Flexural Behaviour and Design of Cold- formed Steel Beams ... · formed Steel Beams with...

Flexural Behaviour and Design of Cold-formed 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 2005

Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges iii

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 iv

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 v

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 vi

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 vii

Publications

Publications in Preparation: 1. Wanniarachchi, KS. and Mahendran, M. (2005) Section Moment Capacities of

Rectangular Hollow Flange Beams. 2. Wanniarachchi, KS. and Mahendran, M. (2005) Experimental Investigation of

Member Buckling Behaviour of Rectangular Hollow Flange Beams 3. Wanniarachchi, KS. and Mahendran, M. (2005) Finite Element Modeling of

Rectangular Hollow Flange Beams 4. Wanniarachchi, KS. and Mahendran, M. (2005) Development of Design Models

for Local Buckling of Rectangular Hollow Flange Beams 5. Wanniarachchi, KS. and Mahendran, M. (2005) 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 Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges viii

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Flexural Behaviour and Design of Cold-formed Steel Beam with Rectangular Hollow Flanges xix

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Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxii

Notation

Abbreviations AISI = American Iron and Steel Institute

AS 4100 = Australian Standard for the Design of Steel Structures

AS/NZS 4600 = Australian Standard for the Design of Cold-formed Steel Structures

ASD = allowable stress design

BHP = BlueScope Steel Products

BMT = based metal thickness

BSI = British Standards Institution

C3D8 = eight node linear brick element

CHS = circular hollow section

COV = covariance

CSA = Canadian Standard Association

FEA = finite element analysis

HFB = hollow flange beam

LRFD = load and resistance factor design specification

LSB = light steel beam

MPC = multipoint constraint

PDT = potentiometric displacement transducers

PTM = Palmer Tube Mills

QUAD4 = quadrilateral shell element

RHFB = rectangular hollow flange beam

RHS = rectangular hollow section

S4 = quadrilateral general purpose shell element with four nodes and six degrees of freedom per node

S4R4 = quadrilateral thin shell element with four nodes, reduced integration, and five degrees of freedom per node

SHS = square hollow section

SPC = single point constraint

TCT = total coated thickness

TRIA3 = triangular shell element

UC = universal column

Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxiii

Symbols ν = poisons ratio

λ = half-wavelength at distortional buckling, or non-dimensional slenderness

ρ = effective width factor

ρnom = measured longitudinal stress (tensile coupon tests)

ρtrue = true longitudinal stress (modified stress)

δ = global imperfection

δl = local imperfection

βo = target reliability index

λd = non-dimensional member slenderness for distortional buckling

λe = plate element slenderness

αm ,cb = moment modification factor

λs = section slenderness

αs = slenderness reduction factor for lateral tortional buckling

αsd = slenderness reduction factor for lateral distortional buckling

λsp = section plastic slenderness

λsy = section yield slenderness

εm = measured longitudinal strain

εn = measured longitudinal strain (tensile coupon test)

εp(ln) = true longitudinal strain (modified strain)

φ = capacity reduction factor

A = cross section area

B = element overall width (HFB)

b = element flat width (general)

be = effective width of flat plate

bf = flange width (RHFB)

Cp = correction factor

d = web depth (HFB)

D = overall depth (HFB)

E = young’s modulus of elasticity

fc , σc = critical stress

fcr = elastic critical stress for local buckling

Fm = mean of the fabrication factor

fmax = maximum edge stress

Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxiv

fod = distortional buckling stress

fol = local buckling stress

fu = tensile strength

fy ,σy = material yield stress

f* = applied stress

G =shear modulus

hf = flange height (RHFB)

hl = lip height (RHFB)

hw = web height (RHFB)

I = second moment of area

Ix = second moment of area about major principal axis

Iy = second moment of area about minor principal axis

Iw = warping constant

Iy = moment of inertia about minor axis

J = polar moment of inertia

Je = effective polar moment of inertia

k = local buckling coefficient

kφ = rotational spring stiffness

L = member length

Le , le = effective length

Lla = initial lever arm length

m = degree of freedom

M = applied moment

M* = design bending moment

Mb = nominal member moment capacity

Mm = mean of the material factor

Mc = critical moment

Mo =flexural torsional buckling moment resistance

Mod = distortional buckling moment resistance

Ms = nominal section moment capacity

MTH = Thin-wall buckling moment

Mu = ultimate moment capacity

My = yield moment

n = number of tests

P = applied jack load

Flexural Behavior and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxv

Pm = mean value of the tested to predicted load ratio

Py = squash load

Ro = corner radius (HFB)

t = steel thickness

tf = thickness (RHFB)

tw = thickness of web (RHFB)

VF = covariance of the fabrication factor

Vm = covariance of the material factor

Vp = Covariance of the tested to predicted load ratio

VQ = covariance of load effect

w = plate width

Z, Zx or Zf = full section modulus about major axis

Ze = effective section modulus

Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges xxvi

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 1-1

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 cold-

formed 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.


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.

1.2 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, 2000)

These sections are commonly in used, but they are more susceptible to structural

instabilities due to their geometrical shapes. The characteristics due to mono-

symmetric 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


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 1.2 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 1.2)

improved their buckling behaviour significantly. They found that this improvement


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 so-

called 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 1997. 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 250 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 (2001) 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


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 cold-

formed 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.

(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

bf

hf

hw

hf

tf

tw

hl


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?

2. 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?


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 1.5.1 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.

1.5.2 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.

2. 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 2), and use them to investigate the local, lateral distortional

and flexural (lateral) torsional buckling modes of failures.


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 2 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 (2000) and

Yuan (2005) 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.


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 2: A summary of current literature relating to various aspects of cold-

formed 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.


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.


CHAPTER 2 Literature Review

2.1 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 so-

called 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.

2.2 Special Characteristics and Design Considerations of Cold- formed 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.

2.2.1 Methods of Forming

In general, two manufacturing methods are used to produce various shapes of cold-

formed steel sections (see Figures 2.1 and 2.2), and they are cold roll-forming and

press brake operations.

2.2.1.1 Cold Roll-forming

The cold roll-forming process consists of feeding a continuous steel strip through a

series of opposing rolls (see Figure 2.1a) to deform the steel plastically to form the

desired shapes. The process involved in cold-forming a Z- section is illustrated in

Figure 2.1b. 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, 2000). This method is


usually used to produce cold-formed steel sections where a large quantity of a given

shape is required.

(a) Cold Roll-forming Machines (Yu, 2000) (b) Roll-forming Sequence for a Z- Section (Hancock, 1998)

Figure 2.1: 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.

2.2.1.2 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 2.2). 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).

2.2.2 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


Figure 2.2 Press Braking (Karren, 1967)

such as I, L, C and Z, which are commonly used in engineering practices of cold-

formed steel construction. However with the improvement of industrial cold-

forming processes, more complex section types are possible (see Figure 2.3) 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, 2000).

Figure 2.3: Various Shapes of Cold-formed Steel Sections (Yu, 2000)


2.2.3 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 2.4).

Figure 2.4: 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 2.5.


Figure 2.5: 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.


2.2.4 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 cold-

formed 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 1974. 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 ASCA1-1968 (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 1988. 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 S136-94 (CSA, 1994). The corresponding design approach of steel

structural design standards, AS 4100 and AS/NZS 4600, will be discussed later in

this chapter.


2.2.5 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 cold-

formed 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.

2.2.5.1 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 2.6(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 2.3.21.

(a) Box Girder (b) Plate Girder

Figure 2.6: Local Buckling of Compression Flanges (SCI, 1998)


2.2.5.2 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

2.7. The eccentricity of loads from the shear centre axis will generally produce

considerable torsional deformation in the thin-walled beams as a result of flexural-

torsional buckling (see Fig. 2.7). The torsional rigidity of an open section is

proportional to t3 (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 cold-

formed steel sections to achieve an economic solution.

Figure 2.7: Torsional Deformations in Eccentrically Loaded Channel Beam (Hancock, 1998)

2.2.5.3 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 flexural-

torsional buckling, may undergo a mode of buckling commonly called distortional

buckling (Hancock, 1997). The previous research studies (Ellifritt et al., 1992,

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,


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 2.8 (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 2.3.2.2 of this chapter gives a

comprehensive review of distortional buckling.

(a) Distortional Buckling (b) Lateral Distortional Buckling

Figure 2.8: Buckling of a Channel Section and a Hollow Flange Beam (SA, 1998b)

2.2.5.4 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 2.9).

(a) Clinching (b) Screw Fastener (c) Bolt Fastener

Figure 2.9: Generally Used Cold-formed Steel Fasteners


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,

2000, see Figure 2.10). 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 2.10: Cross section of a Self-piercing Rivet (Voelkner, 2000)

2.3 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 cold-

formed 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.


2.3.1 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, 1992). 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 2.11 represent the buckling stress versus buckle half-

wavelengths 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 2.11 clearly demonstrates that at short half-wavelengths (50 mm-500 mm),

the changed buckling mode from local buckling in the unattached flange element in

HBS2 changes to local buckling of the top flange at a higher stress in HBS1. At long

half-wavelengths (2 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 (HBS2) is a

conventional flexural-torsional buckle. The flexural-torsional buckling mode of the

open HBS2 section involves longitudinal displacements of the cross-section (called

warping displacements) such that the longitudinal displacements at the free edges of-


Figure 2.11: 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 HBS2. 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 2.11 (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.

2.3.1.1 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 2.7). The local buckling failure in thin-


walled sections can occur under compression, bending or shear loading. Previous

researchers (Bleich, 1952: 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 ( crf ) of a plate element is determined

using Bryan’s (1891) differential Equation (2.1) based on small deflection theory;

2

2

4

4

22

4

4

4

2

3

2)1(12 x

wtf

yw

yxw

xwEt

x ∂∂−=�

�

��

�

∂∂+

∂∂∂+

∂∂

−ν (2.1)

Bryan’s differential equation has been developed based on a rectangular plate of

width w, length a and thickness t, with in plane stress fx acting on the plate as shown

in Figure 2.12. The solution of Bryan’s differential equation for the elastic critical

local buckling stress (fcr) is given by;

2

2

2

)1(12 ��

��

�

−=

wtEk

f cr νπ

(2.2)

Figure 2.12: Rectangular Plate Subjected to Compression Stress (Hancock, 1998)

The elastic critical local buckling stress (fcr) 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


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 (fcr) calculated using Equation 2.2 is

less than the material yield stress (fy). A slender section will buckle locally before

the squash load (Py) or the yield moment (My) is reached. If the elastic critical

buckling stress (fcr) exceeds the yield stress fy, the compression element will buckle

in the inelastic range (Yu, 2000).

Equation 2.2 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, 2000). The bending buckling coefficient, k for long plates

was found to be 23.9 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 2.13. Bulson (1969) also showed

the influence of bending stress ratio fc /ft 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 half-

wavelengths 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 2.1.

Figure 2.13 Bending Buckling Coefficient k Vs Aspect Ratio, a/h (Gerard and Becker, 1957)


Table 2.1: 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 2.14. 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).


Figure 2.14: 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 2.15 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 2.15: Different Buckling Modes and Buckling Stresses for a Triangular HFB Subjected to Bending (Dempsey, 1990)


By rearranging Equation 2.2 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 2.2 shows the values of k calculated using Equation

2.3 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 2.2 Local Buckling Coefficients of Flange for Major Axis Bending (Dempsey, 1990)

Designation B t R b/t fol k

300HFB43 90 4.3 9 16.7 2396 3.72

300HFB38 90 3.8 9 18.9 1980 3.93

300HFB33 90 3.3 9 21.8 1588 4.18

300HFB28 90 2.8 9 25.7 1216 4.45

250HFB28 90 2.8 9 25.7 1219 4.46

250HFB23 90 2.3 9 31.3 868 4.71

200HFB23 90 2.3 9 31.3 868 4.71

200HFB18 90 1.8 9 40.0 552 4.89

Note: B, t, and R - geometric parameters of triangular HFBs (see Figure 1.3) fol - local buckling stress in MPa and k – plate buckling coefficient

2.3.1.2 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


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 2.2.5.3). 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.

(2003) studied distortional buckling of channel beam-columns.

The formulae to predict the elastic distortional buckling stress (fod) 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 2.16a. 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 2.16b.

(a) Compression (Lau and Hancock, 1987) (b) Flexure (Hancock, 1997)

Figure 2.16: Analytical Models for Distortional Buckling in Compression and Flexure


The rotational spring stiffness, k� is given in Lau and Hancock (1987) as:

( ) ��

�

�

��

�

�

��

�

�

�

+−

+=

2

22

2

2

3 '11.11

06.046.5 λλ

λφw

wod

w b

b

Et

fb

Etk (2.4)

where E is the modulus of elasticity, t is the thickness, bw 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 2.4, � is the half-wavelength of the distortional buckling, and

is given by:

25.0

3

2

8.4��

�

�

�=

t

bbI wfxfλ (2.5)

in which, Ixf and bf 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 2.17a,

it is treated as a simply supported beam in flexure (see Figure 2.18a). The rotational

stiffness at the end would then be 2EI/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

2.17b, it is treated as a beam simply supported at one end and built in at the other

(see Figure 2.18b). 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

2.17a to Figure 2.17b will approximately double the torsional restraint stiffness k�

(Hancock, 1997).

(a) (b)

Figure 2.17: Buckling of a C- section Under Compression


(a) (b)

Figure 2.18: Symmetric and Asymmetric Restrained Bending (Hancock, 1997)

On this basis, Equations 2.4 and 2.5, which were developed for the compression

members by Lau and Hancock (1987), were later revised by Hancock (1997) for

flexural members as:

( ) ��

�

��

�

��

�

�

�

++−

+=

2244

24

2

3

39.13192.256.12

'11.11

06.046.52

wdwd

dwod

dw bb

b

Et

fb

Etk

λλλ

λφ (2-6)

25.0

3

2

28.4

��

�

�

�=

t

bbI wfxfdλ (2-7)

Clause 3.3.3.3 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 fod is calculated using equations provided in

Appendix D of the AS/NZS 4600. These formulae are based on Hancock’s (1997)

distortional buckling formulae for C-section flexural members, and so that they may


only cover the ‘Part a’ of Clause 3.3.3.3. AS/NZS 4600 has recommended the

equations given in Appendix D to be used in calculating the distortional buckling

stress (fod) in the case of lateral distortional buckling (Part b of Clause 3.3.3.3) 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. (2000), 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 4600.

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 (1992) 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 2.8). 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 half-

wavelengths (see Figure 2.15). At long wavelengths, the buckling curve is similar to,

but lower than the flexural-torsional buckling curve which is shown as a dashed line


in Figure 2.15. 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, 1982). 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, 1982) 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 Mo was modified by

Pi and Trahair (1997) by introducing an effective torsional rigidity GJe in place of

the nominal torsional rigidity GJ to calculate the lateral distortional buckling moment

resistance Mod.


The elastic flexural-torsional buckling moment resistance without web distortion is

given as:

��

�

�+= 2

2

2

2

LEI

GJL

EIM wy

o

ππ (2.8)

in which, EIy = minor axis flexural rigidity; EIw = warping rigidity; and L = span

length. This formula is revised to include the effect of web distortion as:

��

�

�+= 2

2

2

2

LEI

GJL

EIM w

ey

od

ππ (2.9)

The approximate lateral distortional buckling moment resistance, Mod is obtained by

using the approximate effective torsional rigidity GJe given by;

( )

��

�

�+

��

�

�

=

cF

cF

e

dLEt

GJ

dLEt

GJGJ

2

23

2

23

91.02

91.02

π

π (2.10)

in which, GJF = torsional rigidity of a hollow flange about its own axis; E = Young’s

modulus of elasticity, dc = clear web depth and t = web thickness.

Myz, Myzd – Flexural torsional and lateral distortional Buckling Moment, MTW – Thin-wall values

Figure 2.19 Lateral Distortional Buckling Moments (Pi and Trahair, 1997)


The elastic lateral distortional buckling moments, predicted by Thin-wall (MTW) and

obtained from Equation 2.9 (Mod) were compared with flexural torsional buckling

moment (Mo) for two triangular HFB sections as shown in Figure 2.19. It can be

seen that the approximate values Mod are in close agreement with the accurate Thin-

wall values MTW, and also these lateral-distortional buckling values are significantly

lower than the flexural-torsional buckling moments Mo.

2.4. Design Procedures for HFBs

Current specifications for the design of flexural steel members are based on semi-

empirical 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 2.4.1 and 2.4.2. However, shear, bearing, flange curling and web crippling

are not considered in this study as they are outside the scope of this research.

2.4.1. Design procedures of AS 4100

The nominal section moment capacity (Ms) is defined in Clause 5.2.1 (SA, 1998) as

follows:

Ms = Zefy (2.11)

The effective section modulus (Ze) shall be either plastic section modulus or reduced

section modulus to allow for flexural local buckling. The effective section modulus is

specified in Clauses 5.2.3, 5.2.4, or 5.2.5 (SA, 1998) as follows:

sps λλ ≤ =eZ Lesser of S or 1.5Z (2.12)


syssp λλλ ≤< ( )��

�

��

�−

��

�

�

�

−−

+= ZZZZ cspsy

ssye λλ

λλ (2.13)

sys λλ > ��

�

�=

s

sye ZZ

λλ

(2.14)

where S and Z are the plastic and elastic section modulii, respectively. Ze is the

effective section modulus as specified in Clause 5.2.3 (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 5.2.2 (SA, 1998) as a function of

the element clear width (b), thickness (t), and yield stress (fy):

E

f

tb y

e =λ (2.15)

The section plasticity and yield slenderness limits (�sp, �sy) are taken as the values of

the element slenderness limits (�ep, �ey) given in Table 5.2.2 (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 (Mb) of members with full lateral restraint is

specified in Clause 5.2.1 (SA, 1998) as equal to the nominal section moment

capacity of the critical section. The critical section is defined in Clause 5.3.3 (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 (Ms).

The nominal member moment capacity (Mb) of a beam without full lateral restraint

has been specified in Clause 5.6.1 (SA, 1998) as:

sssmb MMM ≤= αα (2.16)


where the moment modification factor (�m) shall be determined from one of the

methods described in Clause 5.6.1.1 (SA, 1998). For uniform bending moment

distribution �m =1.0. The slenderness reduction factor (�s) is defined in Clause

5.6.1.1 (SA, 1998) as:

��

�

�

��

�

�

−��

�

�

��

�

�+��

�

�

�=

o

s

o

ss M

MMM

36.02

α (2.17)

where, reference buckling moment (Mo) is defined in Clause 5.6.1.1 (SA, 1998) as:

��

�

�

��

�

�

��

�

��

�

��

�

�

�+

��

�

�

�= 2

2

2

2

e

w

e

yo

L

EIGJ

L

EIM

ππ (2.18)

Therefore, the member capacity of a beam subjected to a uniform bending moment

can be rewritten as:

sso

s

o

sb MM

MM

MM

M ≤��

�

�

��

�

�−+��

�

�

�= 36.0

2

(2.19)

Avery et al. (2000) pointed out that Equation 2.19 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 (1992) 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 (Mo) in Equation 2.19

is replaced with the elastic distortional buckling moment (Mod), 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 2.9 and 2.10 to

determine Mod for use with AS 4100 procedure. Their research showed that Equation

2.19 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.


sso

s

o

sb MM

MM

MM

M ≤��

�

�

��

�

�−+��

�

�

�= 8.26.0

2

(2.20)

2.4.2. Design procedures of AS/NZS 4600

The nominal section moment capacity (Ms) is specified in Clause 3.3.2 (SA, 1996) in

a similar fashion to AS 4100 as follows:

yes fZM = (2.21)

However, unlike AS 4100, the effective section modulus (Ze) 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 (be) 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 (fmax) acting over a fictitious effective width

(be) as shown in Figure 2.20.

Figure 2.20: Stress Distribution in Stiffened Compression Element (SA, 1998b)

The effective width concept was first introduced by von Karman et al. (1932) and

since then extensive investigations on light-gauge, cold-formed steel sections have

been carried out. The following equations to calculate effective width (be) was

developed by Winter (1946) for a stiffened compression element simply supported at

both longitudinal edges.


bb e ρ= (2.22)

where, � is the effective width factor defined in Clause 2.2.1.2 (SA, 1996) as:

0.1

22.01

≤��

�

� −=

λλρ (2.23)

where � is the slenderness ratio and is calculated as:

Ef

tb

kmax052.1

��

�

��

�

�=λ (2.24)

In Equation 2.24, k is the plate buckling coefficient and depends on edge boundary

conditions and type of stress (see Section 2.3.2.1). For nominal section moment

capacity (Ms) calculations, fmax is assumed equal to yield stress, fy, for the extreme

flange element

The nominal member moment capacity (Mb) is specified in Clause 3.3.3 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 3.3.3.2 (SA,

1996) as follows:

��

�

�

�=

f

ceb Z

MZM (2.25)

For hollow flange beams, it is appropriate to determine the effective section modulus

Ze at a stress level Mc/Zf, where Mc is the critical moment as defined in Equations

2.26 and 2.27.

For 44.1<dλ ��

�

��

�

��

�

�

�−=

41

2d

yc MMλ

(2.26)


For 44.1≥dλ ��

�

�

�=

2

1

d

yc MMλ

(2.27)

The non-dimensional member slenderness (�d) is given by:

od

yd M

M=λ (2.28)

Avery et al. (2000) 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.

2.4.3. 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. (2000) as a more accurate

and reliable alternative to the AS 4600 design methods. Trahair (1997) equations for

flexural member capacity are given next.

ssnd

b MMc

babM ≤

��

�

��

�

+−+=

21 λ; ob MM ≥ (2.29)

The non-dimensional member slenderness (�d) is given by;

od

sd M

M=λ (2.30)

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. Mu/Ms) and the

normalized design capacity (i.e. Mb/Ms). The results indicated an unacceptable


maximum unconservative error of more than 10 percent. Therefore the coefficients

were established by Avery et al. (2000) for separate group of sections with the same

thickness as given in Table 2.3. The member capacity predicted by Avery et al.

(2000) using the modified Trahair equations was found to be significantly more

accurate and precise than the AS 4100 predictions.

Table 2.3: Coefficients for Trahair’s (1997) Member Capacity Equation

Coefficient t=3.8 t=3.3 t=2.8 t=2.3 Overall

a 1.006 0.999 0.997 0.997 1.000

b 0.024 0.012 0.000 0.000 0.000

c 0.448 0.377 0.321 0.273 0.424

n 1.350 1.407 1.429 1.469 1.196

2.5 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.

2. 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),


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 post-

processing 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

(2003) gave an overview of possible errors, which might occur during linear and

non-linear finite element analysis. Table 2.4 shows a summary of errors that can

occur during finite element modelling.

Table 2.4 Overview of Possible Errors During FEA (Bakker and Pekoz, 2003)

Reality Idealization error

Mechanical model PREPROCESSING Input error Discretization error: equilibrium approximated Geometry error: Geometry approximated Shortcoming in element formulation Program bugs Finite element model SOLUTION Solution error Convergence error Program bugs

Nodal displacements Program bugs

Derive results according to finite element model 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


2.5.1 Analysis Types

2.5.1.1 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 2.21.

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 large-

displacement problems can themselves experience stress-stiffening effects, in which

case the geometry stiffness matrix must also be included in the large-displacement

solution.


Figure 2.21: Material Non-linearity

2.5.1.2 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. (2003) 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.


2.5.2 Initial Conditions

2.5.2.1 Geometric Imperfections

Geometrical imperfections refer to the deviation of a member from perfect geometry

(see Figure 2.22). 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 < 200, and Type 2 imperfections for w/t < 100 (see Figure 2.21).

For Type 1 imperfections an approximate expression d1 ≈ 0.006w 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.

tted 2

1 6 −≈ (d1 and t in mm)

For Type 2 imperfections the maximum deviation is approximately equal to the plate

thickness:

d2 ≈ t

(a) Type 1 (b) Type 2

Figure 2.22: Definition of Geometric Imperfections (Schafer and Pekoz, 1998b)


When precise data of the distribution of geometric imperfection is not available,

three approaches have been used (Pekoz et al., 2003). 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

2.23.

(a) Local buckling (b) Distortional buckling (c) Imperfection

Figure 2.23: Geometric Imperfection (Pekoz et al., 2003)

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. (2000) 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 2.24.


Figure 2.24: Local Imperfections (Avery et al., 2000)

2.5.2.2 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 2.25a). 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

2.25b.


(a) Definition of Flexural and (b) Average Flexural Residual Stress as % fy Membrane Residual Stresses

Figure 2.25: 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 2.26). The same residual stress

model was used by Avery et. al. (2000), 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 2.26: Bending Residual Stress Distribution for Outside Fibre, Expressed as a Percentage of the Yield Stress (fy) with Tension Positive (Doan and Mahendran, 1996)

2.5.3 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., 2000, Pi and


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. (2000)

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 2.27(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.


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 2.5.2.1 and 2.5.2.2). 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 2.27: Finite Element Models of HFBs (Avery et al., 2000)


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

2.28.

Figure 2.28: 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. (2000) 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


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 (1992). It

consists of linear variation in lateral displacement for all the nodes on the cross-

section, 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 2.29.

Three types of material properties for the flanges, web and corners were used and the

nonlinear analyses included bending residual stresses.

Figure 2.29: Physical Model and Typical Finite Element Mesh (Wilkinson and Hancock, 1999)


2.6 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 × 25 mm × 2.5 mm. Spans were varied from 2000 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 2.30a. The

support system used in this study (see Figure 2.30b) 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 2.30: Lateral Buckling Test of RHF Beams (Zhao et al., 1995)


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 (x1 – x1).

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 (y1-y1) (see Figure 2.30). 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 2.31. 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.


Figure 2.31: 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


used to validate the finite element model developed by Avery et al. (2000). 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 2.32.

Figure 2.32: 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

0.55 m 0.55 m 4.5 m


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 2.33). 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 2.33: Lateral Buckling Tests of Cold-formed C- and Z- section Beams (Put et al., 1999)


(c) Support System for Z- section

Figure 2.33: Lateral Buckling Tests of Cold-formed C- and Z- section Beams (Put et al., 1999)

2.7 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 cold-

formed 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.


• 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

2.2.4) allows for this effect by introducing average yield strength (fya) for cold-

formed 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 cold-

formed steel design is important to achieve more structurally efficient cold-

formed structures in an economical manner (see Section 2.2.5.1).

• 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 (t3), 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 2.2.5.3 and 2.3.3.2).

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


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 2.4.1). 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 (Mo) given in AS 4100 is replaced by elastic lateral

distortional buckling moment (Mod), 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


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.


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.

3.2 Material Property Tests

3.2.1 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,


were chosen and their nominal yield strengths (fyn) are 300 MPa, 500 MPa and 550

MPa, respectively, and the nominal tensile strengths (fun) 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, 2001).

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, 1992). 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 (1992) 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, 1992)

Chemical composition (%) Steel Grade

Nominal thickness

(mm) C P Mn Si S Ni Cr Mo Cu Al Ti Nb

0.55 .050 .015 .210 .005 .017 .024 .014 .002 .006 .032 .003 .001

0.80 .050 .013 .200 .005 .010 .023 .018 .002 .012 .036 .003 .001

1.20 .060 .007 .220 .005 .013 .025 .010 .002 .006 .031 .003 .001 G300

1.95 .150 .015 .750 .015 .010 .005 - - - .048 - -

0.55 .060 .012 .210 .005 .014 .026 .011 .002 .006 .038 .003 .001

0.75 .053 .010 .250 .010 .010 .004 - - - .045 - - G550

0.95 .055 .012 .210 .005 .014 .029 .017 .002 .008 .040 .003 .001

G500 1.15 .053 .010 .250 .010 .010 .004 - - - .045 - -

Note: Additional data is presented in Appendix 3A


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

(1982) 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, 25 mm × 100 mm were

cut from each steel grade and thickness giving a total of 24 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 3.2.

(a) Specimens in hydrochloric acid

Figure 3.1: Base Metal Thickness Measurement


(b) Specimens after coating wash-off

Figure 3.1: Base Metal Thickness Measurement

Table 3.2: Metallic Coating Details and Measured Thicknesses of Steel Sheets

Measured (mm) Steel Grade

Nominal BMT (mm) Coating type

TCT BMT Calculated CT

(mm)

0.55 Zincalume (AZ150) 0.603 0.543 0.060

0.80 Zincalume (AZ150) 0.860 0.800 0.060

1.20 Zincalume (AZ150) 1.255 1.192 0.063 G300

1.90 Galvanized (Z275) 1.923 1.882 0.041

0.55 Zincalume (AZ150) 0.617 0.553 0.064

0.75 Galvanized (Z350) 0.800 0.748 0.052 G550

0.95 Zincalume (AZ150) 1.012 0.947 0.065

G500 1.15 Galvanized (Z350) 1.190 1.148 0.042 Note: Coating Thickness, CT = TCT – BMT

3.2.2 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


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 3.2 (a)

whereas Figure 3.2 (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.

(a) Nominal Dimensions of a Typical Tensile Test Specimen

(b) Strain Gauged Tensile Test Specimens

Figure 3.2: Tensile Test Specimens

13 mm R20

40 mm 40 mm 80 mm

200 mm

25 m

m


Table 3.3: Measured Dimensions of Tensile Test Specimens

Measured Thickness (mm) No Steel

grade

Nominal Thickness

(mm) TCT BMT Width

B (mm)

1 0.55 0.603 0.543 12.83

2 0.55 0.605 0.544 12.71

3 0.80 0.864 0.802 12.77

4 0.80 0.860 0.800 12.76

5 1.20 1.251 1.191 12.79

6 1.20 1.255 1.194 12.77

7 1.90 1.920 1.882 12.76

8

G300

1.90 1.925 1.885 12.76

9 0.55 0.616 0.552 12.73

10 0.55 0.620 0.551 12.76

11 0.75 0.802 0.747 12.74

12 0.75 0.805 0.752 12.77

13 0.95 1.013 0.943 12.73

14

G550

0.95 1.018 0.945 12.78

15 1.15 1.192 1.146 12.75

16 G500

1.15 1.190 1.150 12.76

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 2.5 × 10-4 s-1 to 2.5

× 10-3 s-1 and aimed at a target value of 8 × 10-4 s-1. A 2 mm strain gauge installed at

the mid-height of the test specimens (see Figure 3.2(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.


Figure 3.3: Tensile Test Set-up

3.2.3 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.

50 mm extensometer

Top jig

Bottom jig

Specimen mounting handle

Specimen

Strain gauge

Grip jig


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 7.9 × 10-2 mms-1 that gave a target strain rate of about 8 × 10-4 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.

3.2.4 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.

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25 30 35

% Strain

Stre

ss (M

Pa)

G300-0.55 mm

G300-0.80 mm

G300-1.20 mm

G300-1.90 mm

(a) Stress-Strain Curves for G300 Steels

Figure 3.4: Typical Tensile Stress versus Strain Curves for Different Steel Grades and

Thicknesses


0

100

200

300

400

500

600

700

0 1 2 3 4 5 6 7 8

% Strain

Stre

ss (M

Pa)

G550_0.55 mm

G550_0.75 mm

G550_0.95 mm

G500_1.15 mm

(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

fy (MPa) fu (MPa) E (GPa) Test

No Grade Nominal

Thickness (mm) Msd. Ave. Msd. Ave. Msd. Ave.

fu/ fy

1 0.55 357 409 207 1.15

2 0.55 351 354

395 402

207 207

1.13

3 0.80 336 392 203 1.16

4 0.80 328 332

386 389

203 203

1.17

5 1.20 327 378 201 1.16

6 1.20 313 320

375 377

201 201

1.19

7 1.90 298 374 207 1.25

8

G300

1.90 294 296

373 373

207 207

1.26

9 0.55 647 659 226 1.02

10 0.55 658 652

677 668

226 226

1.03

11 0.75 648 653 224 1.01

12 0.75 653 650

672 663

224 224

1.03

13 0.95 618 656 217 1.06

14

G550

0.95 610 614

639 648

217 217

1.05

15 1.15 590 621 223 1.05

16 G500

1.15 575 583

601 611

221 222

1.05 Note: Msd. – Measured Ave – Average


0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25 30 35

% Strain

Stre

ss (M

pa)

Lower yield point, fy = 320 MPa, e = 0.00149

×

Tensile strength fu=380 MPa

Fracture point

Slope = E =215GPa

(a) 1.2 mm G300 steel

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6 7 8

% Strain

Stre

ss (M

pa)

0.2% offset yield, fy = 610 MPa

Slope = E = 217 GPa

Tensile strength, fu =639 MPa

Fracture point

(b) 0.95 mm G550 steel

Figure 3.5: Illustration of Basic Material Properties

Stre

ss (M

Pa)

Stre

ss (M

Pa)


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 (fy), ultimate

tensile strength (fu), 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 0.2% 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 fu/fy, 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 (fu/fy) 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.


(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 4100.

Clause 1.5.1.1 of AS/NZS 4600 (SA, 2005) recommends that the structural steel

shall comply with one of the following standards: AS 1163 (SA, 1991), AS 1397

(SA, 2001), AS/NZS 1594 (SA, 2002), AS/NZS 1595 (SA, 1998) and AS/NZS 3678

(SA, 1996), as appropriate.

Previous investigations (CASE, 2002) at the University of Sydney on certain cross-

sections have shown that they do not comply with the above standards due to the

manufacturing process used. For those situations where Clause 1.5.1.1 is not

satisfied, AS/NZS 4600 allows the use of other steels, the properties and suitability

of which are in accordance with Clause 1.5.1.5 of AS/NZS 4600 (SA, 2005).

According to Clause 1.5.1.5 (b) of AS/NZS 4600 (SA, 2005), G550 steels with

thickness less than 0.9 mm, the yield stress (fy) and the tensile strength (fu) 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 (fy) and the tensile strength (fu) used in design are taken as 75% of the

corresponding specified values or 410 MPa whichever is the lesser.

Fracture point

Fracture point


3.3 Section Capacity Tests

3.3.1 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 22

section capacity tests in this investigation.

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 10-16×16 mm at 50 mm and 100 mm spacings. The

rectangular hollow flanges of sizes 50 mm × 25 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, 1.20 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.

Figure 3.7: RHFB Cross-section

hw

hf

bf

tw

tf

Screw spacing along the beam “s”

hl

hf


The specimen label consists of the beam type (i.e. RHFB) followed by a series of

numbers and scripts. For example, the label “RHFB-120tf -055tw -150hw -G300-50s”

defines a rectangular hollow flange beam (RHFB) specimen made of 1.2 mm flange

thickness (120tf), 0.55 mm web thickness (055tw), 150 mm web height (150hw),

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, 25, 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 22 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

RHFB–120tf – 055tw–150hw –G300–50s

Rectangular Hollow Flange Beam

Flange Thickness = 1.20 mm

Web Thickness = 0.55 mm Web Depth = 150 mm

Screw spacing = 50 mm

Steel grade = G300

Figure 3.8: Specimen Labelling


(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

Measured dimensions

Flange Web Beam RHFB Designation bf

(mm) hf

(mm) hl

(mm) tf

(mm) hw

(mm) tw

(mm)

Maximum global

Imperfection δ (mm)

1 RHFB-120tf-055tw-100hw-G300-50s 52.1 29.2 13.9 1.192 94.5 0.543 1.1

2 RHFB-120tf-055tw-100hw-G300-100s 52.3 29.3 13.7 1.192 94.2 0.543 1.2

3 RHFB-080tf-080tw-150hw-G300-50s 52.6 31.5 14.4 0.800 146.4 0.800 1.2

4 RHFB-080tf-080tw-150hw-G300-100s 52.1 31.1 14.0 0.800 145.3 0.800 1.3

5 RHFB-120tf-120tw-150hw-G300-50s 52.7 30.4 13.8 1.192 146.1 1.192 2.5

6 RHFB-120tf-120tw-150hw-G300-100s 53.1 30.8 14.2 1.192 146.5 1.192 2.3

7 RHFB-080tf-190tw-150hw-G300-50s 51.0 31.2 13.5 0.800 145.4 1.882 1.3

8 RHFB-080tf-190tw-150hw-G300-100s 51.4 31.7 13.9 0.800 145.2 1.882 1.3

9 RHFB-120tf-055tw-150hw-G300-50s 52.2 31.8 13.7 1.192 147.8 0.543 1.1

10 RHFB-120tf-055tw-150hw-G300-100s 52.4 32.4 14.3 1.192 147.3 0.543 1.3

11 RHFB-075tf-075tw-100hw-G550-50s 51.8 30.3 14.2 0.748 96.3 0.748 2.8

12 RHFB-075tf-075tw-100hw-G550-100s 52.3 29.5 13.1 0.748 95.5 0.748 2.4

13 RHFB-075tf-075tw-150hw-G550-50s 52.5 30.2 14.2 0.748 146.7 0.748 1.2

14 RHFB-075tf-075tw-150hw-G550-100s 51.8 30.3 13.9 0.748 147.2 0.748 1.0

15 RHFB-115tf-115tw-150hw-G500-50s 52.8 30.5 13.4 1.148 147.3 1.148 1.5

16 RHFB-115tf-115tw-150hw-G500-100s 53.2 30.7 13.5 1.148 147.2 1.148 1.4

17 RHFB-075tf-115tw-150hw-G550-50s 53.4 29.8 13.4 0.748 147.2 1.148 1.1

18 RHFB-075tf-115tw-150hw-G550-100s 53.2 30.4 14.0 0.748 146.8 1.148 1.4

19 RHFB-115tf-075tw-150hw-G500-50s 53.4 31.2 14.2 1.148 146.5 0.748 1.3

20 RHFB-115tf-075tw-150hw-G500-100s 52.8 30.8 13.5 1.148 147.6 0.748 1.2

21 RHFB-095tf-055tw-150hw-G550-50s 52.6 31.0 14.2 0.947 146.3 0.553 1.0

22 RHFB-095tf-055tw-150hw-G550-100s 52.8 31.1 14.2 0.947 146.8 0.553 1.3

100 mm screw spacing 50 mm screw spacing


3.3.2 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 Ixx, Iyy and

Zx 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, Iw 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 3.3.5.

3.3.3 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.


Table 3.6: Section Properties Based on Measured Cross-section Dimensions

Beam Specimen Designation M kg/m

A (mm2)

Ixx (× 106 mm4)

Zx (× 104 mm3)

J (× 104 mm4)

Iw (× 108 mm6)

1 RHFB-120tf-055tw-100hw-G300-50s 4.00 516 1.72 2.25 10.00 4.86 2 RHFB-120tf-055tw-100hw-G300-100s 3.98 516 1.72 2.25 10.00 4.86 3 RHFB-080tf-080tw-150hw-G300-50s 3.45 448 2.69 2.57 6.52 6.23 4 RHFB-080tf-080tw-150hw-G300-100s 3.42 443 2.61 2.52 6.52 6.23 5 RHFB-120tf-120tw-150hw-G300-50s 5.06 660 3.90 3.74 10.03 9.54 6 RHFB-120tf-120tw-150hw-G300-100s 5.03 666 3.97 3.82 10.10 9.61 7 RHFB-080tf-190tw-150hw-G300-50s 4.74 618 3.00 2.88 6.58 6.23 8 RHFB-080tf-190tw-150hw-G300-100s 4.72 621 3.03 2.90 6.59 6.23 9 RHFB-120tf-055tw-150hw-G300-50s 4.11 557 3.81 3.61 10.00 9.54

10 RHFB-120tf-055tw-150hw-G300-100s 4.08 563 3.86 3.64 10.04 9.54 11 RHFB-075tf-075tw-100hw-G550-50s 3.10 375 1.20 1.53 6.18 3.02 12 RHFB-075tf-075tw-100hw-G550-100s 3.08 370 1.16 1.50 6.14 3.02 13 RHFB-075tf-075tw-150hw-G550-50s 3.36 415 2.45 2.37 6.18 5.91 14 RHFB-075tf-075tw-150hw-G550-100s 3.33 412 2.45 2.36 6.18 5.91 15 RHFB-115tf-115tw-150hw-G500-50s 5.02 636 3.82 3.66 9.56 9.10 16 RHFB-115tf-115tw-150hw-G500-100s 4.99 639 3.84 3.68 9.58 9.10 17 RHFB-075tf-115tw-150hw-G550-50s 3.88 481 2.62 2.53 6.19 5.91 18 RHFB-075tf-115tw-150hw-G550-100s 3.85 484 2.64 2.54 6.19 5.91 19 RHFB-115tf-075tw-150hw-G500-50s 4.46 578 3.71 3.55 9.54 9.10 20 RHFB-115tf-075tw-150hw-G500-100s 4.43 571 3.70 3.53 9.51 9.10 21 RHFB-095tf-055tw-150hw-G550-50s 3.53 462 2.99 2.87 7.77 7.43 22 RHFB-095tf-055tw-150hw-G550-100s 3.51 464 3.02 2.89 7.81 7.43

Note: M– Mass per metre length A – Gross cross-section area Ixx – Second moment of area Zx – Section modulus J– Torsion constant Iw – Warping constant


Figure 3.10: Imperfection Measuring Device

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0 200 400 600 800 1000

Distance (mm)

Impe

rfec

tion

(mm

)

'Top-Flange' Web-Central Bottom-Flange

Figure 3.11: Measured Imperfection along the Length of a Typical Test Specimen

3.3.4 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

Maximum, δ =1.2 mm

Specimen

Laser sensor and data logger

Travelator

Foot screw level


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 3.12 (a) –(c), respectively.

(b) Load Application

Figure 3.12: Test Set-up for Section Moment Capacity Tests

(a) Schematic View

Spreader beam

Loading Component

Spherical head

Rollers

Load transferring device

Transducer

Support box frame

Timber planks

Test Specimen Roller bearings

hw/2

hw/2

Loading arm

Tension flange

Steel plate

Compression flange


(c) Overall View

Figure 3.12: 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 3.12 (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 3.12 (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).

Spreader beam

Loading arm

Support box frame

Spherical head

Steel rollers

Timber planks

Loading

C clamps


The loads were applied to the neutral axis of the test beam through the steel rollers

and loading arms (see Figure 3.12 (b)) attached to the beam’s web using three M12

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 100 ×

150 × 10 mm steel plates were fixed on both sides of the beam web at the loading

and support points (see Figure 3.12 (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 cross-

head of the testing machine was moved at a constant rate of 1.0 mm/min until the

specimen failed.

3.3.5 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 × Lla (3.1)

where P is the applied load and Lla is the initial lever arm length as shown in Figure

3.13.

Figure 3.13: Deformed Shape of Test Specimen

P Lla = L/3

R=P

P Lla = L/3

R=P

Span, L = 1070 mm


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. Lla = 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.

3.3.5.1 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.

0

2

4

6

8

10

12

14

0 4 8 12 16 20 24

Vertical Deflection (mm)

Mom

ent (

kNm

)

(a) Moment versus Vertical Deflection (G300 Steel)

Figure 3.14: Moment versus Vertical Deflection Curves (tf = tw)

RHFB-080tf-080tw-150hw-G300-50s

RHFB-120tf-120tw-150hw-G300-50s

Mom

ent (

kNm

)


0

2

4

6

8

10

12

14

16

18

0 4 8 12 16 20


Mom

ent (

knm

)

(b) Moment versus Vertical Deflection (G500 and G550 Steels)

Figure 3.14: Moment versus Vertical Deflection Curves (tf = tw)

0.0

1.5

3.0

4.5

6.0

7.5

9.0

0 3 6 9 12 15 18 21 24 27


Mom

ent (

knm

)


Figure 3.15: Moment versus Vertical Deflection Curves (tf > tw)

RHFB-115tf-115tw-150hw-G500-50s

RHFB-075tf-075tw-150hw-G550-50s

RHFB-120tf-055tw-150hw-G300-50s

RHFB-120tf-055tw-100hw-G300-50s

Mom

ent (

kNm

) M

omen

t (kN

m)


0

2

4

6

8

10

12

14

0 3 6 9 12 15 18 21 24 27


Mom

ent (

knm

)

(b) Moment versus Vertical Deflection (G500 and G550 Steels)

Figure 3.15: Moment versus Vertical Deflection Curves (tf > tw)

0

2

4

6

8

10

12

0 2 4 6 8 10


Mom

ent (

knm

)


Figure 3.16: Moment versus Vertical Deflection Curves (tf < tw)

RHFB-115tf-075tw-150hw-G500-50s

RHFB-095tf-055tw-150hw-G550-50s

RHFB-080tf-190tw-150hw-G300-50s

Mom

ent (

kNm

) M

omen

t (kN

m)


0

2

4

6

8

10

12

0 2 4 6 8 10


Mom

ent (

knm

)

(b) Moment versus Vertical Deflection (G550 Steel)

Figure 3.16: Moment versus Vertical Deflection Curves (tf < tw)

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 0.65-0.99 and 0.29-0.69

of theoretical first yield moment My 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

RHFB-075tf-115tw-150hw-G550-50s

Mom

ent (

kNm

)


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 12 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.

3.3.5.2 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 fc was calculated first.

mc Ef ε= (3.2)

where E is the measured elastic modulus of steel and εm is the average measured

longitudinal strains in the extreme fibres at mid-span.

Applied uniform moment: xc ZfM = (3.3)

where Zx 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 fc was taken as the measured

yield stress fy.

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.


0.0

2.5

5.0

7.5

10.0

12.5

15.0

-4500 -3000 -1500 0 1500 3000 4500

Strain (microstrain)

Mom

ent (

kNm

)

S1-LB-T S1-LB-C S2-LB-T S2-LB-C S1-SB_T

S1-SB-C S2-SB-T S2-SB-C

S1 – RHFB-120tf-120tw-150hw-G300-50s S2 – RHFB-080tf-080tw-150hw-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-120tf-120tw-150hw-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 3.17.

First yield moment (SB) First yield moment (SB)

Moment corresponding to yield strain (LB)

Actual first yield moment (LB)

Moment corresponding to yield strain (LB)


3.3.5.3 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.

(a) Overall view

(b) Close-up view of failure region

Figure 3.18: Typical Failures of Test Specimens

Local buckling and yielding of flange

Tested specimens

Local buckling and yielding of flange


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.

(a) 50 mm Screw Spacing

(b) 100 mm Screw Spacing

Figure 3.19: Opening of Web and Flange Lips between Screw Fasteners

Gap openings

Failure section

50 mm screw spacing

Failure section

Gap openings

100 mm screw spacing


0

2

4

6

8

10

12

14

0 4 8 12 16 20 24


Mom

ent (

kNm

)

Figure 3.20: Graphical Illustration of Failure Behaviour of a RHFB

Figure 3.20 gives a graphical illustration of the failure behaviour of a selected RHFB

(i.e. RHFB-120tf-120tw-150hw-G300-50s) during the section moment capacity tests.

Figure 3.20 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 3.20 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 3.20. 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.

O

A

B

D

E RHFB-120tf-120tw-150hw-G300-50s

C


3.3.5.4 Comparison of Section Moment Capacities with Predictions from the Current Design Rules

AS 4100 (SA, 1998) Design Method

The section moment capacity (Ms) is defined in Clause 5.2.1 of AS 4100 (SA, 1998)

as follows:

eys ZfM = (3-4)

where the effective section modulus (Ze) allows for the effects of local buckling and

the calculation of Ze is dependent on the section classification recommended in AS

4100. 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 4600.

Table 3.7: Section Classification

Section Classification Beam Beam Designation AS 4100 AS/NZS 4600

1 RHFB-120tf-055tw-100hw-G300 Slender (web) Slender (flange, web)

2 RHFB-080tf-080tw-150hw-G300 Slender (flange) Slender (flange) 3 RHFB-120tf-120tw-150hw-G300 Slender (flange) Slender (flange) 4 RHFB-080tf-190tw-150hw-G300 Slender (flange) Slender (flange) 5 RHFB-120tf-055tw-150hw-G300 Slender (web) Slender (flange, web) 6 RHFB-075tf-075tw-100hw-G550 Slender (flange) Slender (flange) 7 RHFB-075tf-075tw-150hw-G550 Slender (flange) Slender (flange, web) 8 RHFB-115tf-115tw-150hw-G500 Slender (flange) Slender (flange) 9 RHFB-075tf-115tw-150hw-G550 Slender (flange) Slender (flange)

10 RHFB-115tf-075tw-150hw-G500 Slender (web) Slender (flange, web) 11 RHFB-095tf-055tw-150hw-G550 Slender (web) Slender (flange, web)

The effective section modulus is defined in Clauses 5.2.3 to 5.2.5 of AS 4100 (SA,

1998) as follows:

sps λλ ≤ : ZSZ e 5.1<=

syssp λλλ ≤< : ( )��

�

��

�

−−

−+=spsy

ssye ZSZZ

λλλλ

(3-5)

sys λλ > : ��

��

�=

s

sye ZZ

λλ


The section slenderness (λs) is taken as the value of the plate element slenderness

(λe) for the element of the cross-section, which has the greatest value of (λe/λey). The

plate element slenderness (λe) is defined in Clause 5.2.2 (SA, 1998) as a function of

the element clear width (b), thickness (t), and yield stress (fy):

250y

e tb σ

λ = (3-6)

The section plasticity and yield slenderness limits (λsp, λsy) are taken as the values of

the element slenderness limits (λep, λey) given in Table 5.2 of AS 4100 (SA, 1998)

for the element of the cross-section which has the greatest value of λe/λey. The cold-

formed (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 (Ms) is defined in Clause 3.3.2 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 (Ze) 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 (be) 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.


bbbe ≤

��

�

�

��

�

� −=

λλ22.0

1 (3-7)

Ef

tb

k

*052.1��

��

�=λ (3-8)

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:

)1(2)1(24 3 ψψ −+−+=k (3-9)

*1

*2

f

f=ψ

where f1* is compression (+) and f2

* 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 1.5.1.5 of AS/NZS 4600 (SA, 2005) recommends the use of a reduced yield

stress for G550 steels to allow for the reduced ductility in the steels: 0.90fy for 0.6

mm ≤ thickness <0.9 mm and 0.75fy 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.


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 λey 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 (Mu) 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 0.24, 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 0.24, respectively (see Mu/Ms 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 Mu/Ms ratios are about 0.95. 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.


Table 3.8: Comparison of Section Moment Capacities of RHFBs

AS/NZS 4600 AS 4100 Ratio Mu/Ms Beam Section Designation Exp. Mu

(kNm)

Z (or Zf) (×104) mm3

My (kNm) Ze

(×104 mm3) Ms

(kNm) Ze

( ×104 mm3) Ms

(kNm) AS/NZS

4600 AS

4100 1 RHFB-120tf-055tw-100hw-G300-50s 5.91 2.25 7.20 2.12 6.78 2.17 6.94 0.87 0.85 2 RHFB-120tf-055tw-100hw-G300-100s 5.50 2.25 7.20 2.12 6.78 2.16 6.91 0.81 0.80 3 RHFB-080tf-080tw-150hw-G300-50s 7.01 2.57 8.53 2.17 7.20 2.33 7.74 0.97 0.91 4 RHFB-080tf-080tw-150hw-G300-100s 6.96 2.52 8.37 2.15 7.14 2.28 7.57 0.97 0.92 5 RHFB-120tf-120tw-150hw-G300-50s 11.83 3.74 11.97 3.56 11.39 3.54 11.32 1.03 1.01 6 RHFB-120tf-120tw-150hw-G300-100s 11.81 3.82 12.23 3.64 11.64 3.69 11.81 1.01 1.00 7 RHFB-080tf-190tw-150hw-G300-50s 10.16 2.88 9.56 2.58 8.57 2.69 8.93 1.19 1.14 8 RHFB-080tf-190tw-150hw-G300-100s 9.29 2.90 9.63 2.58 8.57 2.71 9.00 1.08 1.03 9 RHFB-120tf-055tw-150hw-G300-50s 7.74 3.61 11.55 3.25 10.40 3.45 11.04 0.74 0.70

10 RHFB-120tf-055tw-150hw-G300-100s 7.83 3.64 11.64 3.29 10.52 3.48 11.14 0.74 0.70 Mean 0.94 0.91 G300 Steel COV 0.16 0.16

11 RHFB-075tf-075tw-100hw-G550-50s 6.44 1.53 7.53 1.03 5.10 1.29 8.39 1.26 0.77 12 RHFB-075tf-075tw-100hw-G550-100s 6.17 1.50 7.43 1.03 5.10 1.27 8.26 1.21 0.75 13 RHFB-075tf-075tw-150hw-G550-50s 9.43 2.37 11.73 1.55 7.68 1.99 12.94 1.23 0.73 14 RHFB-075tf-075tw-150hw-G550-100s 8.30 2.36 11.68 1.55 7.68 1.99 12.94 1.08 0.64 15 RHFB-115tf-115tw-150hw-G500-50s 15.64 3.66 20.13 3.23 15.99 3.35 19.53 0.98 0.80 16 RHFB-115tf-115tw-150hw-G500-100s 13.93 3.68 20.24 3.24 16.04 3.36 19.59 0.87 0.71 17 RHFB-075tf-115tw-150hw-G550-50s 9.82 2.53 12.52 1.94 9.60 2.15 13.98 1.02 0.70 18 RHFB-075tf-115tw-150hw-G550-100s 8.76 2.54 12.57 1.94 9.60 2.17 14.11 0.91 0.62 19 RHFB-115tf-075tw-150hw-G500-50s 11.73 3.55 19.53 2.91 14.41 3.22 18.77 0.81 0.62 20 RHFB-115tf-075tw-150hw-G500-100s 10.80 3.53 19.42 2.89 14.31 3.20 18.66 0.75 0.58 21 RHFB-095tf-055tw-150hw-G550-50s 6.49 2.87 15.79 1.91 9.46 2.52 16.38 0.69 0.40 22 RHFB-095tf-055tw-150hw-G550-100s 5.13 2.89 15.90 1.91 9.46 2.53 16.45 0.54 0.31

Mean 0.95 0.64 G500/G550 Steel COV 0.24 0.23


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 Mu/Ms 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.


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 22 RHFB sections made from G300 steel (10) and

G500/G550 steels (12). 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 4100.


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 22 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.

4.2 Section Geometry and Specimen Sizes

As discussed in Chapter 2, 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

2. 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.


The basic parameters for a typical RHFB section are: flange width (bf), flange height

(hf), web height (hw), flange thickness (tf), web thickness (tw), flange lip height (hl),

support span of the beam (l), steel grade (G), and screw spacing (s) (see Figure 4.1).

Figure 4.1: Cross-section of a Typical RHFB

The element’s width to thickness ratio (b/t) is an important parameter for cold-

formed 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 200.

Table 4.1: Selected Material Thicknesses from Three Steel Grades

Steel Grade Nominal steel thicknesses (mm)

G300 0.55 0.80 - 1.20 1.90 G500 - - - 1.15 - G550 0.55 0.75 0.95 - -

hw

hf

bf

tw

tf

Screw spacing along the beam “s”

hl

hf


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 × 25 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 × 25 mm flanges with

varying material thicknesses, web heights and span lengths are presented in Table 4.2

and Figures 4.2 (a) to (c).

Table 4.2: Elastic Buckling Analysis Results

Flange Web Buckling Stress (MPa)

hw = 100 mm hw = 150 mm Stee

l G

rade

No bf

(mm) hf

(mm) tf

(mm) tw

(mm) LBS 2 m 3 m LBS 2 m 3 m 1 50 25 0.55 0.55 122f 297 239 117w 205 139 2 50 25 0.55 1.20 122f 402 345 122f 248 198 3 50 25 1.20 1.20 576f 501 427 556w 299 242 4 50 25 1.20 0.55 331w 270 203 128w 205 124 5 50 25 0.80 0.80 257f 377 321 247w 239 180 6 50 25 0.80 1.90 257f 528 441 257f 314 256 7 50 25 1.90 1.90 1432f 678 546 1392w 398 322

G300

8 50 25 1.90 0.80 685w 325 365 271w 226 154 9 50 25 0.55 0.55 122f 297 239 117w 205 139

10 50 25 0.55 0.95 122f 384 329 122f 240 188 11 50 25 0.95 0.95 362f 425 365 349w 261 205 12 50 25 0.95 0.55 317w 281 218 126w 206 130 13 50 25 0.75 0.75 326f 361 306 217w 232 172

G550

14 50 25 0.75 1.15 226f 459 393 226f 277 224 15 50 25 1.15 1.15 530-f 487 416 511w 291 235

G500 16 50 25 1.15 0.75 529f 343 287 233w 229 163

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 4.2 (a) to (c).


(a) Buckling Plot for RHFB-120tf-120tw-150hw Section

(b) Buckling Plot for RHFB-120tf-055tw-150hw Section

Figure 4.2: Different Buckling Modes of RHFBs


(c) Buckling Plot for RHFB-080tf-190tw-150hw Section

Figure 4.2: Different Buckling Modes of RHFBs

Figures 4.2 (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 4.2 (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 half-

wavelength 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 4.2 (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


mm. However, lateral torsional buckling occurs beyond 8500 mm. In Figure 4.2 (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 2000 mm. Pure lateral distortional buckling occurs when the half-

wavelength exceeds 2000 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 4.2 and buckling plots such as Figures 4.2 (a) to (c). For

instance, section No. 3 of Grade 300 steel with a web height, hw = 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 2 m and 3 m span lengths. Therefore this

particular section (No. 3 in Table 4.2) is expected to fail by pure distortional

buckling at span lengths of 2 m and 3 m. Similarly, the failure modes of all other

sections were decided based on the results presented in Table 4.2.

Thirty RHFB specimens with different section sizes, lengths and screw spacings

were selected based on the elastic buckling analysis results presented in Table 4.2 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 ×

25 mm flange size, different combinations of flange and web thicknesses and web

heights are given in Table 4.3 for span lengths of 2 m and 3 m. For example, the

expected failure mode of section RHFB-120tf-120tw-150hw is pure lateral distortional

buckling (LD) at 3 m span (see Figure 4.2 (a)). The specimens were chosen only for

the sections highlighted in Table 4.3.


Table 4.3: Lateral Buckling Test Program

Test Specimens and Failure Modes

Web Height hw = 100 mm Web Height hw = 150 mm Steel Grade Flange

Thickness tf

Web Thickness

tw Span 2 m Span 3 m Span 2 m Span 3 m

0.55 0.55 LBF LBF LBF LBF + LD

0.55 1.20 LBF (1) LBF LBF LBF

1.20 1.20 Y (1) Y LD LD (2)

1.20 0.55 LD LD (1) LBW (1) LBW, LD (2)

0.80 0.80 LBF LBF LBF + LD LD (2)

0.80 1.90 LBF LBF LBF LBF, LD (1)

1.90 1.90 Y Y Y Y (1)

G30

0 st

eel

1.90 0.80 Y LD LD LD

0.55 0.55 LBF LBF LBF (1) LBF + LD

0.55 0.95 LBF LBF LBF LBF (1)

0.95 0.95 LBF LBF + LD LD (2) LD

0.95 0.55 LD LD LBW (2) LBW, LD (1)

0.75 0.75 LB LBF, LD (2) LBF, LD (1) LD (2)

G55

0 st

eel

0.75 1.15 LBF LBF LBF LBF, LD (2)

1.15 1.15 LD LD LD LD (2)

G50

0 st

eel

1.15 0.75 LD LD LBW + LD LD (2)

Sub-Total 2 3 7 18

Grand-Total 30

Note: LBF, LBW - Local Buckling of Flange and Web LDB – Lateral Distortional Buckling

Y – Material yielding (1) – 50 mm screw spacing (2) – 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, 12 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 25 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.


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


(b) Partially Bent Flange being Forced Inwards

(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. 10-16 × 16 Hexagon head self-

drilling 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.

Hydraulic pump

Hydraulic jacks

Vertical actuator

Vertical supports

Horizontal actuators

Main Beam

Secondary beam

Vertical actuator

Horizontal actuators

Specimen


(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, tf and tw, presented in Table 4.4 the are measured based metal

thicknesses for each steel grade and thickness (see Section 3.2.1 in Chapter 3).


(a): Measurement of Initial Geometric Imperfections Device

-4

-3

-2

-1

0

1

2

0 50 100 150 200 250 300

Distance (cm)

Impe

rfec

tion

(mm

)

'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

Test beam

Laser beam component

Data acquisition system

Levelling device


Note: Specimen designation is the same as defined in Chapter 3, but span length is also added (eg. 3L means 3 m length)

Flange Web Beam Specimen Designation bf

(mm) hf

(mm) hl

(mm) tf

(mm) hw

(mm) tw

(mm)

Imperfection δ (mm)

1 RHFB-120tf-120tw-150hw-3L-G300-50s 53.5 30.5 14.5 1.192 145.5 1.192 3.5 2 RHFB-120tf-055tw-150hw-3L-G300-50s 52.7 32.2 14.2 1.192 148.8 0.543 2.4 3 RHFB-080tf-080tw-150hw-3L-G300-50s 52.1 31.1 14.0 0.800 145.3 0.800 1.8 4 RHFB-080tf-190tw-150hw-3L-G300-50s 51.8 31.7 13.9 0.800 145.0 1.882 2.7 5 RHFB-190tf-190tw-150hw-3L-G300-50s 54.3 33.7 14.6 1.882 146.0 1.882 3.2 6 RHFB-120tf-055tw-100hw-3L-G300-50s 52.2 30.4 14.5 1.192 94.8 0.543 3.4 7 RHFB-120tf-055tw-150hw-2L-G300-50s 52.6 29.5 14.0 1.192 146.0 0.543 1.2 8 RHFB-055tf-120tw-100hw-2L-G300-50s 52.6 31.6 14.3 0.543 95.2 1.192 1.1 9 RHFB-120tf-120tw-100hw-2L-G300-50s 52.7 30.4 14.4 1.192 95.3 1.192 1.1

10 RHFB-055tf-095tw-150hw-3L-G550-50s 52.4 31.0 13.8 0.553 147.3 0.947 2.8 11 RHFB-095tf-055tw-150hw-3L-G550-50s 52.8 30.3 13.6 0.947 147.0 0.553 1.6 12 RHFB-075tf-075tw-150hw-3L-G550-50s 52.5 29.8 13.5 0.748 147.3 0.748 2.5 13 RHFB-075tf-115tw-150hw-3L-G550-50s 53.0 29.1 13.7 0.748 147.4 1.148 2.7 14 RHFB-115tf-115tw-150hw-3L-G500-50s 53.3 30.3 13.8 1.148 147.7 1.148 3.0 15 RHFB-115tf-075tw-150hw-3L-G500-50s 52.7 31.6 13.5 1.148 146.2 0.748 2.2 16 RHFB-075tf-075tw-100hw-3L-G550-50s 52.5 30.0 13.7 0.748 95.8 0.748 5.7 17 RHFB-055tf-055tw-150hw-2L-G550-50s 52.4 30.8 13.4 0.553 145 0.553 2.4 18 RHFB-095tf-095tw-150hw-2L-G550-50s 53.4 30.7 13.6 0.947 146.8 0.947 0.9 19 RHFB-095tf-055tw-150hw-2L-G550-50s 52.6 30.4 13.5 0.947 146.3 0.553 1.3 20 RHFB-075tf-075tw-150hw-2L-G550-50s 52.2 30.3 13.4 0.748 146.6 0.748 4.7 21 RHFB-080tf-080tw-150hw-3L-G300-100s 52.1 31.1 14.0 0.800 145.3 0.800 1.8 22 RHFB-120tf-120tw-150hw-3L-G300-100s 53.5 30.5 14.5 1.192 145.5 1.192 3.5 23 RHFB-120tf-055tw-150hw-3L-G300-100s 52.7 32.2 14.2 1.192 148.8 0.543 2.4 24 RHFB-075tf-075tw-150hw-3L-G550-100s 52.5 29.8 13.5 0.748 147.3 0.748 2.5 25 RHFB-075tf-075tw-100hw-3L-G550-100s 52.5 30.0 13.7 0.748 95.8 0.748 5.7 26 RHFB-115tf-115tw-150hw-3L-G500-100s 53.3 30.3 13.8 1.148 147.7 1.148 3.0 27 RHFB-075tf-115tw-150hw-3L-G550-100s 53.0 29.1 13.7 0.748 147.4 1.148 2.7 28 RHFB-115tf-075tw-150hw-3L-G500-100s 52.7 31.6 13.5 1.148 146.2 0.748 2.2 29 RHFB-095tf-095tw-150hw-2L-G550-100s 53.4 30.7 13.6 0.947 146.8 0.947 0.9 30 RHFB-095tf-055tw-150hw-2L-G550-100s 52.6 30.4 13.5 0.947 146.3 0.553 1.3

Table 4.4: Measured Section Dimensions and Geometric Imperfections of Test Specimens


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 (250 UC 89.5) and four columns (250 UC 89.5) located at 5 m × 1.8 m grid

points. The main beams were positioned horizontally at 2 m height between each pair

of long columns. The support system included two frames made of 150 UC 37.2 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.

(a) Schematic View

Figure 4.6: Lateral Buckling Test Set-up

P

L

P

L/4 L/4


(b) Overall View

Figure 4.6: Lateral Buckling Test Set-up

4.5.1 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.

Main columns Main beams Wheel system

Support frames

Test beam PDTs

SHS beam


(a) Schematic View

(b) Overall View

Figure 4.7: Support System for Buckling Tests

S1

0.00 m

1.20 m

500 mm 1.80 m

S2

B1

Bearing B3

Thrust Bearing B2

Box-frame S2

Box-frame S1

Travelator

Steel bar

Bearing B1

UC columns


The new support system included two pairs of 150 UC 37.2 columns. Each pair of

columns was connected together by two 50 × 50 × 5 SHS members. The important

components of the support system were the box-frames S1 and S2 (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 90 × 90 × 6 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 1.2 m from the floor in order to avoid any axial forces due to inclined

applied loads. The 1.2 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 S2, 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 B2 was placed on the top plate of the box-

frame S1, whereas a roller bearing B3 was fixed to its bottom plate at a position

vertically below B2 as shown in Figure 4.7 (b). Two 40 mm diameter shafts were

inserted into the bearings B2 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 S2. Thus, the bearings B2 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 S2 was designed to accommodate all the test specimens having

different section sizes. It was required to make S2 in two symmetric halves in order

to insert test specimen conveniently in the S2 box frame (see Figure (4.7 (b)). Test

specimen was fixed inside the S2 box frame using four bolts located symmetrically

about the neutral axis of test beam as shown in Figure 4.7 (b).


4.5.2 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


cantilever loads was not suitable for the RHFB used in this test program. Therefore

the quarter point loading method was adopted.

(a) Wheel System

(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, P2 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).

Hydraulic ram

Load cell

Bearing B4

Pivot 1

Pivot 2

Steel plate

Connector

Wheels to travel longitudinally

SHS beams

Wheels to travel transversely

Hydraulic ram


(c) Overall View

Figure 4.8: Loading System

4.5.3 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.

Wheels to travel longitudinally Wheels to travel

transversely

SHS beams

Pivoting bolts Load cell

Hydraulic rams

Test beam Loading arm


(a) Overall View of Measuring System

(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.

Load cells

PDTs Strain gauges

PDT and load cell connectors

Strain gauge connectors


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 S2 and clamped with the connector in

S2 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 0.2 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-of-

plane. 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 4.10.


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 × Lla (4.1)

where P is the applied jack load and Lla is the initial lever arm length equal to span/4

as illustrated in Figure 4.11.

Figure 4.11: Deformed Shape of Test Specimen

P P Lla = span/4

P P

Lla = span/4


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.

4.7.1 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.

4.7.1.1 RHFBs with Equal Flange and Web Thicknesses (tf = tw)

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 4.12 (a) – (d) for the tf = tw category. The

moment versus deflection graphs for other tests are presented in Appendix 4B.

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Verticale Deflection (mm)

Mom

ent (

kNm

)


Figure 4.12: Moment versus Deflection Curves (tf = tw)

RHFB-190tf-190tw-150hw-G300-3L-50s

RHFB-120tf-120tw-150hw-G300-3L-50s

RHFB-080tf-080tw-150hw-G300-3L-50s


0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0 5 10 15 20 25 30 35

Horizontal Deflection (mm)

Mom

ent (

kNm

)

(b) Moment versus Horizontal Deflection (G300 Steel)

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ent (

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)

(c) Moment versus Vertical Deflection (G550 Steel)

Figure 4.12: Moment versus Deflection Curves (tf = tw)

RHFB-190tf-190tw-150hw-G300-3L-50s

RHFB-120tf-120tw-150hw-G300-3L-50s

RHFB-080tf-080tw-150hw-G300-3L-50s

RHFB-115tf-115tw-150hw-G500-3L-50s

RHFB-075tf-075tw-150hw-G550-3L-50s


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 5 10 15 20 25 30 35 40


Mom

ent (

kNm

)

(d) Moment versus Horizontal Deflection (G550 Steel)

Figure 4.12: Moment versus Deflection Curves (tf = tw) Figures 4.12 (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 out-

of-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 4.12 (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-120tf-120tw-150hw-50s) whereas the maximum out-of-plane

deflection was achieved with the more slender beam sections (except RHFB-075tf-

RHFB-115tf-115tw-150hw-G500-3L-50s

RHFB-075tf-075tw-150hw-G550-3L-50s


075tw-150hw-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-075tf-075tw-150hw-3L-50s’.

Figure 4.13: Typical Lateral Distortional Buckling Failure of RHFBs

4.7.1.2 RHFBs with Flange Thickness Larger than Web Thickness (tf > tw)

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 tf > tw as shown in Figures 4.14 (a) – (d). The moment

versus in-plane and out-of-plane deflection graphs for other tests with tf > tw are

presented in Appendix 4B.

Lateral deflection of compression flange (bottom)


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ent (

kNm

)


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 25 30


Mom

ent (

kNm

)


Figure 4.14: Moment versus Deflection Curves (tf>tw)

RHFB-120tf-055tw-150hw-G300-3L-50s

RHFB-120tf-055tw-150hw-G300-3L-50s


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Mom

ent (

kNm

)


0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 5 10 15 20 25 30


Mom

ent (

kNm

)


Figure 4.14: Moment versus Deflection Curves (tf>tw)

RHFB-115tf-075tw-150hw-G550-3L-50s

RHFB-095tf-055tw-150hw-G550-3L-50s

RHFB-115tf-075tw-150hw-G550-3L-50s

RHFB-095tf-055tw-150hw-G550-3L-50s


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 4.7.1.1. 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-

095tf-055tw-150hw-3L-50s’.

Figure 4.15: Local Buckling of Slender Web near the Loading Points

4.7.1.3 RHFB with Flange Thickness Smaller than Web Thickness (tf < tw)

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 tf < tw as shown in Figures 4.16 (a) – (d). The

moment versus in-plane and out-of-plane deflection graphs for other tests including tf

< tw 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 4.7.1.1 and 4.7.1.2. The

test on section ‘RHFB-080tf-190tw-150hw-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.


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ent (

kNm

)


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

-5 0 5 10 15 20 25 30 35 40 45


Mom

ent (

kNm

)


Figure 4.16: Moment versus Deflection Curves (tf < tw)

RHFB-080tf -190tw-150hw-G550-3L-50s

RHFB-080tf -190tw-150hw-G550-3L-50s


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Mom

ent (

kNm

)


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 5 10 15 20 25 30 35 40 45


Mom

ent (

kNm

)


Figure 4.16: Moment versus Deflection Curves (tf < tw)

RHFB-075tf -115tw-150hw-G550-3L-50s

RHFB-055tf -095tw-150hw-G550-3L-50s

RHFB-075tf -115tw-150hw-G550-3L-50s

RHFB-055tf -095tw-150hw-G550-3L-50s


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-075tf-115tw-150hw-3L-

50s) should have higher in-plane stiffness than a more slender beam section (i.e.

RHFB-055tf-095tw-150hw-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.

4.7.2 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 σc was calculated first.

mc Eεσ = (4.2)

where E is the elastic modulus of steel assumed to be 200000 MPa, and εm is the

average measured longitudinal strain in the extreme fibres at mid-span.

Applied uniform moment, fc ZM ×= σ (Zf -full section modulus) (4.3)

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.


0

2

4

6

8

10

12

14

16

18

��

Strain (Microstrain)

Ben

ding

Mom

ent (

kNm

)

1 – RHFB-190tf-190tw-150hw-3L-50s 2 - RHFB-120tf-120tw-150hw-3L-50s 3 - RHFB-080tf-080tw-150hw-3L-50s

(a) G300 Steel

0

1

2

3

4

5

6

7

8

��

Strain (Microstrain)

Ben

ding

Mom

ent (

kNm

)

1 – RHFB-1.15-1.15-150-50s 2 - RHFB-0.75-0.75-150-50s

(b) G500 and G550 Steels

Figure 4.17: Moment versus Longitudinal Strain Curves

Tension side Compression side

Based on load cell Based on load cell

Based on strain Based on strain 1

2

1

2

Tension side Compression side

Based on load cell Based on load cell

Based on strain Based on strain

2

3

1

2

3

1 B

endi

ng M

omen

t (kN

m)

Ben

ding

Mom

ent (

kNm

)


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

4.7.2). 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.

4.7.3 Comparison of Test Results with Predictions from the Current Design Rules

The ultimate failure moments (Mu) 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 non-

uniform 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 Mu from the tests were divided by a moment distribution factor

(αm) of 1.09 as recommended by AS 4100. AS/NZS 4600 provides only an

approximate equation to calculate the moment distribution factor (Cb), 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.


Table 4.5: Comparison of Experimental Moment Capacities of RHFBs with Predictions from the Current Design Rules

Member Moment Capacities Mb (kNm) Mu/Mb

Bea

m N

o

Specimen Designation

Exp

erim

enta

l M

u (k

Nm

)

AS

4100

(1

998)

AS/

NZ

S 46

00

(199

6)

Pi T

raha

ir

(199

7)

Ave

ry e

t al

(199

9)

Mah

aara

chch

i an

d M

ahen

dran

(2

005)

AS

4100

(1

998)

AS

4600

(1

996)

Pi a

nd T

raha

ir

(199

7)

Ave

ry e

t al

(200

0)

Mah

arac

hchi

an

d

Mah

endr

an

(200

5)

1 RHFB-120tf-120tw-150hw-3L-G300-50s 6.30 7.70 7.89 5.72 7.30 6.16 0.75 0.76 1.01 0.79 0.96

2 RHFB-120tf-055tw-150hw-3L-G300-50s 3.47 7.49 4.68 3.54 5.04 4.37 0.42 0.75 0.90 0.63 0.76

3 RHFB-080tf-080tw-150hw-3L-G300-50s 3.00 5.17 4.23 3.18 4.27 3.47 0.53 0.71 0.87 0.64 0.79

4 RHFB-080tf-190tw-150hw-3L-G300-50s 5.20 5.72 5.60 4.22 5.51 4.41 0.83 0.89 1.13 0.87 1.06

5 RHFB-190tf-190tw-150hw-3L-G300-50s 12.00 12.54 14.11 10.80 13.22 11.06 0.88 0.78 1.02 0.83 0.98

6 RHFB-120tf-055tw-100hw-3L-G300-50s 4.77 5.47 4.95 3.62 4.57 3.89 0.80 0.93 1.21 0.96 1.16

7 RHFB-120tf-055tw-150hw-2L-G300-50s 4.55 8.25 6.37 4.50 5.98 5.05 0.51 0.72 0.93 0.70 0.86

8 RHFB-055tf-120tw-100hw-2L-G300-50s 3.50 3.17 2.59 2.28 2.75 2.09 1.01 1.35 1.41 1.17 1.36

9 RHFB-120tf-120tw-100hw-2L-G300-50s 7.30 6.23 6.32 5.23 5.97 5.73 1.08 1.10 1.28 1.12 1.17

10 RHFB-055tf-095tw-150hw-3L-G550-50s 2.57 5.11 2.84 2.58 3.86 2.84 0.46 1.05 0.91 0.61 0.72

11 RHFB-095tf-055tw-150hw-3L-G550-50s 2.53 8.19 3.79 3.13 4.81 3.79 0.28 0.90 0.74 0.48 0.60

12 RHFB-075tf-075tw-150hw-3L-G550-50s 3.00 6.70 3.78 3.23 4.83 3.78 0.41 1.02 0.85 0.57 0.67

13 RHFB-075tf-115tw-150hw-3L-G550-50s 3.82 6.90 4.74 4.02 5.87 4.74 0.51 0.89 0.87 0.60 0.70

14 RHFB-115tf-115tw-150hw-3L-G500-50s 5.47 10.24 8.34 6.49 9.23 7.80 0.49 0.66 0.77 0.54 0.65

15 RHFB-115tf-075tw-150hw-3L-G500-50s 5.14 10.02 5.99 4.77 7.11 5.99 0.47 0.95 0.99 0.66 0.78

16 RHFB-075tf-075tw-100hw-3L-G550-50s 3.99 5.41 4.17 3.51 4.71 3.57 0.68 1.06 1.04 0.78 0.95

17 RHFB-055tf-055tw-150hw-2L-G550-50s 3.00 5.98 2.76 2.54 3.73 2.76 0.46 1.65 1.08 0.74 0.86


Member Moment Capacities Mb (kNm) Mu/Mb

Bea

m N

o

Specimen Designation

Exp

erim

enta

l M

u (kN

m)

A

S 41

00

(199

8)

AS/

NZ

S 46

00

(199

6)

Pi &

Tra

hair

(1

997)

Ave

ry e

t al

(200

0)

Mah

arac

hchi

an

d M

ahen

dran

(2

005)

AS

4100

(1

998)

AS4

600

(199

6)

Pi &

Tra

hair

(1

997)

Ave

ry e

t al

(200

0)

Mah

arac

hchi

an

d M

ahen

dran

(2

005)

18 RHFB-095tf-095tw-150hw-2L-G550-50s 6.30 10.52 7.11 5.69 8.00 6.58 0.55 0.94 1.02 0.72 0.87

19 RHFB-095tf-055tw-150hw-2L-G550-50s 4.60 10.11 5.19 4.26 6.27 5.19 0.42 1.16 0.99 0.67 0.78

20 RHFB-075tf-075tw-150hw-2L-G550-50s 4.20 8.34 4.66 3.97 5.72 4.66 0.46 1.12 0.97 0.67 0.80

21 RHFB-080tf-080tw-150hw-3L-G300-100s 3.07 5.17 4.23 3.18 4.27 3.47 0.55 0.73 0.89 0.66 0.81

22 RHFB-120tf-120tw-150hw-3L-G300-100s 6.00 7.70 7.89 5.72 7.30 6.16 0.71 0.73 0.96 0.75 0.92

23 RHFB-120tf-055tw-150hw-3L-G300-100s 3.63 7.49 4.68 3.54 5.04 4.37 0.44 0.79 0.94 0.66 0.79

24 RHFB-075tf-075tw-150hw-3L-G550-100s 3.14 6.70 3.78 3.23 4.83 3.78 0.43 1.07 0.89 0.60 0.70

25 RHFB-075tf-075tw-100hw-3L-G550-100s 3.68 5.48 4.17 3.55 4.78 3.57 0.62 0.98 0.95 0.71 0.87

26 RHFB-115tf-115tw-150hw-3L-G500-100s 5.80 10.24 8.34 6.49 9.23 7.80 0.52 0.70 0.82 0.58 0.69

27 RHFB-075tf-115tw-150hw-3L-G550-100s 3.72 6.90 4.74 4.02 5.87 4.74 0.49 0.87 0.85 0.58 0.68

28 RHFB-115tf-075tw-150hw-3L-G500-100s 4.61 10.02 5.99 4.77 7.11 5.99 0.42 0.85 0.89 0.59 0.70

29 RHFB-095tf-095tw-150hw-2L-G550-100s 6.50 10.52 7.11 5.69 8.00 6.58 0.57 0.97 1.05 0.75 0.90

30 RHFB-095tf-055tw-150hw-2L-G550-100s 4.70 10.11 5.19 4.26 6.27 5.19 0.43 1.18 1.01 0.69 0.80

Mean 0.57 0.78 0.97 0.71 0.87

COV 0.33 0.17 0.15 0.22 0.22

Note: Experimental Mu values given in the table were divided by αm of 1.09 for comparison with Mb values

Table 4.5: Comparison of Experimental Moment Capacities of RHFBs with Predictions from the Current Design Rules (cont..)


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.

4.7.3.1 Member Moment Capacity Based on AS 4100 (SA, 1998)

The nominal member moment capacity (Mb) 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 (Mb)

is given by:

sssmb MMM ≤= αα (4.4)

where

�m = a moment modification factor �s = a slenderness reduction factor Ms = the nominal section moment capacity

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

= 1.09. The slenderness reduction factor (�s) is defined by:

��

�

�

��

�

�

��

�

�−+��

�

�

�=

o

s

o

ss M

MMM

36.02

α (4.5)


where Mo is the reference buckling moment and is determined by

��

�

��

�+

��

�

�

�=

2

2

2

2

e

w

e

yo

l

EIGJ

l

EIM

ππ (4.6)

where

E = Elastic modulus G = Shear modulus Iy = Second moment of area about minor principal axis Iw = Warping constant J = Torsion constant le = 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

0.33. 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 4.18. The test beam capacity Mu was plotted as Mu/Ms on the

vertical axis whereas the non-dimensional member slenderness λ was plotted on the

horizontal axis. The nominal section moment capacity Ms 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).

yes fZM = (4.7)


where Ze is the effective section modulus calculated with the extreme compression or

tension fibre at yield stress f y. A sample calculation of Ze is presented in Appendix

3C.

The non-dimensional member slenderness λ was calculated from Equation 4.8.

o

s

MM

=λ (4.8)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

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 Mo, 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).

Slenderness (λλλλ)

Mu/M

s , M

b/Ms


4.7.3.2 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.

sssdmbd MMM ≤= αα (4.9)

where Ms is the section moment capacity and αsd is the modified slenderness

reduction factor given by

0.18.26.02

≤ �

�

�

�

�

�

−��

�

�

��

�

�+��

�

�

�=

od

s

od

ssd M

MMMα (4.10)

αm is the moment modification factor (αm = 1 for uniform moment distribution) and

Mod is the lateral distortional buckling moment calculated by

odod ZfM = (Z – full section modulus) (4.11)

where fod 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 0.15. These results therefore show that the design method


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 4.19. The

nominal section moment capacity Ms was calculated as explained in Section 4.7.3.1

while λd was calculated using Equation 4.8, but with Mo replaced by Mod. 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.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

Test Pi & Trahair (1997)-curve

Figure 4.19: Comparison of Experimental Failure Moments with Predictions using Pi and Trahair’s (1997) method

4.7.3.3 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 2, RHFBs comprising two

torsionally rigid hollow flanges and a slender web are susceptible to lateral

Slenderness (λλλλd)

Mu,M

b/Ms


distortional buckling effects. Hence Clause 3.3.3.3 (b) of AS/NZS 4600 was used to

determine the member moment capacity of RHFBs subjected to lateral distortional

buckling. Clause 3.3.3.3 (b) defines distortional buckling as that involving transverse

bending of a vertical web with lateral displacement of the compression flange. The

member moment capacity (Mb) is given as:

��

�

��

�=

f

ccb Z

MZM (4.12)

where Zc is the effective section modulus calculated at a stress level of fc=Mc/Zf in

the extreme compression fibres. The critical moment (Mc) is calculated from Clause

3.3.3.3 (b) of AS/NZS 4600 as follows:

For λd < 1.414: ��

�

��

�

��

�

�

�−=

41

2d

yc MMλ

(4.13)

For λd ≥ 1.414: ��

�

��

�=

2

1

d

yc MMλ

(4.14)

where λd is the non-dimensional slenderness parameter and is determined from the

next equation.

od

yd M

M=λ (4.15)

where My is the first yield moment and Mod is the elastic distortional buckling

moment and they are given by

yfy fZM = (4.16)

odfod fZM = (4.17)

in which, fod is the elastic distortional buckling stress obtained from the Thin-wall

computer program. Ze is the full section modulus.


The results presented in Table 4.5 for the member moment capacities based on

AS/NZS 4600 method were calculated using Equations 4.12 to 4.17. 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.

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.50 1.00 1.50 2.00 2.50

Experiment

AS/NZS 4600

Figure 4.20: 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 4.20. The

modified member slenderness λd was calculated as explained in Sections 4.7.3.1 and

4.7.3.2. Since AS/NZS 4600 design rules do not have a single equation for member

capacity, Mb, the design curve (Mb/My) 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 4.20, predicted member moment capacities of RHFBs


Mu/M

y , M

b/My


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.

4.7.3.4 Member Moment Capacity Based on Avery et al.’s (2000) 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. (2000, 1999b). Design curves for HFBs

were derived based on the finite element analysis results of Avery et al. (2000,

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 4̀600 design methods.

Trahair (1997) proposed a design curve based on the following equations:

sybobssnb MMMMMMc

babM ≥≥≤�

�

�

�

+−+= ;;

1 2λ (4.18)


od

sd M

M=λ (4.19)

The suitable coefficients (a, b, c, and n) were established using the least square

method. Values of a = 1.0, b = 0.0, c = 0.424, and n = 1.196 were found to minimise

the total error for Trahair’s (1997) design equations (see Table 2.3). However, this

approach resulted in an unacceptable maximum unconservative error of more than 10

percent for HFB sections. Therefore Avery et al. (2000) has derived separate

coefficients for each of the different thickness of the HFB sections. Even though this


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, 2002). Therefore in this study Equations

(4.18) and (4.19) with the coefficients a, b, c and n (1.0, 0.0, 0.424, 1.196)

determined by Avery et al. (2000) 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 4.21.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

Test Avery et. al. (2000)-curve

Figure 4.21: Comparison of Experimental Failure Moments with Predictions using Avery et al.’s (2000) Method

Figure 4.21 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 4.21 it is

quite conservative for beams of low slenderness while being unconservative for

beams of intermediate slenderness (inelastic buckling region).


Mu,

Mb/M

s


The results presented in Table 4.5 for the member moment capacities based on Avery

et al.’s (2000) 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 0.22.

4.7.3.5 Member Capacity Based on Mahaarachchi and Mahendran’s

(2005c) Method

Mahaarachchi and Mahendran (2005a and 2005b) 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 (2005c), which were verified against the lateral

distortional buckling test results of Mahaarachchi and Mahendran (2005a). Equations

4.20 (a) – (c) have been recommended by Mahaarachchi and Mahendran (2005c) for

three regions of member slenderness separating yielding/local buckling, inelastic

lateral distortional buckling, and elastic lateral buckling.

For 59.0≤dλ yc MM = (4.20(a))

For 7.159.0 << dλ ��

�

�=

dyc MM

λ59.0

(4.20(b))

For 7.1≥dλ ��

�

�

�= 2

1

dyc MM

λ (4.20(c))

Comparison of the predicted moment capacities using Equations 4.20(a) to (c) with

the results of 30 lateral buckling tests of RHFB is shown in Figure 4.22. The

predicted member moment capacities using Mahaarachchi and Mahendran (2005)

design method were calculated for all the test specimens as explained in Appendix

4A. As explained in the section 4.7.3.3 on the comparison with AS/NZS 4600


predictions, Mahaarachchi and Mahendran’s predictions cannot be plotted as a

design curve. Instead individual test capacities are compared with corresponding

predictions in Figure 4.22. Figure 4.22 shows that the predicted moment capacities

based on Mahaarachchi and Mahendran’s (2005c) 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 4.22, it is quite conservative for

beams of low slenderness while being unconservative for beams of intermediate

slenderness (inelastic buckling region).

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5

Test

Mahaarachchi & Mahendran

Figure 4.22: Comparison of Experimental Failure Moments with Predictions using

Mahaarachchi and Mahendran’s (2005c) Method The results presented in Table 4.5 for the member moment capacities based on

Mahaarachchi and Mahendran’s (2005c) method were calculated using Equations

4.20 (a) – (c) (see Appendix 4A). The results indicated that the correlation of

predicted moment capacities based on Mahaarachchi and Mahendran’s (2005c)

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 (2005c) design moment

capacities was 0.87 with a coefficient of variation (COV) of 0.22.


Mu/M

y , M

b/My


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 20 different section geometries of RHFBs, two screw spacings 50 mm and

100 mm, and two spans 2 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. (2000, 1999b) and Mahaarachchi and Mahendran (2005)

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.


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 20 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,

2002) 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.


5.2 Development of the Finite Element Models

5.2.1 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 Cb, respectively, in the member capacity equations. For

the four point loading system, �m =1.09 and Cb = 1.0.

Note:

• L = 2m and 3m

• Support 1: free to rotate in-plane and out-of-plane (i.e. about Z-axis and Y-axis)

• Support 2: 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

Y P

L /2 L /4 L /4

21 P

X

Y


properties, and therefore the support conditions and the loading method are not

significant.

Note:

• L = 1070 mm

• Supports 1 and 2: 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. bf = 50 mm, hf = 25 mm,

and hl = 15 mm), but the depth of the web (hw), thickness of the flanges (tf), and the

thickness of web (tw) were changed as in the laboratory tests. The axis system for the

beams was as shown in Figures 5.1(a) and (b).

5.2.2 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 5.2 (a) to all the nodes at the mid-span of the beam. The X-axis

translation was prevented at the mid-span cross section.

L /3 L /3 L /3

P Y

21

P

X


Figure 5.2: 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 5.2 (b) to all nodes at the mid-span of the

beam. The X-axis translation was prevented at the mid-span cross section.

(a) Member Capacity Tests

M

P

u1 = 0 θ2 = 0 θ3 = 0

u2 = 0 u3 = 0 θ1 = 0

1 (X)

2 (Y)

3 Z

Experimental model

Ideal model

(b) Section Capacity Tests

u1 = 0 u3 = 0 θ1 = 0 θ2 = 0 θ3 = 0

u2 = 0 u3 = 0 θ1 = 0 θ2 = 0

1

2

3

P

M u2 = 0 u3 = 0 θ1 = 0

u1 = 0 θ2 = 0 θ3 = 0

Experimental model

Ideal model


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.

The minor principal axis (i.e. Y-axis) was the 2-axis in the finite element model.

Although the beam’s geometry and loading were symmetric about the 2-axis, the

deformation patterns of the beam result from the lateral distortional or lateral

torsional buckling distort the symmetrical condition about the 2-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).

Figure 5.3: Unsymmetrical Local Buckling Behaviour about Z-axis

(a) Flange local buckling

Y

Z

(b) Web local buckling

Y

Z


(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 5.2 (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. 2, 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.

Y

X

Y

X


Figure 5.5: Experimental Finite Element Model simulating Member Capacity Tests

Contact definition Between flange lip and web

Rigid MPC to connect web and web stiffening plate

SPC (2 3 4)

Two steel plates attached to the web

Rigid MPC to connect web-flange

(a) Support Boundary Condition

(b) Mid-Span Boundary Condition

Contact definition Between flange lip and web

Rigid MPC to connect web and flanges

Restrained 1 5 6 on whole section


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 (2, 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. ‘Rf’, 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 20 mm

away from the web (i.e. Rw, 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


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.

5.2.3 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

‘Rigid Beam’ type MPC

Explicit type MPC ‘UX’ and ‘UZ’ to link web nodes and Rigid Beam

‘Pin’ type MPC

Explicit type MPC ‘UX’ to link flange nodes and Rigid Beam

Figure 5.6: Support Boundary Conditions of Ideal Model

SPC (2 3 4)

Rf

Rw

‘Tie’ MPC

X

Z

Y


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, 2002). 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, 2003). 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 1/15th of element length for

which transverse shear deformation is important and Kirchoff constraint is satisfied

analytically (Yuan, 2004). 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

273. Therefore, only few sections were found to have element thickness greater than


1/15th 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.

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

N (SPOS) ← Surface normal positive direction

S 4 R 5 Stress/displacement shell (s)

5 degrees of freedom

Number of nodes Reduced integration

Figure 5.7: General Definition of S4R5 Shell Element

Shell elements follow mid-thickness line of RHFB


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.

5.2.4 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 5.2.1, 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


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.

Support

Applied load

Contact modelling

Mid-span cross section

(a) Physical Model (b) Experimental FE Model

Figure 5.10: Loading Method of Section Capacity Tests

Rigid MPC

Top-Flange

Steel plate

Bolts

Screws

Bottom-Flange

Web

Figure 5.9: Loading Method of Member Capacity Tests

(a) Physical Model (b) Experimental FE Model


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 5.11. A 20 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.

5.2.5 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

5.12). 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, C2,

C3 and C4, see Figure 5.12). This allows any interface movements of two surfaces

M

Figure 5.11: Loading Method of Ideal FE Model


in contact during the deformation. A smooth surface interaction (i.e. zero friction)

was assumed for the contact surfaces in the model.

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.

Master Surface

Master Surface

Slave Surfaces C1

C2

Slave Surfaces

C4

C3

Figure 5.12: Contact Surface Definition


5.2.6 Material Properties

Different material properties can be included in the numerical analysis. The

ABAQUS classical metal plasticity model (HKS, 2002) 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, 2002):

)1( nomnomtrue εσσ += (5.1)

Etrue

nomp

σεε −+= )1ln((ln) (5.2)

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, 2002) was used in all the components of

the ideal finite element model.


5.2.7 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 cold-

formed 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 2). 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 cold-

formed 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 5.13.

The residual stresses of 17% yield strength (fy) 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 cold-

formed C-section, since their forming process is similar to the web of a C-section.


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.

5.2.8 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:

17 % fy

17 % fy

8 % fy 8 % fy

17 % fy


• 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. δg, 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.


t

l te2

6−

=δ (5.3) where t is the element thickness and δlf, δlw are as shown in Figures 5.14 (a) and (b). (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.

5.2.9 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 2004 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

2. Mesh all the web and flange surfaces

δlf

δlw

δg


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 2004 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 200 mm deep RHFB of 3 m length

(i.e. half length = 1.5 m) required 16285 elements and 19595 nodes in the

experimental finite element model. Each analysis usually consisted of 20 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. 2-axis) direction showed that the sum of three nodal

loads at the loading point equalled the vertical reaction at the support for the half-

span 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


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.


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.

2. 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.

5.3.1.1 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. tf =

tw), flange thickness greater than web thickness (i.e. tf > tw) and flange thickness less

than web thickness (i.e. tf < tw) 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 2 m and 3 m beam spans and this provided a suitable range of

member slenderness.


0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0 2 4 6 8 10 12 14 16 18


Mom

ent (

kNm

)

RHFB-1.2tf-1.2tw-150hw-G300-50s

RHFB-1.9tf-1.9tw-150hw-G300-50s

RHFB-0.8tf-0.8tw-150hw-G300-50s

(a) Moment versus Vertical Deflection

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0 5 10 15 20 25 30 35 40


Mom

ent (

kNm

)

RHFB-1.9tf-1.9tw-150hw-G300-50s

RHFB-1.2tf-1.2tw-150hw-G300-50s

RHFB-0.8tf-0.8tw-150hw-G300-50s

(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. tf = tw)

Expt

FEA

Expt

FEA

RHFB-190tf-190tw-150hw-3L-G300-50s

RHFB-120tf-120tw-150hw-3L-G300-50s

RHFB-080tf-080tw-150hw-3L-G300-50s

RHFB-190tf-190tw-150hw-3L-G300-50s

RHFB-120tf-120tw-150hw-3L-G300-50s

RHFB-080tf-080tw-150hw-3L-G300-50s


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26


Mom

ent (

kNm

)

RHFB-1.15tf-0.75tw-150hw-G500-50s

RHFB-0.95tf-0.55tw-150hw-G550-50s


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 5 10 15 20 25 30 35 40 45 50 55


Mom

ent (

kNm

) RHFB-1.15tf-0.75tw-150hw-G500-50s

RHFB-0.95tf-0.55tw-150hw-G550-50s


Figure 5.16 Moment-Deflection Curves for a group of RHFB Specimens with Flange Thickness Greater than Web Thickness (i.e. tf > tw)

Expt

FEA

Expt

FEA

RHFB-115tf-075tw-150hw-3L-G550-50s

RHFB-095tf-055tw-150hw-3L-G550-50s

RHFB-115tf-075tw-150hw-3L-G550-50s

RHFB-095tf-055tw-150hw-3L-G550-50s


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26


Mom

ent (

kNm

) RHFB-0.75tf-1.15tw-150hw-G550-50s


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 25 30 35 40 45 50 55 60


Mom

ent (

kNm

)

RHFB-0.75tf-1.15tw-150hw-G550-50s


Figure 5.17 Moment-Deflection Curves for a RHFB Specimen with Flange Thickness Less than Web Thickness (i.e. tf < tw)

Expt

FEA

Expt

FEA

RHFB-075tf-115tw-150hw-3L-G550-50s

RHFB-075tf-115tw-150hw-3L-G550-50s


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 (Mu) 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 0.07. 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 2 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.


Table 5.1: Comparison of Experimental and FEA Member Moment Capacities

Beam Specimen Designation Exp Mu (kNm)

FEA Mu (kNm)

Exp. / FEA Mu

1 RHFB-120tf-120tw-150hw-3L-G300-50s 6.30 5.99 1.05

2 RHFB-120tf-055tw-150hw-3L-G300-50s 3.47 3.21 1.08

3 RHFB-080tf-080tw-150hw-3L-G300-50s 3.00 3.20 0.94

4 RHFB-080tf-190tw-150hw-3L-G300-50s 5.20 4.31 1.21

5 RHFB-190tf-190tw-150hw-3L-G300-50s 12.00 12.18 0.99

6 RHFB-120tf-055tw-100hw-3L-G300-50s 4.77 4.23 1.13

7 RHFB-120tf-055tw-150hw-2L-G300-50s 4.55 4.44 1.02

8 RHFB-055tf-120tw-100hw-2L-G300-50s 3.50 3.32 1.05

9 RHFB-120tf-120tw-100hw-2L-G300-50s 7.30 7.28 1.00

10 RHFB-055tf-095tw-150hw-3L-G550-50s 2.57 2.33 1.10

11 RHFB-095tf-055tw-150hw-3L-G550-50s 2.53 2.31 1.09

12 RHFB-075tf-075tw-150hw-3L-G550-50s 3.00 2.94 1.02

13 RHFB-075tf-115tw-150hw-3L-G550-50s 3.82 3.85 0.99

14 RHFB-115tf-115tw-150hw-3L-G500-50s 5.47 6.09 0.90

15 RHFB-115tf-075tw-150hw-3L-G500-50s 5.14 4.64 1.11

16 RHFB-075tf-075tw-100hw-3L-G550-50s 3.99 3.87 1.03

17 RHFB-055tf-055tw-150hw-2L-G550-50s 3.00 2.79 1.07

18 RHFB-095tf-095tw-150hw-2L-G550-50s 6.30 5.76 1.09

19 RHFB-095tf-055tw-150hw-2L-G550-50s 4.60 4.21 1.09

20 RHFB-075tf-075tw-150hw-2L-G550-50s 4.20 4.23 0.99

21 RHFB-080tf-080tw-150hw-3L-G300-100s 3.07 3.10 0.99

22 RHFB-120tf-120tw-150hw-3L-G300-100s 6.00 5.81 1.03

23 RHFB-120tf-055tw-150hw-3L-G300-100s 3.63 3.11 1.17

24 RHFB-075tf-075tw-150hw-3L-G550-100s 3.14 2.85 1.10

25 RHFB-075tf-075tw-100hw-3L-G550-100s 3.68 3.80 0.97

26 RHFB-115tf-115tw-150hw-3L-G500-100s 5.80 5.91 0.98

27 RHFB-075tf-115tw-150hw-3L-G550-100s 3.72 3.73 1.00

28 RHFB-115tf-075tw-150hw-3L-G500-100s 4.61 4.50 1.02

29 RHFB-095tf-095tw-150hw-2L-G550-100s 6.50 5.71 1.14

30 RHFB-095tf-055tw-150hw-2L-G550-100s 4.70 4.17 1.13

Mean 1.05

COV 0.07


0

2

4

6

8

10

12

14

-2000 -1500 -1000 -500 0 500 1000 1500 2000

Strain ( Micro Strain)

Mom

ent

(kN

m)

1 1

2 2

3 3

FEA

Exp

FEA

Exp

1 = RHFB-080tf-080tw-150hw-G300-50s 2 = RHFB-120tf-120tw-150hw-G300-50s 3 = RHFB-080tf-080tw-150hw-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


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 5.20 (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 5.20 (a) and (b).

The stress contours shown in Figure 5.20 (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 5.20: Elastic and Ultimate Lateral Distortional Buckling Failure Mode and Stress Contours of a Typical RHFB model from FEA


5.3.1.2 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. tf = tw), flange thickness greater than web thickness (i.e. tf >

tw) and flange thickness less than web thickness (i.e. tf < tw) are shown in Figures

5.21 to 5.23 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 5.21 to 5.23 represent the deflection

at the bottom flange of the mid-span. These curves show that the experimental and

analytical results are in reasonable agreement.

Figure 5.21: Moment versus Vertical Deflection Curves for RHFB Specimens with Equal Flange and Web Thicknesses (i.e. tf = tw)

0

2

4

6

8

10

12

14

0 3 6 9 12 15 18 21 24 27 30


Mom

ent (

kNm

)

RHFB-120tf-120tw-150hw-G300-50s

RHFB-080tf-080tw-150hw-G300-50s Expt

FEA


Figure 5.22: Moment versus Vertical Deflection Curves for RHFB Specimens with

Flange Thickness Greater than Web Thickness (i.e. tf > tw)

Figure 5.23: Moment versus Vertical Deflection Curves for RHFB Specimens with Flange Thickness Less than Web Thickness (i.e. tf < tw)

The analytical and experimental curves for the section moment capacity of RHFBs

presented in Figures 5.21 and 5.23 are in reasonable agreement whereas experimental

and analytical section moment capacity curves presented in Figure 5.22 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

0

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8 9


Mom

ent (

kNm

)

0

3

6

9

12

15

18

0 3 6 9 12 15 18 21 24 27


Mom

ent (

kNm

)

RHFB-120tf-055tw-150hw-G300-50s

RHFB-115tf-075tw-150hw-G300-50s

RHFB-080tf-190tw-150hw-G300-50s

Expt

FEA

Expt

FEA


(marked ‘*’ in Table 5.2) 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 5.2 contains a summary of

ultimate section moment capacity (Ms) 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 0.18. These ratios have means of 0.89, 0.79 and 0.85 and COVs

of 0.20, 0.11 and 0.18 for G300, G500 and G550 steels, respectively.

Table 5.2: Comparison of Experimental and FEA Section Moment Capacities

Section Moment Capacity (kNm) Beam Specimen Designation

Expt FEA Expt./FEA

1 RHFB-120tf-055tw-100hw-G300-50s 5.91 8.40 0.70

2 RHFB-120tf-055tw-100hw-G300-100s 5.50 8.28 0.66

3 RHFB-080tf-080tw-150hw-G300-50s 7.01 6.60 1.06

4 RHFB-080tf-080tw-150hw-G300-100s 6.96 6.53 1.07

5 RHFB-120tf-120tw-150hw-G300-50s 11.83 11.70 1.01

6 RHFB-120tf-120tw-150hw-G300-100s 11.81 11.48 1.03

7 RHFB-080tf-190tw-150hw-G300-50s 10.16 9.96 1.02

8 RHFB-080tf-190tw-150hw-G300-100s 9.29 9.84 0.94

9 RHFB-120tf-055tw-150hw-G300-50s 7.74 11.30 0.68

10 RHFB-120tf-055tw-150hw-G300-100s 7.83 11.21 0.70

11 RHFB-075tf-075tw-100hw-G550-50s 6.44 8.72 0.74

12 RHFB-075tf-075tw-100hw-G550-100s 6.17 8.68 0.71

13 RHFB-075tf-075tw-150hw-G550-50s 9.43 8.56 1.10

14 RHFB-075tf-075tw-150hw-G550-100s 8.30 8.38 0.99

15 RHFB-115tf-115tw-150hw-G500-50s 15.64 17.84 0.88

16 RHFB-115tf-115tw-150hw-G500-100s 13.93 17.73 0.79

17 RHFB-075tf-115tw-150hw-G550-50sp 9.82 11.80 0.83

18 RHFB-075tf-115tw-150hw-G550-100s 8.76 11.71 0.75

19 RHFB-115tf-075tw-150hw-G500-50s 11.73 16.64 0.71

20 RHFB-115tf-075tw-150hw-G500-100s 10.80 16.53 0.65 *

21 RHFB-095tf-055tw-150hw-G550-50s 6.49 13.30 0.49 *

22 RHFB-095tf-055tw-150hw-G550-100s 5.13 12.92 0.40 *

Mean 0.86

COV 0.18


Typical moment versus longitudinal strain graphs for a group of specimens tested for

the section capacity are shown in Figure 5.24 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 2

mm strain gauges.

0

2

4

6

8

10

12

14

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000


Mom

ent (

kNm

)

Expt1-c

Expt1-t

FEA1-t

FEA1-c

Expt2-t

Expt2-c

FEA2-t

FEA2-c

1 = RHFB-120tf-120tw-150hw-G300-50s 2 = RHFB-080tf-080tw-150hw-G300-50s

Figure 5.24: 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 5.25 and

5.26 illustrate the typical experimental and FEA failure modes of RHFB (RHFB-

075tf-075tw-150hw-G550-50s), respectively. As shown in Figures 5.25 and 5.26, local

buckling appeared on both the top flange plate element and web in this particular

beam section.


Figure 5.25: Local Buckling Failure Mode of RHFB Tested for Section Capacity

Figure 5.26: Local buckling Failure Mode of RHFB from Analytical results

5.3.2 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


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 5.11.

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 5.27. As

shown in Table 5.3 and Figure 5.27, 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 Thin-

wall (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 Thin-

wall 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.


Table 5.3: Comparison of Elastic Buckling Moments from FEA (Ideal Model) with Thin-wall Analysis

Note: FEA = Finite element analyses TW = Thin-wall

Buckling Moment (kNm) at Different Beam Spans (mm) 1000 2000 3000 5000 8000 Flange

Size Specimen Designation

FEA TW % Diff FEA TW % Diff FEA TW % Diff FEA TW % Diff FEA TW % Diff

RHFB-200tf-200tw-150hw-G300-50s 41.1 43.3 5.4 23.2 24.4 5.2 18.3 19.2 4.9 13.0 13.4 3.1 8.7 8.9 2.3

RHFB-300tf-300tw-150hw-G300-50s 73.0 76.8 5.2 43.2 45.0 4.2 32.3 33.3 3.1 21.1 21.6 2.4 13.8 13.9 0.7

RHFB-400tf-400tw-150hw-G300-50s 112.0 117.0 4.5 64.8 67.1 3.5 46.7 48.0 2.8 29.6 30.1 1.7 19.0 19.2 1.1

RHFB-500tf-500tw-150hw-G300-50s 155.6 162.0 4.1 87.5 90.1 3.0 61.8 63.2 2.3 38.6 39.0 1.0 24.5 24.6 0.4

RHFB-200tf-200tw-100hw-G300-50s 40.6 43.2 6.4 26.8 28.1 4.9 20.6 21.4 3.9 13.4 14.1 5.2 8.7 9.1 4.6

RHFB-300tf-300tw-100hw-G300-50s 76.3 80.6 5.6 47.1 49.2 4.5 34.2 35.3 3.2 21.6 22.2 2.8 13.9 14.1 1.4

RHFB-400tf-400tw-100hw-G300-50s 117.4 123.7 5.4 68.1 71.2 4.6 48.0 49.5 3.1 29.4 30.5 3.7 18.6 19.2 3.2

50×25

RHFB-500tf-500tw-100hw-G300-50s 132.1 136.4 3.3 73.4 75.2 2.5 50.3 51.5 2.4 31.2 31.4 0.6 19.6 19.7 0.5

RHFB-300tf-300tw-160hw-G300-50s 371.6 382.0 2.8 169.4 174.0 2.7 136.4 138.0 1.2 102.8 104.0 1.2 72.1 73.3 1.7

RHFB-400tf-400tw-160hw-G300-50s 523.4 542.0 3.6 267.6 274.0 2.4 210.3 215.0 2.2 148.6 152.0 2.3 101.3 103.0 1.7 90×45

RHFB-500tf-500tw-160hw-G300-50s 701.3 722.0 3.0 376.4 386.0 2.6 290.0 295.0 1.7 194.6 200.0 2.8 129.6 132.0 1.9

RHFB-200tf-200tw-120hw-G300-50s 11.5 12.3 7.0 6.9 7.3 5.8 5.0 5.2 4.0 3.1 3.2 3.2 2.1 2.1 0

RHFB-300tf-300tw-120hw-G300-50s 20.6 22.3 8.3 11.6 12.3 6.0 7.9 8.4 6.3 5.0 5.1 2.0 3.1 3.2 3.2

RHFB-400tf-400tw-120hw-G300-50s 31.3 33.6 7.3 16.4 17.7 7.9 11.3 12.0 6.2 6.9 7.2 4.3 4.4 4.5 2.3 30×15

RHFB-500tf-500tw-120hw-G300-50s 43.0 46.3 7.7 22.5 23.9 6.2 15.3 16.1 5.2 9.2 9.7 5.4 5.8 6.1 5.2


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 5.27 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.

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Span (mm)

Buc

klin

g om

ent (

kNm

)

S1-FEA

S1-TW

S2-FEA

S2-TW

S3-FEA

S3-TW

S4-FEA

S4-TW

Figure 5.27: Comparison of FEA and Thin-wall Elastic Buckling Moment

Capacities

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 5.2) to simulate the experimental tests for the


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 5.2) 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.


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.

6.2 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:


• 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.

6.2.1 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


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 (be) 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

(fy) and elastic modulus (E) of steel, and plate buckling coefficient (k).

The geometric parameters (see Figure 6.1) and the material property fy 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 200 mm was assumed to be

sufficient to give the section moment capacities of RHFBs from the FEA.

Figure 6.1: Cross-section parameters of a Typical RHFB

h w

h f

b f

tw

tf

h l

h f


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 fy,

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.

6.2.1.1 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 6.2)

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 (Ms) obtained from the finite element analyses for different b/t ratios of the

top plate element ranging from 25 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 (Ms) 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.

Figure 6.2: Local Buckling of the Hollow Flange’s Top Plate Element

Local buckling in Flange’ top plate


Table 6.1 (a): Comparison of FEA Results with the Predictions from Current Steel Design Standards for G300 Steel

Element

Compactness Section Moment Capacity Ms

(kNm) Ms Ratio

AS 4100 AS 4100 /FEA

tp (mm) b/t

AS 4100

AS/NZS 4600 Actual Modified

AS/NZS 4600 FEA

Actual Modified

AS4600/FEA

0.5 100 S S 5.66 13.56 13.74 19.70 0.29 0.69 0.70

0.7 71 S S 8.14 13.97 14.18 20.10 0.40 0.70 0.71

1.0 50 S S 11.87 14.84 15.01 20.70 0.57 0.72 0.73

1.4 36 NC S 17.36 17.36 16.33 21.30 0.82 0.82 0.77

1.6 31 NC S 19.83 19.83 16.92 21.30 0.93 0.93 0.79

2.0 25 C NS 22.17 22.17 17.83 22.20 1.00 1.00 0.80

bf = 50 mm, hf = 25 mm, hw = 150 mm, t = 2mm thickness of all the elements except top flange plate, which is tp and it is varied as shown in Column 1, fy = 300 MPa

Table 6.1 (b): Comparison of FEA Results with the Predictions from Current

Steel Design Standards for G550 Steel

Element Compactness

Section Moment Capacity Ms

(kNm) Ms Ratio

AS 4100 AS 4100 /FEA

tp (mm) b/t AS

4100


AS/NZS 4600 FEA

Actual Modified

AS4600/FEA

0.5 100 S S 7.71 24.65 24.97 35.70 0.22 0.69 0.70

0.7 71 S S 11.00 25.20 25.59 36.30 0.30 0.69 0.70

1.0 50 S S 16.17 26.39 26.82 37.20 0.43 0.71 0.72

1.4 36 S S 23.41 28.63 28.91 38.30 0.61 0.75 0.75

1.6 31 S S 27.45 30.02 30.09 38.90 0.71 0.77 0.77

2.0 25 NC NS 35.06 35.06 32.60 39.80 0.88 0.88 0.82

bf = 50 mm, hf = 25 mm, hw = 150 mm, t = 2mm for all the elements except top flange plate, which is tp and it was varied as shown in Column 1, fy = 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 Ms 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


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 7.

6.2.1.2 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 6.2 (a) and (b) show the section moment

capacities (Ms) obtained from the finite element analyses for different b/t ratios of

flange’s web element ranging from 12.5 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 (Ms) 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.


Figure 6.3: Local Buckling of the Hollow Flange’s Web Element


Element


(kNm) Ms Ratio

AS4100 AS 4100 / FEA

tp (mm) b/t AS

4100


AS/NZS 4600 FEA

Actual Modified

AS4600/FEA

0.5 50.0 S S 11.76 15.02 15.06 20.00 0.59 0.75 0.75

0.7 35.7 NC S 16.33 16.33 15.34 20.30 0.80 0.80 0.76

1.0 25.0 C NS 20.84 20.84 16.68 20.90 1.00 1.00 0.80

1.4 17.9 C NS 21.37 21.37 17.14 21.40 1.00 1.00 0.80

1.6 15.6 C NS 21.63 21.63 17.37 21.70 1.00 1.00 0.80

2.0 12.5 C NS 22.16 22.16 17.83 22.20 1.00 1.00 0.80

bf = 50 mm, hf = 25 mm, hw = 150 mm, t = 2mm for all the elements except hollow flange’s web element, which is tp and it was varied as shown in Column 1, fy = 3000 MPa

The comparison of predicted section moment capacities and FEA results given in

Table 6.2 (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.

Local buckling in Flange web


Table 6.2 (b): Comparison of FEA Results with the Predictions from Current Steel Design Standards for G550 Steel

Element


(kNm) Ratio

AS4100 AS 4100 / FEA

tp (mm) b/t AS

4100


AS/NZS 4600 FEA

Actual Modified

AS4600/FEA

0.5 50.0 S S 15.92 27.69 27.72 36.20 0.44 0.76 0.77

0.7 35.7 S S 22.61 28.07 28.08 36.80 0.61 0.76 0.76

1.0 25.0 NC NS 32.87 32.87 30.58 37.50 0.88 0.88 0.82

1.4 17.9 C NS 39.17 39.17 31.32 38.50 1.02 1.02 0.81

1.6 15.6 C NS 39.66 39.66 31.85 39.00 1.02 1.02 0.82

2.0 12.5 C NS 40.62 40.62 32.70 39.80 1.02 1.02 0.82

bf = 50 mm, hf = 25 mm, hw = 150 mm, t = 2mm for all the elements except hollow flange’s web element, which is tp and it was varied as shown in Column 1, fy = 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 2%, 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 8.7.

6.2.1.3 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


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

(Ms) obtained from the finite element analyses for different b/t ratios of web element

ranging from 60 to 240 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 (Ms) 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.

Figure 6.4: Local Buckling of RHFB’s Web Element


Element


(kNm) Ms Ratio

AS4100 AS 4100 / FEA

tp (mm) b/t

AS 4100


AS/NZS 4600 FEA

Actual Modified

AS4600/FEA

0.5 240 S S 7.52 13.22 15.63 19.00 0.40 0.70 0.82

0.7 171 S NS 10.58 14.49 17.27 19.40 0.55 0.75 0.89

1.0 120 S NS 15.22 16.48 17.40 20.10 0.76 0.82 0.87

1.4 86 NC NS 20.88 20.88 17.57 20.90 1.00 1.00 0.84

1.6 75 C NS 21.94 21.94 17.66 21.30 1.03 1.03 0.83

2.0 60 C NS 22.16 22.16 17.83 22.20 1.00 1.00 0.80

bf = 50 mm, hf = 25 mm, hw = 150 mm, t = 2mm for all the elements except web element, which is tp and it was varied as shown in Column 1, fy = 300 MPa

Local buckling in beam web


Table 6.3 (b): Comparison of FEA Results with the Predictions from Current Steel Design Standards for G550 Steel

Element


(kNm) Ms Ratio

AS4100 AS 4100 / FEA

tp (mm) b/t

AS 4100


AS/NZS 4600 FEA

Actual Modified

AS4600/FEA

0.5 240 S S 10.18 22.79 26.25 34.50 0.30 0.66 0.76

0.7 171 S S 14.32 24.50 29.07 35.20 0.41 0.70 0.83

1.0 120 S NS 20.61 27.15 31.90 36.30 0.57 0.75 0.88

1.4 86 S NS 29.15 30.89 32.22 37.70 0.77 0.82 0.85

1.6 75 NC NS 33.55 33.55 32.38 38.40 0.87 0.87 0.84

2.0 60 C NS 40.62 40.62 32.70 39.80 1.02 1.02 0.82

bf = 50 mm, hf = 25 mm, hw = 150 mm, t = 2mm for all the elements except web element, which is tp and it was varied as shown in Column 1, fy = 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 24 could also be another reason. It must be noted that the

FEA gave a ‘k’ value of 52.

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.


6.2.2 Local and Lateral Distortional Buckling Behaviour of RHFB

As described in Section 6.2.1, 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, 2005c) .

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. (2000)

further investigated the behaviour of such HFBs using finite element analyses and

developed new design rules. Mahaarachchi and Mahendran (2005a, 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.


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 (200 mm) to very long lengths (8000 mm) of RHFBs and thus

both local and lateral buckling effects.

6.2.2.1 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

S1 – Slender section S2 – Compact section Span (mm)

Perfect RS % Reduction Perfect RS %

Reduction 200 14.50 12.80 11.7 39.84 34.5 13.4

500 13.20 11.90 9.8 37.49 32.8 12.5

1000 12.00 11.20 6.7 32.02 29.5 7.9

2000 7.76 7.40 4.6 27.82 26.3 5.5

3000 6.38 6.20 2.9 24.19 23.5 2.9

5000 5.09 5.00 1.8 17.30 16.9 2.3

8000 4.06 4.00 1.5 12.10 12.0 0.8

Note: S1 – RHFB-120tf-120tw-150hw-G300-50s S2 - RHFB-300tf-300tw-150hw-G300-50s

RS – Including residual stresses


Table 6.4b: Effect of Initial Geometric Imperfections on RHFB Moment Capacities

S1 – Slender section S2 – Compact section Span mm

Perfect GI % Reduction Perfect GI %

Reduction 200 14.50 14.20 2.1 39.84 38.50 3.4

500 13.20 13.00 1.5 37.49 35.30 5.8

1000 12.00 11.60 3.3 32.02 31.10 2.9

2000 7.76 7.50 3.4 27.82 25.80 7.2

3000 6.38 5.90 7.6 24.19 22.80 5.8

5000 5.09 4.80 5.8 17.30 16.30 5.8

8000 4.06 3.72 8.4 12.10 11.20 7.4 Note: – S1 - RHFB-120tf-120tw-150hw-G300-50s S2 - RHFB-300tf-300tw-150hw-G300-50s GI – Including geometric imperfection

Table 6.4c: Combined Effects of Residual Stresses and Initial Geometric Imperfections on RHFB Moment Capacities

S1 – Slender section S2 – Compact section Span mm

Perfect RS + GI % Reduction Perfect RS + GI %

Reduction 200 14.50 12.20 15.9 39.84 33.2 16.7

500 13.20 11.30 14.4 37.49 31.5 16.0

1000 12.00 10.50 12.5 32.02 27.6 13.8

2000 7.76 6.93 10.7 27.82 24.4 12.3

3000 6.38 5.70 8.1 24.19 21.6 10.7

5000 5.09 4.63 9.1 17.30 15.9 8.1

8000 4.06 3.69 9.1 12.10 11.2 7.4

Note: – S1 - RHFB-120tf-120tw-150hw-G300-50s S2 - RHFB-300tf-300tw-150hw-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.


0

5

10

15

20

25

30

35

40

45

0 2000 4000 6000 8000

Span (mm)

Mom

ent (

kNm

)S1-perfect

S1-RS

S1-GI

S1-RS+GI

S2-Perfect

S2-RS

S2-GI

S2-RS+GI

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.

6.2.2.2 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 (2005) investigated the compression behaviour of similar cold-

formed 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

S1 – RHFB-120tf-120tw-150hw-G300

S2 – RHFB-300tf-300tw-150hw-G300


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.



(c) Continuous Fastening

Figure 6.6: Effect of Fastening Arrangement on the Deformation Shape of RHFB


Figures 6.6 (a) to (c) show the FEA deformation shapes of RHFB-080tf-080tw-

150hw-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-080tf-080tw-150hw-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.



Figure 6.7: Stress Contours of RHFB for Different Fastening Arrangements


(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 (2 slender

and 2 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.


Table 6.5 (a): Comparison of Moment Capacity Results of RHFB-080tf-080tw- 150hw-G300 (slender) for Different Fastening Arrangements

Screw spacing % Reduction compared to continuous connection Span

(mm) Continuous (Mu, kNm)

50 mm 100 mm 50 mm 100 mm

200 8.50 8.10 7.86 4.7 7.6

500 8.40 8.00 7.60 4.8 9.5

1000 7.40 6.50 6.18 12.2 16.6

2000 4.80 3.96 3.60 17.5 24.9

3000 3.45 3.05 2.90 11.6 16.0

5000 2.55 2.41 2.34 5.5 8.3

8000 2.10 2.02 1.98 3.8 5.7

Table 6.5 (b): Comparison of Moment Capacity Results of RHFB-120tf-120tw- 150hw-G300 (slender) for Different Fastening Arrangements



50 mm 100 mm 50 mm 100 mm

200 12.50 12.20 12 2.4 4.0

500 11.80 11.30 11 4.2 6.8

1000 11.00 10.50 10.2 4.5 7.3

2000 8.00 6.93 6.5 13.4 18.8

3000 6.40 5.70 5.4 10.9 15.6

5000 5.00 4.63 4.5 7.4 10.0

8000 3.80 3.69 3.6 2.9 5.3

Table 6.5 (c): Comparison of Moment Capacity Results of RHFB-300tf-300tw- 150hw-G300 (compact) for Different Fastening Arrangements



50 mm 100 mm 50 mm 100 mm

200 34.6 33.3 32.2 3.8 6.9

500 32.8 31.5 30.6 4.0 6.7

1000 29.8 27.6 26.2 7.4 12.1

2000 27.3 24.6 23.1 9.9 15.4

3000 23.8 21.6 20.3 9.2 14.7

5000 16.8 15.9 15.2 5.4 9.5

8000 11.7 11.2 10.9 4.3 6.8


Table 6.5 (d): Comparison of Moment Capacity Results of RHFB-500tf-500tw- 150hw-G300 (compact) for Different Fastening Arrangements



50 mm 100 mm 50 mm 100 mm

200 55.9 54.5 53.1 2.5 5.0

500 53.8 52.1 51 3.2 5.2

1000 50.5 48.3 46.2 4.4 8.5

2000 47.2 44.1 42.2 6.6 10.6

3000 42.8 39.3 37.8 8.2 11.7

5000 30.5 28.5 27.5 6.6 9.8

8000 20.8 19.6 19 5.8 8.7

Tables 6.5 (a) to (d) and Figures 6.9 (a) to (b) indicate that the moment capacities of

RHFB with continuous connection of web and flanges are slightly higher than those

of RHFBs connected by screw fasteners at 50 mm and 100 mm spacing along the

beam. The percentage reduction in moment capacities due to screw fastening at 50

mm and 100 mm spacings were also calculated and presented in Tables 6.5 (a) to (d).

The highest percentage reductions in moment capacities in the slender RHFB

sections were 17.5% and 24.9% for the 50 mm and 100 mm screw spacing,

respectively, whereas for the compact sections, they were 9.9% and 15.4%,

respectively. The comparisons of analytical results also demonstrate that the

influence of screw fastening for the intermediate beam spans is larger than those for

the shorter and longer beam spans. Therefore it shows that the discontinuity between

web and flanges is more sensitive in the lateral distortional buckling region than local

and lateral torsional buckling regions.


0

2

4

6

8

10

12

14

0 2000 4000 6000 8000 10000

Span (mm)

Mom

ent (

kNm

)

S1-Cont'

S1-50 mm

S1-100 mm

S2-Cont'

S2-50 mm

S2-100 mm

Figure 6.9 (a): Comparison of Moment Capacities of Slender RHFBs with Different

Fastening Arrangement

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000

Span (mm)

Mom

ent (

kNm

)

S3-Cont'

S3-50 mm

S3-100 mm

S4-Cont'

S4-50 mm

S4-100 mm

Figure 6.9 (b): Comparison of Moment Capacities of Compact RHFBs with Different Fastening Arrangement

The results in Tables 6.5 (a) to (c) were obtained from the finite element analyses

using the same residual stress model developed to simulate the initial stress

conditions of a press-braked RHFB section. However, in practice, a continuous web

to flange connection means welding, which would have induced very high residual

stresses in the RHFB member. Hence the moment capacities of continuously welded

RHFB could be even below that of the screw fastened RHFB if an accurate residual

stress model was used to simulate welding effects. Pi and Trahair (1997) reported

S2 – RHFB-120tf-120tw-150hw-G300

S1 – RHFB-080tf-080tw-150hw-G300

S3 – RHFB-120tf-120tw-150hw-G300

S4 – RHFB-120tf-120tw-150hw-G300


that the strengths of triangular HFBs (i.e. made of using electric resistance welding

method) without residual stresses are higher than those of HFRBs with residual

stresses, except at high and low slenderness. In this research, two continuously

welded RHFB sections (one compact and the other slender) were analysed using a

higher residual stress model developed for elastic resistance welded LSB sections. A

span of 2000 mm was chosen to simulate lateral distortional buckling. The results

indicated that the reduction in moment capacity of 9.2 and 8.1% for compact and

slender RHFB sections, respectively. When compared with the moment capacity

results of press-braked RHFBs with lower residual stresses (4.6% and 5.5% in Table

4.6a) it is clear that RHFB moment capacities will be further reduced for

continuously welded RHFBs. Therefore screw fastening at a closer spacing along the

RHFB member is likely to achieve the moment capacity (or even higher) of the

continuously welded RHFB. This study therefore considered only 50 mm screw

spacing in the analyses to obtain the ultimate moment capacities in the development

of suitable design rules for RHFBs.

6.2.3 Detailed FEA to Review the Current Design Rules

In order to undertake a thorough review of the current design rules and to develop

new design rules for RHFBs, it is important that a comprehensive moment capacity

data base is available for RHFBs with varying geometry, slenderness and steel

grades. Member capacity data from experimental analyses was limited and in some

cases were inaccurate due to the difficulties in manufacturing the RHFB specimens.

Therefore detailed finite element analyses using the validated ideal finite element

model were undertaken for 64 RHFB sections with spans from 200 mm to 8000 mm

and two steel grades (yield stresses of 300 MPa and 550 MPa). The RHFB sections

were chosen so that all the possible buckling failure modes could be achieved.

Primarily, the specimens were categorized into two groups; slender and compact (or

non-slender) RHFBs based on the AS/NZS 4600 design provisions. The moment

capacity data obtained from the compact sections were used in the development of

design curves to avoid multiple buckling modes in the same analysis. Tables 6.6 (a)

and (b) present the member moment capacity results from FEA whereas Figures 6.10

(a) and (b) present the member moment capacity curves in terms of moment capacity

versus beam spans.


Table 6.6 (a): Member Moment Capacities of Compact RHFB Sections from FEA

Analytical Moment Capacities Mu (kNm) at Various Span Lengths L (mm) No Flange Size Beam Designation

200 500 1000 2000 3000 5000 8000

S1 RHFB-200tf -200tw -150hw -G300-50s 21.6 20.9 17.8 14.3 12.30 9.59 6.99

S2 RHFB-300tf -300tw -150hw -G300-50s 33.2 31.5 27.6 24.4 21.60 15.90 11.20

S3 RHFB-400tf -400tw -150hw -G300-50s 44.4 42.5 37.9 34.4 30.50 22.20 15.40

S4 RHFB-500tf -500tw -150hw -G300-50s 54.5 52.1 48.3 44.1 39.30 28.50 19.60

S5 RHFB-200tf -200tw -100hw -G300-50s 14.7 14.1 12.6 11.8 10.30 8.80 7.20

S6 RHFB-300tf -300tw -100hw -G300-50s 22.0 21.6 19.7 18.7 16.30 12.80 10.30

S7 RHFB-400tf -400tw -100hw -G300-50s 29.4 28.8 26.9 25.6 22.60 18.50 14.30

S8

50 × 25

RHFB-500tf -500tw -100hw -G300-50s 36.7 35.9 34.0 32.4 28.80 22.20 15.80

S9 RHFB-300tf -300tw -160hw –G300-50s 60.0 59.8 58.9 53.8 51.90 49.90 46.50

S10 RHFB-400tf -400tw -160hw –G300-50s 80.6 79.2 76.5 73.4 71.50 68.80 63.50

S11

90 × 45

RHFB-500tf -500tw -160hw –G300-50s 101.0 101.0 101.0 96.5 91.40 87.30 80.20

S12 RHFB-200tf -200tw -120hw -G300-50s 11.30 10.10 7.47 4.99 3.65 2.33 1.52

S13 RHFB-300tf -300tw -120hw -G300-50s 17.00 15.30 12.20 8.32 5.96 3.73 2.43

S14

30 ×15

RHFB-400tf -400tw -120hw -G300-50s 22.70 20.60 17.00 11.80 8.36 5.22 3.34

S15 RHFB-200tf -200tw -150hw –G550-50s 38.2 35.4 27.9 16.7 14.20 10.10 7.24

S16 RHFB-300tf -300tw -150hw –G550-50s 59.9 55.0 43.8 30.6 24.30 16.70 11.50

S17 RHFB-400tf -400tw -150hw –G550-50s 80.7 73.2 61.2 45.5 35.10 25.30 15.90

S18 RHFB-500tf -500tw -150hw –G550-50s 95.4 88.6 76.6 60.6 46.00 30.00 20.20

S19 RHFB-200tf -200tw -100hw –G550-50s 26.4 24.2 20.5 16.6 13.30 10.10 7.40

S20 RHFB-300tf -300tw -100hw –G550-50s 40.1 37.2 32.6 28.7 21.70 16.30 12.30

S21 RHFB-400tf -400tw -100hw –G550-50s 53.5 50.4 45.1 40.6 30.40 21.40 16.50

S22

50 × 25

RHFB-500tf -500tw -100hw –G550-50s 66.9 62.5 57.8 52.2 39.40 26.80 17.60


Table 6.6 (a): Member Moment Capacities of Compact RHFB Sections from FEA…cont’

Analytical Moment Capacities Mu (kNm) at Various Span Lengths L (mm) No Flange Size Beam Designation

200 500 1000 2000 3000 5000 8000

S23 RHFB-300tf -300tw -160hw –G550-50s 108.0 107.0 105.0 89.0 79.2 69.6 55.6

S24 RHFB-400tf -400tw -160hw –G550-50s 146.0 144.0 140.3 121.4 114.0 99.8 75.9

S25

90 × 45

RHFB-500tf -500tw -160hw –G550-50s 183.0 182.0 179.0 158.0 148.0 123.0 93.5

S26 RHFB-200tf -200tw -120hw –G550-50s 20.60 16.00 9.05 5.31 3.83 2.49 1.63

S27 RHFB-300tf -300tw -120hw –G550-50s 30.90 24.30 15.50 8.90 6.24 3.95 2.53

S28 RHFB-400tf -400tw -120hw –G550-50s 41.30 32.90 22.50 12.70 8.77 5.51 3.54

S29 RHFB-500tf -500tw -120hw –G300-50s 28.30 26.10 21.90 15.40 11.00 6.83 4.37

S30

30 × 15

RHFB-500tf -500tw -120hw –G550-50s 51.60 45.20 29.60 16.60 11.50 7.20 4.67


Table 6.6 (b): Member Moment Capacities of Slender RHFB Sections from FEA

Analytical Moment Capacities Mu (kNm) at Various Span Lengths L (mm) No Beam Designation 200 500 1000 2000 2500 3000 4000 5000 6000 8000

S1 RHFB-080tf -080tw -150hw -G300-50s 8.10 8.00 6.50 3.96 3.35 3.05 2.65 2.41 2.22 2.02

S2 RHFB-100tf -100tw -150hw -G300-50s 9.84 9.10 8.50 5.40 4.65 4.30 3.82 3.48 3.20 2.82

S3 RHFB-120tf -120tw -150hw -G300-50s 12.20 11.30 10.50 6.93 6.15 5.70 5.13 4.63 4.25 3.69

S4 RHFB-190tf -190tw -150hw -G300-50s 20.50 20.00 16.90 13.30 12.40 11.60 10.20 8.94 8.11 6.57

S5 RHFB-080tf -100tw -150hw -G300-50s 8.06 7.82 7.33 4.37 3.75 3.43 3.01 2.73 2.54 2.22

S6 RHFB-080tf -120tw -150hw -G300-50s 8.42 8.05 7.69 4.63 4.06 3.68 3.25 2.94 2.75 2.33

S7 RHFB-080tf -190tw -150hw -G300-50s 9.61 9.10 8.43 5.19 4.63 4.10 3.61 3.26 3.10 2.60

S8 RHFB-100tf -120tw -150hw -G300-50s 10.40 9.85 9.19 5.84 5.16 4.80 4.23 3.82 3.52 2.98

S9 RHFB-100tf -190tw -150hw -G300-50s 11.70 11.32 9.50 6.83 6.20 5.57 4.92 4.38 4.06 3.34

S10 RHFB-120tf -190tw -150hw -G300-50s 13.60 12.87 10.80 8.41 7.70 7.03 6.19 5.47 5.01 4.08

S11 RHFB-100tf -080tw -150hw -G300-50s 9.33 9.11 8.86 4.92 4.06 3.72 3.23 2.96 2.69 2.41

S12 RHFB-120tf -080tw -150hw -G300-50s 11.80 11.48 9.50 5.77 4.70 4.28 3.70 3.43 3.11 2.82

S13 RHFB-190tf -080tw -150hw -G300-50s 18.10 17.70 13.00 8.54 6.75 5.87 5.03 4.70 4.25 4.02

S14 RHFB-120tf -100tw -150hw -G300-50s 12.30 12.15 10.50 6.34 5.45 5.10 4.50 4.13 3.80 3.29

S15 RHFB-190tf -100tw -150hw -G300-50s 18.70 17.40 15.00 9.36 7.82 7.14 6.35 5.92 5.38 4.87

S16 RHFB-190tf -120tw -150hw -G300-50s 19.20 17.85 15.10 10.30 8.99 8.50 7.57 6.96 6.31 5.53

S17 RHFB-055tf -055tw -150hw –G550-50s 6.80 6.70 6.62 3.00 1.94 1.70 1.35 1.26 1.11 0.99

S18 RHFB-075tf -075tw -150hw –G550-50s 11.00 10.70 9.45 3.63 3.34 2.80 2.40 2.19 2.23 1.78

S19 RHFB-095tf -095tw -150hw –G550-50s 14.80 13.40 12.10 5.62 4.73 4.15 3.92 3.29 3.02 2.65

S20 RHFB-115tf -115tw -150hw –G550-50s 20.30 Not available

Not available 7.34 6.28 5.80 5.01 4.50 4.11 3.50


Table 6.6 (b): Member Moment Capacities of Slender RHFB Sections from FEA…cont’

Analytical Moment Capacities Mu (kNm) at Various Span Lengths L (mm) No Beam Designation 200 500 1000 2000 2500 3000 4000 5000 6000 8000

S21 RHFB-055tf -075tw -150hw –G550-50s 6.90 6.70 6.30 3.75 2.32 1.98 1.78 1.60 1.43 1.27

S22 RHFB-055tf -095tw -150hw –G550-50s 7.75 7.67 7.43 3.50 2.52 2.22 1.94 1.67 1.62 1.42

S23 RHFB-055tf -115tw -150hw –G550-50s 8.87 8.73 8.50 3.61 2.64 2.27 2.01 1.86 1.72 1.51 S24 RHFB-075tf -095tw -150hw –G550-50s 10.20 Not

available Not

available 4.45 3.76 3.41 3.01 2.56 2.35 2.07

S25 RHFB-075tf -115tw -150hw –G550-50s 12.30 11.40 9.12 4.73 4.06 3.67 3.14 2.90 2.65 2.24

S26 RHFB-095tf -115tw -150hw –G550-50s 15.80 13.85 11.32 6.11 5.23 4.86 4.31 3.70 3.38 2.91

S27 RHFB-075tf -055tw -150hw –G550-50s 8.68 8.68 8.69 3.86 3.13 2.03 1.63 2.01 1.45 1.40

S28 RHFB-095tf -055tw -150hw –G550-50s 12.00 11.87 11.32 4.75 3.60 2.78 2.23 1.86 1.78 1.75

S29 RHFB-115tf -055tw -150hw –G550-50s 15.00 13.46 10.79 4.81 3.42 3.35 2.80 2.39 2.24 2.21

S30 RHFB-095tf -075tw -150hw –G550-50s 14.50 14.20 12.20 4.90 3.94 3.50 3.01 2.65 2.50 2.25

S31 RHFB-115tf -075tw -150hw –G550-50s 17.90 17.30 14.10 5.76 4.76 4.42 3.51 3.24 2.94 2.65

S32 RHFB-115tf -095tw -150hw –G550-50s 18.50 17.92 14.70 6.66 5.61 5.04 4.67 3.98 3.68 3.18


0

20

40

60

80

100

120

140

160

180

200

0 1000 2000 3000 4000 5000 6000 7000 8000

S1 S2

S3 S4

S5 S6

S7 S8

S9 S10

S11 S12

S13 S14

S15 S16

S17 S18

S19 S20

S21 S22

S23 S24

S25 S26

S27 S28

S29 S30

Span (mm)

Mu

(kN

m)

Figure 6.10 (a): Moment Capacity Curves for Compact RHFB Sections from FEA


0.0

5.0

10.0

15.0

20.0

25.0

0 1000 2000 3000 4000 5000 6000 7000 8000

S1 S2

S3 S4

S5 S6

S7 S8

S9 S10

S11 S12

S13 S14

S15 S16

S17 S18

S19 S20

S21 S22

S23 S24

S25 S26

S27 S28

S29 S30

S31 S32

Span (mm)

Mu

(kN

m)

Figure 6.10 (b): Moment Capacity Curves for Slender RHFB Sections from FEA


6.3 Review of Current Design Rules

In this section, the FEA moment capacities presented in Section 6.2 are compared

with the section and member design moment capacities obtained using the Australian

Standards for the design of steel structures, AS 4100 (SA, 1998) and cold-formed

steel structures, AS/NZS 4600 (SA, 1996). Predicted moment capacities from other

alternative design procedures are also compared with the FEA moment capacities.

For the sake of completeness, the current design rules for section and member

moment capacities are presented again.

6.3.1 AS 4100 Moment Capacity

6.3.1.1 Section Moment Capacity

The section moment capacity (Ms) is defined as follows:

yes fZM = (6.1)

The effective section modulus (Ze) is defined as follows:

( )

��

��

�=>

��

�

��

�

−−

−+=≤<

=≤

s

syesys

spsy

ssyesyssp

esps

ZZ

ZSZZ

SZ

λλ

λλ

λλλλ

λλλ

λλ

:

:

:

(6.2)

The section slenderness (λs) is taken as the value of the plate element slenderness

(λe) for the element of the cross-section which has the greatest value of (λe/λey). The

plate element slenderness (λe) is defined as a function of the element clear width (b),

thickness (t), and yield stress (fy).


250y

e

f

tb=λ (6.3)

In the moment capacity calculations, nominal section dimensions and yield stresses

were used. The yield stress was taken as 300 MPa and 550 MPa for the steel grades

G300 and G550, respectively, as used in the finite element parametric studies.

The section plasticity and yield slenderness limits (λsp, λsy) are taken as the values of

the element slenderness limits (λep, λey) given in Table 5.2 (SA, 1998) for the

element of the cross-section which has the greatest value of λe/λey and for the cold-

formed case.

The section moment capacities of 30 compact RHFB sections were calculated using

AS 4100 design equations described above. However, a modified AS 4100 design

method was used to calculate the section moment capacities of 29 slender RHFB

sections as described in Section 6.2 and Appendix 3C. A comparison of the FEA and

AS 4100 design section moment capacities is provided in Tables 6.7 (a) and (b) for

compact and slender RHFB sections, respectively.

As shown in Table 6.7 (a), the section moment capacities of the compact sections are

predicted accurately with a mean of 1.00 and a COV of 0.02. However, AS 4100

section moment capacity rules conservatively estimate the reduction in capacity due

to local buckling in slender RHFB sections. The results in Table 6.7 (b) show that AS

4100 design rules (modified) predict the section moment capacities of slender RHFB

sections conservatively with a mean of 0.75 and a COV of 0.08. The maximum

unconservative error for compact RHFB sections is an acceptable 5%, and therefore

it is safe to use the AS 4100 specifications for section capacity design checks of

RHFB beam sections subject to pure bending moment.


Table 6.7 (a) Comparison of FEA and AS 4100 Section Moment Capacities for Compact RHFB Sections

Hollow Flange Size

(mm×mm)

Section Designation

Sect

ion

com

pact

ness

FEA

Ms

(kN

m)

AS

4100

Ms

(kN

m)

AS

4100

/ FE

A

RHFB-200tf-200tw-150hw-G300-50s C 21.60 22.16 1.03

RHFB-300tf-300tw-150hw-G300-50s C 33.20 32.96 0.99

RHFB-400tf-400tw-150hw-G300-50s C 44.40 43.59 0.98

RHFB-500tf-500tw-150hw-G300-50s C 54.50 54.04 0.99

RHFB-200tf-200tw-100hw-G300-50s C 14.70 14.70 1.00

RHFB-300tf-300tw-100hw-G300-50s C 22.00 21.87 0.99

RHFB-400tf-400tw-100hw-G300-50s C 29.40 28.92 0.98

50×25

RHFB-500tf-500tw-100hw-G300-50s C 36.70 35.85 0.98

RHFB-300tf-300tw-160hw-G300-50s C 60.00 61.28 1.02

RHFB-400tf-400tw-160hw-G300-50s C 80.60 81.32 1.01 90×45

RHFB-500tf-500tw-160hw-G300-50s C 101.00 101.18 1.00

RHFB-200tf-200tw-120hw-G300-50s C 11.30 11.26 1.00

RHFB-300tf-300tw-120hw-G300-50s C 17.00 16.68 0.98

RHFB-400tf-400tw-120hw-G300-50s C 22.70 21.95 0.97 30×15

RHFB-500tf-500tw-120hw-G300-50s C 28.30 27.08 0.97

RHFB-200tf-200tw-150hw-G550-50s C 38.20 40.62 1.05

RHFB-300tf-300tw-150hw-G550-50s C 59.90 60.43 1.01

RHFB-400tf-400tw-150hw-G550-50s C 80.70 79.92 0.99

RHFB-500tf-500tw-150hw-G550-50s C 99.2 99.07 1.00

RHFB-200tf-200tw-100hw-G550-50s C 26.40 26.95 1.02

RHFB-300tf-300tw-100hw-G550-50s C 40.10 40.10 1.00

RHFB-400tf-400tw-100hw-G550-50s C 53.50 53.02 0.99

50×25

RHFB-500tf-500tw-100hw-G550-50s C 66.90 65.73 0.98

RHFB-300tf-300tw-160hw-G550-50s C 108.00 112.4 1.04

RHFB-400tf-400tw-160hw-G550-50s C 146.00 149.10 1.02 90×45

RHFB-500tf-500tw-160hw-G550-50s C 183.00 185.5 1.01

RHFB-200tf-200tw-120hw-G550-50s C 20.60 20.65 1.00

RHFB-300tf-300tw-120hw-G550-50s C 30.90 30.57 0.99

RHFB-400tf-400tw-120hw-G550-50s C 41.30 40.24 0.97 30×15

RHFB-500tf-500tw-120hw-G550-50s C 51.60 49.64 0.96

Mean 1.00

COV 0.02


Table 6.7 (b) Comparison of FEA and AS 4100 Section Moment Capacities for Slender RHFB Sections

Hollow Flange Size

(mm×mm)

Section Designation

Sect

ion

com

pact

ness

FEA

Ms

(kN

m)

AS

4100

Ms

(kN

m)

AS

4100

/ FE

A

RHFB-080tf-080tw-150hw-G300-50s S (F, W) 8.10 5.54 0.68


RHFB-080tf-120tw-150hw-G300-50s S (F) 8.42 6.92 0.82

RHFB-080tf-190tw-150hw-G300-50s S (F) 9.61 7.79 0.81



RHFB-100tf-120tw-150hw-G300-50s S (F) 10.40 8.60 0.83

RHFB-100tf-190tw-150hw-G300-50s S (F) 11.70 9.45 0.81



RHFB-120tf-120tw-150hw-G300-50s S (F) 12.20 10.42 0.85

RHFB-120tf-190tw-150hw-G300-50s S (F) 13.60 11.26 0.83

RHFB-190tf-080tw-150hw-G300-50s S (W) 18.10 13.73 0.76

RHFB-190tf-100tw-150hw-G300-50s S (W) 18.70 15.07 0.81




RHFB-055tf-115tw-150hw-G550-50s S (F) 8.87 6.48 0.73




RHFB-075tf-115tw-150hw-G550-50s S (F) 12.30 9.10 0.74




RHFB-095tf-115tw-150hw-G550-50s S (F) 15.80 12.31 0.78

RHFB-115tf-055tw-150hw-G550-50s S (W) 15.00 11.34 0.76

RHFB-115tf-075tw-150hw-G550-50s S (W) 17.90 12.52 0.70

50×25

RHFB-115tf-095tw-150hw-G550-50s S (W) 18.50 13.74 0.74

Mean 0.75

COV 0.08

Note: F – Slender flange (i.e. Flange top plate element) W – Slender web element


6.3.1.2 Member Moment Capacity The member capacity of a beam subject to a uniform bending moment is defined as follows:

sso

s

o

sb MM

MM

MM

M ≤��

�

�

�−+��

�

�

�= 36.0

2

(6.4)

Equation 6.4 was modified by Pi and Trahair (1997) to include the lateral distortional buckling effects.

ssod

s

od

sbd MM

MM

MM

M ≤��

�

�

�−+��

�

�

�= 8.26.0

2

(6.5)

where, od

sd M

M=λ

Pi and Trahair (1997) also provided equations to estimate the elastic distortional

buckling moment (Mod) using an approximate effective torsional rigidity (GJe).

��

�

�+= 2

2

2

2

LEI

GJL

EIM w

ey

od

ππ (6.6)

dLEt

GJ

dLEt

GJGJ

F

F

e

2

23

2

23

91.02

91.02

π

π

+= (6.7)

For compact RHFB sections, the Mod values from Thin-wall program were on

average 4% more than the accurate Mod values obtained from FEA (see Table 5.3

which shows a maximum difference of 8.3% only in some cases). This difference

was expected as Thin-wall program assumes continuous connection between web

and flange lips. Due to this small difference in Mod values and hence even smaller

differences in the resulting Mb values, the simpler method of using Thin-wall was

adopted here. However, for slender RHFB sections, FEA Mod values were used.


A comparison of the predicted dimensionless moment capacity results (Mb/Ms) with

finite element analyses results is provided in Figures 6.11 (a) and (b) and 6.12 (a) and

(b). The member capacity Mu (from FEA) and Mb (from AS 4100) is plotted as

Mb/Ms on the vertical axis whereas the member slenderness �d on the horizontal axis.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

AS4600 S1

S2 S3

S4 S5

S6 S7

S8 S9

S10 S11

S12 S13

S14 S15

S16 S17

S18 S19

S20 S21

S22 S23

S24 S25

S26 S27

S28 S29

S30


Mu/

My

(a) Compact RHFB Sections

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

AS4100 Sec1

Sec2 Sec3

Sec4 Sec5

Sec6 Sec7

Sec8 Sec9

Sec10 Sec11

Sec12 Sec13

Sec14 Sec15

Sec16 Sec18

Sec19 Sec20

Sec21 Sec22

Sec23 Sec24

Sec25 Sec26

Sec27 Sec28

Sec29 Sec30

Sec31 Sec32


Mb/M

s

(b) Slender RHFB Sections

Figure 6.11: Comparison of Moments Capacities with AS 4100 Predictions

Mb/M

s


AS 4100

Mb/M

s

Slenderness (λd)

Mb/M

s

Slenderness (λd)


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

P&T S1

S2 S3

S4 S5

S6 S7

S8 S9

S10 S11

S12 S13

S14 S15

S16 S17

S18 S19

S20 S21

S22 S23

S24 S25

S26 S27

S28 S29

S30


Mu/

My

(a) Compact Sections

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

P&T Sec1

Sec2 Sec3

Sec4 Sec5

Sec6 Sec7

Sec8 Sec9

Sec10 Sec11

Sec12 Sec13

Sec14 Sec15

Sec16 Sec18

Sec19 Sec20

Sec21 Sec22

Sec23 Sec24

Sec25 Sec26

Sec27 Sec28

Sec29 Sec30

Sec31 Sec32


Mb/M

s

(b) Slender Sections

Figure 6.12: Comparison of Moment Capacities with Pi and Trahair’s (1997)

Predictions

Mb/M

s


Mb/M

s


Mb/M

s

Slenderness (λd)

Mb/M

s

Slenderness (λd)


The comparisons provided in Figures 6.11 (a) and (b) and 6.12 (a) and (b)

demonstrate that the moment capacity predicted by AS 4100 (Equation 6.4) and Pi

and Trahair (1997) (Equation 6.5) are conservative in general. The average

conservative error for all RHFB slender sections is 15% (Mean = 0.85) for AS 4100

method and 17% (Mean = 0.83) for the Pi and Trahair (1997) method with a COV of

0.12 and 0.11 for the two methods, respectively, whereas for all RHFB compact

sections, the average conservative error is 4% (Mean = 0.96) for the AS 4100 method

and 8% (Mean = 0.92) for the Pi and Trahair (1997) method with COV of 0.07 for

both cases. The comparison of moment capacities given in Figures 6.11 (a) and 6.12

(a) further demonstrate that both AS 4100 and Pi and Trahair (1997) methods are

incapable of predicting the moment capacities of slender RHFBs accurately in the

low slenderness region whereas they could predict the moment capacities of compact

RHFBs in the low slenderness region quiet accurately. This is due to a combination

of the conservative section capacity prediction and the conservative slenderness

reduction function (αs) based on the lower bound fit of experimental results for

compact, hot-rolled I-sections. However, the maximum unconservative error for

slender RHFB sections was 4% for AS 4100 while it was 14% and 7% for AS 4100

and Pi and Trahair (1997) methods, respectively for compact RHFB sections.

6.3.2 AS/NZS 4600 Moment Capacity

6.3.2.1 Section Moment Capacity

The section moment capacity (Ms) is defined in AS/NZS 4600 in a similar fashion to

AS 4100 (see Equation 6.1). However, unlike AS 4100, the effective section

modulus (Ze) is based on the initiation of yielding in the extreme compression fibre.

Unlike AS 4100, the plate element slenderness is a function of the applied stress (f*),

as shown in Equation 6.9.

bbbe ≤−=λ

λ22.01 (6.8)

Ef

tb

k

*052.1��

�

�=λ (6.9)


Table 6.8 (a) Comparison of FEA and AS/NZS 4600 Section Moment Capacities for Compact RHFB Sections

Hollow Flange Size

(mm×mm)

Section Designation

Sect

ion

com

pact

ness

FEA

Ms

(kN

m)

AS

4600

Ms

(kN

m)

AS

4600

/ FE

A

RHFB-200tf-200tw-150hw-G300-50s C 21.60 17.83 0.83

RHFB-300tf-300tw-150hw-G300-50s C 33.20 26.55 0.80

RHFB-400tf-400tw-150hw-G300-50s C 44.40 35.13 0.79

RHFB-500tf-500tw-150hw-G300-50s C 54.50 43.58 0.80

RHFB-200tf-200tw-100hw-G300-50s C 14.70 11.54 0.79

RHFB-300tf-300tw-100hw-G300-50s C 22.00 17.18 0.78

RHFB-400tf-400tw-100hw-G300-50s C 29.40 22.75 0.77

50×25

RHFB-500tf-500tw-100hw-G300-50s C 36.70 28.25 0.77

RHFB-300tf-300tw-160hw-G300-50s C 60.00 48.17 0.80

RHFB-400tf-400tw-160hw-G300-50s C 80.60 63.99 0.79 90×45

RHFB-500tf-500tw-160hw-G300-50s C 101.00 79.68 0.79

RHFB-200tf-200tw-120hw-G300-50s C 11.30 9.19 0.81

RHFB-300tf-300tw-120hw-G300-50s C 17.00 13.62 0.80

RHFB-400tf-400tw-120hw-G300-50s C 22.70 17.93 0.79 30×15

RHFB-500tf-500tw-120hw-G300-50s C 28.30 22.12 0.78

RHFB-200tf-200tw-150hw-G550-50s C 38.20 32.70 0.86

RHFB-300tf-300tw-150hw-G550-50s C 59.90 48.67 0.81

RHFB-400tf-400tw-150hw-G550-50s C 80.70 64.40 0.80

RHFB-500tf-500tw-150hw-G550-50s C 99.2 79.89 0.81

RHFB-200tf-200tw-100hw-G550-50s C 26.40 21.15 0.80

RHFB-300tf-300tw-100hw-G550-50s C 40.10 31.51 0.79

RHFB-400tf-400tw-100hw-G550-50s C 53.50 41.72 0.78

50×25

RHFB-500tf-500tw-100hw-G550-50s C 66.90 51.78 0.77

RHFB-300tf-300tw-160hw-G550-50s C 108.00 88.32 0.82

RHFB-400tf-400tw-160hw-G550-50s C 146.00 117.31 0.80 90×45

RHFB-500tf-500tw-160hw-G550-50s C 183.00 146.08 0.80

RHFB-200tf-200tw-120hw-G550-50s C 20.60 16.85 0.82

RHFB-300tf-300tw-120hw-G550-50s C 30.90 24.97 0.81

RHFB-400tf-400tw-120hw-G550-50s C 41.30 32.87 0.80 30×15

RHFB-500tf-500tw-120hw-G550-50s C 51.60 40.56 0.79

Mean 0.80

COV 0.02


Table 6.8 (b) Comparison of FEA and AS/NZS 4600 Section Moment Capacities for Slender RHFB Sections

Hollow Flange Size

(mm×mm)

Section Designation

Sect

ion

com

pact

ness

FEA

Ms

(kN

m)

AS

4600

Ms

(kN

m)

AS

4600

/ FE

A

RHFB-080tf-080tw-150hw-G300-50s S (F) 8.10 6.30 0.78

RHFB-080tf-100tw-150hw-G300-50s S (F) 8.06 6.57 0.82

RHFB-080tf-120tw-150hw-G300-50s S (F) 8.42 6.81 0.81

RHFB-080tf-190tw-150hw-G300-50s S (F) 9.61 7.71 0.80

RHFB-100tf-080tw-150hw-G300-50s S (F) 9.33 8.07 0.86

RHFB-100tf-100tw-150hw-G300-50s S (F) 9.84 8.31 0.84

RHFB-100tf-120tw-150hw-G300-50s S (F) 10.40 8.58 0.83

RHFB-100tf-190tw-150hw-G300-50s S (F) 11.70 9.42 0.81

RHFB-120tf-080tw-150hw-G300-50s S (F) 11.80 9.87 0.84

RHFB-120tf-100tw-150hw-G300-50s S (F) 12.30 10.11 0.82

RHFB-120tf-120tw-150hw-G300-50s S (F) 12.20 10.35 0.85

RHFB-120tf-190tw-150hw-G300-50s S (F) 13.60 11.22 0.83


RHFB-055tf-075tw-150hw-G550-50s S (F) 6.90 6.40 0.93

RHFB-055tf-095tw-150hw-G550-50s S (F) 7.75 7.35 0.95

RHFB-055tf-115tw-150hw-G550-50s S (F) 8.87 7.92 0.89


RHFB-075tf-075tw-150hw-G550-50s S (F) 11.00 9.15 0.83

RHFB-075tf-095tw-150hw-G550-50s S (F) 10.20 10.16 1.00

RHFB-075tf-115tw-150hw-G550-50s S (F) 12.30 10.69 0.87


RHFB-095tf-075tw-150hw-G550-50s S (F) 14.50 12.19 0.84

RHFB-095tf-095tw-150hw-G550-50s S (F) 14.80 13.28 0.90

RHFB-095tf-115tw-150hw-G550-50s S (F) 15.80 13.77 0.87


RHFB-115tf-075tw-150hw-G550-50s S (F) 17.90 15.17 0.85

50×25

RHFB-115tf-095tw-150hw-G550-50s S (F) 18.50 16.35 0.88

Mean 0.86

COV 0.06

Note: F – Slender flange (i.e. Flange top plate) W – Slender web


The section moment capacities of all the RHFB sections were calculated using the

AS/NZS 4600 design equations given above with the local buckling coefficients (k)

equal to 4 for the uniformly compressed stiffened elements and

)1(2)1(24 3 ψψ −+−+=k for the stiffened elements with a stress gradient (where,

ψ = f2*/f1

*, f1* is compression and f2

* is either tension or lower compression stress) .

A comparison of section moment capacities from FEA and AS/NZS 4600 is provided

in Tables 6.8 (a) and (b).

The AS/NZS 4600 section capacity method more accurately estimates the reduction

in section moment capacity due to local buckling in slender sections, compared with

the AS 4100 method. However, it does not permit the use of inelastic reserve

capacity and hence the section moment capacities of compact RHFB sections are

about 1.25 times the yield moment capacity. The AS/NZS 4600 prediction is always

conservative, therefore it is safe to use the AS/NZS 4600 specifications for section

moment capacity design checks of RHFB members subject to pure bending moment.

6.3.2.2 Member Moment Capacity

Unlike AS 4100, AS/NZS 4600 provides equations specifically intended for the

design of members subject to distortional buckling that involves transverse bending

of a vertical web with lateral displacement of the compression flange. The member

capacity of a beam subject to this type of distortional buckling is defined as follows:

��

�

�=Z

MZM c

cb (6.10)

For RHFBs, it is appropriate to determine the effective section modulus (Zc) at a

stress corresponding to Mc/Z, where Mc is the critical moment as defined bellow.

��

�

�=≥

��

�

�−=<

2

2

1:414.1

41:414.1

dycd

dycd

MMFor

MMFor

λλ

λλ (6.11)

The non-dimensional member slenderness (λd) is given by:


od

yd M

M=λ (6.12)

The elastic lateral distortional buckling moment (Mod) were determined using

Equations 6.6 and 6.7 or a buckling analysis program such as Thin-wall for compact

RHFBs (see section 6.3.1.2 for reasons). For slender RHFBs, FEA Mod values were

used. This approach was also used when comparing with other design methods.

A comparison of member moment capacities from FEA and AS/NZS 4600 for RHFB

compact and slender sections is given in Figures 6.13 (a) and (b), respectively. In

these Figures, Mu values from FEA are compared with the Mc curve from AS/NZS

4600 (not Mb). For compact sections of any span, Mb is equal to Mc. However, for

slender sections with short spans, Mb will be less than Mc due to local buckling

effects. Therefore AS/NZS 4600 design curve presented in Figure 6.13 (b) cannot be

compared with some of the FEA points with local buckling effects. However, many

FEA points of slender sections may not have local buckling effects at longer or even

intermediate beam spans. Therefore a graphical comparison was also provided for

slender sections for the sake of completeness.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

AS4600 S1

S2 S3

S4 S5

S6 S7

S8 S9

S10 S11

S12 S13

S14 S15

S16 S17

S18 S19

S20 S21

S22 S23

S24 S25

S26 S27

S28 S29

S30


Mu/

My


Figure 6.13: Comparison of Moment capacities with AS/NZS 4600 Predictions

Mc/M

y

Slenderness (λd)


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

AS4600 Sec1

Sec2 Sec3

Sec4 Sec5

Sec6 Sec7

Sec8 Sec9

Sec10 Sec11

Sec12 Sec13

Sec14 Sec15

Sec16 Sec17

Sec18 Sec19

Sec20 Sec21

Sec22 Sec23

Sec24 Sec25

Sec26 Sec27

Sec28 Sec29

Sec30 Sec31

Sec32


Mc/M

y


Figure 6.13: Comparison of Moment capacities with AS/NZS 4600 Predictions

Figures 6.13 (a) and (b) show that the current AS/NZS 4600 design rule for lateral

distortional buckling (Equation 6.11) is not suitable as it is quite conservative for

beams of low slenderness while being unconservative for beams of intermediate

slenderness (inelastic buckling region). Comparison based on the flexural torsional

slenderness show that the strengths Mu/Ms of beams with web distortion are

significantly lower than those without web distortion. This suggests that the

detrimental effects of web distortion are not accurately accounted for. It is adequate

for elastic lateral torsional buckling region as expected for long spans.

6.3.3 Member Moment Capacity Proposed by Avery et al. (2000)

Alternative member moment capacity equations proposed by Trahair (1997) were

used by Avery et al. (2000) to develop design equations for triangular hollow flange

beam (HFB) flexural members. They are:

sybobssnb MMMMMMc

babM ≥≥≤�

�

�

�

+−+= ;;

1 2λ (6.13)


Mc/M

y

Slenderness (λd)


od

sd M

M=λ (6.14)

Values of a = 1.0, b = 0.0, c = 0.424, and n = 1.196 were found to minimise the total

error for the Trahair (1997) design equations. However, this approach resulted in an

unacceptable maximum unconservative error of more than 10% for HFB sections and

Avery et al. (2000) developed separate design curves by varying a, b, c and

depending on the HFB groups. Equations (6.13) and (6.14) with the coefficients a, b,

c and n determined for the entire HFB groups were used to predict the moment

capacities of RHFBs and the comparison of the predicted moment capacities with the

FEA results for compact and slender sections is shown in Figures 6.14 (a) and (b).

As observed in Figures 6.14 (a) and (b), the predictions based on Avery et al.’s

(2000) method for both compact and slender RHFB sections are similar to AS 4100

and Pi and Trahair (1997) predictions for lower slenderness, but for intermediate and

higher slenderness, they are quiet similar to AS/NZS 4600 predictions. Therefore,

Avery et al.’s (2000) method is also quite conservative for beams with low

slenderness while being unconservative for beams with intermediate slenderness

(inelastic buckling region).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

Avery et al S1

S2 S3

S4 S5

S6 S7

S8 S9

S10 S11

S12 S13

S14 S15

S16 S17

S18 S19

S20 S21

S22 S23

S24 S25

S26 S27

S28 S29

S30


Mu/

Ms


Figure 6.14: Comparison of Moment Capacities with Avery et al. (2000) Predictions

Mb/M

s

Slenderness (λd)


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

Avery et al Sec1

Sec2 Sec3

Sec4 Sec5

Sec6 Sec7

Sec8 Sec9

Sec10 Sec11

Sec12 Sec13

Sec14 Sec15

Sec16 Sec18

Sec19 Sec20

Sec21 Sec22

Sec23 Sec24

Sec25 Sec26

Sec27 Sec28

Sec29 Sec30

Sec31 Sec32


Mu/

Ms


Figure 6.14: Comparison of Moment Capacities with Avery et al.’s (2000) Predictions

6.3.4 Moment Capacity Proposed by Maharachchi and Mahendran (2005)

Mahaarachchi and Mahendran (2005c) produced alternative design formulae for the

new hollow flange channel sections known as LiteSteel Beams (LSB). Equations

6.15 (a) to (c) were recommended for the three regions of member slenderness

separating yielding/local buckling, inelastic lateral distortional buckling, and elastic

lateral buckling.

For 59.0≤dλ yc MM = (6.15(a))

For 7.159.0 << dλ ��

�

�=

dyc MM

λ59.0

(6.15(b))

For 7.1≥dλ ��

�

�

�= 2

1

dyc MM

λ (6.15(c))

Comparison of the predicted moment capacities using Equations 6.15(a) to (c) with

FEA results of compact and slender RHFB sections are shown in Figures 6.15 (a)

and (b). As described earlier in Section 6.3.2 in the case of AS/NZS 4600 method,

the FEA results cannot be compared directly with AS/NZS 4600 predictions in a

graphical form due to the presence of local buckling effects.

Mb/M

s

Slenderness (λd)


Therefore Mahaarachchi and Mahendran’s (2005) design curve presented in Figure

6.15 (b) can only be compared with FEA points of RHFBs that do not have local

buckling effects. Even for the slender RHFBs, local buckling effects are not present

for longer or even intermediate beam spans. Therefore a graphical comparison was

also provided for slender sections for the sake of completeness.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

M & M S1

S2 S3

S4 S5

S6 S7

S8 S9

S10 S11

S12 S13

S14 S15

S16 S17

S18 S19

S20 S21

S22 S23

S24 S25

S26 S27

S28 S29

S30


Mu/

My


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

M & M Sec1

Sec2 Sec3

Sec4 Sec5

Sec6 Sec7

Sec8 Sec9

Sec10 Sec11

Sec12 Sec13

Sec14 Sec15

Sec16 Sec17

Sec18 Sec19

Sec20 Sec21

Sec22 Sec23

Sec24 Sec25

Sec26 Sec27

Sec28 Sec29

Sec30 Sec31

Sec32


Mc/M

y


Figure 6.15: Comparison of Moment Capacities with Mahaarachchi and Mahendran’s (2005) Predictions

Mc/M

y

Slenderness (λd)

Mc/M

y

Slenderness (λd)


Figures 6.15 (a) and (b) show that the predictions from Mahaarachchi and

Mahendran’s (2005c) method are generally conservative for both compact and

slender RHFB sections in the lower part of the intermediate slenderness region (i.e.

local and distortional buckling region), whereas in the higher slenderness region (i.e.

lateral torsional buckling region), their predictions closely agreed for both compact

and slender RHFB sections. However, in the upper part of the intermediate

slenderness region (i.e. lateral distortional buckling region) their predictions are quiet

unconservative. This could be due to the fact that the design curves developed by

Mahaarachchi and Mahendran (2005c) were based on experimental and analytical

results of continuously welded, singly symmetric LSB sections. However, the RHFB

sections are intermittently screw fastened and therefore their lateral distortional

buckling resistance could be reduced due to the lack of continuity between the flange

and the web. This effect is particularly higher in the lateral distortional buckling

region (see Section 6.2.2.2).

6.4 Development of Moment Capacity Rules for RHFBs As shown in Section 6.3, no design methods provide accurate prediction of the

moment capacity of RHFB sections. Further no provisions are given for lateral

distortional buckling except AS/NZS 4600 which has been found to be

unconservative. The comparison of design methods implies that in some situations

competing methods may exist. However, the fact that one method may be used to

determine the strength prediction in a given situation does not imply an increased

accuracy and resistance factors are needed to be within their target reliability.

An objective of this research was to derive and verify appropriate design formulae

for RHFB sections. The finite element analysis results were used to derive suitable

design equations. Thin-walled nature and intermittent screw fastening of RHFB

sections complicates their behaviour and design. Experimental and finite element

analyses reveal the presence of at least three buckling modes, namely, local, lateral

distortional and lateral torsional. Most of the design recommendations have only

addressed local and lateral torsional buckling modes and therefore new design rules

for lateral distortional buckling are required to design the new RHFBs accurately.

Following sections will describe the procedures for the development of new design

rules and their applicability.


6.4.1 Development of Moment Capacity Rules for Compact RHFB Sections

The moment capacities of fully effective or compact sections are generally well

predicted by the current design methods. The lateral torsional buckling strength of

compact sections follows the same trend predicted by AS/NZS 4600. A new design

rule was therefore developed for the inelastic buckling region that is based on the

mean of all the compact section results. The equation was established by solving for

minimum total error for 32 compact sections and 7 beam spans ranging from 200 mm

to 8000 mm. This was achieved by minimising the square of the difference between

the normalised analytical capacity (i.e. Mu/Ms) and the normalised design capacity

(Mb/Ms). The established design rule is given by Equation 6.16, and Figure 6.16

shows the comparison of the design curve based on this equation with FEA results.

The FEA results were spread around the developed design curve with a mean test to

predicted ratio of 1.00 and a COV of 0.08. A suitable capacity reduction factor was

calculated for this equation using the AISI procedure (see Section 6.4.3) and it was

found to be 0.82. Meanwhile Equation 6.16(c) represents the elastic buckling curve

as for the AS/NZS 4600 design rule using a revised member slenderness limit as

shown in Figure 6.16, and it is adequate.

71.0≤dλ yc MM = (6.16(a))

70.171.0 << dλ ��

�

��

� += 210

55.1

d

dyc MM

λλ

(6.16(b))

70.1≥dλ ��

�

��

�=

2

1

d

yc MMλ

(6.16(c))

where

od

yd M

M=λ

yfy fZM = fZ and yf are full section modulus and yield stress

odfod fZM = odf is obtained from Thin-wall program


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

S1 S2

S3 S4

S5 S6

S7 S8

S9 S10

S11 S12

S13 S14

S15 S16

S17 S18

S19 S20

S21 S22

S23 S24

S25 S26

S27 S28

S29 S30

Wanni-I


Mu/

My

Figure 6.16: Comparisons of Moment Capacities Predicted by Equation 6.16 with

FEA Results

The member slenderness values that separate the three regions (yielding/local

buckling and inelastic lateral distortional buckling, and elastic lateral torsional

buckling) were determined from the equations representing the three regions, and

they are 0.71 and 1.70. Since the capacity reduction factor for the design rule based

on Equation 6.16 is 0.82, attempts were made to develop a design equation that gives

the currently used capacity reduction factor of 0.9, and it is given in Equation 6.17.

For this equation, the mean test to predicted ratio was 1.09 with a COV of 0.08.

Figure 6.17 compares the predicted moment capacities from Equations 6.16 and 6.17

and FEA results.

65.0≤dλ yc MM = (6.17(a))

80.165.0 << dλ ��

�

��

� += 210

51

d

dyc MM

λλ

(6.17(b))

80.1≥dλ ��

�

��

�=

2

1

d

yc MMλ

(6.17(c))

Mc/M

y

Slenderness (λd)


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

Eq.6.17 S1

S2 S3

S4 S5

S6 S7

S8 S9

S10 S11

S12 S13

S14 S15

S16 S17

S18 S19

S20 S21

S22 S23

S24 S25

S26 S27

S28 S29

S30 Eq.6.16


Mu/

My

Figure 6.17: Comparison of Moment Capacities Predicted by Equations 6.16 and

6.17 and FEA Results

The comparison of moment capacities predicted by Equation 6.17 with Equation

6.15 of Mahaarachchi and Mahendran (2005), Equation 6.11 of AS/NZS 4600

(1996) and FEA results are provided in Figure 6.18.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

Eq.6.17 S1

S2 S3

S4 S5

S6 S7

S8 S9

S10 S11

S12 S13

S14 S15

S16 S17

S18 S19

S20 S21

S22 S23

S24 S25

S26 S27

S28 S29

S30 M & M

AS 4600


Mc

/ My

Figure 6.18: Comparison of Moment Capacities Predicted by Equation 6.17 and

other Existing Design Rules and FEA Results

Mc/M

y

Slenderness (λd)

Mc/M

y

Slenderness (λd)


The comparisons provided in Figure 6.18 demonstrate that the member capacity

predicted by the new design formula (i.e. Equation 6.17) is more accurate than the

available design recommendations. However, the design rules presented by

Mahaarachchi and Mahendran (2005) (i.e. Equation 6.15) could also predict moment

capacities of RHFBs quite accurately. The mean test to predicted ratio using their

design formula was 1.12 with COV of 0.11. Therefore, the predicted capacity by

Equation 6.17 and Mahaarachchi and Mahendran (2005) can be used with the

AS/NZS 4600 (1996) capacity reduction factor of 0.9 to produce adequate safety for

design. Therefore Equations 6.17 and 6.15 (Mahaarachchi and Mahendran, 2005) are

recommended for the design of RHFB members subject to uniform bending.

6.4.2 Verification of the New Design Formula for Slender RHFB Sections

The new design formulae were developed for RHFBs using the finite element

analysis results of compact sections and are given by Equations 6.17 (a) to (c).

Therefore their applicability to slender RHFBs needs to be confirmed. The FEA

results of slender RHFB sections allow for capacity reduction due to local buckling

effects. Hence the new design formula (Equation 6.17 (b)) was used for a number of

slender RHFB sections to calculate member moment capacity (Mb), and those

calculated (i.e. predicted) values were compared with the FEA results as shown in

Tables 6.9(a) and (b) for G300 and G550 steels, respectively. The beam spans

provided in Tables 6.9 (a) and (b) are corresponding to the lateral distortional

buckling region defined for the new design formula. Figure 6.19 gives a graphical

comparison of predicted moment capacities (Note that Mb allows for local buckling

effects) and FEA results for both G300 and G550 steels. A sample calculation of

member moment capacity using the new design formula is given in Appendix 6A. As

noted in earlier sections, the Mod values required for λd calculations were based on

accurate FEA models for slender sections.


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.60 0.80 1.00 1.20 1.40 1.60 1.80


Mb,M

u/M

y

Pred-G300

FEA-G300

Pred-G550

FEA-G550

Figure 6.19: Comparison of Predicted Moment Capacities and FEA Results for

G300 and G550 Steel Slender RFHBs

The graphical comparison of predicted moment capacities with FEA results for

slender RHFBs indicates that the predictions from this new design formula for

slender RHFB are more conservative in the lower end of the intermediate slenderness

region while in the upper end, it agrees well . This could be due to the fact that this

new design formula was developed using compact RHFB sections for which the

local buckling was not a consideration. However, in terms of slender RHFB sections,

local buckling could be a prominent criterion in the lower end of the intermediate

slenderness region, whereas in the upper end, local buckling has less effects. Figure

6.19 further demonstrates that the predicted moment capacities are more conservative

for higher grade steels than lower grade steels in the lower end of intermediate

slenderness region.

The comparisons given in Tables 6.9 (a) and (b), and Figure 6.19 indicate that the

developed design formula for lateral distortional buckling can also be safely used to

design the slender RHFBs as well. The mean ratio between the FEA to predicted

moment capacity for G300 steel is 1.06 with a COV of 0.08, whereas for G550

steels, they are 1.18 and 0.19, respectively. Therefore this comparison indicates that

this new design formula can be used more consistently although more safely for

lower grade steel than the higher grade steel slender RHFBs.


Table 6.9(a): Comparison of Predicted Moment Capacities using New Design Formula with FEA Results for G300 Steel Slender RHFBs

Beam Span λd Mb Mu Mb/Mu

2000 1.13 3.56 3.96 1.11 3000 1.33 3.02 3.05 1.01 RHFB-080tf-080tw-150hw-G300-50s 5000 1.63 2.45 2.41 0.98 2000 1.09 4.83 5.40 1.12 3000 1.24 4.20 4.30 1.02 RHFB-100tf-100tw-150hw-G300-50s 5000 1.47 3.47 3.48 1.00 2000 1.04 6.17 6.93 1.12 3000 1.18 5.33 5.70 1.07 5000 1.41 4.36 4.63 1.06

RHFB-120tf-120tw-150hw-G300-50s

8000 1.65 3.66 3.69 1.01 2000 0.93 11.07 13.30 1.20 3000 1.05 9.61 11.60 1.21 5000 1.29 7.59 8.94 1.18

RHFB-190tf-190tw-150hw-G300-50s

8000 1.56 6.13 6.57 1.07 2000 1.11 3.76 4.37 1.16 3000 1.28 3.26 3.43 1.05 RHFB-080tf-100tw-150hw-G300-50s 5000 1.51 2.73 2.73 1.00 2000 1.09 3.96 4.63 1.17 3000 1.25 3.46 3.68 1.06 RHFB-080tf-120tw-150hw-G300-50s 5000 1.50 2.84 2.94 1.03 2000 1.09 4.44 5.19 1.17 3000 1.25 3.83 4.10 1.07 RHFB-080tf-190tw-150hw-G300-50s 5000 1.48 3.20 3.26 1.02 2000 1.06 5.07 5.84 1.15 3000 1.21 4.40 4.80 1.09 RHFB-100tf-120tw-150hw-G300-50s 5000 1.43 3.67 3.82 1.04 2000 1.03 5.72 6.83 1.20 3000 1.17 4.98 5.57 1.12 RHFB-100tf-190tw-150hw-G300-50s 5000 1.41 4.07 4.38 1.08 2000 0.99 7.04 8.41 1.19 3000 1.12 6.10 7.03 1.15 RHFB-120tf-190tw-150hw-G300-50s 5000 1.37 4.85 5.47 1.13 2000 1.13 4.51 4.92 1.09 3000 1.34 3.75 3.72 0.99 RHFB-100tf-080tw-150hw-G300-50s 5000 1.64 2.99 2.96 0.99 2000 1.13 5.36 5.77 1.08 3000 1.34 4.41 4.28 0.97 RHFB-120tf-080tw-150hw-G300-50s 5000 1.58 3.67 3.43 0.93 2000 1.14 8.08 8.54 1.06 3000 1.47 6.06 5.87 0.97 RHFB-190tf-080tw-150hw-G300-50s 5000 1.70 5.15 4.70 0.91 2000 1.08 5.78 6.34 1.10 3000 1.23 4.98 5.10 1.02 5000 1.44 4.16 4.13 0.99

RHFB-120tf-100tw-150hw-G300-50s

8000 1.68 3.51 3.29 0.94 2000 1.08 8.73 9.36 1.07 3000 1.24 7.45 7.14 0.96 5000 1.42 6.39 5.92 0.93

RHFB-190tf-100tw-150hw-G300-50s

8000 1.69 5.26 4.87 0.93 2000 1.03 9.36 10.30 1.10 3000 1.16 8.16 8.50 1.04 5000 1.36 6.81 6.96 1.02

RHFB-190tf-120tw-150hw-G300-50s

8000 1.59 5.71 5.53 0.97 Mean 1.06 COV 0.08


Table 6.9(b): Comparison of Predicted Moment Capacities using New Design Formula with FEA Results for G550 Steel Slender RHFBs

Beam Span λd Mb Mu Mb/Mu 1000 0.98 3.91 6.62 1.69 RHFB-055tf-055tw-150hw-G300-50s 2000 1.70 2.54 3.00 1.18 1000 0.97 6.54 9.45 1.44 RHFB-075tf-075tw-150hw-G300-50s 2000 1.70 3.74 3.63 0.97 1000 0.95 8.97 12.10 1.35 RHFB-095tf-095tw-150hw-G300-50s 2000 1.49 5.72 5.62 0.98 1000 1.43 7.39 7.34 0.99 RHFB-115tf-115tw-150hw-G300-50s 2000 1.68 6.30 5.80 0.92 1000 1.00 4.46 6.30 1.41 RHFB-055tf-075tw-150hw-G300-50s 2000 1.63 2.82 3.75 1.33 1000 1.01 4.74 7.43 1.57 RHFB-055tf-095tw-150hw-G300-50s 2000 1.60 3.02 3.50 1.16 1000 0.88 5.81 8.50 1.46 RHFB-055tf-115tw-150hw-G300-50s 2000 1.59 3.21 3.61 1.12 1000 1.52 4.34 4.45 1.02 RHFB-075tf-095tw-150hw-G300-50s 2000 1.80 3.68 3.41 0.93 1000 0.98 7.09 9.12 1.29 2000 1.50 4.56 4.73 1.04 RHFB-075tf-115tw-150hw-G300-50s 3000 1.77 3.88 3.67 0.95 1000 0.96 9.11 11.32 1.24 2000 1.45 6.00 6.11 1.02 RHFB-095tf-115tw-150hw-G300-50s 3000 1.66 5.27 4.86 0.92 1000 0.97 5.79 8.69 1.50 RHFB-075tf-055tw-150hw-G300-50s 2000 1.68 3.66 3.86 1.06 1000 0.95 7.72 11.32 1.47 RHFB-095tf-055tw-150hw-G300-50s 2000 1.69 4.74 4.75 1.00 1000 0.95 8.62 12.20 1.41 RHFB-095tf-075tw-150hw-G300-50s 2000 1.58 5.21 4.90 0.94 1000 0.95 10.70 14.10 1.32 RHFB-115tf-075tw-150hw-G300-50s 2000 1.60 6.28 5.76 0.92 1000 0.94 11.21 14.70 1.31 RHFB-115tf-095tw-150hw-G300-50s 2000 1.48 7.02 6.66 0.95

Mean 1.18 COV 0.19

In the above calculations for slender RHFB sections, Mod was obtained from the

elastic buckling finite element analyses to improve the accuracy of comparison.

However, in practice other simple methods such as Thin-wall buckling analysis

program or Pi and Trahair’s equations (Equations 6.6 and 6.7) may used to obtain the

elastic buckling moment. For compact RHFB sections, it has been shown that the

difference in Mod and the resulting Mb values is negligible for design purposes. To

investigate the effect of using Mod values from Thin-wall or Pi and Trahair’s

equations on the member moment capacity Mb, two critical slender RHFB sections

were selected (one from G300 and other from G550) that have the highest difference

in FEA and Thin-wall buckling moments (Mod). The results show that for the G300

steel section (ie. RHFB-080tf-080tw-150hw-G300-3L-50s) the Mb prediction was


5.6% unconservative whereas for the G550 steel section (i.e. RHFB-055tf-055tw-

150hw-G550-3L-50s) it was 6%. It is likely that the average difference between Mb

values will be less than 5%. Therefore Thin-wall buckling program or Pi and

Trahair’s equations could be used to obtain the elastic distortional buckling moment

Mod for member moment capacity calculations.

6.4.3 Calculation of Capacity Reduction Factor (φφφφ)

The proposed design equation in Section 6.4.1 of this chapter could predict the

moment capacities of RHFBs, however, it was derived based on limited FEA data.

Therefore the actual moment capacity of a RHFB used in a real steel structure could

be considerably less than the value predicted by these equations due to expected

variations in material, fabrication, and loading effects. Therefore a capacity reduction

factor, which is commonly used in the current design codes, is recommended for the

strength predicted by the proposed design formula.

The American cold-formed steel structures code (AISI, 1996) recommends a

statistical model for the determination of capacity reduction factors. This model

accounts for the variations in material, fabrication and load effects. This model is

used in the Australian cold-formed steel structures code AS/NZS 4600 (SA, 1996)

and this reduction factor accounts for the uncertainties and variabilities associated

with loads, analysis, the limit state model, material properties, geometry and

fabrication. Based on this model, the capacity reduction factor φ is given by the

following equation.

QPpFM VVCVV

mmm ePFM2222

)(5.1 +++−= βφ (6.18)

where

Mm, Vm = Mean and coefficient of variation of the material factor

= 1.0, 1.0 (this is the ratio of actual material property to that specified)

Fm, Vf = Mean and coefficient of variation of the fabrication factor

= 1.0, 1.0 (this is the ratio of actual geometric property, e.g. thickness, to that specified)

Vq = Coefficient of variation of load effect


βo = Target reliability index = 2.5 for lateral buckling in flexure

Cp = Correction factor depending on the number of tests = (1+1/n)(m/(m- 2)

Pm = Mean value of the tested to predicted moment capacity ratio

Vp = Covariance of the tested to predicted moment capacity ratio

n = Number of tests

m = Degree of freedom = n-1

The values for Pm and Vp have to be determined from experiments or analyses. In

this study, Pm and Vp are the mean values and coefficient of variation of the ratio of

FEA moment capacity, to that predicted by the design equation. Other values are

taken from the American cold-formed steel structures code (AISI, 1996) and are

considered to be conservative for most flexural members subjected to lateral

buckling. Mahaarachchi and Mahendran (2005c) used the same values in their

investigation. The substitution of these assumed values leads to the following

equation.

20502.05.25.1 ppVC

m eP +−=φ (6.19a)

)2(

)1

1(

−

+=

m

mnC p (6.19b)

Equations 6.19 (a) and (b) were used to calculate the reduction factor φ for the design

formula (Equation 6.16 and 6.17) derived in Section 6.4.1. The capacity reduction

factor φ for Equation 6.17 was determined using the values for Pm and Vp

corresponding to the minimum of the square difference between the normalised

analytical moment capacities (i.e. Mu/Ms) and the normalised design moment

capacities (Mb/Ms). The calculated φ factor for Equation 6.16 was 0.82,

corresponding to the Pm and Vp values of 1.0 and 0.08, respectively. The capacity

reduction factor φ = 0.9 is used in other existing steel design standards and therefore

attempts were made to raise the φ factor to 0.9 by modifying Equation 6.16 to 6.17.

The Pm and Vp values corresponding to Equation 6.17 are 1.09 and 0.08. It is shown


in Sections 6.4.1 and 6.4.2 that the proposed design Equation 6.17 can be reliably

used to predict the moment capacity of RHFB, with a reduction factor of 0.9. It

should be noted that the Australian cold-formed steel structures code also

recommends a capacity reduction factor φ of 0.9 for flexural members. Therefore, the

recommended reduction factor is acceptable within the current design provisions.

6.5 Summary

This chapter has presented the details of an extensive parametric study into the

structural behaviour of RHFBs subjected to bending actions, and the procedures of

development of new design rules for RHFBs. For this purpose, finite element

analyses were used based on the validated ideal finite element model developed in

Chapter 5. The results show that the current design recommendations are not suitable

for RHFBs except for the method proposed by Mahaarachchi and Mahendran

(2005c). The proposed design formula in this research more accurately predicts the

moment capacities of RHFB subjected to lateral distortional buckling than those in

AS/NZS 4600 (SA, 1996), AS 4100 (1998), Pi and Trhair (1997) and Avery et al.

(2000). Mahaarachchi and Mahendran’s (2005c) design method could also predict

the moment capacities of RHFBs reasonably well, however it shows that the

proposed design formula is the most accurate for the RHFBs. In this study, the

current design rules given in AS/NZS 4600 to predict the section moment capacity

and the lateral torsional buckling moment capacity were found to be sufficient.

Therefore a new design rule was developed only for the inelastic lateral distortional

buckling region. The AS/NZS 4600 capacity reduction factor of 0.9 can be used with

the proposed new design formula. The new design formula is therefore

recommended to use in the design of RHFBs subjected to lateral distortional

buckling.


CHAPTER 7 Conclusions and Recommendations

7.1 Conclusions

7.1.1 General

This thesis has described a detailed investigation into the structural behaviour of

Rectangular Hollow Flange Beams (RHFB), subjected to flexural action. Buckling

and ultimate failure behaviour of the new RHFBs was investigated using both

experimental and finite element analyses to gain a thorough understanding of the

structural behaviour of this new beam with torsionally rigid rectangular hollow

flanges.

In the first phase of this research, experimental investigation of RHFBs using thirty

full scale lateral buckling tests and twenty two section moment capacity tests were

conducted in two separate and specially made test rigs to simulate the required

loading and support conditions. In addition to these full scale member capacity

(lateral buckling) and section capacity tests, a series of tensile coupon tests was also

conducted to obtain the material properties of steels that were used to make the

RHFB test specimens.

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 and validated by

comparing the failure loads, the moment-deflection curves and the failure modes

with corresponding results from the full scale tests. Apart from these experimental

finite element models, ideal finite element models simulating ideal simply supported

boundary conditions and a uniform moment loading were also developed for use in a

detailed parametric study. Both finite element models included all significant effects


that may influence the ultimate moment capacity of RHFBs, including material

inelasticity, local buckling, lataeral distortional buckling and lateral torsional

buckling deformations, member instability, intermittent screw fastening and

associated discontinuity along the beam, residual stresses and initial imperfections.

The parametric study results were used to review the current design formulae and to

develop new design formulae for RHFBs subjected to local, lateral distortional and

lateral torsional buckling effects.

Following important conclusions and recommendations have been drawn, based on

the research presented in this thesis.

7.1.2 Experimental Investigation

Section moment capacity tests reported in Chapter 3 showed that the slender RHFB

sections experienced local buckling effects in either flange or web, or both depending

on the their element slenderness. When the flange or web buckled locally, the gap

between flange lips and web opened up between the intermittent screw fasteners,

mainly after the ultimate moment was reached.

The tests results showed that the predicted section moment capacities based on AS

4100 (1998) and AS/NZS 4600 (1996) are unconservative when compared with the

experimental section moment capacities in some cases. 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. Therefore 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 experiment to predicted (i.e. Mu/Ms) section moment capacity ratios is

about 0.95, but the coefficient of variation is as high as 0.24 due to lack of adequate

quality control in test specimen fabrication.

The section capacity behaviour of such RHFB sections fabricated by using

intermittent screw fastening was not observed or investigated by previous researchers

for flexural actions, and therefore these observations and findings are important and


useful to the researchers, manufactures and designers expecting to use RHFBs in

steel buildings and structures.

The lateral buckling test results reported in Chapter 4 showed that the new RHFBs

failed by lateral distortional buckling at intermediate beam slenderness. The use of

different combinations of thicknesses of flanges and web indicated that increasing

the flange thickness is more effective in enhancing the RHFB flexural capacity than

web thickness.

The comparison of lateral buckling test results with the predictions of member

capacities calculated using the current steel design codes AS 4100 and AS/NZS

4600, and the design methods proposed by Pi and Trahair (1997), Avery et al (1999)

and Mahaarachchi and Mahendran (2005c) using non-dimensionalised moment and

slenderness results showed that the member moment capacities predicted by all the

design methods for lateral distortional buckling were generally unconservative. The

predicted moment capacities of AS 4100 (1998) were extremely higher than those of

test moment capacities whereas AS/NZS 4600 (1996) also predicted moment

capacities that were higher than test moment capacities. In contrast, the predicted

moment capacities of Pi and Trahair’s (1997) and Mahaarachchi and Mahendran’s

(2005) design methods were more close to the test moment capacities.

7.1.3 Finite Element Analyses and Parametric studies

Comparison of experimental and finite element analysis results presented in Chapter

5 showed that the buckling and ultimate failure behaviour of RHFBs can be

simulated well using appropriate finite element models. This confirms that finite

element analysis is an excellent tool for use in the investigations of flexural

behaviour of complex beam types such as RHFBs, and it can be successfully used to

contribute towards the development of design models and rules when existing design

methods are found to be inadequate. This will reduce the reliance on time consuming

and expensive experimental testing while allowing the incorporation a wide range of

possible influential parameters and enhancing the overall reliability of final results.


Finite element analysis into the effect of intermittent screw fastening on flexural

capacity of RHFBs indicates that the discontinuity due to screw fastening has a

significant influence within the intermediate beam spans (i.e. lateral distortional

buckling region). However, this effect is minimal at short and longer beam spans.

Parametric study into the effect of residual stresses on the moment capacities of

RHFBs indicated that the residual stress effect is only significant at very short beam

spans (about 13%), but they are only marginal at longer span beams (about 2%).

Similarly, the initial geometric imperfection has little influence (about 5%) on the

moment capacities of RHFBs at shorter spans while its influence at longer span is

little higher (about 9%).

Comparison of finite element analysis results with the predictions from the current

design rules AS 4100 (1998) and AS/NZS 4600 (1996) indicates that the design rules

given for section moment capacity design within the current steel design standards

are sufficient. However, the results further indicate that the section moment capacity

prediction for slender RHFB sections is more accurate using AS/NZS 4600 than AS

4100, whereas for compact RHFB sections, AS 4100 is more accurate as it allows for

the non-elastic reserve capacity .

Finite element analysis investigations into the lateral distortional buckling of RHFBs

indicate that lateral distortional buckling dominates the failure in the intermediate

member slenderness region (intermediate beam spans). This failure mode of RHFBs

is affected by intermittent screw spaning and therefore a minimum of 50 mm screw

spacing was adopted in this study to minimize the moment capacity reduction due to

larger screw spacing. The comparison of finite element analysis results with the

predictions from the current design rules indicates that the current design rules in AS

4100 (1998) and AS/NZS 4600 (1996) are not suitable for the lateral distortional

buckling design of RHFBs. The comparison of finite element analysis results with

other design methods indicates that Mahaarachchi and Mahendran’s (2005c) design

method could closely predict the lateral distortional buckling moment capacity of

RHFBs.

However this thesis developed a new design rule to predict the lateral distortional

buckling of RHFBs more accurately. An extensive series of finite element analyses


was conducted incorporating all significant parameters such as residual stresses,

geometric imperfections, contact surfaces and nonlinear material behaviour. The

developed design formula is most suitable for use in the design of RHFBs. This new

design formula was first developed using finite element analyses results obtained

from the compact RHFB sections, however its validity for slender sections was also

verified. Finite element analysis results demonstrated that the current design formula

for lateral torsional buckling is adequate. However, the non-dimensionalised

slenderness limit suggested by Mahaarachchi and Mahendran for lateral torsional

buckling was slightly varied in the new design formula. Therefore the same design

formula can be used in the design of RHFBs for lateral torsional buckling within the

limit defined in this thesis.

7.2 Recommendations

In the design of RHFBs subjected to flexural action, it is recommended that the

current design rules given in AS 4100 and AS/NZS 4600 are used to calculate the

section moment capacities of RHFBs depending on the section slenderness. AS/NZS

4600 design method was found to be more suitable for slender sections while AS

4100 design methods was more suitable for compact sections. However, since RHFB

is a light gauge (thickness < 3 mm), high strength (yield stress > 450 MPa) and cold-

formed steel section, it is recommended that AS/NZs 4600 design rules are used for

RHFBs.

In the design of RHFBs for lateral distortional buckling, it is recommended that the

new design formula developed in this thesis is used to more accurately design

RHFBs subjected to lateral distortional buckling. However, Mahaarachchi and

Mahendran’s (2005c) design formula for lateral distortional buckling can also be

used for this purpose.

In the design of RHFBs for lateral torsional buckling, it is recommended that the

existing design rules provided in AS/NZS 4600 (1996) can be used with the new

non-dimensionalized slenderness limit given as part of the new design formula

developed in this thesis.


It is recommended that the test set-up used in the lateral buckling test program can be

accurately used to simulate the ideal simply support conditions in any future research

project on the flexural behaviour of steel beams. The finite element model developed

in this thesis to investigate the flexural behaviour of RHFB is also recommended for

any future research project aimed at investigating the buckling and ultimate strength

behaviour of similar hollow flange beams.

The research undertaken in this thesis has demonstrated that the innovative RHFB

sections can perform well as economically sound and structurally efficient flexural

members. It has shown the effects of various combinations of web and flange

thicknesses on the RHFB flexural capacity. 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.

7.3 Future Work

Although other hollow flange beam types such as triangular hollow flange beams and

rectangular hollow flange channel sections are used as mainstream structural

components in buildings and other steel structures in Australia, the new RHFB has

not yet been introduced into the industry. Therefore further research should be

undertaken to enhance the understanding of the other behavioural aspects of RHFBs

(for example, shear buckling, web bearing and connections) under specific

Australian conditions.

In this research project intermittent screw spacing was selected (50 mm) to minimise

the moment capacity reduction due to discontinuity at flange-web connection.

However, there is a possibility to find an optimum screw spacing if a comprehensive

research study is carried out incorporating the possible influential parameters such as

steel thickness, steel grade, section and beam slenderness.


The interactive buckling behaviour of RHFBs under flexural action was not

specifically investigated in this study; however the interactive buckling effects may

have an influence on the overall performance of RHFBs. Therefore a thorough

understanding is required to make conclusions on this effect.

Only rectangular hollow flanges and screw fastening method were considered in this

study. However research can be extended to other hollow flange shapes with

different connection methods to optimize the performance of hollow flange beams.

In this research, the web was extended only 10 mm from the flange-web junction into

the hollow flanges to minimize the self-weight of RHFB. However, it can be

extended to reach the top flange plate. This is likely to enhance the bearing strength

of the RHFB sections (at the supports and under concentrated loads) while also

facilitating easier fabrication.

Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges R-1

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Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3A-1

Appendix 3A: Steel Properties and Chemical Compositions Provided by Steel Supplier (BHPSTEEL)

PRODUCT: ZNCALUME G300 AZ150 SPECIFICATION: AS 1397

CHEMICAL ANALYSS (Basic Oxygen Steelmaking Process)

Dimension Chemical Composition (%)

C P Mn Si S Ni Cr Mo Cu Al Ti Nb 0.55 × 1200

0.05 0.015 0.21 <0.005 0.017 0.024 0.014 0.002 0.006 0.032 <0.003 0.001

MECHANICAL TESTING

Tensile AS 1391

Elongation Thickness (mm) Yield Strength

(MPa)

Tensile Strength

(MPa) L0 (mm) %

Hardness HR30T

0.55 349 408 80 26 56






0.05 0.013 0.20 <0.005 0.010 0.023 0.018 0.002 0.012 0.036 <0.003 0.001

MECHANICAL TESTING

Tensile AS 1391


(MPa)

Tensile Strength

(MPa) L0 (mm) %

Hardness HR30T

0.80 351 409 80 29 59






0.06 0.007 0.22 0.005 0.013 0.025 0.010 0.002 0.006 0.031 <0.003 0.001

MECHANICAL TESTING

Tensile AS 1391


(MPa)

Tensile Strength

(MPa) L0 (mm) %

Hardness HR30T

1.20 332 389 80 30 66






0.06 0.012 0.21 0.005 0.014 0.026 0.011 0.002 0.006 0.038 <0.003 0.001

MECHANICAL TESTING

Tensile AS 1391


(MPa)

Tensile Strength

(MPa) L0 (mm) %

Hardness HR30T

0.55 698 698 80 2 80






0.055 0.012 0.21 0.005 0.014 0.029 0.017 0.002 0.008 0.040 <0.003 0.001

MECHANICAL TESTING

Tensile AS 1391


(MPa)

Tensile Strength

(MPa) L0 (mm) %

Hardness HR30T

0.95 643 645 80 10 79

Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3B-1

Appendix 3B: Stress versus Strain Graphs from Tensile Tests

0.55 mm G300 steel (Specimen 1)

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25 30 35

% Strain

Stre

ss (M

Pa)


0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25 30 35

% Strain

Stre

ss (M

Pa)


0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

% Strain

Stre

ss (M

Pa)



0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

% Strain

Stre

ss (M

Pa)


0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30 35

% Strain

Stre

ss (M

Pa)


0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30 35

% Strain

Stre

ss (M

Pa)



0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30

% Strain

Stre

ss (M

Pa)


0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30 35

% Strain

Stre

ss (M

Pa)


0

100

200

300

400

500

600

700

0 1 2 3 4 5 6

% Strain

Stre

ss (M

Pa)



0

100

200

300

400

500

600

700

0 1 2 3 4

% Strain

Stre

ss (M

Pa)


0

100

200

300

400

500

600

700

800

0 1 2 3 4 5 6

% Strain

Stre

ss (

MPa

)


0

100

200

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400

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600

700

0 1 2 3 4 5

% Strain

Stre

ss (M

Pa)



0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14

% Strain

Stre

ss (

MPa

)


0

100

200

300

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500

600

700

0 2 4 6 8 10 12 14

% Strain

Stre

ss (

MPa

)


0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14

% Strain

Stre

ss (

MPa

)



0

100

200

300

400

500

600

700

0 1 2 3 4 5 6 7 8

% Strain

Stre

ss (M

Pa)

Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3C-1

APPENDIX 3C: Calculation of Section Properties and Section Moment Capacities Based on the Current Design Rules

I. AS 4100 Design Method

Section moment capacity Ms is defined in AS 4100 (SA, 1998) as:

Ms = Ze fy (3C-1)

where Ze is the effective section modulus and fy is the yield stress.

Unlike hot-rolled steel sections, cold-formed steel sections such as RHFB may

include many slender elements. Therefore, the effective section modulus (Ze) defined

in AS 4100 (1998) for slender sections is not quiet suitable. Following procedure was

therefore used to calculate the effective section modulus (Ze) for RHFB sections.

Calculation of effective section modulus Ze based on AS 4100 (SA, 1998) 1

2 2 3 3

4 4

5

7

6

8

9 9 10 10

11 11

12

13

Figure 3C-1: Element Definition

Elements of a typical RHFB section were defined as shown in Figure 3C-1.

For example 1, consider RHFB-120tf-120tw-150hw-G300-50s section:

Centreline dimensions, yield stress and steel thicknesses of the above cross-section

are:


Element No. 1,8 2,9 3,10 4,6,11,13 5,12 7

Dimensions (mm) 52.7 25.2 30.4 13.8 10.0 118.5

Note: element 7 dimension is the clear distance, allowing for elements 4 and 11. fy = 320 MPa, flange thickness tf = 1.192 mm, and web thickness tw = 1.192 mm

Effective widths of the elements were determined as follows.

250

ye

f

tb=λ (3C-2)

For example 1, consider element 1;

250320

192.17.52=eλ =50

bbe

eye ��

�

��

�=

λλ

= 5.535040��

��

� = 42.2 mm

Similarly, the effective widths of other elements were calculated and tabulated in

Table 3C-1. In these calculations the use of λey for both longitudinal edges supported

element case might have caused some inaccuracies for elements 2 and 7 as those

elements are only intermittently screw-fastened. This could have led to a slight

overestimation of Ms.

Table 3C-1: Element slenderness and effective width

Element λe λey be 1 50 40 42.2 2 24 40 25.2 3 29 40 to 115 30.4 4 13 22 13.8 5 9 22 10.0 6 13 - 13.8 7 112 115 118.5

All other elements are in tension and hence they are not subjected to local buckling.


Calculation of centroid distance, y from top fibre of compression flange

y = [2×25.2×1.192×30.4 + 2×30.4×1.192×15.2 + 2×13.8×1.192×(30.4+6.9) + 10×1.192

×(30.4-5) + 13.8×1.192×(30.4+6.9) + 118.5×1.192×(30.4+13.8+59.25) +52.7×1.192

×(2×30.4+2×13.8+118.5) + 2×25.2×1.192×(30.4+13.8+118.5+13.8) + 2×30.4×1.192

×(30.4.+2×13.8+118.5+15.2) + 2×13.8×1.192×(30.4+13.8+118.5+6.9) + 10×1.192

×(30.4×2×13.8+118.5+5) + 13.8×1.192×(30.4+13.8+118.5+6.9)]/643 (A=643 mm2)

y =105.2 mm

Calculation of second moment of area, Ix based on the effective section

Ix = 1/12×42.2×1.1923 + 42.2×1.192×1052 + 2×1/12×25.2×1.1923 + 2×25.2×1.192×(105-

30.4)2 + 2×1/12×1.192×30.43 + 2×30.4×1.192×(105-15.2)2 + 2×1/12×1.192×13.83 +

2×1.192×13.8×(105-30.4-6.9)2 + 1/12×1.192×103 + 1.192×10×(105-30.4+5)2 + 1/12

×1.192×13.83 + 1.192 ×13.8×(105-30.4-7.2)2 + 1/12×1.192×59.253 + 1.192×59.25×

(105-30.4-13.8-59.25)2 + 1/12×1.192×59.253 + 1.192×59.25×(101-30.4-13.8-59.25)2

1/12×52.7×1.1923 + 52.7×1.192×1012 + 2×1/12×25.2×1.1923 + 2×25.2×1.192×

(101-30.4)2 + 2×1/12×1.192×30.43 + 2×1.192×30.4×(101-15.2)2 + 2×1/12×1.192×

13.83 + 2 ×1.192×13.8×(101-30.4-6.9)2 + 1/12×1.192×103 +1.192×10×(101-30.4+5)2

+ 1/12×1.192×13.83 + 1.192×13.8×(101-30.4-6.9)2

= 3819894 mm4

∴ Ze = 3731647/105 = 3.54×104 mm3

Ms = Zefy = 3.54×104 × 320 = 11.32 kNm

For example 2, consider RHFB-075tf-075tw-150hw-G550-50s section:

Centreline dimensions, yield stress and steel thicknesses of the above cross-section

are:

Element No. 1,8 2,9 3,10 4,6,11,13 5,12 7

Dimensions (mm) 52.5 25.1 30.2 14.2 10.0 118.5

fy = 650 MPa, flange thickness tf = 0.748 mm, and web thickness tw = 0.748 mm

Effective widths of the elements were determined as follows.


250

ye

f

tb=λ (3C-2)

consider element 1;

250650

748.05.52=eλ =113

bbe

eye ��

�

��

�=

λλ

= 5.5211340

��

��

� = 19 mm

Calculation procedures for other elements are similar to element 1. Centroid distance

(y) and effective section modulus (Ze) were calculated as for example 1. The

calculated values are;

y = 94.6 mm , Ix = 1879705 mm4 , Ze = 1879705/94.6 = 1.99×104 mm3

Ms = Zefy = 1.99×104 × 650 = 12.94 kNm

II. AS/NZS 4600 Design Method

Figure 3C-2: Element Definition and Stress Distribution Centreline dimensions of RHFB-120tf-120tw-150hw-G300-50s beam section

Element No. 1,8 2,9 3,10 4,6,11,13 5,12 7

tw

tf 1

2 2 3

3

4 4

5

7c

6

7t

8

9 9 10 10

11 11

12

13

320

257 226 183


Dimensions (mm) 52.7 25.2 30.4 13.8 10.0 118.5

fy = 320 MPa, flange thickness tf = 1.192 mm, and web thickness tw = 1.192 mm First yield moment, My

yfy fZM = (3C-3)

Distance to neutral axis in the full section = 206.9/2 = 103.5 mm

Element Ixi (mm4)

1 23 5.103192.17.52192.17.52121 ××+×× 672284

2 2× ]1.73192.12.25192.12.25121

[ 23 ××+×× 320061

3 2× ])2.155.103(192.14.304.30192.1121

[ 23 −××+×× 570122

4 2× ])3.375.103(192.18.138.13192.1121

[ 23 −××+×× 144494

5 23 )54.305.103(192.11010192.1121 +−××+×× 72713

6 23 )3.375.103(192.18.138.13192.1121 −××+×× 72241

7c 23 )7.292.445.103(192.13.593.59192.1121 −−××+×× 82563

�Ixi 1934477

∴ 3868954193447722 =×== � fxifx II

341074.35.103

3868954mmZ f ×==

My = Zf × fy = 3.74 × 104 × 320 = 11.97 kNm Section Moment Capacity Based on AS/NZS 4600 (SA, 1996)

yes fZM = (3C-3)

where Ze is the effective section modulus and fy is the yield stress.

Calculation of effective section modulus Ze based on AS/NZS 4600 (SA, 1996),

Effective widths of elements be; bbe ρ= (3C-4)

λλρ

22.01−

= (3C-5)


and Ef

tb

k

*052.1��

��

�=λ (3C-6)

where f * is the measured and modified yield stress at extreme compression fibbers

for G300 and G550 steels, respectively (see Section 3.3.5.4). Assume web is fully

effective.

Element 1

Ef

tb

k

*052.1��

��

�=λ = 201000

320192.1

7.52

4

052.1��

��

�=λ = 0.93

k=4 for stiffened uniform compression element

λ > 0.673 ∴ be = ρb

λλρ

22.01−

= =93.0

93.022.0

1−=0.82

∴ be = 0.82 × 52.7 = 43.3 mm

Element 2

201000226

192.12.25

4

052.1��

��

�=λ = 0.38 ∴ λ < 0.673

be = b = 25.2 mm

Element 3

320226

*1

*2 ==

f

fψ = 0.70

)1(2)1(24 3 ψψ −+−+=k = 4 + 2(1-0.7)3 + 2(1-0.7) = 4.65

201000320

192.14.30

65.4

052.1��

��

�=λ = 0.50 ∴λ < 0.673

be = b = 30.4 mm

7.034.30

31 −=

−=

ψe

e

bb =13.2 mm

ψ > -0.236 ∴ 12 eee bbb −= = 17.2 mm, be1 + be2 = 30.4 mm Element 4


226183

*1

*2 ==

f

fψ = 0.81

k = 0.43 unstiffened element in bending

201000226

192.18.13

43.0

052.1��

��

�=λ = 0.64 ∴λ < 0.673

be = b = 13.8 mm

81.038.13

31 −=

−=

ψe

e

bb = 6.3 mm

ψ > -0.236 ∴ 12 eee bbb −= = 7.5 mm

be1 + be2 = 13.8 mm

Element 5

257226

*1

*2 ==

f

fψ = 0.88

201000257

192.110

43.0

052.1��

��

�=λ = 0.48 ∴λ < 0.673

be = b = 10 mm

88.0310

31 −=

−=

ψe

e

bb = 4.7 mm

ψ > -0.236 ∴ 12 eee bbb −= = 5.3 mm

be1 + be2 = 10 mm

Element 6: Element 6 is fully effective Check web Element 7

183183

*1

*2 −==

f

fψ = -1

)1(2)1(24 3 ψψ −+−+=k = 4 + 2(1+1)3 + 2(1+1) = 24


201000183

192.15.118

24

052.1��

��

�=λ = 0.64 ∴λ < 0.673

be = b = 118.5 mm

135.118

31 +=

−=

ψe

e

bb = 29.6 mm

ψ < -0.236 ∴ 22e

e

bb = = 59.3 mm

be1 + be2 = 88.9 mm > b/2 ∴ be = 59.3 mm

Therefore web is fully effective. No iteration required.

In the above calculations, k values of stiffened elements were used for 2 and 7, which

were only intermittently screw-fastened and hence could have caused some

inaccuracies, ie. overestimated Ms slightly.

Element 8, 9, 10, 11, 12, 13 All are in tension and therefore fully effective Distance to the neutral axis in effective section

Element Effective width be (mm)

Thickness (mm)

Effective area Ai (mm2)

Distance to centroid fro top Yi (mm)

AiYi (mm3)

1 43.3 1.192 51.6 0 0 2 25.2 1.192 60.0 30.4 1823 3 30.4 1.192 72.5 15.2 1115 4 13.8 1.192 32.9 37.3 1102 5 10.0 1.192 11.9 25.4 303 6 13.8 1.192 16.4 37.3 614 7c 59.3 1.192 70.6 73.8 5214 7t 59.3 1.192 70.6 133.1 9399 8 52.7 1.192 62.8 206.9 12997 9 25.2 1.192 60.0 176.5 10586

10 30.4 1.192 72.5 191.7 13893 11 13.8 1.192 32.9 169.6 5580 12 10.0 1.192 11.9 181.5 2163 13 13.8 1.192 16.4 169.6 2790

Σ 643 67690 Distance to neutral axis from top fibre 67690/643 = 105.2mm


Distance to neutral axis from bottom fibre 206.9 – 105.2 = 101.7 mm Second moment of area Ix for effective section

Element Ixi (mm4)

1 23 2.105192.12.43192.12.43121 ××+×× 572111

2 2× ]8.74192.12.25192.12.25121

[ 23 ××+×× 335998

3 2× ])2.152.105(192.14.304.30192.1121

[ 23 −××+×× 593338

4 2× ])3.372.105(192.18.138.13192.1121

[ 23 −××+×× 152420

5 23 )54.302.105(192.11010192.1121 +−××+×× 76095

6 23 )3.372.105(192.18.138.13192.1121 −××+×× 76204

7c 23 )7.292.442.105(192.13.593.59192.1121 −−××+×× 90309

7t 23 )7.292.447.101(192.13.593.59192.1121 −−××+×× 75273

8 23 7.101192.17.52192.17.52121 ××+×× 649136

9 2× ])4.307.101(192.12.25192.12.25121

[ 23 −××+×× 304511

10 2× ])2.157.101(192.14.304.30192.1121

[ 23 −××+×× 547374

11 2× ])3.377.101(192.18.138.13192.1121

[ 23 −××+×× 136780

12 23 )54.307.101(192.11010192.1121 +−××+×× 69409

13 23 )3.377.101(192.18.138.13192.1121 −××+×× 68384

�Ix 3747340 mm4

∴ 2.105

3747340=exZ = 3.56 × 104 mm3

Ms = Ze × fy = 3.56 × 104 × 320 = 11.39 kNm


For example 2,

Figure 3C-3: Element Definition and Stress Distribution Full section modulus Zf was calculated similar to example 1, fy = 495 MPa (lesser of 0.9fy = 585 MPa or 495 MPa) Distance to neutral axis in the full section = 207.1/2 = 103.6 mm

Ixf = 2454490 mm4 Zf = 2454490/103.6 =23692 mm3 My = Zf × fy = 2.37 × 104 × 495 = 11.73 kNm

Effective section modulus Ze assuming web is fully effective. Effective widths of all the elements were calculated as for example 1.

Check web: Element 7

283283

*1

*2 −==

f

fψ = -1

)1(2)1(24 3 ψψ −+−+=k = 4 + 2(1+1)3 + 2(1+1) = 24

tw

tf 1

2 2 3

3

4 4

5

7c

6

7t

8

9 9 10 10

11 11

12

13

495

398 351 283


224000283

748.03.118

24

052.1��

��

�=λ = 0.68 ∴λ > 0.673

be = ρb = 0.68 × 118.3 = 80.4 mm

134.80

31 +=

−=

ψe

e

bb = 20.1 mm

ψ < -0.236 ∴ 22e

e

bb = = 40.2 mm

be1 + be2 = 60.3 mm > b/2 ∴ be = 59.3 mm

Therefore web is fully effective. No iteration required.

Calculation of centroid distance and effective section modulus is similar to example

1. The corresponding values are;

The distance to the neutral axis from top fibres, ycg = 104.2 mm Second moment of area, Ix = 1615112 mm4

2.1041615112=exZ = 1.55 × 104 mm3

Ms = Ze × fy = 1.55 × 104 × 495 = 7.68 kNm

Flexural Behaviour and Design of Cold-formed Steel Beams with Rectangular Hollow Flanges 3D-1

Appendix 3D: Moment versus Deflection and Strain Graphs

Moment versus Deflection Graphs

1 - RHFB-120tf-0550tw-100hw-G300-50s

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30


Mom

ent (

kNm

)

2 - RHFB-120tf-055tw-100hw-G300-100s

0

1

2

3

4

5

6

0 5 10 15 20 25


Mom

ent (

kNm

)

3 - RHFB-080tf-080tw-150hw-G300-50s

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12


Mom

ent (

kNm

)


4 - RHFB-080tf-080tw-150hw-G300-100s

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16


Mom

ent (

kNm

)

5 - RHFB-120tf-120tw-150hw-G300-50s

0

2

4

6

8

10

12

14

0 4 8 12 16 20 24


Mom

ent (

kNm

)

6 - RHFB-120tf-120tw-150hw-G300-100s

0

2

4

6

8

10

12

14

0 4 8 12 16 20 24


Mom

ent (

kNm

)


7 - RHFB-080tf-190tw-150hw-G300-50s

0

2

4

6

8

10

12

0 2 4 6 8 10 12


Mom

ent (

kNm

)

8 - RHFB-080tf-190tw-150hw-G300-100s

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10 12


Mom

ent (

kNm

)

9 - RHFB-120tf-055tw-150hw-G300-50s

0

1

2

3

4

5

6

7

8

9

0 4 8 12 16 20 24


Mom

ent (

kNm

)


10 - RHFB-120tf-055tw-150hw-G300-100s

0

1

2

3

4

5

6

7

8

9

0 4 8 12 16 20 24 28


Mom

ent (

kNm

)

11 - RHFB-075tf-075tw-100hw-G550-50s

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12


Mom

ent (

kNm

)

12 - RHFB-075tf-075tw-100hw-G550-100s

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12


Mom

ent (

kNm

)


13 - RHFB-075tf-075tw-150hw-G550-50s

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10 12 14


Mom

ent (

kNm

)

14 - RHFB-075tf-075tw-150hw-G550-100s

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10 12 14 16


Mom

ent (

kNm

)

15 - RHFB-115tf-115tw-150hw-G500-50s

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12 14 16 18 20


Mom

ent (

kNm

)


16 - RHFB-115tf-115tw-150hw-G500-100s

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16 18 20


Mom

ent (

kNm

)

17 - RHFB-075tf-115tw-150hw-G550-50s

0

2

4

6

8

10

12

0 2 4 6 8 10


Mom

ent (

kNm

)

18 - RHFB-075tf-115tw-150hw-G550-100s

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10


Mom

ent (

kNm

)


19 - RHFB-115tf-075tw-150hw-G550-50s

0

2

4

6

8

10

12

14

0 4 8 12 16 20


Mom

ent (

kNm

)

20- RHFB-115tf-075tw-150hw-G550-100s

0

2

4

6

8

10

12

0 4 8 12 16 20 24


Mom

ent (

kNm

)

21- RHFB-095tf-055tw-150hw-G550-50s

0

1

2

3

4

5

6

7

0 4 8 12 16 20 24


Mom

ent (

kNm

)


22- RHFB-095tf-055tw-150hw-G550-100s

0

1

2

3

4

5

6

0 4 8 12 16 20 24


Mom

ent (

kNm

)

Moment versus Strain Graphs

1 - RHFB-120tf-0550tw-100hw-G300-50s

0

1

2

3

4

5

6

7

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

2 - RHFB-120tf-0550tw-100hw-G300-100s

0

1

2

3

4

5

6

-3000 -2000 -1000 0 1000 2000 3000


Mom

ent (

kNm

)


3 - RHFB-080tf-080tw-150hw-G300-50s

0

1

2

3

4

5

6

7

8

-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

4 - RHFB-080tf-080tw-150hw-G300-100s

0

1

2

3

4

5

6

7

8

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

5 - RHFB-120tf-120tw-150hw-G300-50s

0

2

4

6

8

10

12

14

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)


6 - RHFB-120tf-120tw-150hw-G300-100s

0

2

4

6

8

10

12

14

-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

7 - RHFB-080tf-190tw-150hw-G300-50s

0

2

4

6

8

10

12

-3000 -2000 -1000 0 1000 2000 3000


Mom

ent (

kNm

)

8 - RHFB-080tf-190tw-150hw-G300-100s

0

2

4

6

8

10

12

-3000 -2000 -1000 0 1000 2000 3000


Mom

ent (

kNm

)


9 - RHFB-120tf-055tw-150hw-G300-50s

0

1

2

3

4

5

6

7

8

9

-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

10 - RHFB-120tf-055tw-150hw-G300-100s

0

1

2

3

4

5

6

7

8

9

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

11 - RHFB-075tf-075tw-100hw-G550-50s

0

1

2

3

4

5

6

7

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)


12 - RHFB-075tf-075tw-100hw-G550-100s

0

1

2

3

4

5

6

7

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

13 - RHFB-075tf-075tw-150hw-G550-50s

0

1

2

3

4

5

6

7

8

9

10

-4000 -3000 -2000 -1000 0 1000 2000 3000


Mom

ent (

kNm

)

14 - RHFB-075tf-075tw-150hw-G550-100s

0

1

2

3

4

5

6

7

8

9

-4000 -3000 -2000 -1000 0 1000 2000 3000


Mom

ent (

kNm

)


15 - RHFB-115tf-115tw-150hw-G500-50s

0

2

4

6

8

10

12

14

16

18

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

16 - RHFB-115tf-115tw-150hw-G500-100s

0

2

4

6

8

10

12

14

16

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

17 - RHFB-075tf-115tw-150hw-G550-50s

0

2

4

6

8

10

12

-3000 -2000 -1000 0 1000 2000 3000


Mom

ent (

kNm

)


18 - RHFB-075tf-115tw-150hw-G550-100s

0

1

2

3

4

5

6

7

8

9

10

-3000 -2000 -1000 0 1000 2000 3000


Mom

ent (

kNm

)

19 - RHFB-115tf-075tw-150hw-G550-50s

0

2

4

6

8

10

12

14

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

20 - RHFB-115tf-075tw-150hw-G550-100s

0

2

4

6

8

10

12

-4500 -3500 -2500 -1500 -500 500 1500 2500 3500


Mom

ent (

kNm

)


21- RHFB-095tf-055tw-150hw-G550-50s

0

1

2

3

4

5

6

7

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)

22- RHFB-095tf-055tw-150hw-G550-100s

0

1

2

3

4

5

6

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000


Mom

ent (

kNm

)


APPENDIX 4A: Calculation of Member Moment Capacities Based on

the Current Design Rules

In the calculations here, it is assumed that the hollow flange and web element are

continuously connected, ignoring the effect of intermittent screw fastening.

I. AS 4100 Design Method

Member moment capacity (Mb) is calculated from Equation 4A-1.

Mb = αm αs Ms (4A-1)

where αm = 1 (standardised for uniform moment distribution), αs and Ms were calculated

according to AS 4100 design provisions.

��

�

�

��

�

�

��

�

�−+��

�

�

�=

o

s

o

ss M

MMM

36.02

α (4A-2)

Ms = Ze fy (4A-3)

Calculation procedures of effective section modulus (Ze) using AS 4100 (SA, 1998)

design rules are explained in Appendix 3C. The calculated effective section modulus

(Ze) for the section RHFB-120tf-120tw-150hw-3m-G300-50s is 3.67×104 mm3.

Ms = Zefy = 3.67×104 × 320 Nmm = 11.74 kNm

��

�

�

�+= 2

2

2

2

e

w

e

yo

l

EIGJ

l

EIM

ππ (4A-4)


where

E =201 GPa (Measured)

Iy = 1.24×105 mm4 (From Thin-wall program using measured dimensions)

J = 1 ×105 mm3 (From Thin-wall program using measured dimensions)

Iw = 9.52×108 mm4 (From Thin-wall program using measured dimensions)

le = 3000 mm

G = 80 GPa

Mo =15 kNm

∴ αs = 0.67 (From Equation 4A-2)

Mb = αm αs Ms = 1×0.67×11.74 = 7.87 kNm

II. AS/NZS 4600 Design Method

According to AS/NZS 4600 (SA, 1996) specifications, member moment capacity of

cold-formed steel beams, subjected to lateral distortional buckling, is calculated using;

��

�

�

�=

f

ccb Z

MZM (4A-5)

when distortional buckling involves transverse bending of a vertical web with lateral

displacement of the compression flange

λd < 1.414: yd

c MM��

�

��

�−=

41

2λ (4A-6)

λd ≥ 1.414: y

d

c MM��

�

��

�= 2

1λ

(4A-7)


where

od

yd M

M=λ (4A-8)

odfod fZM = (4A-9)

Zf – Excel program, same calculation procedure as in Appendix 3C with new beam

dimensions corresponding to this beam

∴ Zf = 3.77 ×104 mm3

fod = 240 MPa (Elastic buckling analyses using Thin-wall)

Mod = 3.77 × 104 × 240 = 9.05 kNm (From Equation 4A-9)

My = Zf fy = 3.77 × 104 × 320 = 12.06 kNm

od

yd M

M=λ =1.15 < 1.414

∴ yd

c MM ��

�

�

�−=

41

2λ= 8.04 kNm

4

6

1077.31004.8

××==

f

cc Z

Mf = 213 MPa

Zc – Excel program, same procedures as in Appendix 3C to determine Ze, but fy was

replaced with fc = 213 MPa

∴ Zc = 3.70 ×104 mm3

��

�

�

�=

f

ccb Z

MZM = 7.88 kNm


III. Pi and Trahair’s (1997) Design Method

ssmbd MM αα= (4A-10)

�m =1 (for uniform moment distribution)

��

�

�

��

�

�

��

�

�−+��

�

�

�=

od

s

od

ssd M

MMM

8.26.02

α (4A-11)

yes fZM = =3.67 × 104 × 320 = 11.74 kNm

��

�

�

�+= 2

2

2

2

e

we

e

yod

l

EIGJ

l

EIM

ππ= fod Zf fod – from Thin-wall program

fod = 240 MPa (Elastic buckling analyses, Thin-wall)

Zf = 3.77 × 104 mm3

Mod = 9.05 kNm

�sd = 0.49

Mbd = 1 × 0.49 × 11.74 = 5.8 kNm (From Equation 4A-10)

IV. Avery et al.’s (2000) Design Method

snd

b Mc

babM

��

�

��

�

+−+=

21 λ (4A-12)

Coefficients a, b, c and n are suggested by Avery et al. (2000) as follows

a = 1.0, b = 0, c = 0.424 and n = 1.196


∴ s

d

b MM��

�

��

�

+= 392.2424.01

1λ

od

sd M

M=λ Ms = 11.74 kNm and Mod = 9.05 kNm

λd = 1.14

Mb = 0.63 × 11.74 = 7.44 kNm (From Equation 4A-12)

V. Mahaarachchi and Mahendran’s (2005c) Design Method

��

�

�

�=

f

ccb Z

MZM (4A-13)

λd ≤ 0.59 yc MM =

0.59 < λd ≤ 1.7 ��

�

�=

dyc MM

λ59.0

(4A-14)

λd ≥ 1.7 ��

�

�

�= 2

1

d

yc MMλ

od

yd M

M=λ = 1.15 (see AS/NZS 4600 design method)

∴ 0.59 < λd < 1.7 ��

�

�=

dyc MM

λ59.0

= 6.17 kNm (From Equation 4A-14)

∴f

cc Z

Mf = = 164 MPa ∴Zc = Zf = 3.77 × 104 mm3 (from Excel program)

��

�

�

�=

f

ccb Z

MZM = 6.17 kNm (From Equation 4A-13)


Appendix 4B: Moment versus Deflection and Strain Graphs from Lateral Buckling Tests

Moment versus Deflection Graphs

1 - RHFB-120tf-120tw-150hw-3L-G300-50s

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 11 12


Mom

ent (

kNm

)

0.0

1.5

3.0

4.5

6.0

7.5

0 3 6 9 12 15 18 21 24 27 30


Mom

ent (

kNm

)


2 - RHFB-120tf-055tw-150hw-3L-G300-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 2 3 5 6 8 9 11 12 14 15


Mom

ent (

kNm

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 3 6 9 12 15 18 21 24 27 30


Mom

ent (

kNm

)

3 - RHFB-080tf-080tw-150hw-3L-G300-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10 12 14 16 18


Mom

ent (

kNm

)



3 - RHFB-080tf-080tw-150hw-3L-G300-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 3 6 9 12 15 18 21 24 27 30


Mom

ent (

kNm

)

4 - RHFB-080tf-190tw-150hw-3L-G300-50s

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16


Mom

ent (

kNm

)

0

1

2

3

4

5

6

-6 -3 0 3 6 9 12 15 18 21 24 27 30


Mom

ent (

kNm

)




5 - RHFB-190tf-190tw-150hw-3L-G300-50s

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14 16 18


Mom

ent (

kNm

)

0

2

4

6

8

10

12

14

0 3 5 8 10 13 15 18 20 23 25


Mom

ent (

kNm

)

6 - RHFB-120tf-055tw-100hw-3L-G300-50s

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18 20 22


Mom

ent (

kNm

)



6 - RHFB-120tf-055tw-100hw-3L-G300-50s

0

1

2

3

4

5

6

0 3 5 8 10 13 15 18 20 23 25


Mom

ent (

kNm

)

7 - RHFB-120tf-055tw-150hw-2L-G300-50s

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10


Mom

ent (

kNm

)

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8


Mom

ent (

kNm

)




8 - RHFB-055tf-120tw-100hw-2L-G300-50s

0

1

2

3

4

0 2 4 6 8 10 12


Mom

ent (

kNm

)

0

1

2

3

4

0 1 2 3 4 5 6 7 8


Mom

ent (

kNm

)

9- RHFB-120tf-120tw-100hw-2L-G300-50s

0

1

2

3

4

5

6

7

8

0 3 5 8 10 13 15 18 20 23 25


Mom

ent (

kNm

)



9- RHFB-120tf-120tw-100hw-2L-G300-50s

0

1

2

3

4

5

6

7

8

0 3 6 9 12 15 18 21 24 27 30


Mom

ent (

kNm

)

10- RHFB-055tf-095tw-150hw-3L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14 16 18 20


Mom

ent (

kNm

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 4 8 12 16 20 24 28 32 36 40


Mom

ent (

kNm

)




11- RHFB-095tf-055tw-150hw-3L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 1 2 3 4 5 6 7 8


Mom

ent (

kNm

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14 16


Mom

ent (

kNm

)

12- RHFB-075tf-075tw-150hw-3L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10 12 14 16 18 20 22 24


Mom

ent (

kNm

)



12- RHFB-075tf-075tw-150hw-3L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 4 8 12 16 20 24 28 32 36


Mom

ent (

kNm

)

13- RHFB-075tf-115tw-150hw-3L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


Mom

ent (

kNm

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28


Mom

ent (

kNm

)




14- RHFB-115tf-115tw-150hw-3L-G500-50s

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18 20


Mom

ent (

kNm

)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 4 8 12 16 20 24 28 32 36


Mom

ent (

kNm

)

15- RHFB-115tf-075tw-150hw-3L-G500-50s

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18


Mom

ent (

kNm

)



15- RHFB-115tf-075tw-150hw-3L-G500-50s

0

1

2

3

4

5

6

0 4 8 12 16 20 24 28 32


Mom

ent (

kNm

)

16- RHFB-075tf-075tw-100hw-3L-G500-50s

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16 18 20 22 24


Mom

ent (

kNm

)

0

1

2

3

4

5

0 4 8 12 16 20 24 28 32


Mom

ent (

kNm

)




17- RHFB-055tf-055tw-150hw-2L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 1 2 3 4 5 6 7 8 9 10 11 12


Mom

ent (

kNm

)

0

1

2

3

4

0 2 4 6 8 10 12 14 16


Mom

ent (

kNm

)

18- RHFB-095tf-095tw-150hw-2L-G550-50s

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20


Mom

ent (

kNm

)



18- RHFB-095tf-095tw-150hw-2L-G550-50s

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20 22 24


Mom

ent (

kNm

)

19- RHFB-095tf-055tw-150hw-2L-G550-50s

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10


Mom

ent (

kNm

)

0

1

2

3

4

5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0


Mom

ent (

kNm

)




20- RHFB-075tf-075tw-150hw-2L-G550-50s

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16


Mom

ent (

kNm

)

0

1

2

3

4

5

0 3 6 9 12 15 18 21 24


Mom

ent (

kNm

)

21- RHFB-080tf-080tw-150hw-3L-G300-100s

0

1

2

3

4

0 2 4 6 8 10 12 14 16 18 20


Mom

ent (

kNm

)


21- RHFB-080tf-080tw-150hw-3L-G300-100s

0

1

2

3

4

0 4 8 12 16 20 24 28 32 36 40


Mom

ent (

kNm

)

22- RHFB-120tf-120tw-150hw-3L-G300-100s

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20


Mom

ent (

kNm

)

0

1

2

3

4

5

6

7

0 4 8 12 16 20 24 28 32 36 40


Mom

ent (

kNm

)


23- RHFB-120tf-055tw-150hw-3L-G300-100s

0

1

2

3

4

0 2 4 6 8 10 12 14 16


Mom

ent (

kNm

)

0

1

2

3

4

0 3 6 9 12 15 18 21 24


Mom

ent (

kNm

)

24- RHFB-075tf-075tw-150hw-3L-G550-100s

0

1

2

3

4

0 3 6 9 12 15 18 21 24 27 30


Mom

ent (

kNm

)


24- RHFB-075tf-075tw-150hw-3L-G550-100s

0

1

2

3

4

0 6 12 18 24 30 36 42 48 54


Mom

ent (

kNm

)

25- RHFB-075tf-075tw-100hw-3L-G550-100s

0

1

2

3

4

0 3 6 9 12 15 18 21 24 27 30


Mom

ent (

kNm

)

0

1

2

3

4

-4 0 4 8 12 16 20 24 28 32


Mom

ent (

kNm

)


26- RHFB-115tf-115tw-150hw-3L-G500-100s

0

1

2

3

4

5

6

7

0 3 6 9 12 15 18 21 24


Mom

ent (

kNm

)

0

1

2

3

4

5

6

7

-4 0 4 8 12 16 20 24 28 32


Mom

ent (

kNm

)

27- RHFB-075tf-115tw-150hw-3L-G550-100s

0

1

2

3

4

0 3 6 9 12 15 18 21


Mom

ent (

kNm

)


27- RHFB-075tf-115tw-150hw-3L-G550-100s

0

1

2

3

4

-4 0 4 8 12 16 20 24 28 32


Mom

ent (

kNm

)

28- RHFB-115tf-075tw-150hw-3L-G500-100s

0

1

2

3

4

5

0 3 6 9 12 15 18 21 24


Mom

ent (

kNm

)

0

1

2

3

4

5

0 4 8 12 16 20 24 28 32


Mom

ent (

kNm

)


29- RHFB-095tf-095tw-150hw-2L-G550-100s

0

1

2

3

4

5

6

7

0 3 6 9 12 15 18 21


Mom

ent (

kNm

)

0

1

2

3

4

5

6

7

0 4 8 12 16 20 24 28 32


Mom

ent (

kNm

)

30- RHFB-095tf-055tw-150hw-2L-G550-100s

0

1

2

3

4

5

0 2 4 6 8 10 12


Mom

ent (

kNm

)


30- RHFB-095tf-055tw-150hw-2L-G550-100s

0

1

2

3

4

5

0 1 2 3 4 5 6


Mom

ent (

kNm

)

Moment versus Strain Graphs

1 - RHFB-120tf-120tw-150hw-3L-G300-50s

0

1

2

3

4

5

6

7

-1500 -1200 -900 -600 -300 0 300 600 900 1200 1500


Mom

ent (

kNm

)


2 - RHFB-120tf-055tw-150hw-3L-G300-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-750 -450 -150 150 450 750


Mom

ent (

kNm

)

3 - RHFB-080tf-080tw-150hw-3L-G300-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)

4 - RHFB-080tf-190tw-150hw-3L-G300-50s

0

1

2

3

4

5

6

-1500 -1250 -1000 -750 -500 -250 0 250 500 750 1000 1250 1500


Mom

ent (

kNm

)


5 - RHFB-190tf-190tw-150hw-3L-G300-50s

0

2

4

6

8

10

12

14

-1800 -1500 -1200 -900 -600 -300 0 300 600 900 1200 1500 1800


Mom

ent (

kNm

)

6 - RHFB-120tf-055tw-100hw-3L-G300-50s

0

1

2

3

4

5

6

-1800 -1500 -1200 -900 -600 -300 0 300 600 900 1200 1500 1800


Mom

ent (

kNm

)

7 - RHFB-120tf-055tw-150hw-2L-G300-50s

0

1

2

3

4

5

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200


Mom

ent (

kNm

)


8 - RHFB-055tf-120tw-100hw-2L-G300-50s

0

1

2

3

4

-2000 -1600 -1200 -800 -400 0 400 800 1200 1600 2000


Mom

ent (

kNm

)

9 - RHFB-120tf-120tw-100hw-2L-G300-50s

0

1

2

3

4

5

6

7

8

-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500


Mom

ent (

kNm

)

10 - RHFB-055tf-095tw-150hw-3L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200


Mom

ent (

kNm

)


11 - RHFB-095tf-055tw-150hw-3L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)

12 - RHFB-075tf-075tw-150hw-3L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)

13 - RHFB-075tf-115tw-150hw-3L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)


14 - RHFB-115tf-115tw-150hw-3L-G500-50s

0

1

2

3

4

5

6

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200


Mom

ent (

kNm

)

15 - RHFB-115tf-075tw-150hw-3L-G500-50s

0

1

2

3

4

5

6

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200


Mom

ent (

kNm

)

16 - RHFB-075tf-075tw-100hw-3L-G500-50s

0

1

2

3

4

5

-1800 -1500 -1200 -900 -600 -300 0 300 600 900 1200 1500 1800


Mom

ent (

kNm

)


17 - RHFB-055tf-055tw-150hw-2L-G550-50s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)

18 - RHFB-095tf-095tw-150hw-2L-G550-50s

0

1

2

3

4

5

6

7

-1500 -1250 -1000 -750 -500 -250 0 250 500 750 1000 1250 1500


Mom

ent (

kNm

)

19 - RHFB-095tf-055tw-150hw-2L-G550-50s

0

1

2

3

4

5

-1250 -1000 -750 -500 -250 0 250 500 750 1000 1250


Mom

ent (

kNm

)


20 - RHFB-075tf-075tw-150hw-2L-G550-50s

0

1

2

3

4

5

-1250 -1000 -750 -500 -250 0 250 500 750 1000 1250


Mom

ent (

kNm

)

21 - RHFB-080tf-080tw-150hw-3L-G300-100s

0

1

2

3

4

-1250 -1000 -750 -500 -250 0 250 500 750 1000 1250


Mom

ent (

kNm

)

22 - RHFB-120tf-120tw-150hw-3L-G300-100s

0

1

2

3

4

5

6

7

-1250 -1000 -750 -500 -250 0 250 500 750 1000 1250


Mom

ent (

kNm

)


23 - RHFB-120tf-055tw-150hw-3L-G300-100s

0

1

2

3

4

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)

24 - RHFB-075tf-075tw-150hw-3L-G550-100s

0

1

2

3

4

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)

25 - RHFB-075tf-075tw-100hw-3L-G550-100s

0

1

2

3

4

-1500 -1250 -1000 -750 -500 -250 0 250 500 750 1000 1250 1500


Mom

ent (

kNm

)


26 - RHFB-115tf-115tw-150hw-3L-G500-100s

0

1

2

3

4

5

6

7

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)

27 - RHFB-075tf-115tw-150hw-3L-G550-100s

0

1

2

3

4

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)

28 - RHFB-115tf-075tw-150hw-3L-G500-100s

0

1

2

3

4

5

-1000 -800 -600 -400 -200 0 200 400 600 800 1000


Mom

ent (

kNm

)


29 - RHFB-095tf-095tw-150hw-2L-G550-100s

0

1

2

3

4

5

6

7

-1800 -1500 -1200 -900 -600 -300 0 300 600 900 1200 1500 1800


Mom

ent (

kNm

)

30 - RHFB-095tf-055tw-150hw-2L-G550-100s

0

1

2

3

4

5

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200


Mom

ent (

kNm

)


APPENDIX 5A: ABAQUS Subroutine Used to Include Residual Stresses in RHFBs SUBROUTINE SIGINI(SIGMA,COORDS,NTENS,NCRDS,NOEL,NPT,LAYER,KSPT) C INCLUDE 'ABA_PARAM.INC' C REAL X,Y,Z,nipt,ipt,sigmaout,Fy DIMENSION SIGMA(NTENS), COORDS(NCRDS) C X=COORDS(1) Y=COORDS(2) Z=COORDS(3) SIGMA(2)=0. SIGMA(3)=0. nipt=5. Fy=800. C IF(KSPT.EQ.1.) THEN ipt=1. ENDIF IF(KSPT.EQ.2.) THEN ipt=2. ENDIF IF(KSPT.EQ.3.) THEN ipt=3. ENDIF IF(KSPT.EQ.4.) THEN ipt=4. ENDIF IF(KSPT.EQ.5.) THEN ipt=5. ENDIF C C RESIDUAL STRESS IF((NOEL.LE.200.).AND.(NOEL.GE.1.)) THEN sigmaout=0.17*Fy ELSEIF((NOEL.LE.1200.).AND.(NOEL.GE.601.))THEN sigmaout=0.17*Fy ELSEIF((NOEL.LE.600.).AND.(NOEL.GE.201.))THEN sigmaout=0.08*Fy C C ENDIF C C ALONG THE THICKNESS VARIATION IF(sigmaout.NE.0.) THEN SIGMA(1)=sigmaout*(1.-2.*(nipt-ipt)/(nipt-1.)) ELSE SIGMA(1)=0. ENDIF C RETURN END

Note: Same residual stress model was used in all the analyses with the change of element numbers


APPENDIX 6A: Calculation of Member Moment Capacities Based on New Design Formula

As defined in the new design formula, member moment capacity of cold-formed

steel beams, subjected to lateral distortional buckling, is calculated using;

��

�

�

��

�

�=

f

ccb Z

MZM (6A-1)

when distortional buckling involves transverse bending of a vertical web with lateral

displacement of the compression flange

λd ≤ 0.65: yc MM =

0.65 ≤ λd ≤ 1.80: y

d

dc MM

��

�

� +=

210

51

λλ

(6A-2)

λd ≥ 1.80: y

d

c MM��

�

�=

2

1λ

where

od

yd M

M=λ (6A-3)

odfod fZM = (6A-4)

Zf – Excel program, same calculation procedure as in Appendix 3C with new beam

dimensions corresponding to ideal RHFB-120tf-120tw-150hw-G300-3L-50s

∴ Zf = 3.59 ×104 mm3

fod = 214 MPa (Elastic buckling analyses using Thin-wall)

Mod = 3.59 × 104 × 214 = 7.7 kNm (From equation 6A-4)

My = Zf fy = 3.59 × 104 × 300 = 10.77 kNm


od

yd M

M=λ =1.18 < 1.80

∴ y

d

dc MM �

�

�

�

��

�

� +=

210

51

λλ

= 5.33 kNm

4

6

1059.31033.5

××==

f

cc Z

Mf = 148 MPa

Zc – Excel program, same procedures as in Appendix 3C to determine Ze, but fy was

replaced with fc = 148 MPa

∴ Zc = 3.59 ×104 mm3

��

�

�

��

�

�=

f

ccb Z

MZM =5.33 kNm

Flexural Behaviour and Design of Cold- formed Steel Beams ... · formed Steel Beams with...

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