imperfections. The equipment comprises a level table with guided rails with an accuracy of 0.01 mm, a laser sensor, and a travelator. The laser sensor was attached to the travelator which could move in-plane and normal to the plane. The specimen was positioned and levelled using the adjustable screws of the table and clamped. The laser sensor was then moved along the specimen while taking the readings at every 100 mm intervals. The readings were taken along three lines in the longitudinal direction of the specimen in order to determine the maximum initial crookedness along the web and flanges for each specimen. The maximum initial crookedness values (δ) for each test specimen are given in Table 3.5. Although the imperfection magnitudes were measured along the entire length of the test specimens, only the central L/3 region was considered to obtain the maximum imperfection, δ. This is because the central region was critical and failure occurred in this region. Figure 3.11 illustrates the variation of imperfection magnitudes for a typical test specimen. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-16

102 Beam Specimen Designation Table 3.6: Section Properties Based on Measured Cross-section Dimensions M kg/m A (mm ) I xx ( 10 6 mm 4 ) Z x ( 10 4 mm 3 ) J ( 10 4 mm 4 ) I w ( 10 8 mm 6 ) 1 RHFB-10t f -055t w -100h w -G300-50s RHFB-10t f -055t w -100h w -G s RHFB-080t f -080t w -150h w -G300-50s RHFB-080t f -080t w -150h w -G s RHFB-10t f -10t w -150h w -G300-50s RHFB-10t f -10t w -150h w -G s RHFB-080t f -190t w -150h w -G300-50s RHFB-080t f -190t w -150h w -G s RHFB-10t f -055t w -150h w -G300-50s RHFB-10t f -055t w -150h w -G s RHFB-075t f -075t w -100h w -G550-50s RHFB-075t f -075t w -100h w -G s RHFB-075t f -075t w -150h w -G550-50s RHFB-075t f -075t w -150h w -G s RHFB-115t f -115t w -150h w -G500-50s RHFB-115t f -115t w -150h w -G s RHFB-075t f -115t w -150h w -G550-50s RHFB-075t f -115t w -150h w -G s RHFB-115t f -075t w -150h w -G500-50s RHFB-115t f -075t w -150h w -G s RHFB-095t f -055t w -150h w -G550-50s RHFB-095t f -055t w -150h w -G s Note: M Mass per metre length A Gross cross-section area I xx Second moment of area Z x Section modulus J Torsion constant I w Warping constant Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-17

103 Specimen Travelator Laser sensor and data logger Foot screw level Figure 3.10: Imperfection Measuring Device Imperfection (mm) Maximum, δ =1. mm Distance (mm) 'Top-Flange' Web-Central Bottom-Flange Figure 3.11: Measured Imperfection along the Length of a Typical Test Specimen Test Set-up and Instrumentation The section capacities of RHFBs were determined based on the bending tests of short and fully laterally restrained RHFB sections. The tests were undertaken using a 300 kn capacity Tinious Olsen testing machine in the Structures Laboratory at the Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-18

104 Queensland University of Technology. Relatively short and fully laterally restrained RHFB specimens were tested to failure using a four point bending test set-up. A schematic view, load application and overall view of the test set-up are shown in Figures 3.1 (a) (c), respectively. Loading Component Spreader beam Load transferring device Spherical head Support box frame Rollers Timber planks Test Specimen Transducer Roller bearings (a) Schematic View Steel plate Compression flange h w / Loading arm h w / Tension flange (b) Load Application Figure 3.1: Test Set-up for Section Moment Capacity Tests Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-19

105 Loading Spreader beam Spherical head Support box frame Steel rollers C clamps Loading arm Timber planks (c) Overall View Figure 3.1: Test Set-up for Section Moment Capacity Tests The test specimens were supported on half rounds placed on a ball bearing as shown in Figure 3.1 (c). The bottom surfaces of the half rounds and alloy balls were machine ground and polished to a high degree of smoothness, and smooth ball bearing surfaces were lubricated to further facilitate the sliding of the half rounds on the ball bearing when the beam deflected under load. The ends of the beam were free to rotate upon the half rounds. Thus it was considered that simply supported conditions were simulated accurately at the end supports. The simply supported beam specimens were tested by loading them symmetrically at two points on the span, through a spreader beam that was loaded centrally by the ram of the testing machine (see Figure 3.1 (c)). This four point loading arrangement provided a uniform bending moment and zero shear force within the central region of the test beam. The tests were conducted with loading points at a distance of span/3 from the supports as shown in Figure 3.13 (span = 1070 mm). Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-0

106 The loads were applied to the neutral axis of the test beam through the steel rollers and loading arms (see Figure 3.1 (b)) attached to the beam s web using three M1 bolts at 30 mm spacings. Timber plates were fixed on both sides of the beam web between the supports and loading points using a set of C-clamps, whereas two mm steel plates were fixed on both sides of the beam web at the loading and support points (see Figure 3.1 (b)) to avoid premature failure due to web bearing, crippling and shear. During the tests, the bending strains were measured using two strain gauges located on the top and bottom flanges of the specimen at midspan whereas the vertical deflections were measured using three linear displacement transducers located at midspan and loading points. The EDCAR data acquisition system was used to record all the strain and deflection data until the specimen was loaded to failure. The crosshead of the testing machine was moved at a constant rate of 1.0 mm/min until the specimen failed Test Results and Discussion Since the end spans of the test specimen were reinforced using web stiffeners, the sections in the mid-span region failed during bending tests. The verticality of the applied loads was maintained throughout the test and therefore the applied uniform moment (M) to the test beam between the loading points was calculated using; M = P L la (3.1) where P is the applied load and L la is the initial lever arm length as shown in Figure L la = L/3 P P L la = L/3 R=P R=P Span, L = 1070 mm Figure 3.13: Deformed Shape of Test Specimen Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-1

107 The applied loads (P) at the loading points of the test specimen were equal to half of the load reading from the Tinious Olsen testing machine. The applied uniform moment (M) between the loading points was therefore calculated from Equation 3.1 using half of the load reading (i.e. P) and the initial lever arm length (i.e. L la = 357 mm). The applied moment was also calculated using the top and bottom flange strain gauge readings at the mid-span of test beam. The close agreement between the two moments verified the accuracy of load readings and the applied uniform moment values. The following sections will present and discuss the details of section moment capacity test results Moment versus In-plane Vertical Deflection Curves This section presents the experimental curves of applied moment versus in-plane vertical deflection at the mid-span cross section of the test beam for selected tests. The moment versus deflection graphs for other tests are presented in Appendix 3D Moment (knm) RHFB-10t f -10t w -150h w -G300-50s RHFB-080t f -080t w -150h w -G300-50s Vertical Deflection (mm) (a) Moment versus Vertical Deflection (G300 Steel) Figure 3.14: Moment versus Vertical Deflection Curves (t f = t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-

108 Moment (knm) (knm) RHFB-115t f -115t w -150h w -G500-50s RHFB-075t f -075t w -150h w -G550-50s Vertical Deflection (mm) (b) Moment versus Vertical Deflection (G500 and G550 Steels) Figure 3.14: Moment versus Vertical Deflection Curves (t f = t w ) 9.0 Moment (knm) (knm) RHFB-10t f -055t w -150h w -G300-50s RHFB-10t f -055t w -100h w -G300-50s Vertical Deflection (mm) (a) Moment versus Vertical Deflection (G300 Steel) Figure 3.15: Moment versus Vertical Deflection Curves (t f > t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-3

109 14 1 Moment (knm) (knm) RHFB-115t f -075t w -150h w -G500-50s RHFB-095t f -055t w -150h w -G550-50s Vertical Deflection (mm) (b) Moment versus Vertical Deflection (G500 and G550 Steels) Figure 3.15: Moment versus Vertical Deflection Curves (t f > t w ) 1 10 Moment (knm) (knm) RHFB-080t f -190t w -150h w -G300-50s Vertical Deflection (mm) (a) Moment versus Vertical Deflection (G300 Steel) Figure 3.16: Moment versus Vertical Deflection Curves (t f < t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-4

110 1 10 Moment (knm) (knm) RHFB-075t f -115t w -150h w -G550-50s Vertical Deflection (mm) (b) Moment versus Vertical Deflection (G550 Steel) Figure 3.16: Moment versus Vertical Deflection Curves (t f < t w ) Figures 3.14 to 3.16 show some of the moment versus vertical deflection graphs from the section capacity tests. They demonstrate a linear response during the initial stage of the tests irrespective of the steel grade and the thickness. In theory, nonlinearity commences with the commencement of yielding, i.e. when the bending moment reaches the first yield moment. In practice, yielding may be initiated before the ideal first yield moment because of the residual stresses present in the sections due to the cold-forming process used during the specimen fabrication (Hasan and Hancock, 1988). However, the extent to which the residual stresses affected the behaviour of RHFB sections need to be further investigated. Nonlinearity could also commence early due to initial geometric imperfections in the section. Available results show that the first yield moment of the RHFBs was in the range of and of theoretical first yield moment M y for G300 and G500/G550 steels, respectively. Following the departure from elastic linearity, the bending moment continued to increase upon further application of load. This is because of strain hardening and inelastic reserve capacity of the section. Essentially, similar moment-deflection behaviour was observed for each of the 10 specimens made of G300 steels and they Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-5

111 exhibited a plateau associated with increasing deflection in the ultimate moment region whereas the moment-deflection behaviour of G500/G550 steel specimens appears to display an instant failure for each of the 1 specimens associated with increasing deflection (see Figures 3.14 to 3.16). Hence it is evident from the test results that the material behaviour has a significant influence on the flexural behaviour of RHFB sections for short span beams Moment versus Longitudinal Strain Curves In each test the longitudinal strains were measured in the compression and tension flanges of the test beam at mid-span to verify the measured load readings from the Tinious Olsen testing machine. The applied uniform moment (M) was calculated based on the measured longitudinal strains for the elastic region. The longitudinal stress in the extreme fibres f c was calculated first. f c = Eε (3.) m where E is the measured elastic modulus of steel and ε m is the average measured longitudinal strains in the extreme fibres at mid-span. Applied uniform moment: M = f Z (3.3) c x where Z x is Section modulus as given in Table 3.6. Based on the measured longitudinal strain readings and using the knowledge of steel yield stress as given in Table 3.4, the first yield point was determined, i.e. the point when the measured strain reaches the yield strain (yield stress/e). The first yield moment was then calculated using Equation 3.3 where f c was taken as the measured yield stress f y. Figure 3.17 shows the moment versus longitudinal strain curves for two G300 steel sections using moments calculated based on the load readings and strain gauge readings. The moment versus longitudinal strain curves for other sections are presented in Appendix 3D. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-6

112 15.0 First yield moment (SB) 1.5 First yield moment (SB) Moment (knm) Moment corresponding to yield strain (LB) Moment corresponding to yield strain (LB) Actual first yield moment (LB) Strain (microstrain) S1-LB-T S1-LB-C S-LB-T S-LB-C S1-SB_T S1-SB-C S-SB-T S-SB-C S1 RHFB-10t f -10t w -150h w -G300-50s S RHFB-080t f -080t w -150h w -G300-50s LB Load based moment SB Strain based moment T Tension C - Compression Figure 3.17: Typical Moment versus Longitudinal Strain Graphs The curves presented in Figure 3.17 were plotted using calculated uniform moments from the load and strain gauge measurements. As shown in these figures, the moment (based on load and strain gauge measurements) versus longitudinal strain (measured) curves agree closely in the elastic region, and thus verify the accuracy of the load measurements from the testing machine. Figure 3.17 further illustrates the differences between the first yield moments calculated from the load and strain gauge measurements for the RHFB-10t f -10t w -150h w -G300-50s section. According to Figure 3.17, actual first yield moments based on the load measurements were less than that calculated from the strain gauge measurements. This difference between the calculated first yield moments from the load and strain gauge measurements may be due to the residual stresses present in the test beam that would have caused premature yielding and hence nonlinearity began earlier as shown in Figure Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-7

113 Failure Modes of RHFBs in Section Moment Capacity Tests All the specimens included in this test program were classified as slender sections according to AS 4100 specifications, whereas they included at least one slender element according to AS/NZS 4600 specifications. Hence most of the tested specimens were expected to experience either flange or web local buckling before they reached the first yield moment. When the top flange plate buckled, sympathetic rotation at the flange-web corner led to deformation of the web. The local buckling formation in the flange and web was observed closer to the midspan than the loading points of the specimens. Some of the tested RHFB specimens and a typical locally buckled specimen are shown in Figures 3.18 (a) and (b), respectively. There was no lateral deformation of test specimens during the tests and no specimen was observed to fail suddenly. Local buckling and yielding of flange Tested specimens (a) Overall view Local buckling and yielding of flange (b) Close-up view of failure region Figure 3.18: Typical Failures of Test Specimens Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-8

114 The web and flanges were connected intermittently using screws at equal spacings of 50 mm and 100 mm as illustrated in Figure 3.9, and therefore there was a discontinuity in the web and flange connection between the screws. When the top flange plate buckled, a gap opened between the unconnected web and flange lips at the failure region due to sympathetic rotation at the flange and web corner. Figures 3.19 (a) and (b) show this occurrence for the screw spacings of 50 mm and 100 mm, respectively. As expected, the comparison of Figures 3.19 (a) and (b) indicates that the web distortion at the failure region is severe for larger screw spacing. Failure section 50 mm screw spacing Gap openings (a) 50 mm Screw Spacing Failure section Gap openings 100 mm screw spacing (b) 100 mm Screw Spacing Figure 3.19: Opening of Web and Flange Lips between Screw Fasteners Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-9

115 14 1 B D Moment (knm) A C RHFB-10t f -10t w -150h w -G300-50s E 0 O Vertical Deflection (mm) Figure 3.0: Graphical Illustration of Failure Behaviour of a RHFB Figure 3.0 gives a graphical illustration of the failure behaviour of a selected RHFB (i.e. RHFB-10t f -10t w -150h w -G300-50s) during the section moment capacity tests. Figure 3.0 indicates that the beam behaved linearly until the applied moment reached point A, and then it behaved nonlinearly until point B. The comparison of Figures 3.17 and 3.0 confirmed the premature nonlinearity at about 7 knm, possibly due to the presence of residual stresses. Although there was not any distinctive sign of local buckling in the beam until the applied moment reached point B, local buckling might have occured earlier in the compression flange as in the case of yielding and which might have eventually led to a sudden change in the top plate of compression flange at point B (see Figure 3.18 (a)). Sympathetic rotation of flange web corner occurred after top flange plate buckled locally at point B, which led to expanding the gap between the unconnected web and flange lips within the region CD. This sudden failure in the compression flange and the opening of gap between the unconnected web and flange lips resulted in the moment drop from point B to C as shown in Figure 3.0. However further moment gain was observed until point D after the top plate of compression flange buckled and yielded and the moment dropped to point E. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-30

116 Comparison of Section Moment Capacities with Predictions from the Current Design Rules AS 4100 (SA, 1998) Design Method The section moment capacity (M s ) is defined in Clause 5..1 of AS 4100 (SA, 1998) as follows: M = f Z (3-4) s y e where the effective section modulus (Z e ) allows for the effects of local buckling and the calculation of Z e is dependent on the section classification recommended in AS Table 3.7 presents the details of section classification for test specimens used in this test program based on both AS 4100 and AS/NZS Table 3.7: Section Classification Beam Beam Designation Section Classification AS 4100 AS/NZS RHFB-10t f -055t w -100h w -G300 Slender (web) Slender (flange, web) RHFB-080t f -080t w -150h w -G300 Slender (flange) Slender (flange) 3 RHFB-10t f -10t w -150h w -G300 Slender (flange) Slender (flange) 4 RHFB-080t f -190t w -150h w -G300 Slender (flange) Slender (flange) 5 RHFB-10t f -055t w -150h w -G300 Slender (web) Slender (flange, web) 6 RHFB-075t f -075t w -100h w -G550 Slender (flange) Slender (flange) 7 RHFB-075t f -075t w -150h w -G550 Slender (flange) Slender (flange, web) 8 RHFB-115t f -115t w -150h w -G500 Slender (flange) Slender (flange) 9 RHFB-075t f -115t w -150h w -G550 Slender (flange) Slender (flange) 10 RHFB-115t f -075t w -150h w -G500 Slender (web) Slender (flange, web) 11 RHFB-095t f -055t w -150h w -G550 Slender (web) Slender (flange, web) The effective section modulus is defined in Clauses 5..3 to 5..5 of AS 4100 (SA, 1998) as follows: λ : Z e = S < 1. 5Z s λ sp λ sy λ s λsp < λs λsy : Z e = Z + ( S Z ) λ sy λ sp λ sy λ s > λ sy : Z e = Z λ s (3-5) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-31

117 The section slenderness (λ s ) is taken as the value of the plate element slenderness (λ e ) for the element of the cross-section, which has the greatest value of (λ e /λ ey ). The plate element slenderness (λ e ) is defined in Clause 5.. (SA, 1998) as a function of the element clear width (b), thickness (t), and yield stress (f y ): b σ y λ e = (3-6) t 50 The section plasticity and yield slenderness limits (λ sp, λ sy ) are taken as the values of the element slenderness limits (λ ep, λ ey ) given in Table 5. of AS 4100 (SA, 1998) for the element of the cross-section which has the greatest value of λ e /λ ey. The coldformed (CF) element slenderness limits were considered to be the most appropriate for RHFB sections (see Appendix 3C). These slenderness limits were established from lower bound fits to the experimental local buckling resistances of plate elements in uniform compression and flexure. The section moment capacity values based on AS 4100 design rules are given in Table 3.8. Appendix 3C shows the example calculations of section moment capacity of a RHFB based on AS 4100 (SA, 1998) design rules. Measured yield stresses were used in all the calculations for G300, G500 and G550 steels. AS/NZS 4600 (SA, 1996) Design Method The section moment capacity (M s ) is defined in Clause 3.3. of AS/NZS 4600 (SA, 1996) in a similar manner to AS 4100 (see Equation 3-4). However, unlike AS 4100, the effective section modulus (Z e ) is based on the initiation of yielding in the extreme compression fibre and therefore does not allow for the inelastic reserve capacity of the section. The effects of local buckling in the slender elements in compression are accounted for by using effective widths (b e ) in the calculation of their effective section modulus (see Equation 3-7). Unlike AS 4100, the plate element slenderness (λ) is a function of the applied stress (f*), as shown in Equation 3-8. This accounts for the severity of local buckling effects with increasing member slenderness. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-3

118 b e = 1 0. λ λ b b (3-7) * 1.05 b f λ = k t (3-8) E where k is the local buckling coefficient, and k = 4 for uniformly compressed stiffened elements, whereas for the stiffened elements with a stress gradient, k is determined from the following equation: 3 k = 4 + (1 ψ ) + (1 ψ ) (3-9) ψ = f f * * 1 where f 1 * is compression (+) and f * can be either tension (-) or compression. The section capacities of all the RHFB sections were calculated using the AS/NZS 4600 method described above, with the local buckling coefficient (k) equals to 4 for the stiffened elements with uniform compression and using Equation 3-9 for the elements with stress gradient. Clause of AS/NZS 4600 (SA, 005) recommends the use of a reduced yield stress for G550 steels to allow for the reduced ductility in the steels: 0.90f y for 0.6 mm thickness <0.9 mm and 0.75f y for thickness < 0.6 mm or 495 MPa, whichever is lesser. Therefore the measured (actual) and modified (reduced) yield stresses were used in the calculations of plate element slenderness using Equation 3-8, and the section moment capacities of RHFBs using Equation 3-4 for G300 and G550 steels, respectively. Measured cross-section dimensions given in Table 3.5 were used in these calculations. The section moment capacity results based on AS/NZS 4600 (SA, 1996) are given in Table 3.8. Appendix 3C shows the example calculations of section moment capacity of a RHFB based on AS/NZS 4600 (SA, 1996) design rules. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-33

119 In the design capacity calculations, the plate elements of RHFBs were assumed to be either stiffened (both longitudinal edges supported) or unstiffened (one longitudinal edge supported) elements and accordingly the corresponding λ ey and k values were used. However, some of the plate elements (mainly the web and bottom flange elements) were held together through intermittent screw fastening. Hence the above assumption may not be accurate and could have led to slight overestimation of the section moment capacities. Finite element analyses reported in Chapter 6 investigate this effect in detail. Comparisons The maximum bending moment (M u ) achieved by each test specimen is listed in Table 3.8 and is compared with the predictions based on the steel structures standard, AS 4100 (SA, 1998) and the cold-formed steel structures standard, AS/NZS 4600 (SA, 1996). The comparison of predicted moment capacities based on AS 4100 and AS/NZS 4600 with experimental moment capacities showed that both design methods are conservative in general. However, AS/NZS 4600 section capacity method estimates comparatively more accurately the reduction in section moment capacity due to local buckling effects in slender RHFB sections than the AS 4100 method. AS/NZS 4600 overestimates the failure moment of G300 and G500/G550 steel specimens by 6% (mean = 0.94) with a COV of 0.16, and 5% (mean = 0.95) with a COV of 0.4, respectively, while AS 4100 predictions were 9% (mean =0.91) and 36% (mean = 0.64) higher than the test section moment capacities with COVs of 0.16 and 0.4, respectively (see M u /M s ratio in Table 3.8). From this comparison, it is apparent that both AS 4100 and AS/NZS 4600 overestimate the section moment capacities of RHFBs. However, AS/NZS 4600 design rules may be used for both G300 and G550 steel RHFB sections to predict their section moment capacities as the overall M u /M s ratios are about In contrast AS 4100 design rules can only be used for G300 steel RHFB since they overestimate the section moment capacities by about 36% for G550 steel RHFB. Appendix 3C shows example calculations of section moment capacities based on both AS 4100 and AS/NZS 4600 design rules. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-34

120 Table 3.8: Comparison of Section Moment Capacities of RHFBs Beam Section Designation Z (or Z AS/NZS 4600 AS 4100 Ratio M u /M Exp. M f ) s u ( 10 (knm) M ) y mm 3 (knm) Z e M s Z e M s AS/NZS AS ( 10 4 mm 3 ) (knm) ( 10 4 mm 3 ) (knm) RHFB-10t f -055t w -100h w -G300-50s RHFB-10t f -055t w -100h w -G s RHFB-080t f -080t w -150h w -G300-50s RHFB-080t f -080t w -150h w -G s RHFB-10t f -10t w -150h w -G300-50s RHFB-10t f -10t w -150h w -G s RHFB-080t f -190t w -150h w -G300-50s RHFB-080t f -190t w -150h w -G s RHFB-10t f -055t w -150h w -G300-50s RHFB-10t f -055t w -150h w -G s G300 Steel Mean COV RHFB-075t f -075t w -100h w -G550-50s RHFB-075t f -075t w -100h w -G s RHFB-075t f -075t w -150h w -G550-50s RHFB-075t f -075t w -150h w -G s RHFB-115t f -115t w -150h w -G500-50s RHFB-115t f -115t w -150h w -G s RHFB-075t f -115t w -150h w -G550-50s RHFB-075t f -115t w -150h w -G s RHFB-115t f -075t w -150h w -G500-50s RHFB-115t f -075t w -150h w -G s RHFB-095t f -055t w -150h w -G550-50s RHFB-095t f -055t w -150h w -G s G500/G550 Steel Mean COV Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-35

121 The lower experimental failure moments compared with predicted design capacities could be attributed to several factors including residual stresses and initial geometric imperfections that were present in the test specimens. The test specimens were fabricated manually (see Chapter 4) and therefore only limited control on the shape and size of the specimens could be achieved during the forming process. Thus, the specimens had considerable amount of irregularities in the shape (i.e. geometric imperfections, see Figure 3.11) and the size, which could decrease the section moment capacities of RHFBs. The curvature of flange top plate was not considered in the moment capacity calculations, ie. Flat plate assumption. The corner radii were also assumed to be negligible. All these assumptions could also have lead to the overestimation of moment capacities. The discontinuity between the web and lower flange lip elements due to intermittent screw fastening could also reduce the section moment capacities of RHFBs. As illustrated in Figures 3.19 (a) and (b), the gap between flange lips and web opened up between screw fasteners when the flange buckled locally. Depending on the b/t ratio of web and flange lip elements, there is a tendency of local buckling in the web and flange lips between screw fastener locations. However, this effect could not be accounted for in either AS 4100 or AS/NZS 4600 when the element slenderness was calculated. Instead of intermittent screw connections between the web and flange lips, a continuous connection was assumed when the local buckling coefficient k was calculated to determine the section moment capacities using AS/NZS 4600 and AS 4100 design rules. This could lead to higher predicted moment capacities than the tested failure moments as listed in Table 3.8. Higher predictions from AS 4100 are partly due to the use of measured yield stress, and not the reduced yield stress as in AS/NZS 4600 (compare the M u /M s ratios 0.64 with 0.95 in Table 3.8). This observation justifies the use of reduced yield stress in AS/NZS 4600 to allow for the reduced ductility in such steels. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-36

122 3.4 Summary This chapter has presented the details of an experimental investigation of the material properties of steels and the section moment capacities of the new cold-formed rectangular hollow flange beam (RHFB) sections and the results. Four point bending tests were conducted for a total of RHFB sections made from G300 steel (10) and G500/G550 steels (1). Test results are presented in the form of bending moment versus vertical deflection and longitudinal strains for each section. The maximum bending moment attained by each test specimen was listed and compared with design capacity predictions from the current steel design standards based on the measured cross-section dimensions and material properties. The test results indicated that the predicted section moment capacities from AS/NZS 4600 and AS 4100 design rules are unconservative, and therefore they may not be safe to use in the section capacity calculations of RHFB. However, there is significant potential for the use of the very efficient RHFBs if a well controlled manufacturing method is used and a more representative design approach is adopted by modifying the current design rules specified in AS/NZS 4600 and AS Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3-37

123 CHAPTER 4 Experimental Studies on Flexural Behaviour of RHFB Members 4.1 General This research was aimed at investigating the flexural member behaviour of rectangular hollow flange beams (RHFB) and to verify the adequacy of the existing design rules based on the behaviour of RHFBs. For this purpose 30 full scale lateral buckling tests and section moment capacity tests were conducted using typical RHFBs to failure. This chapter presents the details of the full scale lateral buckling tests and the results relating to the flexural member behaviour of RHFBs. 4. Section Geometry and Specimen Sizes As discussed in Chapter, cold-formed steel beams comprising rectangular hollow flanges and a slender web (see Figure 4.1) are susceptible to various buckling modes under flexural action. They are: 1. Local buckling of flanges. Local buckling of web 3. Lateral distortional buckling 4. Lateral torsional buckling 5. Lateral distortional buckling and local buckling of flanges 6. Lateral distortional buckling and local buckling of web 7. Material yielding Essentially, the above failure modes are governed by the material and geometric properties of RHFB, and therefore it was important to choose the relevant key parameters carefully in order to investigate and fully understand all the possible failure modes of RHFB using a series of full scale lateral buckling tests. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-1

124 The basic parameters for a typical RHFB section are: flange width (b f ), flange height (h f ), web height (h w ), flange thickness (t f ), web thickness (t w ), flange lip height (h l ), support span of the beam (l), steel grade (G), and screw spacing (s) (see Figure 4.1). b f t f h f Screw spacing along the beam s t w h w h l h f Figure 4.1: Cross-section of a Typical RHFB The element s width to thickness ratio (b/t) is an important parameter for coldformed steel sections under compression or bending action. Hence the upper limit of flange width, flange height and web depth were decided based on the maximum b/t ratio values recommended in AS/NZS 4600 (SA, 1996). For this purpose, the available steel thicknesses from both lower (G300) and higher (G500 and G550) grade steels were used and their details are given in Table 4.1. In AS/NZS 4600, the maximum overall flat width-to-thickness ratio for stiffened compression elements with both longitudinal edges connected to other stiffened elements is given as 500 and the maximum depth-to-thickness ratio for unreinforced webs is given as 00. Table 4.1: Selected Material Thicknesses from Three Steel Grades Steel Grade Nominal steel thicknesses (mm) G G G Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-

125 Preliminary elastic buckling analyses were conducted using a finite strip program Thin-wall to decide suitable cross-section sizes of RHFB that would fail by different buckling modes. Many RHFB sections comprising different flange sizes, material thicknesses and web heights were analysed. This computer program gives elastic buckling loads at different buckling half-wavelengths with corresponding buckling failure modes. The results showed that 50 mm 5 mm flange was the most suitable one to investigate all the possible failure modes of RHFBs using available steel thicknesses in the industry. It is also the most suitable flange from a practical application viewpoint. The elastic buckling analysis results of Thin-wall computer program for a number of RHFB sections comprising 50 mm 5 mm flanges with varying material thicknesses, web heights and span lengths are presented in Table 4. and Figures 4. (a) to (c). Table 4.: Elastic Buckling Analysis Results Steel Grade G300 G550 G500 No b f (mm) Flange Web Buckling Stress (MPa) h f (mm) t f (mm) t w h w = 100 mm h w = 150 mm (mm) LBS m 3 m LBS m 3 m f w f f f w w w f w f f f w w w f w f f f w w w f w f f f w f w Note: Section parameters are defined in Figure 4.1 LBS Local Buckling Stress In the buckling analyses, an idealized RHFB with no flange lips and full continuity between web and flange elements was assumed as shown in Figures 4. (a) to (c). Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-3

126 (a) Buckling Plot for RHFB-10t f -10t w -150h w Section (b) Buckling Plot for RHFB-10t f -055t w -150h w Section Figure 4.: Different Buckling Modes of RHFBs Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-4

127 (c) Buckling Plot for RHFB-080t f -190t w -150h w Section Figure 4.: Different Buckling Modes of RHFBs Figures 4. (a) to (c) illustrate the change of buckling failure modes at different buckling half wavelengths for three RHFB sections using the buckling plots obtained from the Thin-wall buckling analyses. The graphs represent the variation of maximum stress in the section at buckling with different half-wavelengths. According to Figure 4. (a), local flange buckling changes to local web buckling at a buckling half wavelength of 90 mm, whereas at a half-wavelength of 350 mm, local web buckling changes to lateral distortional buckling. Interactive lateral distortional and local web buckling occurs for half-wavelengths in the range of 350 mm to 1000 mm approximately. Pure lateral distortional buckling occurs when the halfwavelength exceeds 1000 mm. However, lateral torsional buckling occurs beyond about 6500 mm. Yield strength of material should be compared with the buckling stress to check whether yielding occurs before buckling. In Figure 4. (b), local web buckling occurs for buckling half wavelengths up to 400 mm, but changes to lateral distortional buckling beyond 400 mm. Interactive lateral distortional and local web buckling occurs for half-wavelengths in the range of about 400 mm to 1600 mm. Pure lateral distortional buckling occurs when the half-wavelength exceeds 1600 Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-5

128 mm. However, lateral torsional buckling occurs beyond 8500 mm. In Figure 4. (c), local flange buckling occurs for buckling half wavelengths up to 300 mm and changes to lateral distortional buckling beyond 300 mm. Interactive lateral distortional and local flange buckling occurs for half-wavelengths in the range of about 300 mm to 000 mm. Pure lateral distortional buckling occurs when the halfwavelength exceeds 000 mm. However, lateral torsional buckling occurs beyond 6000 mm. The cross-section sizes and the specimen lengths of RHFB to simulate different buckling failure modes during full scale bending tests were decided based on the results given in Table 4. and buckling plots such as Figures 4. (a) to (c). For instance, section No. 3 of Grade 300 steel with a web height, h w = 150 mm has local buckling stress of 556 (w) MPa, in which w indicates that the section experiences local buckling in the web. In this case, the local buckling stress is greater than the lateral distortional buckling stresses at m and 3 m span lengths. Therefore this particular section (No. 3 in Table 4.) is expected to fail by pure distortional buckling at span lengths of m and 3 m. Similarly, the failure modes of all other sections were decided based on the results presented in Table 4.. Thirty RHFB specimens with different section sizes, lengths and screw spacings were selected based on the elastic buckling analysis results presented in Table 4. to simulate different failure modes in full scale bending tests. Details of bending tests are given in the next section. 4.3 Test Program Table 4.3 shows the lateral buckling test program for RHFBs using G300, G500 and G550 steels. Expected failure modes of RHFB test specimens comprising 50 mm 5 mm flange size, different combinations of flange and web thicknesses and web heights are given in Table 4.3 for span lengths of m and 3 m. For example, the expected failure mode of section RHFB-10t f -10t w -150h w is pure lateral distortional buckling (LD) at 3 m span (see Figure 4. (a)). The specimens were chosen only for the sections highlighted in Table 4.3. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-6

129 Table 4.3: Lateral Buckling Test Program Steel Grade Flange Thickness t f Web Thickness t w Test Specimens and Failure Modes Web Height h w = 100 mm Web Height h w = 150 mm Span m Span 3 m Span m Span 3 m LBF LBF LBF LBF + LD LBF (1) LBF LBF LBF Y (1) Y LD LD () G300 steel LD LD (1) LBW (1) LBW, LD () LBF LBF LBF + LD LD () LBF LBF LBF LBF, LD (1) Y Y Y Y (1) Y LD LD LD LBF LBF LBF (1) LBF + LD LBF LBF LBF LBF (1) G550 steel LBF LBF + LD LD () LD LD LD LBW () LBW, LD (1) LB LBF, LD () LBF, LD (1) LD () LBF LBF LBF LBF, LD () G500 steel LD LD LD LD () LD LD LBW + LD LD () Sub-Total Grand-Total 30 Note: LBF, LBW - Local Buckling of Flange and Web LDB Lateral Distortional Buckling Y Material yielding (1) 50 mm screw spacing () 50 mm and 100 mm screw spacings The specimens were chosen to maintain an even distribution of test specimens within each category. For this purpose, 1 specimens were selected from G300 steel while 18 specimens were selected from G500 and G550 steels. Five specimens were from the sections with a web height of 100 mm while 5 specimens were for a section with 150 mm web height. Six local buckling failure modes, thirteen lateral distortional buckling failure modes, nine interactive buckling failure modes and two material yielding failure modes were selected as the expected failure modes. Twenty specimens were made with 50 mm screw spacing while ten specimens were made using 100 mm screw spacing. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-7

130 4.4 Test Specimens Test specimens were fabricated by screw fastening two rectangular hollow flanges and a web plate at equal spacing (50 mm or 100 mm) along the length. The flanges were cold-formed first using press-braking method, but the required rectangular shape could not be achieved due to difficulties in fitting the press-braking machine tools in the fully folded rectangular hollow flanges during the cold-forming process. Hence the flanges were folded to a certain level first as shown in Figure 4.3 (a), and were then forced into the required shape using a set of six hydraulic jacks as illustrated in Figure 4.3 (b). Three jacks were used on each side of the side timber and I-section supports to force the flange inward while the flange being held in position firmly by the vertical timber and steel plate supports located above the flange. Figure 4.3 (c) shows the final shape of the test specimens. (a) Partially Bent Flange Figure 4.3 Fabrication of Rectangular Hollow Flanges Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-8

131 Secondary beam Main Beam Hydraulic jacks Vertical supports Vertical actuator Horizontal actuators Hydraulic pump (b) Partially Bent Flange being Forced Inwards Vertical actuator Specimen Horizontal actuators (c) Final Shape of Flange Figure 4.3 Fabrication of Rectangular Hollow Flanges Once all the rectangular hollow flanges were made of the required sizes and thicknesses, they were first clamped together with the corresponding web plate (see Figure 4.4 (a)) and then connected together using No Hexagon head selfdrilling screw fasteners at equal spacings of 50 mm and 100 mm (see Figure 4.4 (b)). Figures 4.4 (a) to (d) show the screw fastening process and the final RHFB specimen. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-9

132 (a) Flanges and Web Clamped Together (b) Flanges and Web Screwed Together (c) Built-in RHFB (d) Schematic View of RHFB Figure 4.4: Assembling Process and Final Shape of a Typical RHFB The combined press-braking and forced bending process used to make the rectangular hollow flanges could have generated additional residual stresses and geometric imperfections in the test specimens. Therefore the initial geometric imperfections and residual stresses are important parameters and should be measured. The initial geometric imperfections of built-up RHFB sections were measured in the laboratory as illustrated the in Figure 4.5 (a), whereas Figure 4.5 (b) illustrates variation of initial bow-out imperfection along a typical RHFB section. However, residual stresses could not be measured due to time constraints. The measured section dimensions were used to calculate the centreline dimensions of the RHFB cross-sections, and the calculated centreline dimensions and the maximum measured bow-out imperfections (δ) for each test specimen are given in Table 4.4. The thickness values, t f and t w, presented in Table 4.4 the are measured based metal thicknesses for each steel grade and thickness (see Section 3..1 in Chapter 3). Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-10

133 Data acquisition system Laser beam component Test beam Levelling device (a): Measurement of Initial Geometric Imperfections Device Imperfection (mm) Distance (cm) 'Top-Flange' Web-Central Bottom-Flange 0 (b): Variation of Initial Geometric Imperfections along a RHFB Specimen Figure 4.5: Initial Geometric Imperfections and RHFB Specimens Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-11

134 Table 4.4: Measured Section Dimensions and Geometric Imperfections of Test Specimens Beam Specimen Designation Flange Web Imperfection δ b f h f h l t f h w t w (mm) (mm) (mm) (mm) (mm) (mm) (mm) 1 RHFB-10t f -10t w -150h w -3L-G300-50s RHFB-10t f -055t w -150h w -3L-G300-50s RHFB-080t f -080t w -150h w -3L-G300-50s RHFB-080t f -190t w -150h w -3L-G300-50s RHFB-190t f -190t w -150h w -3L-G300-50s RHFB-10t f -055t w -100h w -3L-G300-50s RHFB-10t f -055t w -150h w -L-G300-50s RHFB-055t f -10t w -100h w -L-G300-50s RHFB-10t f -10t w -100h w -L-G300-50s RHFB-055t f -095t w -150h w -3L-G550-50s RHFB-095t f -055t w -150h w -3L-G550-50s RHFB-075t f -075t w -150h w -3L-G550-50s RHFB-075t f -115t w -150h w -3L-G550-50s RHFB-115t f -115t w -150h w -3L-G500-50s RHFB-115t f -075t w -150h w -3L-G500-50s RHFB-075t f -075t w -100h w -3L-G550-50s RHFB-055t f -055t w -150h w -L-G550-50s RHFB-095t f -095t w -150h w -L-G550-50s RHFB-095t f -055t w -150h w -L-G550-50s RHFB-075t f -075t w -150h w -L-G550-50s RHFB-080t f -080t w -150h w -3L-G s RHFB-10t f -10t w -150h w -3L-G s RHFB-10t f -055t w -150h w -3L-G s RHFB-075t f -075t w -150h w -3L-G s RHFB-075t f -075t w -100h w -3L-G s RHFB-115t f -115t w -150h w -3L-G s RHFB-075t f -115t w -150h w -3L-G s RHFB-115t f -075t w -150h w -3L-G s RHFB-095t f -095t w -150h w -L-G s RHFB-095t f -055t w -150h w -L-G s Note: Specimen designation is the same as defined in Chapter 3, but span length is also added (eg. 3L means 3 m length) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-1

135 4.5 Test Set-up In order to investigate the elastic buckling and ultimate strength behaviour of the RHFB sections used as flexural members, a full-scale bending test rig was designed, fabricated and built in the QUT Structures Laboratory. The test rig required special support conditions that prevented in-plane and out-of-plane deflections and twisting rotation without restraining in-plane and out-of-plane rotations and warping displacements. It also required the load application through the shear centre of the doubly symmetric RHFB sections with no twisting and lateral restraints to the test beam. The test rig used for lateral distortional buckling tests included a support system and a loading system, which were attached to an external frame consisting of two main beams (50 UC 89.5) and four columns (50 UC 89.5) located at 5 m 1.8 m grid points. The main beams were positioned horizontally at m height between each pair of long columns. The support system included two frames made of 150 UC 37. and 50 mm 50 mm 5 mm SHS sections and was set up within the external frame by fixing the top and bottom of the frames to the main beams and the strong floor. The support frames were kept in a vertical position and perpendicular to the longitudinal axis of test beam. The loading system including two hydraulic rams and a manually operated hydraulic pump was suspended from a specially made wheel system that rested on SHS beams positioned on top of the main beams directly over the loading points of the test beams. In addition, a measuring system was set up to record the applied load, and the strain and deflections of the test beam at several locations. Figures 4.6 (a) and (b) show the schematic and overall views of the test set-up. L/4 L/4 P P L (a) Schematic View Figure 4.6: Lateral Buckling Test Set-up Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-13

136 Main columns Wheel system Main beams SHS beam Test beam PDTs Support frames (b) Overall View Figure 4.6: Lateral Buckling Test Set-up Support System The support system was designed to ensure that the test beam was simply supported in-plane and out-of-plane at the ends of the test beam. It was similar to that used by Zhao et al. (1995), Put et al. (1999) and Mahendran and Doan (1999). Based on the required support conditions described by Zhao et al. (1995), the ends of the span were fixed against in-plane vertical deflections, out-of-plane deflections and twist rotations, but they were unrestrained against major and minor axis rotations. In other words, the ends of the span could rotate freely about its in-plane horizontal axis and vertical axis, but did not twist. To achieve the support conditions described by Zhao et al. (1995), the modified support system of Mahendran and Doan (1999) was further improved to achieve more accurate and convenient support conditions for RHFB specimens in this test program. Figures 4.7 (a) and (b) show the schematic and overall view of the new support system used in this lateral buckling test program. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-14

137 1.80 m 500 mm S B1 1.0 m S m (a) Schematic View Thrust Bearing B UC columns Box-frame S Travelator Steel bar Bearing B1 Box-frame S1 Bearing B3 (b) Overall View Figure 4.7: Support System for Buckling Tests Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-15

138 The new support system included two pairs of 150 UC 37. columns. Each pair of columns was connected together by two SHS members. The important components of the support system were the box-frames S1 and S (see Figure 4.7). The box-frame S1 had two 40 mm diameter shafts that were welded in alignment to each other at the middle of its vertical plates. The roller bearings B1 were then inserted into each shaft and were seated on angle supports welded to the inner flanges of the support frame as shown in Figure 4.7. In this way, the box-frame S1 was supported by two roller bearings B1. The angle supports were kept at the same level of 1. m from the floor in order to avoid any axial forces due to inclined applied loads. The 1. m height to the angle supports was selected to facilitate proper access to the beam being tested, so that thorough observations could be made during testing. Two steel strips were welded to the steel angle in the horizontal travel direction of the roller bearings B1 at one pair of columns to prevent the longitudinal movement of the beam. This was not applied at the other pair of columns to allow free longitudinal displacement of the beam. The roller bearing B1 and the steel angles restrained the box-frames S1 and S, and test beam against vertical displacements, but allowed them to rotate freely about the in-plane horizontal axis of the beam sections. A thrust ball bearing B was placed on the top plate of the boxframe S1, whereas a roller bearing B3 was fixed to its bottom plate at a position vertically below B as shown in Figure 4.7 (b). Two 40 mm diameter shafts were inserted into the bearings B and B3 at one end whereas the other ends of these shafts were welded to steel plates that were bolted to the top and bottom plates of the box-frame S. Thus, the bearings B and B3 allowed the test specimens to rotate freely about the vertical axis of the test beams, but did not allow them to twist. The box-frame S was designed to accommodate all the test specimens having different section sizes. It was required to make S in two symmetric halves in order to insert test specimen conveniently in the S box frame (see Figure (4.7 (b)). Test specimen was fixed inside the S box frame using four bolts located symmetrically about the neutral axis of test beam as shown in Figure 4.7 (b). Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-16

139 4.5. Loading System A gravity loading system was used by other researchers in the past (Zhao et. al., 1995, Put et al., 1999) to investigate the lateral buckling of simply supported beams. However, this method was considered tedious and labour intensive and could not load the beam specimen continuously. Mahendran and Doan (1999) used an improved loading system with hydraulic jacks instead of gravity loads. However, this loading system also had a disadvantage of restraining the lateral movement of test beam. It did not allow the continuation of loading into the post-buckling range due to the fact that roller bearings could slip out of position and cause injuries to people and damage the components. Therefore a new loading system was designed to eliminate the above mentioned shortcomings. The new loading system included two hydraulic rams connected to a wheel system, load cell and a series of other components as illustrated in Figure 4.8. The hydraulic rams were operated under displacement control to ensure that the same load was applied at each loading position of the test beam simultaneously. This provided identical bending moments at the two quarter points of the test beam, and a uniform bending moment between them. The load was applied vertically upward at the two quarter points of the test beam and therefore the bottom flange was in compression. Previous researchers have used both the overhang and quarter point loading methods to investigate the lateral buckling behaviour of various section types. Zhao et al. (1995) and Mahendran and Doan (1999) used the overhang loading method, and Put et al. (1999) used quarter point loading method to investigate the lateral buckling of simply supported beams. In the overhang loading method, the cantilever loads are applied to the test beam at a short distance from the supports, which provide a uniform bending moment within the entire span. On the other hand the quarter point loading method provides a uniform bending moment only between the points of load application. Therefore the overhang loading method was preferred as it provides a uniform moment within the entire span, but it has the possible undesirable effect of warping restraints due to the overhang component of the test beam. In addition, the RHFB has the limitation on its length due to fabrication difficulties; hence the overhang loading method which requires longer test beams to accommodate Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-17

140 cantilever loads was not suitable for the RHFB used in this test program. Therefore the quarter point loading method was adopted. Wheels to travel transversely Hydraulic ram Wheels to travel longitudinally SHS beams (a) Wheel System Hydraulic ram Load cell Bearing B4 Pivot 1 Pivot Steel plate Connector (b) Loading Arm Figure 4.8: Loading System The loading system was designed so that there was no restraint on displacements or rotations in any direction from the loading device to the test beam at the loading positions. The wheel system ensured that the loading arm moved in plane when the beam deformed under the loading, whereas two pivots and bearing (i.e. P1, P and B4, see Figure 4.8 (b)) ensured that the load was applied to the test beam without applying a torque and hence the load acted in vertical plane when the beam deformed in plane. Therefore all the six degrees of freedom were considered unrestrained at the loading positions of the test beam. The load was applied through the shear centre of the cross-section (i.e. centroid) to eliminate load height and torsional effects. The overall loading system is shown in Figure 4.8 (c). Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-18

141 Wheels to travel longitudinally Hydraulic rams Wheels to travel transversely SHS beams Load cell Pivoting bolts Test beam Loading arm (c) Overall View Figure 4.8: Loading System Measuring System The loads applied at the quarter points of test beam were measured using two 60 kn load cells attached to each loading arm and hydraulic ram as shown in Figure 4.9 (a). The measuring system was also set up to record the longitudinal strain, the in-plane and out-of-plane deflections of the test beam at midspan, and the vertical deflection under both loading points of the test beam. The EDCAR unit was used to automatically record all these measurements. The unit included a HP3497A DATA acquisition unit, a HP3498A extender and a PC as shown in Figures 4.9 (b) and (c). Tests were conducted with two electrical strain gauges on the top and bottom flanges at the mid-span of each test beam. The in-plane and out-of-plane deflections were measured using five Potentiometric Displacement Transducers (PDTs). The PDTs, load cells and strain gauges were connected to the computer that used the EDCAR data acquisition software to record the data continuously during the tests. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-19

142 Load cells PDTs Strain gauges (a) Overall View of Measuring System Strain gauge connectors PDT and load cell connectors (b) Data Logger (c) Data Acquisition System Figure 4.9: Measurement and Data Acquisition Systems 4.6 Test Procedure Table 4.4 lists the test specimens used in this program while Figure 4.4 (c) shows a typical built-up RHFB specimen. The cross-section dimensions, material thicknesses and geometric imperfections of each test specimen were measured using a vernier calliper, micrometer and an especially designed measuring table for the geometric imperfections. The measured values are presented in Table 4.4. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-0

143 Test specimens were cut 60 mm longer than their intended span since connection assembly needed extra 30 mm at each support. Holes were drilled on the web at each loading and support positions to insert bolts. Strain gauge and deflection measuring points were marked before the beam was positioned and clamped to the test rig. The test beam was inserted within the box frame S and clamped with the connector in S using two support plate stiffeners on the beam web (see Figure 4.7 (b)). These stiffeners were used to avoid web crippling and twisting of the section at the supports. They were not connected to top and bottom flanges so that warping restraints were not introduced. The loading arms were then bolted to the web at each quarter points of the test beam. The strain gauges and wire displacement transducers were mounted at the required positions, and the resistance of each strain gauge was checked using a multimeter to verify that gauges are accurate. The support frame was aligned to avoid any initial twisting while the loading jack and arm were aligned in order to prevent any eccentricities. The jacks were connected in parallel to ensure that equal vertical loads were applied at the shear centre of test beam. The load cells, transducers, and strain gauges were connected to the data logger. Each channel was individually checked to ensure correct operation. A small load was applied first to allow the loading and support systems to settle on wheels and bearings evenly. The measuring system was then initialized with zero values. A trial load of 10% of the expected ultimate capacity was applied and released in order to remove any slack in the system and to ensure functionality. The load was then applied gradually while the test data was recorded continuously at about 0. kn load increments. Load and displacement readings were recorded by the Edcar software at each load increment and the corresponding load-displacement curves were plotted and displayed on the computer screen continuously until the test beam failed. The applied load started to drop off when the test beam buckled out-ofplane. The loading was continued until the test beam failed by out-of-plane buckling, but was not maintained for too long to prevent damages to the test components and injuries to people. The buckling behaviour of the test beam was observed throughout the test and recorded. A typical RHFB specimen after failure is shown in Figure Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-1

144 Figure 4.10: Typical RHFB Specimen after Failure 4.7 Results and Discussions Since the verticality of the applied loads at the quarter points was maintained throughout the test, the applied uniform moment (M) between the quarter points of the test beam was calculated using; M = P L la (4.1) where P is the applied jack load and L la is the initial lever arm length equal to span/4 as illustrated in Figure L la = span/4 P P L la = span/4 P P Figure 4.11: Deformed Shape of Test Specimen Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-

145 The mean value of the load cell readings at the two quarter points was used to calculate the applied uniform moment. The applied moment was also calculated using the top and bottom flange strain gauge readings. The close agreement between the two moments thus verified the accuracy of load cell readings and the applied uniform moment values Moment versus Deflection Curves This section presents the experimental curves of applied moment versus in-plane deflection and out-of-plane deflection at the mid-span cross section of the test beam for selected typical tests RHFBs with Equal Flange and Web Thicknesses (t f = t w ) Five selected test results (Three G300 steel RHFBs and Two G550 steel RHFBs) were used to plot the moment versus in-plane (vertical) and out-of-plane (horizontal) deflection graphs as shown in Figures 4.1 (a) (d) for the t f = t w category. The moment versus deflection graphs for other tests are presented in Appendix 4B. Moment (knm) RHFB-190t f -190t w -150h w -G300-3L-50s RHFB-10t f -10t w -150h w -G300-3L-50s RHFB-080t f -080t w -150h w -G300-3L-50s Verticale Deflection (mm) (a) Moment versus Vertical Deflection (G300 Steel) Figure 4.1: Moment versus Deflection Curves (t f = t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-3

146 Moment (knm) RHFB-190t f -190t w -150h w -G300-3L-50s RHFB-10t f -10t w -150h w -G300-3L-50s RHFB-080t f -080t w -150h w -G300-3L-50s Horizontal Deflection (mm) (b) Moment versus Horizontal Deflection (G300 Steel) Moment (knm) RHFB-115t f -115t w -150h w -G500-3L-50s RHFB-075t f -075t w -150h w -G550-3L-50s Verticale Deflection (mm) (c) Moment versus Vertical Deflection (G550 Steel) Figure 4.1: Moment versus Deflection Curves (t f = t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-4

147 Moment (knm) RHFB-115t f -115t w -150h w -G500-3L-50s RHFB-075t f -075t w -150h w -G550-3L-50s Horizontal Deflection (mm) (d) Moment versus Horizontal Deflection (G550 Steel) Figure 4.1: Moment versus Deflection Curves (t f = t w ) Figures 4.1 (a) (d) illustrate the moment versus deflection behaviour of RHFBs made of same flange and web thicknesses. From these figures, it can be seen that the moment versus in-plane and out-of-plane deflection curves are non-linear. However, there was a linear relationship in the moment versus in-plane deflection up to about 80% of the ultimate failure moment for both steel grades. For the moment versus outof-plane deflections, there was a linear behaviour in the initial stage, however, it was minor and not up to the extent of moment versus in-plane deflections. The lateral buckling test results of cold-formed channel beams presented by Bogdan et al. (1999) and cold-formed RHS beams presented by Zhao et al. (1995) have shown similar relationships between the applied moment and deflections. From Figures 4.1 (a) (d), it can also be observed that the sections with different slenderness have different in-plane and out-of-plane stiffness. In both steel grades, the maximum in-plane deflection was achieved with the less slender beam sections (except RHFB-10t f -10t w -150h w -50s) whereas the maximum out-of-plane deflection was achieved with the more slender beam sections (except RHFB-075t f - Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-5

148 075t w -150h w -50s). This trend was understandable as the less slender beams can resist larger moments than the more slender beams before failing by the lateral distortional buckling (i.e. out-of-plane buckling). On the other hand, the more slender beams had large out-of-plane deflections before they failed by lateral distortional buckling. Figure 4.13 shows the typical lateral distortional buckling failure of a RHFB with the cross section RHFB-075t f -075t w -150h w -3L-50s. Lateral deflection of compression flange (bottom) Figure 4.13: Typical Lateral Distortional Buckling Failure of RHFBs RHFBs with Flange Thickness Larger than Web Thickness (t f > t w ) Three test results (One G300 steel RHFB and Two G550 steel RHFBs) were used to plot the moment versus in-plane (vertical) and out-of-plane (horizontal) deflection graphs for the category of t f > t w as shown in Figures 4.14 (a) (d). The moment versus in-plane and out-of-plane deflection graphs for other tests with t f > t w are presented in Appendix 4B. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-6

149 Moment (knm) RHFB-10t f -055t w -150h w -G300-3L-50s Verticale Deflection (mm) (a) Moment versus Vertical Deflection (G300 Steel) Moment (knm) RHFB-10t f -055t w -150h w -G300-3L-50s Horizontal Deflection (mm) (b) Moment versus Horizontal Deflection (G300 Steel) Figure 4.14: Moment versus Deflection Curves (t f >t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-7

150 Moment (knm) RHFB-115t f -075t w -150h w -G550-3L-50s RHFB-095t f -055t w -150h w -G550-3L-50s Verticale Deflection (mm) (c) Moment versus Vertical Deflection (G550 Steel) 6.0 Moment (knm) RHFB-115t f -075t w -150h w -G550-3L-50s RHFB-095t f -055t w -150h w -G550-3L-50s Horizontal Deflection (mm) (d) Moment versus Horizontal Deflection (G550 Steel) Figure 4.14: Moment versus Deflection Curves (t f >t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-8

151 The moment versus in-plane and out-of-plane deflection curves given in Figures 4.14 (a) (d) are also non-linear as described in Section The two test beams having a web thickness of 0.55 mm failed by interactive lateral distortional and local web buckling. In these two test beams, web buckling was observed near the loading points. Figure 4.15 shows the web buckling observed in the test of section RHFB- 095t f -055t w -150h w -3L-50s. Figure 4.15: Local Buckling of Slender Web near the Loading Points RHFB with Flange Thickness Smaller than Web Thickness (t f < t w ) Three test results (One G300 steel RHFB and Two G550 steel RHFBs) were used to plot the moment versus in-plane (vertical) and out-of-plane (horizontal) deflection curves for the section category of t f < t w as shown in Figures 4.16 (a) (d). The moment versus in-plane and out-of-plane deflection graphs for other tests including t f < t w are presented in Appendix 4B. The moment versus in-plane and out-of-plane deflection curves shown in Figures 4.16 (a) (d) are also non-linear as described in Sections and The test on section RHFB-080t f -190t w -150h w -3L-50s indicated negative out-of-plane deflection in the initial stage as shown in Figure 4.16 (b). The reason for this behaviour could be due to the local imperfection of the web. The local web imperfection could have been straightened during the loading while the entire beam deflected laterally towards the positive direction. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-9

152 Moment (knm) RHFB-080t f -190t w -150h w -G550-3L-50s Verticale Deflection (mm) (a) Moment versus Vertical Deflection (G300 Steel) Moment (knm) RHFB-080t f -190t w -150h w -G550-3L-50s Horizontal Deflection (mm) (b) Moment versus Horizontal Deflection (G300 Steel) Figure 4.16: Moment versus Deflection Curves (t f < t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-30

153 Moment (knm) RHFB-075t f -115t w -150h w -G550-3L-50s RHFB-055t f -095t w -150h w -G550-3L-50s Verticale Deflection (mm) (c) Moment versus Vertical Deflection (G550 Steel) Moment (knm) RHFB-075t f -115t w -150h w -G550-3L-50s RHFB-055t f -095t w -150h w -G550-3L-50s Horizontal Deflection (mm) (d) Moment versus Horizontal Deflection (G550 Steel) Figure 4.16: Moment versus Deflection Curves (t f < t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-31

154 This type of behaviour could result in a moment versus out-of-plane deflection curve as given in Figure 4.16 (b). Comparison of in-plane deflection shown in Figure 4.16 (c) is complicated. A less slender beam section (i.e. RHFB-075t f -115t w -150h w -3L- 50s) should have higher in-plane stiffness than a more slender beam section (i.e. RHFB-055t f -095t w -150h w -3L-50s). This could be due to certain experimental errors. By comparing the moment-deflection behaviour of G300 and G500/G550 grade steel sections, G300 steel RHFB sections clearly demonstrate a peak moment and moment drop off in their corresponding graphs, but G550 steel RHFB sections do not show such a distinct peak moment or moment drop off Moment versus Longitudinal Strain Curves In each test the longitudinal strains were measured in the compression and tension flanges of the test beam at mid-span to verify the measured load cell readings. The applied uniform moment was calculated based on the measured longitudinal strains for the elastic region. The longitudinal stress in the extreme fibres σ c was calculated first. σ c = Eε m (4.) where E is the elastic modulus of steel assumed to be MPa, and ε m is the average measured longitudinal strain in the extreme fibres at mid-span. Applied uniform moment, M = σ Z (Z f -full section modulus) (4.3) c f Figures 4.17 (a) and (b) show the moment versus longitudinal strain curves for a few G300 and G500/G550 grade steel sections, respectively, using the moments calculated based on the load cell and strain gauge measurements. The moment versus longitudinal strain curves for other sections are presented in Appendix 4B. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-3

155 18 Based on load cell 16 Based on load cell Bending Bending Moment (knm) 1 Compression side Based on strain Based on strain 1 3 Tension side 0 Strain (Microstrain) 1 RHFB-190t f -190t w -150h w -3L-50s - RHFB-10t f -10t w -150h w -3L-50s 3 - RHFB-080t f -080t w -150h w -3L-50s (a) G300 Steel 8 Based on load cell 7 Based on load cell Bending Bending Moment (knm) 1 Based on strain Compression side Based on strain 1 Tension side 0 Strain (Microstrain) 1 RHFB s - RHFB s (b) G500 and G550 Steels Figure 4.17: Moment versus Longitudinal Strain Curves Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-33

156 Five test results were chosen for the curves shown in Figures 4.17 (a) and (b). The curves were plotted using the uniform moments calculated based on the load cell measurements (see Section 4.7.1) and the strain gauge measurements (see Section 4.7.). As shown in these figures, the moment versus longitudinal strain curves based on the load cell and strain gauge readings closely follow each other verifying the accuracy of the load cell measurements. These curves show that the uniform moment calculated based on strain gauge measurements are slightly lower than those calculated from the load cell measurements. Figures 4.17 (a) and (b) further demonstrate that the moment (i.e. based on the load cell measurements) versus longitudinal strain (i.e. measured at the compression flange) curves are distinctively non-linear towards the end of the tests. This non-linear behaviour of the compression side of test beams was clearly demonstrated in the G550 steel sections Comparison of Test Results with Predictions from the Current Design Rules The ultimate failure moments (M u ) from 30 lateral buckling tests are given in Table 4.5. Predicted member moment capacities based on the Australian steel structures design standard AS 4100 (SA, 1998) and the Australia/New Zealand cold-formed steel structures standard AS/NZS 4600 (SA, 1996) are also included. The member moment capacities based on a modified design method by Pi and Trahair (1997) is also presented in Table 4.5. In AS 4100 and AS/NZS 4600, the flexural members are checked for their section and member moment capacities whereas they are checked for their member moment capacities in Pi and Trahair s modified design method. The quarter point loading method was used in this test series to eliminate the warping restraints produced by the overhang loading method. However, it produces a nonuniform bending moment distribution within the beam span. Therefore in order to compare failure moments with code predicted moments for a uniform moment case, the failure moments M u from the tests were divided by a moment distribution factor (α m ) of 1.09 as recommended by AS AS/NZS 4600 provides only an approximate equation to calculate the moment distribution factor (C b ), but it gives 1.0 in this case. Therefore test failure moments were divided by 1.09 even for the comparison with AS/NZS 4600 predictions. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-34

157 Table 4.5: Comparison of Experimental Moment Capacities of RHFBs with Predictions from the Current Design Rules Member Moment Capacities M b (knm) M u /M b Beam No Specimen Designation Experimental Mu (knm) AS 4100 (1998) AS/NZS 4600 (1996) Pi Trahair (1997) Avery et al (1999) Mahaarachchi and Mahendran (005) AS 4100 (1998) AS 4600 (1996) Pi and Trahair (1997) Avery et al (000) Maharachchi and Mahendran (005) 1 RHFB-10t f -10t w -150h w -3L-G300-50s RHFB-10t f -055t w -150h w -3L-G300-50s RHFB-080t f -080t w -150h w -3L-G300-50s RHFB-080t f -190t w -150h w -3L-G300-50s RHFB-190t f -190t w -150h w -3L-G300-50s RHFB-10t f -055t w -100h w -3L-G300-50s RHFB-10t f -055t w -150h w -L-G300-50s RHFB-055t f -10t w -100h w -L-G300-50s RHFB-10t f -10t w -100h w -L-G300-50s RHFB-055t f -095t w -150h w -3L-G550-50s RHFB-095t f -055t w -150h w -3L-G550-50s RHFB-075t f -075t w -150h w -3L-G550-50s RHFB-075t f -115t w -150h w -3L-G550-50s RHFB-115t f -115t w -150h w -3L-G500-50s RHFB-115t f -075t w -150h w -3L-G500-50s RHFB-075t f -075t w -100h w -3L-G550-50s RHFB-055t f -055t w -150h w -L-G550-50s Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-35

158 Table 4.5: Comparison of Experimental Moment Capacities of RHFBs with Predictions from the Current Design Rules (cont..) Member Moment Capacities M b (knm) M u /M b Beam No Specimen Designation Experimental Mu (knm) AS 4100 (1998) AS/NZS 4600 (1996) Pi & Trahair (1997) Avery et al (000) Maharachchi and Mahendran (005) AS 4100 (1998) AS4600 (1996) Pi & Trahair (1997) Avery et al (000) Maharachchi and Mahendran (005) 18 RHFB-095t f -095t w -150h w -L-G550-50s RHFB-095t f -055t w -150h w -L-G550-50s RHFB-075t f -075t w -150h w -L-G550-50s RHFB-080t f -080t w -150h w -3L-G s RHFB-10t f -10t w -150h w -3L-G s RHFB-10t f -055t w -150h w -3L-G s RHFB-075t f -075t w -150h w -3L-G s RHFB-075t f -075t w -100h w -3L-G s RHFB-115t f -115t w -150h w -3L-G s RHFB-075t f -115t w -150h w -3L-G s RHFB-115t f -075t w -150h w -3L-G s RHFB-095t f -095t w -150h w -L-G s RHFB-095t f -055t w -150h w -L-G s Mean COV Note: Experimental M u values given in the table were divided by α m of 1.09 for comparison with M b values Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-36

159 As explained in Chapter 3 and Appendix 3C some plate elements (mainly the web and lower flange element) were only intermittently screw-fastened, however, they were assumed to be continuously connected in the moment capacity calculations. The curvature of the flange top plate (see Figure 4.4c) was not considered in these moment capacity calculations, ie. flat plate assumption. The corner radii were also assumed to be negligible. All of these assumptions could have lead to an overestimation of moment capacities Member Moment Capacity Based on AS 4100 (SA, 1998) The nominal member moment capacity (M b ) of hot-rolled steel beams that fail by lateral torsional buckling is given in AS 4100 for different restraint conditions (SA, 1998). However, this chapter considers only the restraint conditions which are most suitable and comparable with the experimental restraint conditions used for the lateral buckling tests of RHFBs. The nominal member moment capacity of segments without full lateral restraint was chosen with both ends fully or partially restrained. For segments of constant cross-section, the nominal member moment capacity (M b ) is given by: M b = α α M M (4.4) m s s s where = a moment modification factor = a slenderness reduction factor M s = the nominal section moment capacity m s The moment modification factor m is defined in AS 4100 for different moment distribution patterns. For uniform moment distribution m = 1, whereas for the quarter point loading condition and its corresponding moment distribution pattern, m = The slenderness reduction factor ( s ) is defined by: M s M s α + s = (4.5) M o M o Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-37

160 where M o is the reference buckling moment and is determined by M o π EI le π EI GJ + le = y w (4.6) where E = Elastic modulus G = Shear modulus I y = Second moment of area about minor principal axis I w = Warping constant J = Torsion constant l e = Effective length (SA, 1998 Clause 5.6.3) The results presented in Table 4.5 for the member moment capacities based on AS 4100 were calculated from Equations 4.4 to 4.6 assuming m = 1 (A sample calculation is presented in Appendix 4A). The results show that the AS 4100 design formulae are unconservative for the design of RHFBs that fail by lateral distortional buckling. The mean of the ratios between experimental moment capacities to the predicted moment capacities of AS 4100 was 0.57 with a coefficient of variation of These results therefore indicate that the design formulae provided in AS 4100 are unsafe for the design of RHFB flexural members. For the purpose of graphical comparison, all the test beam moment capacities from 30 lateral buckling tests and slenderness results were non-dimensionalised and are plotted in Figure The test beam capacity M u was plotted as M u /M s on the vertical axis whereas the non-dimensional member slenderness λ was plotted on the horizontal axis. The nominal section moment capacity M s was calculated from the following equation based on the classification of RHFB test section (i.e. compact, non-compact or slender). All the RHFB sections considered in this test program were found to be slender (see Chapter 3). M = Z f (4.7) s e y Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-38

161 where Z e is the effective section modulus calculated with the extreme compression or tension fibre at yield stress f y. A sample calculation of Z e is presented in Appendix 3C. The non-dimensional member slenderness λ was calculated from Equation 4.8. M s λ = (4.8) M o Mu/Ms, Mb/Ms Slenderness (λ) Test AS4100-curve Figure 4.18: Comparison of Experimental Failure Moments with AS 4100 Predictions Comparison of results in Figure 4.18 shows that the current AS 4100 design rules for lateral buckling (i.e. Equations 4.4 to 4.6) is not suitable as it predicts unconservative member moment capacities. The reason for such a significant overestimation of member capacities from AS 4100 is due to the incorrect use of reference buckling moment M o, which is based on lateral torsional buckling. However, the RHFBs failed by lateral distortional buckling. Therefore Pi and Trahair (1997) modified the AS 4100 design method to suit beams that fail by lateral distortional buckling based on their investigation of lateral distortional buckling of triangular hollow flange beams (HFBs). Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-39

162 Member Moment Capacity Based on Pi and Trahair s (1997) Method Pi and Trahair (1997) developed a nonlinear inelastic method to analyse lateral distortional buckling behaviour of electric resistance welded triangular HFBs. From their analyses they modified the AS 4100 design method for the HFBs and recommended the following member capacity formula to allow for lateral distortional buckling. M bd = α α M M (4.9) m sd s s where M s is the section moment capacity and α sd is the modified slenderness reduction factor given by M s M s α 0.6 sd = (4.10) M od M od α m is the moment modification factor (α m = 1 for uniform moment distribution) and M od is the lateral distortional buckling moment calculated by M od = Zf od (Z full section modulus) (4.11) where f od is obtained from the elastic buckling analyses (Thin-wall program) and Z is the full section modulus. The results presented in Table 4.5 for the member moment capacities based on Pi and Trahair s (1997) method were calculated using Equations 4.9 to 4.11 assuming m = 1 (see Appendix 4A). The results showed that Pi and Trahair s design method predicts member moment capacities of RHFBs more accurately than the current AS 4100 design method. The mean of the ratios between experimental moment capacities to the Pi and Trahair s (1997) design moment capacities was 0.97 with a coefficient of variation of These results therefore show that the design method Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-40

163 developed by Pi and Trahair (1997) to cope with lateral distortional buckling of triangular HFBs can be used for the design of RHFBs more accurately than the AS 4100 design method. The graphical comparison of non-dimentionalised beam moment capacities and slenderness results from 30 lateral buckling tests are given in Figure The nominal section moment capacity M s was calculated as explained in Section while λ d was calculated using Equation 4.8, but with M o replaced by M od. Figure 4.19 shows that Pi and Trahair s (1997) design curve gives better correlation with the experimental member capacities than the AS 4100 design curve Mu,Mb/Ms Slenderness (λ d ) Test Pi & Trahair (1997)-curve Figure 4.19: Comparison of Experimental Failure Moments with Predictions using Pi and Trahair s (1997) method Member Moment Capacity Based on AS/NZS 4600 (SA, 1996) The member moment capacity is defined in AS/NZS 4600 for different buckling modes and beam types (SA, 1996). However, this chapter only presents and discusses the moment capacities based on the buckling behaviour of RHFBs. According to the details provided in Chapters 1 and, RHFBs comprising two torsionally rigid hollow flanges and a slender web are susceptible to lateral Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-41

164 distortional buckling effects. Hence Clause (b) of AS/NZS 4600 was used to determine the member moment capacity of RHFBs subjected to lateral distortional buckling. Clause (b) defines distortional buckling as that involving transverse bending of a vertical web with lateral displacement of the compression flange. The member moment capacity (M b ) is given as: M c M b = Z c (4.1) Z f where Z c is the effective section modulus calculated at a stress level of f c =M c /Z f in the extreme compression fibres. The critical moment (M c ) is calculated from Clause (b) of AS/NZS 4600 as follows: For λ d < 1.414: λ d M c = M y 1 4 (4.13) For λ d 1.414: 1 M c M y λd = (4.14) where λ d is the non-dimensional slenderness parameter and is determined from the next equation. M y λ d = (4.15) M od where M y is the first yield moment and M od is the elastic distortional buckling moment and they are given by M = Z f (4.16) y f y M = Z f (4.17) od f od in which, f od is the elastic distortional buckling stress obtained from the Thin-wall computer program. Z e is the full section modulus. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-4

165 The results presented in Table 4.5 for the member moment capacities based on AS/NZS 4600 method were calculated using Equations 4.1 to Example calculations are given in Appendix 4A. The results showed that the AS/NZS 4600 design formula overestimates the member moment capacities of RHFBs, however, the predictions are more accurate than the AS 4100 design method. The mean value of the ratios between experimental moment capacities to the AS/NZS 4600 design moment capacities was 0.78 with a coefficient of variation of 0.17 (see Table 4.5). These results therefore show that the design formulae in AS/NZS 4600 provide better estimates of RHFB moment capacity than the AS 4100 design method. 1.0 Mu/My, Mb/My Experiment AS/NZS Slenderness (λ d ) Figure 4.0: Comparison of Experimental Failure Moments with AS/NZS 4600 (1996) Predictions The graphical comparison of non-dimensionalised beam moment capacities and slenderness results from 30 lateral buckling tests is given in Figure 4.0. The modified member slenderness λ d was calculated as explained in Sections and Since AS/NZS 4600 design rules do not have a single equation for member capacity, M b, the design curve (M b /M y ) for AS/NZS 4600 cannot be plotted. Instead the member capacities corresponding to test beam capacities are plotted as discrete points. As observed in Figure 4.0, predicted member moment capacities of RHFBs Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-43

166 are quite unconservative for all beam slenderness. Therefore the comparison of test results and the predictions using AS/NZS 4600 design rules for the member moment capacities of RHFB indicates that the AS/NZS 4600 design rules are not safe to use in either the lateral torsional buckling design or for the lateral distortional buckling designs Member Moment Capacity Based on Avery et al. s (000) Method Alternative member moment capacity equations were also proposed by Trahair (1997). The accuracy of these equations for the design of electric resistance welded, triangular HFB flexural members was investigated and lateral distortional buckling design curves were produced by Avery et al. (000, 1999b). Design curves for HFBs were derived based on the finite element analysis results of Avery et al. (000, 1999a), which were verified using the lateral distortional buckling tests of Mahendran and Doan (1999). A design procedure for HFB members based on a modified form of Trahair s equations is more accurate and reliable alternative to the AS 4100 and AS/NZS`4600 design methods. Trahair (1997) proposed a design curve based on the following equations: M b a b = b + M n s M s ; M b M o ; 1+ cλ M b M sy (4.18) The non-dimensional member slenderness ( d ) is given by: M s λ d = (4.19) M od The suitable coefficients (a, b, c, and n) were established using the least square method. Values of a = 1.0, b = 0.0, c = 0.44, and n = were found to minimise the total error for Trahair s (1997) design equations (see Table.3). However, this approach resulted in an unacceptable maximum unconservative error of more than 10 percent for HFB sections. Therefore Avery et al. (000) has derived separate coefficients for each of the different thickness of the HFB sections. Even though this Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-44

167 approach was more accurate for the HFB section range it is very complicated and requires different design curves for each thickness of HFB. It also does not follow the intent of Trahair s formulation, which was to suggest different design curves for certain groups of beams, eg. hot-rolled I-sections, or cold-formed channels, rather than to have different design curves within the same family of cross-sections produced by the same manufacturer (CASE, 00). Therefore in this study Equations (4.18) and (4.19) with the coefficients a, b, c and n (1.0, 0.0, 0.44, 1.196) determined by Avery et al. (000) were used to predict the moment capacity of RHFB. Comparison of the predicted moment capacities with the results of 30 lateral buckling tests of RHFB is shown in Figure Mu, Mb/Ms Slenderness (λ d ) Test Avery et. al. (000)-curve Figure 4.1: Comparison of Experimental Failure Moments with Predictions using Avery et al. s (000) Method Figure 4.1 shows that the predictions based on Avery et al. s (1999) method is similar to AS/NZS 4600 predictions for the lateral distortional buckling region and hence not suitable in the design of RHFB sections. As observed in Figure 4.1 it is quite conservative for beams of low slenderness while being unconservative for beams of intermediate slenderness (inelastic buckling region). Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-45

168 The results presented in Table 4.5 for the member moment capacities based on Avery et al. s (000) method were calculated using Equations 4.18 and 4.19 (see Appendix 4A). However, the results indicated that the correlation of predicted moment capacities based on Avery et al. s (1999) method with the experimental results is poor compared with Pi and Trahair s (1997) method. The mean of the ratios between experimental moment capacities to Avery et al. s (1999) design moment capacities was 0.71 with a coefficient of variation (COV) of Member Capacity Based on Mahaarachchi and Mahendran s (005c) Method Mahaarachchi and Mahendran (005a and 005b) investigated the flexural behaviour of dual electric resistance welded hollow flange channel sections known as LitetSteel Beams (LSB) experimentally and analytically to produce alternative design formulae for LSB. Design curves were derived using the finite element analysis results of Mahaarachchi and Mahendran (005c), which were verified against the lateral distortional buckling test results of Mahaarachchi and Mahendran (005a). Equations 4.0 (a) (c) have been recommended by Mahaarachchi and Mahendran (005c) for three regions of member slenderness separating yielding/local buckling, inelastic lateral distortional buckling, and elastic lateral buckling. For λ d c y M = M (4.0(a)) For 0.59 λ < 1. 7 < d For λ 1. 7 d 0.59 M c = M y λd 1 M c M y λd = (4.0(b)) (4.0(c)) Comparison of the predicted moment capacities using Equations 4.0(a) to (c) with the results of 30 lateral buckling tests of RHFB is shown in Figure 4.. The predicted member moment capacities using Mahaarachchi and Mahendran (005) design method were calculated for all the test specimens as explained in Appendix 4A. As explained in the section on the comparison with AS/NZS 4600 Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-46

169 predictions, Mahaarachchi and Mahendran s predictions cannot be plotted as a design curve. Instead individual test capacities are compared with corresponding predictions in Figure 4.. Figure 4. shows that the predicted moment capacities based on Mahaarachchi and Mahendran s (005c) method are better correlated with the experimental moment capacities than those predicted by AS/NZS 4600 and Avery et al. s (1999) method. As observed in Figure 4., it is quite conservative for beams of low slenderness while being unconservative for beams of intermediate slenderness (inelastic buckling region). 1.0 Mu/My, Mb/My Test Mahaarachchi & Mahendran Slenderness (λ d ) Figure 4.: Comparison of Experimental Failure Moments with Predictions using Mahaarachchi and Mahendran s (005c) Method The results presented in Table 4.5 for the member moment capacities based on Mahaarachchi and Mahendran s (005c) method were calculated using Equations 4.0 (a) (c) (see Appendix 4A). The results indicated that the correlation of predicted moment capacities based on Mahaarachchi and Mahendran s (005c) method with the experimental results of RHFBs is better than all other methods except Pi and Trahair s (1997) method. The mean of the ratios between experimental moment capacities to Maharachchi and Mahendran s (005c) design moment capacities was 0.87 with a coefficient of variation (COV) of 0.. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-47

170 4.8 Summary This chapter has presented the details and results of a series of lateral distortional buckling tests of an innovative cold-formed steel beam with rectangular hollow flanges, known as RHFB. The buckling tests of RHFBs were conducted in a purpose-built test rig. The support and loading systems were specially designed to satisfy the idealised boundary conditions required for such buckling tests. The tests included 0 different section geometries of RHFBs, two screw spacings 50 mm and 100 mm, and two spans m and 3 m, giving a total of 30 lateral buckling tests. The test results showed that the new RHFBs failed by lateral distortional buckling at intermediate beam slenderness. The nonlinear behaviour of RHFB was discussed using moment versus in-plane and out-of-plane deflection plots. The effect of overhang and quarter point loading method including warping effect was also presented and discussed. The lateral buckling test results were compared with the predictions of member capacities calculated using the Australian hot-rolled steel structures design code AS 4100, the Australian/New Zealand cold-formed steel structures design code AS/NZS 4600, and the desgn methods proposed by Pi and Trahair (1997), Avery et al. (000, 1999b) and Mahaarachchi and Mahendran (005) using non-dimensionalised moment and slenderness results. The member moment capacities predicted by all the design methods for lateral distortional buckling were generally unconservative for RHFBs with higher slenderness. The predicted moment capacities of AS 4100 were extremely higher than the test moment capacities (unconservative) because AS 4100 design rules were based on lateral torsional buckling failures of hot-rolled I-section beams. In the case of AS/NZS 4600, predicted moment capacities are also higher than the test moment capacities. In the member capacity calculations using the design methods mentioned above, it was assumed that the hollow flanges and web elements were connected continuously, ignoring the effect of intermittent screw fastening. This assumption could have partly contributed to the overestimation of member capacities. The lateral distortional buckling behaviour of RHFB is further investigated using finite element analyses in Chapter 6 that includes the effects of intermittent screw fastening. All the results would then be used to develop accurate design rules for screw fastened RHFB sections subjected to flexural loading. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 4-48

171 CHAPTER 5 Finite Element Modelling and Analysis of RHFB 5.1 General Chapters 3 and 4 presented the details of section and member capacity tests of Rectangular Hollow Flange Beams (RHFBs) including a series of material property tests. Twenty two section capacity tests and thirty member capacity (lateral distortional buckling) tests were conducted on 0 different section geometries of RHFBs. The section geometries for this test series were chosen to achieve different failure modes, and therefore the sections represented a broad range of web and flange slenderness values, but it is desirable to test a much larger selection of specimens. However, a more extensive test program would have been expensive and time consuming. Numerical or finite element analysis provides a relatively inexpensive, and time efficient alternative to physical experiments. However, it is vital to have a sound set of experimental data upon which to calibrate a finite element model. It is then possible to investigate a wide range of parameters using the model. In order to model the ultimate section and member capacities of RHFBs, the finite element program should include the effects of material and geometric non-linearity, residual stresses, initial geometric imperfections and local buckling. The ABAQUS Version 6.3 (HKS, 00) provided by High Performance Computing and Research Support section of the Queensland University of Technology was used in the numerical analysis. This chapter describes the essential stages in the development of finite element models to simulate the section and member capacity tests of RHFBs. The models were calibrated using experimental data obtained from the tests presented in Chapters 3 and 4. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-1

172 5. Development of the Finite Element Models 5..1 Physical Models The physical situation being modelled has to be considered first. The section capacity and lateral buckling tests to investigate the section and member capacities of RHFBs outlined in Chapter 3 and 4 were modelled. A number of general factors were considered in the finite element model. They are: RHFB itself, method of loading and nature of restraints. Figure 5.1(a) is a simplified diagrammatic representation of the experimental layout of the lateral buckling tests. A four point loading system was used in the physical model to minimize the effect of bending moment distribution on the member moment capacity. AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996) allow for the moment distribution patterns on the member capacity by introducing moment modification factors m and C b, respectively, in the member capacity equations. For the four point loading system, m =1.09 and C b = 1.0. L /4 L / L /4 Y Y 1 X P P Note: L = m and 3m Support 1: free to rotate in-plane and out-of-plane (i.e. about Z-axis and Y-axis) Support : free to rotate in-plane and out-of-plane (i.e. about Z-axis and Y-axis) and free to move along longitudinal axis (i.e. X) Figure 5.1 (a): Physical Model of Lateral Buckling Test Figure 5.1(b) is a simplified diagrammatic representation of the experimental layout of section capacity tests. A four point loading method was used in this test series using an existing test set-up in the QUT structural laboratory. The section capacity of RHFB is only governed by local buckling (i.e. section properties) and the material Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-

173 properties, and therefore the support conditions and the loading method are not significant. Y P P X 1 L /3 L /3 L /3 Note: L = 1070 mm Supports 1 and : free to rotate in-plane (i.e. about Z-axis) and free to move along longitudinal axis (i.e. along X-axis) Figure 5.1(b): Physical Model of Section Moment Capacity Test The cross-section of the beam flanges was not changed (i.e. b f = 50 mm, h f = 5 mm, and h l = 15 mm), but the depth of the web (h w ), thickness of the flanges (t f ), and the thickness of web (t w ) were changed as in the laboratory tests. The axis system for the beams was as shown in Figures 5.1(a) and (b). 5.. Symmetry and Boundary Conditions The size of a finite element model can be reduced significantly by using symmetry in the structure being analysed. The symmetry is considered about a particular axis or a plane of a structure with respect to geometry, boundary conditions and loading patterns before and after the deformations. In the test set-up of lateral buckling tests, the beam itself and the loading system were symmetric about a plane perpendicular to the longitudinal axis (i.e. X-axis) of the beam at its mid-span. The support conditions were almost symmetric about the mid-plane, but only one support provides restraint against X-axis translation. However, it does not violate symmetric condition of the beam about the mid-plane since the beam was not subjected to any lateral loadings. Therefore, it was possible to consider only half the span of the beam, and apply the boundary conditions as shown in Figure 5. (a) to all the nodes at the mid-span of the beam. The X-axis translation was prevented at the mid-span cross section. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-3

174 u 1 = 0 θ = 0 θ 3 = 0 P Experimental model u = 0 u 3 = 0 θ 1 = 0 (Y) M 3 Z 1 (X) Ideal model (a) Member Capacity Tests 3 u 1 = 0 u 3 = 0 θ 1 = 0 θ = 0 θ 3 = 0 1 u 1 = 0 θ = 0 θ 3 = 0 P Experimental model Ideal model u = 0 u 3 = 0 θ 1 = 0 θ = 0 M u = 0 u 3 = 0 θ 1 = 0 (b) Section Capacity Tests Figure 5.: Experimental and Ideal Finite Element Models Similarly, it was possible to consider only half the span of the beam in the section capacity finite element models also by considering the symmetry about a plane perpendicular to the longitudinal axis (i.e. X-axis) at mid-span. The boundary conditions were applied as shown in Figure 5. (b) to all nodes at the mid-span of the beam. The X-axis translation was prevented at the mid-span cross section. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-4

175 The two principal axes of the RHFBs are the axes of symmetry (i.e. Y and Z) as defined in Figures 5.1 (a) and (b). The major principal axis (sometimes referred to as the Z-axis) was the 3-axis in the finite element models, but could not be used as an axis of symmetry in the finite element analysis despite the geometrical symmetry of the beam about the 3-axis. The reason was unsymmetrical flexural behaviour of the beam about the 3-axis, resulting in the top half of cross section in compression and the bottom half in tension. The compression portion of the section could be subjected to local buckling as illustrated in Figure 5.3, which violates the symmetry of the beam about major axis (i.e. Z-axis) in the section and member capacity models. Y Z Y Z (a) Flange local buckling (b) Web local buckling Figure 5.3: Unsymmetrical Local Buckling Behaviour about Z-axis The minor principal axis (i.e. Y-axis) was the -axis in the finite element model. Although the beam s geometry and loading were symmetric about the -axis, the deformation patterns of the beam result from the lateral distortional or lateral torsional buckling distort the symmetrical condition about the -axis in the member capacity models. However, very short beams used in the section capacity models were not susceptible to the lateral distortional or lateral torsional buckling and therefore it would be able to consider the symmetry about the minor principal axis (i.e. Y-axis) in the section capacity models unless local buckling of beam s web occurred. However, symmetry about the minor principal axis was not considered in the section capacity models to maintain uniformity in both section and member capacity models. Figure 5.4 illustrates the unsymmetrical nature of typical global buckling failure modes about the minor axis of the beam (i.e. Y-axis). Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-5

176 Y X Y X (a) Lateral distortional buckling (b) Lateral torsional buckling Figure 5.4: Unsymmetrical Global Buckling Behaviour about Y-axis The support boundary conditions as defined in Figure 5. (a) were provided by Zhao et al. (1995) for the ideal simply supported boundary conditions for the lateral buckling tests. The objective of both experimental and ideal finite element models was to provide these ideal simply supported boundary conditions and thereby use them to produce the design curves suitable for the new beam type, RHFBs. The boundary conditions used in the lateral buckling tests were able to achieve all of the above boundary conditions, with one exception. The twist restraint about the longitudinal axis (i.e. X-axis) at the support was only applied to the beam web, and the flanges were set unrestrained to allow for free warping. This boundary condition was required in the experimental finite element model to simulate accurately the support conditions in the physical model of lateral buckling tests. Figures 5.5 (a) and (b) illustrate the boundary conditions at the support and mid-span sections, respectively, in the experimental finite element model for member capacity. Two steel plates (thickness of 10 mm each) attached to the web at the support to avoid web buckling were modelled using S4R5 shell elements and they were connected to the beam web using MPC rigid beams (centroidal node was considered as independent and all other nodes on the web and steel plates were considered dependant) as shown in Figure 5.5 (a). A single point constraint (SPC) which simulates pinned end boundary conditions was applied to the centroid of the section at the support (i.e., 3 and 4 restrained) as shown in Figure 5.5 (a). The hollow flange s top plates were not perfectly flat, instead had a curvature due to the specimen fabrication method used (see Chapter 4). This effect was simulated in the experimental finite element model by using a curved finite element surface as the top flange plate, whereas a flat finite element surface was used in the ideal finite element model. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-6

177 Rigid MPC to connect web and web stiffening plate Two steel plates attached to the web SPC ( 3 4) Contact definition Between flange lip and web Rigid MPC to connect web-flange (a) Support Boundary Condition Rigid MPC to connect web and flanges Contact definition Between flange lip and web Restrained on whole section (b) Mid-Span Boundary Condition Figure 5.5: Experimental Finite Element Model simulating Member Capacity Tests Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-7

178 The mid-span boundary conditions are the same for both experimental and ideal finite element models for member capacity. In which, the in-plane rotation about Z- axis (i.e. referred to as 6) and out-of-plane rotation about Y-axis (i.e. referred to as 5) and the longitudinal displacement (i.e. referred to as 1) become zero at the mid section of the beam due to symmetry. Therefore, the boundary conditions (1 5 6) shown in Figure 5.5 (b) were applied to the middle plane of the beam used in the experimental finite element model for member capacity. In the experimental finite element model for section capacity, the same boundary conditions at the support were used as defined for the member capacity model (see Figure 5.5 (a)) except the out-of-plane rotation, which was prevented to simulate experimental conditions of section capacity tests. The support boundary conditions of experimental finite element model for section capacity therefore remained the same as for the member capacity model as shown in Figure 5.5 (a) except SPC (, 3, 4, 5). The mid-span boundary conditions were assigned in the experimental finite element model for section capacity as described for the member capacity model shown in Figure 5.5 (b), providing restraints of 1, 3, 4, 5, and 6. The idealised boundary conditions were applied to the ideal finite element models as illustrated in Figure 5.6 so that it can be used to develop design curves. The rigid beam type Multiple Point Constraint (MPC) elements were generated by connecting dummy nodes at 10 mm away from the flange nodes (i.e. R f, see Figure 5.6). The dummy node of the intersection points between inner horizontal flange surface and the web were selected as the common independent nodes, and the dummy nodes of each flange were selected as dependent nodes to create these MPCs in both top and bottom flanges as shown in Figure 5.6. These MPCs were used to distribute the load evenly to each flange node from the intersection point of flange and web. Similarly, another rigid beam type MPC was created by connecting dummy nodes at 0 mm away from the web (i.e. R w, see Figure 5.6). The centre dummy node of the web was used as independent node and other dummy nodes of the web were used as dependant nodes. This MPC was used to transfer the applied moment at the centre dummy node of the web to the flange s nodes through a pin MPC as shown in the Figure 5.6. The pin type MPCs connecting web and flanges were used to allow flanges to rotate independently about the minor axis (i.e. warping restraint Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-8

179 eliminated). The explicit type MPC elements were created linking rigid beams to the corresponding nodes on the edge of the flange and web. For the nodes on the web, only X and Z translational degrees of freedom are linked whereas for the flanges, only X translation was linked. This will allow the web or flanges to expand without distortion at the support, thus eliminating possible warping stress concentrations. The centroid of the section was connected to the independent node at the web rigid beam with a Tie type MPC. The Tie MPC was used to maintain the moment applied about major axis at the centre of web rigid beam uniform within the entire member. The ideal finite element model was common for both section and member capacity models and it predicts the moment capacities of RHFBs based on the beam s span. Explicit type MPC UX to link flange nodes and Rigid Beam SPC ( 3 4) Tie MPC R f Rigid Beam type MPC R w Y X Z Explicit type MPC UX and UZ to link web nodes and Rigid Beam Pin type MPC Figure 5.6: Support Boundary Conditions of Ideal Model 5..3 Choice of Element Type ABAQUS has several element types suitable for numerical analysis: two or three dimensional solid elements, membrane and truss elements, beam elements, and shell Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-9

180 elements are some of them. The primary aim of this analysis was to understand the different buckling failure modes of RHFBs and hence predict the ultimate flexural capacity. Beam, membrane and truss elements were not appropriate for the buckling problems (HKS, 00). A stress-free flat membrane has no stiffness perpendicular to its plane and out-of-plane loading will cause numerical singularities and convergence difficulties. Truss elements do not transmit moments since they have only axial stiffness. Neither local buckling of web and flanges nor distortional buckling of the members can be modelled by beam elements, and thus the failure behaviour can not be modelled accurately using beam elements (Bakker and Pekoz, 003). Therefore the most appropriate element type to model RHFBs for the flexural capacity is shell element and they were used in all the finite element models. The 4-noded shell elements can be used efficiently to model this type of beam geometries. Three types of 4-noded shell elements are available in ABAQUS Standard Version 6.3 (S4, S4R, and S4R5). Both S4 and S4R elements are doubly curved general-purpose, finite membrane strain shell elements, where, R stands for reduced integration with hourglass control. These two elements are often used for modelling shell structures with thickness larger than 1/15 th of element length for which transverse shear deformation is important and Kirchoff constraint is satisfied analytically (Yuan, 004). This element imposes the Kirchoff constraint numerically. In comparison, S4 and S4R elements have six degrees of freedom per node and have multiple integration locations for each element. They will be more accurate than the S4R5 element for thick shell structures, but is significantly more computationally expensive. Hence the most appropriate element type for modelling the RHFBs was found as the S4R5 shell element. The general definition of a S4R5 shell element is shown in Figure 5.7. ABAQUS has two basic types of shell elements: thick shell elements and thin shell elements. The S4R5 element is a thin shear flexible, isoparametric quadrilateral shell with four nodes and five degrees of freedom per node, utilizing reduced integration and bilinear interpolation schemes. The characteristic length is the flange width or the web height for modelling the local buckling of RHFBs as appropriate. In the experimental program, the values of b/t for the flanges were varied from 8 to 91, whereas for the web, it was varied from 53 to 73. Therefore, only few sections were found to have element thickness greater than Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-10

181 1/15 th of the characteristic length limitation. Therefore thin shell elements were acceptable in this analysis. The thin shell elements had zero thickness, but a thickness was assigned as a shell element property within ABAQUS. The shell model followed the mid-thickness line of the real RHFB as illustrated in Figure 5.8. N (SPOS) Surface normal positive direction Stress/displacement shell (s) Number of nodes S 4 R 5 5 degrees of freedom Reduced integration Figure 5.7: General Definition of S4R5 Shell Element Shell elements follow midthickness line of RHFB Figure 5.8: Location of Shell Elements within RHFB Cross-Section In addition to S4R5 shell elements, different types of Multiple Point Constraints (MPCs) were used to create appropriate boundary conditions and loading system in both ideal and experimental finite element models. Rigid beam type MPC elements were used to spread the applied moment at the centroid of the cross-section evenly through the web the flanges. Pinned type MPCs were used to allow flanges Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-11

182 to rotate independently about the minor axes, so that warping restraint was eliminated. Explicit type MPCs were used to provide appropriate boundary conditions (i.e. required degrees of freedom) so that flanges and web can expand without distortion at support, thus eliminating possible stress concentration Loading Method Separate loading systems were used in the experimental and ideal models. The loading system adopted in the experimental finite element models was to simulate the physical conditions in the experimental test set-up whereas an idealized loading system was used in the ideal finite element model so that it can be used to develop design curves for the RHFBs. Figures 5.9 (a) and (b) illustrate the method of loading in the physical and experimental finite element models, respectively, for the lateral buckling tests. The loading method used in the physical model ensured neither rotation nor displacement restraints were put on any direction in the test beam at the loading positions, and therefore the loading point in the finite element model was unrestrained. The point load applied at the quarter point of the beam in the physical model was transferred to the beam web through three bolts located at 30 mm spacing (see Figure 5.9 (a)), and therefore this physical condition was simulated in the experimental finite element model with three nodal loads at similar locations as for the test beam (see Figure 5.9 (b)). The bolts were modelled using three rigid type MPCs at each loading position and thereby the two steel strips in the loading system were connected to the beam web. Similarly, Figures 5.10 (a) and (b) illustrate the method of loading in the physical and experimental finite element model, respectively, for the section capacity tests. As described in Section 5..1, neither support conditions nor loading method are important for the section capacity of RHFBs; only cross-section properties (i.e. local buckling) and the material properties govern the section capacity of a typical RHFB. However, experimental finite element model for the section capacity was developed to simulate experimental conditions closely. Three point loading system used in the physical section capacity model was simulated using a single concentrated load at the Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-1

183 middle bolt as shown in Figure 5.10(b), whereas the ideal finite element model (see Figure 5.11) was common for both section and member capacity tests. Top-Flange Web Rigid MPC Steel plate Bolts Screws Bottom-Flange (a) Physical Model (b) Experimental FE Model Figure 5.9: Loading Method of Member Capacity Tests Applied load Support Mid-span cross section Contact modelling (a) Physical Model (b) Experimental FE Model Figure 5.10: Loading Method of Section Capacity Tests Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-13

184 The ideal finite lement model was loaded using an end moment applied at the centroid of the beam s end section to ensure uniform moment along the entire beam. The applied moment at the centroid of the beam was distributed evenly within the end section using a rigid MPC routine as illustrated in Figure A 0 mm wide elastric strip was applied at the end of the beam to avoid stress concentration at the loading point. Although two equal and opposite end moments were applied in the ideal finite element model, only half the beam was considered due to the symmetric conditions as described in the previous section. M Figure 5.11: Loading Method of Ideal FE Model 5..5 Modelling of Contact Surfaces Since the flange lips and the web were not rigidly connected together, the nodes on the flange lips and elements in the web were modelled as contact pairs (see Figure 5.1). Since both the top and bottom flange lips and the web could come into contact with each other during the loading, they were modelled as contact pairs (i.e. C1, C, C3 and C4, see Figure 5.1). This allows any interface movements of two surfaces Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-14

185 in contact during the deformation. A smooth surface interaction (i.e. zero friction) was assumed for the contact surfaces in the model. C Master Surface C1 Slave Surfaces Master Surface C4 C3 Slave Surfaces Figure 5.1: Contact Surface Definition In contact problems, one surface (or set of elements normal) must be assigned as the MASTER, while the second surface (or set of nodes) selected as the SLAVE. The problem arises when the master penetrates the slave as this does not occur in practice. One solution for this is to use a very fine mesh so that penetrations can be minimized or eliminated. However, a very fine mesh would result in a large number of nodes and elements, and hence the increase of the analysis time. The mesh size adopted in this analysis was 5 mm 5 mm in flanges and hence 900 S4R5 elements in a flange lip (i.e. slave surface) and 10 mm 5 mm in the web and hence 450 S4R5 elements on the corresponding web strip (i.e. master surface). This mesh size was used throughout the entire model, and it was considered adequate to obtain the desired results. The mesh density may have to be reduced even up to 1 mm 1 mm to eliminate penetration completely. However it was not considered appropriate as it would increase the analysis time considerably. ABAQUS requires the slave surface to be of finer mesh than the master surface so that penetration of slave surface is minimal. In other words ABAQUS allows minimal penetration so that the accuracy of the solution is acceptable. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-15

186 5..6 Material Properties Different material properties can be included in the numerical analysis. The ABAQUS classical metal plasticity model (HKS, 00) was used in the ideal finite element model for the nonlinear inelastic analyses. This material model implements the following criteria: The von-mises yield surface to define isotropic yielding. Associated plastic flow theory. That is, as the material yields the inelastic deformation rate is in the direction of the normal to the yield surface (the plastic deformation is volume invariant). This assumption is generally acceptable for most calculations with metals. Either perfect plasticity or isotropic hardening behaviour. Perfect plasticity assumes no strain hardening (i.e. the yield stress does not change with increasing plastic strain). Isotropic hardening allows strain hardening; with the yield surface changing size uniformly in all directions such that the yield stress increases in all stress directions as plastic strain occurs. The material properties obtained from the standard coupon tests (see Chapter 3) were input to the experimental finite element models as a set of points on the stress-strain curves. ABAQUS uses true stress and strain data, and hence the values of engineering stress and strain from the standard coupon tests were modified before being input into the model using the following relationships (HKS, 00): σ = σ 1+ ε ) (5.1) true nom ( nom σtrue ε p(ln) = ln(1 + εnom) (5.) E In the experimental finite element models, RHFB had three major components; top hollow flange, bottom hollow flange and web plate. Each beam was made of the same steel grade with different combinations of flange and web thicknesses. Therefore the corresponding material properties obtained from the tensile coupon tests were assigned to each component of the RHFB. However, the ABAQUS classical elastic perfect plastic model (HKS, 00) was used in all the components of the ideal finite element model. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-16

187 5..7 Residual Stresses During the formation process of cold-formed steel sections, residual stresses are induced within the cross section. While the net effect of residual stresses must be zero for equilibrium, the presence of residual stresses can result in premature yielding of plate elements. Two types of residual stresses are present in the coldformed steel structural members and they are: Membrane residual stresses, which are uniform through the thickness of a plate element. Bending residual stresses, which vary linearly through the thickness Key (1988) investigated the effects of various types of residual stresses using finite strip analyses of cold-formed square hollow section (SHS) columns and found that the membrane residual stresses had an insignificant effect, but the bending residual stresses had the major impact on the behaviour of cold-formed SHS. Schafer et. al. (1996) reviewed the past research on residual stresses and concluded that for the cold-formed C-sections the membrane residual stresses can be ignored, but recommended the inclusion of bending residual stresses (see chapter ). The present analysis therefore incorporated the bending residual stresses, and that was deemed to be sufficient, based on Key s (1988) and Schafer et. al. (1996) findings. The magnitudes of the residual stresses for this study were based on the residual stress model recommended by Schafer et al. (1996) for a cold-formed steel channel section formed by press-braking process. The forming process adopted in the coldformed C-section and the flanges of RHFB can be considered similar in terms of the amount of cold work for each corresponding element in both sections. Hence, the magnitude and distribution pattern of residual stresses in the flanges of RHFBs can be modelled as for the cold-formed C-sections as illustrated in Figure The residual stresses of 17% yield strength (f y ) in bending were applied to the outer horizontal plate and the two vertical plates of the flanges (see Figure 5.13) as for the inner horizontal plate of the flanges since their forming process is similar to the coldformed C-section, since their forming process is similar to the web of a C-section. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-17

188 17 % f y 17 % f y 17 % f y 8 % f y 8 % f y Figure 5.13: Residual Stress Distributions of a Typical Flange of RHFB Similarly the residual stresses of 8% yield strength were assigned to the flange of a C-section. The residual stresses in the two lips were considered negligible since the free end of lips may help to release the stress build up. It was considered that it is necessary to incorporate these bending residual stresses in the finite element models as recommended by Schafer and Pekoz (1998b) to ensure that lower bound ultimate strengths of the RHFBs are obtained from the finite element analyses. The residual stress model derived for this study was used for both the ideal and experimental finite element models. The residual stresses were applied using the ABAQUS commands: * INITIAL CONDITIONS option with TYPE = STRESS, USER. The user defined initial stresses were created using the SIGNI FORTRAN user subroutine, which defines the local component of the initial stresses as a function of global coordinates. An example of coding for the subroutine to include residual stresses is given in Appendix 5A of the thesis Initial Geometric Imperfections Experimental data for geometric imperfections are limited. However, it is known that imperfections must be included in a finite element model to simulate the true shape of the specimen and introduce some inherent instability into the model. In general, two parameters are considered as important in finite element modelling with the inclusion of initial geometric imperfections. They are: Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-18

189 Magnitude of the imperfection Shape of the imperfection These two parameters are considered to have a direct link to the ultimate capacity of a beam member. However, initial imperfections may vary from member to member in steel structures. These random imperfections only initiate the buckling deformations, but the ultimate member capacity is mainly governed by the primary buckling mode. The shape of imperfection was introduced into the finite element model by modifying the nodal coordinates using a field created by scaling the approximate buckling eigenvectors obtained from an elastic-bifurcation buckling analysis of a geometrically perfect specimen. The first two buckling modes obtained from the elastic-bifurcation buckling analysis were applied in the model for non-linear analysis. The magnitudes of member imperfections were measured for each test specimen during the experimental stage (see Chapter 4). The measured values of geometric imperfections were used to define the maximum value of global imperfections in the nonlinear analysis using experimental finite element models (i.e. δ g, see Figure 5.14 (c)). The fabrication tolerance of L/1000 as recommended by AS 4100 for compression members was used as the magnitude of maximum bow-out (i.e. global) imperfection in the ideal finite element model. The imperfection shapes were assigned from either lateral torsional or lateral distortional buckling modes obtained from the elastic bifurcation buckling analysis. The first two buckling modes obtained from the bifurcation buckling analysis were used. The magnitudes of local geometric imperfections for both the ideal and experimental finite element models were estimated using Equation 5.3 as recommended by Schafer et al. (1996). Possible local buckling modes of a typical RHFB under flexural loading are illustrated in Figures 5.14 (a) and (b), where Figure 5.14 (a) is local buckling in the flange top plate and Figure 5.14 (b) is local buckling in the web plate. Figure 5.14 (c) shows a possible global buckling mode. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-19

190 l t δ = 6te (5.3) where t is the element thickness and δ lf, δ lw are as shown in Figures 5.14 (a) and (b). δ lf δ lw δ g (a) Flange Local Buckling (b) Web Local Buckling (c) Lateral Distortional Figure 5.14: Possible Buckling Modes of a RHFB Schafer and Pekoz (1998b) recommended that at least the first two eigenmodes obtained from the elastic-bifurcation buckling analysis should be used in the nonlinear analysis to simulate imperfection shapes accurately. Therefore, in this analysis, the worst possible deformation modes were considered as the first two eigenmodes, and were used in the nonlinear ABAQUS model Pre and Post Processing ABAQUS requires an input file which defines the nodes, elements, material properties, boundary conditions and loading. The input file for ABAQUS analysis was developed by using MSC/Patran 004 Version as a pre-processor. The results were viewed using MSC/Patran post-processing facilities. The pre-processing and post-processing stages included the following steps to generate input file and view the results from ABAQUS analyses. 1. Define geometric surfaces for web and flanges. Mesh all the web and flange surfaces Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-0

191 3. Define loads, simply supported and symmetric boundary conditions, elastic material properties, element properties, contact pairs, bifurcations buckling analysis parameters 4. Generate input file for bifurcation buckling analysis 5. Run bifurcation buckling analysis using ABAQUS (version 6.3) to obtain the first two buckling eigenmodes. 6. Define nonlinear material properties and nonlinear static analysis parameters 7. Generate input file for the nonlinear static analysis 8. Enter initial geometric imperfection to the input file (Step 7) using first two eigenmodes from the bifurcation buckling analysis. 9. Run nonlinear static analysis using input file (Step 8) along with initial stress input subroutine. 10. Import nonlinear static analysis results into the Patran 004 database and view the results using Patran post-processing facilities. The data required for the load-deflection plot were imported from the Patran to the Excel worksheet using FTP (File Transfer Protocol) The finite element analysis generated vast amount of data in the temporary files and the permanent files during the analysis. A typical 00 mm deep RHFB of 3 m length (i.e. half length = 1.5 m) required 1685 elements and nodes in the experimental finite element model. Each analysis usually consisted of 0 to 40 increments. It was possible for the output to contain full details of deformations, stresses and strains in each direction for each node during every increment. However, only a fraction of the available output was required to obtain the load-deflection relationship for the beam being analysed. Equilibrium in the vertical (i.e. -axis) direction showed that the sum of three nodal loads at the loading point equalled the vertical reaction at the support for the halfspan beam being analysed. The bending moment in the central region of the beam (i.e. between the loading point and the symmetric plane at the other end of the beam) was uniform under this load arrangement and calculated from the support reaction. The horizontal deflection at the centroid and vertical deflection at the centre of the compression flange were obtained for the symmetric section similar to experimental measurements so that experimental and numerical results can be Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-1

192 compared. The ABAQUS output file (.rpt) was generated to extract the required deflection values for only the above two nodes, and then the.rpt file was imported to Excel spreadsheet to plot the moment vs vertical deflection and the moment vs horizontal (lateral) deflection curves for the RHFBs. 5.3 Validation of Finite Element Models Since the ultimate aim of this research is to develop suitable design curves for the new RHFB beams type using an extensive parametric study based on the validated finite element models, the validation process of finite element models for the section and member capacities is a significant part of this research. Hence it is essential to ensure the results obtained from the finite element models compares well with those from the experiments as well as other established analytical methods. Two series of comparisons were required to validate both ideal and experimental finite element models developed for the section and member capacities. The first series of comparison involved the use of experimental test results obtained from the section and member capacity tests with the nonlinear analysis of the experimental finite element models in order to simulate the experimental conditions accurately. Visualization of the deformation shape and stress contours was also used to assist with model verification. The second series involved comparison of the local and elastic lateral distortional buckling moments obtained using the ideal finite element model with the corresponding moment solutions obtained from the established finite strip analysis program, Thin-wall (Papangelis, 1994). Two methods of analyses were used, namely, the bifurcation elastic buckling analysis and the non-linear inelastic analyses. Elastic buckling analyses were used to obtain the eigenvectors for the geometric imperfections and to obtain the elastic buckling failure moments. The non-linear inelastic analysis including the material and geometric nonlinearity effects and the residual stresses were then performed to obtain the ultimate section and member capacities of the RHFBs. These results were then used to plot the moment versus deflection curves. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-

193 5.3.1 Experimental Finite Element Models Before using the finite element model to develop the moment capacity curves for RHFB sections subjected to uniform bending moment, it was necessary to validate the model for non-linear analyses. This was achieved by comparing the non-linear analysis results of experimental finite element models with the results obtained from the experimental tests described in Chapter 3 and 4. The accuracy of the experimental finite element models was validated by: 1. experimental results from the testing of RHFBs for section capacities and member capacities. The experimental finite element models were found to give reasonable agreement with the experimental results.. visualisation of the defined geometry and stress contours. An animated sequence of failure mode was generated using the Patran post-processor and the results of non-linear analysis. No significant stress discontinuities across the element boundaries were identified and the deformation behaviour confirmed to the expected behaviour Experimental Finite Element Model for Member Capacity Typical moment versus vertical (in-plane) and horizontal (out-of-plane) deflection curves for a group of selected beams with equal flange and web thicknesses (i.e. t f = t w ), flange thickness greater than web thickness (i.e. t f > t w ) and flange thickness less than web thickness (i.e. t f < t w ) are shown in Figures 5.15 to 5.17 with their corresponding analytical results obtained from the experimental finite element models for the member capacity. The vertical deflection curves represent the deflection at the bottom flange of mid-span whereas the horizontal deflection curves represent the deflection at the web centre of mid-span. Experiments were conducted for a range of RHFB sections. The measured cross-section dimensions are shown in Chapter 4 (refer to Table 4.4). Different combinations of flange and web thicknesses were chosen for m and 3 m beam spans and this provided a suitable range of member slenderness. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-3

194 14.0 Moment (knm) FEA Expt RHFB-190t RHFB-1.9tf-1.9tw-150hw-G300-50s f -190t w -150h w -3L-G300-50s RHFB-10t RHFB-1.tf-1.tw-150hw-G300-50s f -10t w -150h w -3L-G300-50s RHFB-080t RHFB-0.8tf-0.8tw-150hw-G300-50s f -080t w -150h w -3L-G300-50s Vertical Deflection (mm) (a) Moment versus Vertical Deflection Moment (knm) RHFB-190t RHFB-1.9tf-1.9tw-150hw-G300-50s f -190t w -150h w -3L-G300-50s RHFB-10t RHFB-1.tf-1.tw-150hw-G300-50s f -10t w -150h w -3L-G300-50s FEA Expt.0 RHFB-080t RHFB-0.8tf-0.8tw-150hw-G300-50s f -080t w -150h w -3L-G300-50s Horizontal Deflection (mm) (b) Moment versus Horizontal Deflection Figure 5.15: Moment-Deflection Curves for a group of RHFB Specimens with Equal Flange and Web Thicknesses (i.e. t f = t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-4

195 Moment (knm) RHFB-115t RHFB-1.15tf-0.75tw-150hw-G500-50s f -075t w -150h w -3L-G550-50s RHFB-095t RHFB-0.95tf-0.55tw-150hw-G550-50s f -055t w -150h w -3L-G550-50s FEA Expt Vertical Deflection (mm) (a) Moment versus Vertical Deflection Moment (knm) RHFB-115t RHFB-1.15tf-0.75tw-150hw-G500-50s f -075t w -150h w -3L-G550-50s RHFB-095t RHFB-0.95tf-0.55tw-150hw-G550-50s f -055t w -150h w -3L-G550-50s FEA Expt Horizontal Deflection (mm) (b) Moment versus Horizontal Deflection Figure 5.16 Moment-Deflection Curves for a group of RHFB Specimens with Flange Thickness Greater than Web Thickness (i.e. t f > t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-5

196 Moment (knm) RHFB-075t RHFB-0.75tf-1.15tw-150hw-G550-50s f -115t w -150h w -3L-G550-50s FEA Expt Vertical Deflection (mm) (a) Moment versus Vertical Deflection Moment (knm) RHFB-075t RHFB-0.75tf-1.15tw-150hw-G550-50s f -115t w -150h w -3L-G550-50s FEA Expt Horizontal Deflection (mm) (b) Moment versus Horizontal Deflection Figure 5.17 Moment-Deflection Curves for a RHFB Specimen with Flange Thickness Less than Web Thickness (i.e. t f < t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-6

197 As illustrated in Figures 5.15 to 5.17, loading was not continued well beyond the maximum moment in the experimental program to avoid excessive out-of-plane deformations of the test beams and the possible damage to test rig components and injuries to people. However, the non-linear finite element analysis including the Riks method allowed the loading to continue well beyond the ultimate moment. The differences of the lateral displacement between the results from finite element analyses and experiments could have been due to any possible lateral restraints imposed by the hydraulic jacks to the test specimens. Even though the loading system was designed to avoid such lateral restraints, there could have been some friction in the bearings. This was not measured and no attempt was made to include the friction effects. However, it is considered that lateral restraint has minimal effect on the buckling moment. The analytical and experimental curves for the member capacity of RHFBs presented in Figures 5.15 to 5.17 show that they are in reasonable agreement. Table 5.1 contains a summary of the ultimate moment capacity (M u ) results of the nonlinear analyses using the experimental finite element model and a comparison of these results with the experimental test results provided in Chapter 4 for the member capacities of RHFBs. The overall mean of experimental to FEA ultimate member moment capacity ratio was 1.05 with a coefficient of variation (COV) of These capacity ratios were also calculated for G300, G500 and G550 steels separately and were found to be 1.05, 1.00 and 1.06 respectively, with COVs of 0.07, 0.09 and 0.05, respectively. A typical moment versus longitudinal strain curves for a group of RHFB specimens tested for the member capacity are shown in Figure 5.18 with the corresponding analytical curves. The strains were measured on the top and bottom surfaces of the flanges at the mid-span of the test beams using mm strain gauges. Tension strain was considered as + ve and compression strain was considered as - ve. Hence the negative side of Figure 5.18 represents the experimental and analytical moment versus longitudinal strain curves for the compression flange, and positive side represents those curves for the tension flange. These curves show that the experimental and analytical results are in reasonable agreement. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-7

198 Table 5.1: Comparison of Experimental and FEA Member Moment Capacities Beam Specimen Designation Exp M u (knm) FEA M u (knm) Exp. / FEA M u 1 RHFB-10t f -10t w -150h w -3L-G300-50s RHFB-10t f -055t w -150h w -3L-G300-50s RHFB-080t f -080t w -150h w -3L-G300-50s RHFB-080t f -190t w -150h w -3L-G300-50s RHFB-190t f -190t w -150h w -3L-G300-50s RHFB-10t f -055t w -100h w -3L-G300-50s RHFB-10t f -055t w -150h w -L-G300-50s RHFB-055t f -10t w -100h w -L-G300-50s RHFB-10t f -10t w -100h w -L-G300-50s RHFB-055t f -095t w -150h w -3L-G550-50s RHFB-095t f -055t w -150h w -3L-G550-50s RHFB-075t f -075t w -150h w -3L-G550-50s RHFB-075t f -115t w -150h w -3L-G550-50s RHFB-115t f -115t w -150h w -3L-G500-50s RHFB-115t f -075t w -150h w -3L-G500-50s RHFB-075t f -075t w -100h w -3L-G550-50s RHFB-055t f -055t w -150h w -L-G550-50s RHFB-095t f -095t w -150h w -L-G550-50s RHFB-095t f -055t w -150h w -L-G550-50s RHFB-075t f -075t w -150h w -L-G550-50s RHFB-080t f -080t w -150h w -3L-G s RHFB-10t f -10t w -150h w -3L-G s RHFB-10t f -055t w -150h w -3L-G s RHFB-075t f -075t w -150h w -3L-G s RHFB-075t f -075t w -100h w -3L-G s RHFB-115t f -115t w -150h w -3L-G s RHFB-075t f -115t w -150h w -3L-G s RHFB-115t f -075t w -150h w -3L-G s RHFB-095t f -095t w -150h w -L-G s RHFB-095t f -055t w -150h w -L-G s Mean 1.05 COV 0.07 Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-8

199 14 Moment (knm) 3 FEA Exp FEA Exp Strain ( Micro Strain) 1 = RHFB-080t f -080t w -150h w -G300-50s = RHFB-10t f -10t w -150h w -G300-50s 3 = RHFB-080t f -080t w -150h w -G300-50s Figure 5.18: Moment versus Longitudinal Strain Graphs for a Group of RHFB Specimens Figure 5.19: Distortional Buckling Failure Mode of a Typical RHFB Specimen During Tests Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-9

200 The most common failure mode of the RHFBs with intermediate beam spans was the lateral distortional buckling as illustrated in Figures 5.19 (experimental) and 5.0 (a) and (b) (analytical). Deformation shapes for a typical RHFB with 3 m span from experiment and FEA compare well as shown in Figures 5.19 and 5.0 (a) and (b). The stress contours shown in Figure 5.0 (b) illustrate the uniform moment distribution between the quarter point load locations as expected. Stress concentration can be seen at the loading points and the flange and web connection points by MPCs. This was also observed during the lateral buckling tests with some yielding around the screws used to connect the web and flanges. (a) Elastic Buckling Mode (b) Ultimate Failure Mode Figure 5.0: Elastic and Ultimate Lateral Distortional Buckling Failure Mode and Stress Contours of a Typical RHFB model from FEA Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-30

201 Experimental Finite Element Model for Section Moment Capacity The moment versus vertical deflection curves for a set of RHFBs with equal flange and web thicknesses (i.e. t f = t w ), flange thickness greater than web thickness (i.e. t f > t w ) and flange thickness less than web thickness (i.e. t f < t w ) are shown in Figures 5.1 to 5.3 with their corresponding analytical results from experimental finite element model for the section capacity. The section capacity tests were performed on laterally restrained short beams and therefore out-of-plane buckling did not occur. The vertical deflection curves given in Figures 5.1 to 5.3 represent the deflection at the bottom flange of the mid-span. These curves show that the experimental and analytical results are in reasonable agreement Moment (knm) RHFB-080t f -080t w -150h w -G300-50s RHFB-10t f -10t w -150h w -G300-50s FEA Expt Vertical Deflection (mm) Figure 5.1: Moment versus Vertical Deflection Curves for RHFB Specimens with Equal Flange and Web Thicknesses (i.e. t f = t w ) Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-31

202 FEA Expt RHFB-115t f -075t w -150h w -G300-50s Moment (knm) RHFB-10t f -055t w -150h w -G300-50s Vertical Deflection (mm) Figure 5.: Moment versus Vertical Deflection Curves for RHFB Specimens with Flange Thickness Greater than Web Thickness (i.e. t f > t w ) Moment (knm) FEA Expt RHFB-080t f -190t w -150h w -G300-50s Vertical Deflection (mm) Figure 5.3: Moment versus Vertical Deflection Curves for RHFB Specimens with Flange Thickness Less than Web Thickness (i.e. t f < t w ) The analytical and experimental curves for the section moment capacity of RHFBs presented in Figures 5.1 and 5.3 are in reasonable agreement whereas experimental and analytical section moment capacity curves presented in Figure 5. do not agree well. Comparison of test and predicted section moment capacities in Table 3.8 also indicated some significant differences in these test results. The same test results Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-3

203 (marked * in Table 5.) compared similarly with the FEA section moment capacities and therefore it could be assumed that these test results were influenced by some experimental errors. Therefore these test results were omitted in the calculation of the overall mean and coefficient of variation of the ratio of experimental and FEA section moment capacities. Table 5. contains a summary of ultimate section moment capacity (M s ) results from the nonlinear analyses using the experimental finite element model and a comparison of these results with the experimental test results provided in Chapter 3 for the section capacity of RHFBs. The overall mean of the ratio of experimental to FEA section moment capacities is 0.86 with a COV of These ratios have means of 0.89, 0.79 and 0.85 and COVs of 0.0, 0.11 and 0.18 for G300, G500 and G550 steels, respectively. Table 5.: Comparison of Experimental and FEA Section Moment Capacities Beam Specimen Designation Section Moment Capacity (knm) Expt FEA Expt./FEA 1 RHFB-10t f -055t w -100h w -G300-50s RHFB-10t f -055t w -100h w -G s RHFB-080t f -080t w -150h w -G300-50s RHFB-080t f -080t w -150h w -G s RHFB-10t f -10t w -150h w -G300-50s RHFB-10t f -10t w -150h w -G s RHFB-080t f -190t w -150h w -G300-50s RHFB-080t f -190t w -150h w -G s RHFB-10t f -055t w -150h w -G300-50s RHFB-10t f -055t w -150h w -G s RHFB-075t f -075t w -100h w -G550-50s RHFB-075t f -075t w -100h w -G s RHFB-075t f -075t w -150h w -G550-50s RHFB-075t f -075t w -150h w -G s RHFB-115t f -115t w -150h w -G500-50s RHFB-115t f -115t w -150h w -G s RHFB-075t f -115t w -150h w -G550-50sp RHFB-075t f -115t w -150h w -G s RHFB-115t f -075t w -150h w -G500-50s RHFB-115t f -075t w -150h w -G s * 1 RHFB-095t f -055t w -150h w -G550-50s * RHFB-095t f -055t w -150h w -G s * Mean 0.86 COV 0.18 Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-33

204 Typical moment versus longitudinal strain graphs for a group of specimens tested for the section capacity are shown in Figure 5.4 with their corresponding analytical curves from FEA. The strains were measured on the top and bottom surface of the flanges at the mid-span of the test beams. These curves show that the experimental and analytical results are in reasonable agreement. The strains were measured using mm strain gauges. Moment (knm) Strain (microstrain) Expt1-c Expt1-t FEA1-t FEA1-c Expt-t Expt-c FEA-t FEA-c 1 = RHFB-10t f -10t w -150h w -G300-50s = RHFB-080tf-080t w -150h w -G300-50s Figure 5.4: Moment versus Longitudinal Strain Graphs for a Group of RHFB Specimens The most common failure mode of RHFBs at shorter spans was local buckling of top flange plate or web depending on the flange and web slenderness. Figures 5.5 and 5.6 illustrate the typical experimental and FEA failure modes of RHFB (RHFB- 075t f -075t w -150h w -G550-50s), respectively. As shown in Figures 5.5 and 5.6, local buckling appeared on both the top flange plate element and web in this particular beam section. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-34

205 Figure 5.5: Local Buckling Failure Mode of RHFB Tested for Section Capacity Figure 5.6: Local buckling Failure Mode of RHFB from Analytical results 5.3. Validation of Ideal Finite Element Models The validation process of the experimental finite element models for the section and member moment capacities were discussed in Section 5.3.1, and it was shown that the experimental finite element models could predict section and member moment capacities reasonably well. This section presents the details of ideal finite element model validation. Before using the ideal finite element model for non-linear analyses to develop the design curves for RHFBs, it was considered desirable to establish its accuracy for elastic buckling analyses. This was achieved by conducting a series of elastic buckling analyses using the ideal finite element model described in the previous sections to obtain the elastic lateral distortional buckling moments, and compare them with the solutions obtained from the established finite strip analysis Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-35

206 program, Thin-wall (Papangelis, 1994). This comparison was intended to verify the accuracy of the finite element type, mesh density, boundary conditions, and the loading method. However, the main shortcoming here is that the ideal finite element model is for the intermittently screw fastened RHFBs whereas Thin-wall and Pi and Trahair s method assumed that the flange and web are connected continuously. Despite this, this comparison is considered to add to the validation of the ideal finite element model. Since the experimental finite element model was first validated using full scale experimental results, such validation also confirms indirectly the accuracy of similar finite element models such as the ideal finite element model. Fifteen RHFB sections were considered and their designations are given in Table 5.3. Each section was analysed for five different spans ranging from 1000 mm to 8000 mm, using the simply supported configurations provided in Figures 5.6 and The results of these analyses and comparisons with the solutions obtained from the finite strip analysis program, Thin-wall (Papangelis, 1994) are presented in Table 5.3, and graphical comparison for few selected sections is given in Figure 5.7. As shown in Table 5.3 and Figure 5.7, the ideal finite element model described in the previous sections accurately predicts the elastic lateral distortional buckling moments of all RHFB sections for a range of member slenderness. Using the results of Thinwall (Papangelis, 1994) as a basis for comparison, the model used in the present study predicts lateral distortional buckling moments on average by 3.6% less than Thin-wall. The comparison of the two methods suggests that the model used in the present study may in fact be more accurate than the finite strip program Thin-wall due to the fact that the exact configuration of RHFB was not represented in the Thinwall model. Thin-wall assumed continuous connection between flanges and web despite the fact they were screw fastened at 50 mm spacing. The present ideal finite element model took into account this exact connection configuration and is therefore expected to provide accurate results. The finite element model is also considered more accurate due to a substantially finer mesh density and improved boundary conditions at the support. These comparisons verify the suitability and accuracy of the element type, mesh density, geometry, boundary conditions, and the method used to generate the required uniform moment distribution. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-36

207 Table 5.3: Comparison of Elastic Buckling Moments from FEA (Ideal Model) with Thin-wall Analysis Flange Size Specimen Designation Buckling Moment (knm) at Different Beam Spans (mm) FEA TW % Diff FEA TW % Diff FEA TW % Diff FEA TW % Diff FEA TW % Diff RHFB-00t f -00t w -150h w -G300-50s RHFB-300t f -300t w -150h w -G300-50s RHFB-400t f -400t w -150h w -G300-50s RHFB-500t f -500t w -150h w -G300-50s RHFB-00t f -00t w -100h w -G300-50s RHFB-300t f -300t w -100h w -G300-50s RHFB-400t f -400t w -100h w -G300-50s RHFB-500t f -500t w -100h w -G300-50s RHFB-300t f -300t w -160h w -G300-50s RHFB-400t f -400t w -160h w -G300-50s RHFB-500t f -500t w -160h w -G300-50s RHFB-00t f -00t w -10h w -G300-50s RHFB-300t f -300t w -10h w -G300-50s RHFB-400t f -400t w -10h w -G300-50s RHFB-500t f -500t w -10h w -G300-50s Note: FEA = Finite element analyses TW = Thin-wall Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-37

208 As shown in Table 5.3, the present ideal finite element model predicts lesser elastic buckling moment capacities than Thin-wall buckling formula at shorter beam spans, and at the longer beam spans it predicts approximately the same moment capacities. The reduction in moment capacities predicted by the ideal finite element model is due to the fact that the existing discontinuities between the web and flange connections as it may reduce the bending stiffness against lateral distortional buckling at shorter beam spans. This type of structural behaviour could be expected because out-of-plane bending stiffness is decreased with the increased beam span and consequently the effect of discontinuity in the web and flange connection becomes insignificant at longer beam spans. Elastic buckling moment comparison given in Figure 5.7 for selected RHFB sections also indicates that the effect of discontinuity in the web-flange connection is more significant at shorter beam spans than the longer span beams. Buckling oment (knm) S1-FEA S1-TW S-FEA S-TW S3-FEA S3-TW S4-FEA Span (mm) S4-TW Figure 5.7: Comparison of FEA and Thin-wall Elastic Buckling Capacities Moment 5.4 Summary This chapter has described the finite element models developed for the investigation of RHFB flexural behaviour. Two experimental finite element models were developed separately (see Section 5.) to simulate the experimental tests for the Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-38

209 section and member moment capacities of RHFBs, and they were validated (see Section 5.3) using experimental results presented in Chapters 3 and 4. An ideal finite element model was developed incorporating idealized boundary conditions and a uniform moment loading (see Section 5.) to suit both section and member moment capacities of RHFBs, and was validated from the results obtained from an established finite strip analysis program Thin-wall. The models account for all significant behavioural effects, including material inelasticity, local buckling and lateral distortional buckling deformations, member instability, web distortion, residual stresses, and geometric imperfections. The comparison of finite element analysis results with both experimental and other analytical results obtain from an established buckling analysis program (Thin-wall) indicated that these models could accurately predict both elastic lateral distortional buckling moment and non-linear ultimate moment capacities of RHFBs subjected to pure bending. Therefore the ideal finite element model incorporating ideal simply support boundary conditions and uniform moment conditions will be used to conduct an in-depth parametric study (see Chapter 6) to develop a large data base on the flexural characteristics of RHFBs. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 5-39

210 CHAPTER 6 Parametric Studies and Development of Design Rules for RHFB 6.1 General Chapter 5 described the finite element analyses which simulated a series of bending tests of RHFBs for the section and member moment capacities. However, Chapter 5 only considered the stages of finite element model development and validation to simulate experiments. However, also of interest are the results of the numerical simulation which examined the effects of various parameters including material properties, residual stresses, geometric imperfections and section and member slenderness. Earlier chapters based on experimental analyses have not been able to develop accurate design rules for the new RHFBs. This chapter therefore presents the details of a parametric study conducted to understand the effects of various parameters on the section and member moment capacities of RHFBs and to develop new design rules to predict section and member moment capacities of RHFBs. 6. Parametric Study The results presented in Chapter 5 have shown that the experimental finite element models for section and member moment capacities could simulate the observed experimental behaviour of RHFB in the bending tests, whereas the ideal finite element model gave RHFB buckling results that compared well with those from the well known finite strip buckling analysis. The ideal finite element models were particularly developed to simulate ideal simply supported boundary conditions and a uniform moment and hence were used in the parametric study. Ideal RHFB section shape was modelled and the analyses were conducted using nominal values of the material properties. The ideal finite element models also included the following common features: Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-1

211 Half of the beam span with ideal pinned end boundary conditions Elastic and perfect plasticity material behaviour Initial geometric imperfections Residual stresses Contact surfaces Applied end moment to create uniform moment within the span To fully understand the structural behaviour of cold-formed steel beams with rectangular hollow flanges, affected by a range of parameters including section geometry, material properties and initial conditions, a large number of finite element analyses were required. A significant amount of time was required to obtain the results of each model in the pre-processing phase (i.e. the definition of geometry, mesh, loads, and boundary conditions). Therefore PATRAN database journal file, containing instructions for the pre-processor, was used to automatically generate a model. Variables such as geometry (section, span etc.), finite element mesh, loads, boundary conditions, material properties, and analysis input parameters could then be automatically created by rebuilding the journal file. It was therefore possible to generate a large number of models with no user input other than the preliminary creation of the journal file. This process was used to create a large number of ABAQUS input files, which were analysed using the bifurcation buckling solution sequence to obtain the elastic buckling eigenvectors. The local and global geometric imperfections were then incorporated into the nonlinear analysis model and the analysis continued using the nonlinear static solution sequence. Appropriate models were chosen to investigate the various buckling failure modes and parameters of RHFBs and the results are presented in the following sections Local Buckling Behaviour of RHFB The flexural behaviour of cold-formed steel beams comprising torsionally rigid rectangular hollow flanges and a slender web that was made of thin, high strength steel is complex and very much depend on a range of parameters including the Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-

212 section geometry, steel thicknesses and yield stress. Local buckling of compression elements is one of the failure modes of RHFB and which depends on the plate element slenderness (λ). The local buckling behaviour of cold-formed steel structural members comprising different shapes and sizes of compression elements has been investigated by many researchers in the past, and design rules have been established. However, the applicability of these design rules to the new beam type, RHFB, is not known and need to be justified. A comprehensive parametric study was therefore undertaken for the section moment capacity of RHFB. Well known design method adopted to account for the local buckling of thin-walled steel elements is the effective width approach. Australian steel design standards, AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996), have adopted this effective width approach to determine the reduced section moment capacities. According to both design standards, the effective width (b e ) of slender elements depends on the element s slenderness (λ). The element slenderness (λ) is a function of width to thickness ratio (b/t), yield stress (f y ) and elastic modulus (E) of steel, and plate buckling coefficient (k). The geometric parameters (see Figure 6.1) and the material property f y were considered as the important parameters for the local buckling behaviour of RHFB and therefore they were varied in this study to investigate their effects on the section moment capacities of RHFB. A constant beam length of 00 mm was assumed to be sufficient to give the section moment capacities of RHFBs from the FEA. b f tf h f tw h w h l h f Figure 6.1: Cross-section parameters of a Typical RHFB Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-3

213 Width to thickness ratio (b/t) was only varied for the critical compression elements: i.e. flanges top plate, flange web, and the beam web. In the cases of yield stress f y, variations were made even beyond the actual values (i.e. from 300 MPa to 800 MPa) for the sake of completeness and gaining more data from the parametric study. The results are summarised and discussed in the following sections Effect of Local Buckling in the Hollow Flange s Top Plate Element The effect of local buckling in the hollow flange s top plate element (see Figure 6.) on the section moment capacities of RHFB was investigated by choosing different width to thickness ratios (b/t) for the flange top plate. All the other elements in the RHFB section were chosen to be compact (non-slender) according to AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996). Tables 6.1 (a) and (b) show the section moment capacities (M s ) obtained from the finite element analyses for different b/t ratios of the top plate element ranging from 5 to 100 and the predicted section moment capacities from the current steel design standards AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996) The calculation procedure of section moment capacity (M s ) using AS 4100 and AS/NZS 4600 is shown in Appendix 3C. It must be noted that all other geometric dimensions were unchanged, and are given in the same table. The comparison of results indicates that both steel design standards underestimate the section moment capacities of RHFBs when the hollow flange s top plate element experiences local buckling. Local buckling in Flange top plate Figure 6.: Local Buckling of the Hollow Flange s Top Plate Element Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-4

214 Table 6.1 (a): Comparison of FEA Results with the Predictions from Current Steel Design Standards for G300 Steel t p (mm) b/t Element Compactness AS 4100 Section Moment Capacity M s (knm) AS/NZS AS Actual Modified AS/NZS 4600 FEA M s Ratio AS 4100 /FEA Actual Modified AS4600/ FEA S S S S S S NC S NC S C NS b f = 50 mm, h f = 5 mm, h w = 150 mm, t = mm thickness of all the elements except top flange plate, which is t p and it is varied as shown in Column 1, f y = 300 MPa Table 6.1 (b): Comparison of FEA Results with the Predictions from Current Steel Design Standards for G550 Steel t p (mm) b/t Element Compactness AS 4100 Section Moment Capacity M s (knm) AS/NZS AS Actual Modified AS/NZS 4600 FEA M s Ratio AS 4100 /FEA Actual Modified AS4600/ FEA S S S S S S S S S S NC NS b f = 50 mm, h f = 5 mm, h w = 150 mm, t = mm for all the elements except top flange plate, which is t p and it was varied as shown in Column 1, f y = 550 MPa AS 4100 design rules were modified for slender RHFB sections to limit the local buckling effects to slender elements in the calculation of effective section modulus (see Appendix 3C). However existing design rules given in AS 4100 estimates the effective section modulus of slender sections by assuming all the elements to be slender as for the most slender element of the section and therefore it is rather conservative (see actual AS 4100 M s in Table 6.1). The comparison of predicted section moment capacities and FEA results given in Tables 6.1 (a) and (b) demonstrate that both AS 4100 (SA, 1998) and AS/NZS 4600 section moment capacity rules conservatively estimate the reduction in capacity due to local buckling Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-5

215 in flange s top plate. The comparison of results further demonstrates that AS/NZS 4600 design rules predicts section moment capacities of RHFB consisting a slender top flange plate more accurately than AS 4100 design rules, whereas AS 4100 design predictions are more accurate when the flange top plate is either non-compact or compact. This is due to the fact that AS/NZS 4600 only considered first yield of extreme compression fibres in the section capacity calculation. However, the predicted moment capacities by AS 4100 and AS/NZS 4600 design rules are always conservative, and therefore it is safe to use them in the section moment capacity checks of RHFB subjected to pure bending with local buckling in the top flange plate. The conservative predictions of section moment capacities of RHFBs could be due to several reasons when only the flange s top plate element was slender. One reason could be due to the presence of a number of other compact elements in the RHFB section. The finite element analyses allow elastic plastic material behaviour and are therefore it likely to give higher moment capacity due to the possible inelastic reserve capacity of other compact elements in the section. Also the conservative assumption of using local buckling coefficient k as 4.0 could be another reason. It must be noted that the FEA gave a k value of Effect of Local Buckling in the Hollow Flange s Web Element The effect of local buckling in the hollow flange s web element (see Figure 6.3) on the section moment capacities of RHFB was investigated by choosing different width to thickness ratios (b/t) for this element. All the other elements within the RHFB section were chosen to be compact (non-slender) according to AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996). Tables 6. (a) and (b) show the section moment capacities (M s ) obtained from the finite element analyses for different b/t ratios of flange s web element ranging from 1.5 to 50 and the predicted section moment capacities from the current steel design standards AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996). The calculation procedure of section moment capacity (M s ) using AS 4100 and AS/NZS 4600 is presented in Appendix 3C. It must be noted that all other geometric dimensions were unchanged, and are given in the same table. The comparison of results indicates that both steel design standards underestimate the section moment capacities of RHFB when the hollow flange s web plate element experiences local buckling. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-6

216 Local buckling in Flange web Figure 6.3: Local Buckling of the Hollow Flange s Web Element Table 6. (a): Comparison of FEA Results with the Predictions from Current Steel Design Standards for G300 Steel t p (mm) b/t Element Compactness AS 4100 Section Moment Capacity M s (knm) AS/NZS AS Actual Modified AS/NZS 4600 FEA M s Ratio AS 4100 / FEA Actual Modified AS4600 /FEA S S NC S C NS C NS C NS C NS b f = 50 mm, h f = 5 mm, h w = 150 mm, t = mm for all the elements except hollow flange s web element, which is t p and it was varied as shown in Column 1, f y = 3000 MPa The comparison of predicted section moment capacities and FEA results given in Table 6. (a) and (b) demonstrates that both AS 4100 (SA, 1998) and AS/NZS 4600 section moment capacity rules conservatively estimate the reduction in capacity due to local buckling in the hollow flange s web element. The comparison of results further demonstrates that AS/NZS 4600 design rules predict the section moment capacities of RHFB including slender flange webs more accurately than AS 4100 design rules. However, AS 4100 design rules predict the section moment capacities more accurately when the flange s web elements are either non-compact or compact. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-7

217 Table 6. (b): Comparison of FEA Results with the Predictions from Current Steel Design Standards for G550 Steel t p (mm) b/t Element Compactness AS 4100 Section Moment Capacity M s (knm) AS/NZS AS Actual Modified AS/NZS 4600 FEA Ratio AS 4100 / FEA Actual Modified AS4600/ FEA S S S S NC NS C NS C NS C NS b f = 50 mm, h f = 5 mm, h w = 150 mm, t = mm for all the elements except hollow flange s web element, which is t p and it was varied as shown in Column 1, f y = 550 MPa As described before, this is due to the fact that AS/NZS 4600 limits the moment capacity to first yield of extreme compression fibres. However, the predicted moment capacities based on AS 4100 and AS/NZS 4600 design rules are always conservative, except for a few cases where the maximum unconservative error was only %, and therefore it is safe to use them in the section moment capacity checks of RHFB subjected to pure bending with local buckling in its hollow flange s web element. As described for the flange s top plate element, the reason for such conservative predictions of section moment capacities of RHFB with slender web element could be due to several reasons. One reason could again be due to the presence of a number of other compact elements in the RHFB section. Since the finite element analyses allow elastic plastic material behaviour, they could give a higher moment capacity due to the possible inelastic reserve capacity of other compact elements in the section. A conservative assumption of using k as 4.53 local buckling coefficient k could also be another reason. It must be noted that the FEA gave a k value of Effect of Local Buckling in the RHFB s Web Element The effect of local buckling in the RHFB s web element (see Figure 6.4) on the section moment capacities of RHFB was investigated by choosing different width to thickness ratios (b/t) of the web element. All the other elements in the RHFB section Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-8

218 were chosen to be compact (non-slender) according to AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996). Tables 6.3 (a) and (b) show the section moment capacities (M s ) obtained from the finite element analyses for different b/t ratios of web element ranging from 60 to 40 and the predicted section moment capacities from the current steel design standards AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996). The calculation procedure of section moment capacity (M s ) using AS 4100 and AS/NZS 4600 is presented in Appendix 3C. It must be noted that all other geometric dimensions were unchanged, and are given in the same table. The comparison of results indicates that both steel design standards underestimate the section moment capacities of RHFB when the RHFB s web element experiences local buckling. Local buckling in beam web Figure 6.4: Local Buckling of RHFB s Web Element Table 6.3 (a): Comparison of FEA Results with the Predictions from Current Steel Design Standards for G300 Steel t p (mm) b/t Element Compactness AS 4100 Section Moment Capacity M s (knm) AS/NZS AS Actual Modified AS/NZS 4600 FEA M s Ratio AS 4100 / FEA Actual Modified AS4600/ FEA S S S NS S NS NC NS C NS C NS b f = 50 mm, h f = 5 mm, h w = 150 mm, t = mm for all the elements except web element, which is t p and it was varied as shown in Column 1, f y = 300 MPa Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-9

219 Table 6.3 (b): Comparison of FEA Results with the Predictions from Current Steel Design Standards for G550 Steel t p (mm) b/t Element Compactness AS 4100 Section Moment Capacity M s (knm) AS/NZS AS Actual Modified AS/NZS 4600 FEA M s Ratio AS 4100 / FEA Actual Modified AS4600 /FEA S S S S S NS S NS NC NS C NS b f = 50 mm, h f = 5 mm, h w = 150 mm, t = mm for all the elements except web element, which is t p and it was varied as shown in Column 1, f y = 550 MPa The comparison of predicted section moment capacities and FEA results given in Tables 6.3 (a) and (b) also demonstrate that both AS 4100 (SA, 1998) and AS/NZS 4600 section moment capacity rules conservatively estimate the reduction in capacity due to local web buckling. The comparison of results also demonstrates that AS/NZS 4600 design rules predict the section moment capacities of RHFB with slender beam web more accurately than AS 4100 design rules. However, AS 4100 design rules predict the section moment capacities more accurately when the beam web is either non-compact or compact. The predicted moment capacities by AS 4100 and AS/NZS 4600 design rules are always conservative, except in a few cases where the maximum unconservative error was only 3%, and therefore it is safe to use them in the section moment capacity checks for RHFBs subjected to pure bending with local buckling of web element. As described for the hollow flange s top plate and web elements, the reason for such conservative predictions of section moment capacities of RHFBs with slender RHFB web element could also be due to the number of other compact elements in the RHFB section. A conservative assumption of using local buckling coefficient k as 4 could also be another reason. It must be noted that the FEA gave a k value of 5. This investigation shows that the AS/NZS 4600 design rules are conservative for all the possible local buckling cases of RHFB sections and therefore they can be used in the design of RHFBs for section moment capacity. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-10

220 6.. Local and Lateral Distortional Buckling Behaviour of RHFB As described in Section 6..1, the flexural behaviour of RHFBs is complex and very much depend on a range of parameters including section geometry, steel thickness and yield stress. Lateral distortional buckling is a major failure mode in RHFB due to the presence of torsionally stiff flanges and a slender web. However, research into the lateral distortional buckling behaviour of innovative steel sections has been limited and therefore the current Australian Steel Structures Design Standards AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 1996) do not include appropriate design formulae for the lateral distortional buckling in RHFBs. Design formulae are provided in AS/NZS 4600 (SA, 1996) for distortional buckling, but their applicability to various structural sections was proven unsafe by other researchers (Mahaarachchi and Mahendran, 005c). Pi and Trahair (1997) investigated the behaviour of HFBs with triangular flanges using a nonlinear inelastic method to analyse the lateral distortional buckling behaviour and suggested alternative design formulae for them. Avery et al. (000) further investigated the behaviour of such HFBs using finite element analyses and developed new design rules. Mahaarachchi and Mahendran (005a, b, c) investigated the flexural behaviour of hollow flange channel sections experimentally as well as analytically and developed new design rules by modifying the AS/ NZS 4600 (SA, 1996) design formulae. However, the alternative design formulae developed by previous researchers were specifically developed for certain section types and sizes, and therefore their applicability to the new RHFBs is not known. The inelastic buckling and strength of the new RHFB sections has not been investigated and no research has been conducted to investigate the effect of web distortion, initial geometric imperfections, residual stresses, stress-strain characteristics and moment distribution on the inelastic behaviour and buckling strength of RHFB sections. Therefore a detailed parametric study into the flexural behaviour of RHFB sections was undertaken using validated finite element models presented in Chapter 5 to ensure the relevance of available design methods developed for the HFB flexural members, and if necessary, to propose suitable modifications to account for the effects of lateral distortional buckling. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-11

221 Effects of geometric imperfection and residual stresses, and also the lack of continuity of web-flange connections on the moment capacities of RHFBs were investigated and their results are presented in the following sections. Analyses included very short (00 mm) to very long lengths (8000 mm) of RHFBs and thus both local and lateral buckling effects Effect of Initial Geometric Imperfections and Residual Stresses on RHFB Moment capacities Effects of residual stresses and initial geometric imperfections on moment capacities of RHFBs were investigated using selected slender and compact RHFB sections. Tables 6.4 (a) (c) show the comparison of analytical results for two beam sections without residual stresses and geometric imperfections (perfect), with residual stresses only, with geometric imperfections only, and with both residual stresses and geometric imperfections whereas Figure 6.5 illustrates the graphical comparison of analytical results. The moment capacities were compared for various beam spans. Percentage reduction in moment capacities due to the presence of residual stresses, geometric imperfections and their combinations were calculated and presented in Tables 6.4 (a) to (c). Table 6.4a: Effect of Residual Stresses on RHFB Moment Capacities Span (mm) Perfect S1 Slender section RS % Reduction Perfect S Compact section RS % Reduction Note: S1 RHFB-10t f -10t w -150h w -G300-50s S - RHFB-300t f -300t w -150h w -G300-50s RS Including residual stresses Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-1

222 Table 6.4b: Effect of Initial Geometric Imperfections on RHFB Moment Capacities Span mm Perfect S1 Slender section GI % Reduction Perfect S Compact section GI % Reduction Note: S1 - RHFB-10t f -10t w -150h w -G300-50s S - RHFB-300t f -300t w -150h w -G300-50s GI Including geometric imperfection Table 6.4c: Combined Effects of Residual Stresses and Initial Geometric Imperfections on RHFB Moment Capacities Span mm Perfect S1 Slender section RS + GI % Reduction Perfect S Compact section RS + GI % Reduction Note: S1 - RHFB-10t f -10t w -150h w -G300-50s S - RHFB-300t f -300t w -150h w -G300-50s RS + GI Including residual stresses and geometric imperfection Tables 6.4 (a) to (c) and Figure 6.5 show that the effect of residual stresses on moment capacities of RHFBs is higher for slender and compact beam sections with short spans whereas it is small for longer spans. The initial geometric imperfection effect on moment capacities of RHFBs is high for longer span slender and compact beam sections whereas it is small for shorter spans. As shown in Table 6.4 (a), the maximum percentage reductions in moment capacities due to the presence of residual stress are 11.7% and 13.4% for the slender and compact sections, respectively. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-13

223 S1-perfect S1-RS Moment (knm) S RHFB-300t f -300t w -150h w -G300 S1 RHFB-10t f -10t w -150h w -G300 S1-GI S1-RS+GI S-Perfect S-RS S-GI S-RS+GI Span (mm) Figure 6.5: Graphical Comparison of the Effect of Geometric Imperfections and Residual Stresses on Moment Capacities of RHFBs Table 6.4 (b) shows that the maximum reductions in moment capacities due to initial geometric imperfections are 8.4% and 7.4% for the slender and compact sections, respectively. These results implied that the effect of residual stresses on moment capacities of RHFB is greater than that of initial geometric imperfections. Table 6.4 (c) indicates that the maximum reduction of moment capacities due to the combined effect of residual stresses and the initial geometric imperfections are 15.9% and 16.7% for the slender and compact sections, respectively Effects of Screw Fastening on RHFB Moment Capacities Two separately formed rectangular hollow flanges are connected to a single web using screws at equal spacing along the member to produce RHFBs. The effects of screw fastening on moment capacities of RHFBs are not known and need to be investigated. Zhao (005) investigated the compression behaviour of similar coldformed steel sections with rectangular hollow flanges connected by intermittent fastening using screws and spot welds. His findings showed that the compression member capacity of RHFBs is reduced significantly due to increased screw spacing beyond a certain value. However, it is not known whether the same is true for the flexural members. Therefore both slender and compact RHFB sections for various Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-14

224 spans were analysed using different fastening conditions including continuous fastening and screw fastening at 50 mm and 100 mm equal spacings. The screws were modelled by using Rigid Fixed MPC type assuming that sufficient screw diameters are selected to resist induced tensile and shear stresses in the fasteners. (a) 50 mm Screw Spacing (b) 100 mm Screw Spacing (c) Continuous Fastening Figure 6.6: Effect of Fastening Arrangement on the Deformation Shape of RHFB Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-15

225 Figures 6.6 (a) to (c) show the FEA deformation shapes of RHFB-080t f -080t w - 150h w -3L-G300 for different fastening arrangements. The comparison of Figures 6.7 (a) to (c) indicates that the end of the web at compression side of the beam has undergone some deformations due to the discontinuity of flange and web connection due to screw fastening. However, such deformation is not observed in the continuously fastened RHFB. It further shows that the large deformation for higher screw spacing. Figures 6.7 (a) to (c) show the stress contours from the finite element analyses of RHFB-080t f -080t w -150h w -3L-G300 using the ideal simply supported and uniform moment conditions for different fastening arrangements including 50 mm and 100 mm screw spacing and continuous fastening. (a) 50 mm Screw Spacing (b) 100 mm Screw Spacing Figure 6.7: Stress Contours of RHFB for Different Fastening Arrangements Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-16

226 (c) Continuous Fastening Figure 6.7: Stress Contours of RHFB for Different Fastening Arrangements Figures 6.8 (a) and (b) shows a close-up view at a screw location at failure. It shows large stress concentrations in Figure 6.8 (a) whereas Figure 6.8 (b) indicates some deformations around the screw fasteners due to these stresses. (a) Stress Distribution (b) Deformation Figure 6.8: Close-up View at a Screw Location Tables 6.5 (a) to (d) compare the analytical results of four RHFB sections ( slender and compact sections) with different fastening arrangements. The percentage reductions in moment capacities due to screw fastening were also calculated and presented in Tables 6.5 (a) to (d). Figures 6.9 (a) and (b) present a graphical comparison of moment capacity reduction due to screw fastening. Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 6-17