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LOCAL BUCKLING BEHAVIOUR AND DESIGN OF COLD-FORMED
STEEL COMPRESSION MEMBERS AT ELEVATED TEMPERATURES
JUNG HOON LEE
Local Buckling Behaviour and Design of Cold-formed Steel Compression Members
at Elevated Temperatures
By
Jung Hoon Lee
School of Civil Engineering Queensland University of Technology
A THESIS SUBMITTED TO THE SCHOOL OF CIVIL ENGINEERING
QUEENSLAND UNIVERSITY OF TECHNOLOGY IN PARTIAL
FULFILLMENT OF REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
MAY 2004
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
i
ACKNOWLEDGEMENTS
I am sincerely grateful for my supervisor Prof. Mahan Mahendran for his expertise,
guidance and constructive suggestions over the past three and a half years and the
valuable assistance in many ways. Prof. Pentti Makelainen to be mentioned with
thanks so kindly agreeing to serve as an associate supervisor.
I would also like to thank Queensland University of Technology (QUT) and the
Department of Education, Science and Training, Australia for providing financial
support of my research project through the International Postgraduate Research
Scholarship (IPRS). Thanks also to the School of Civil Engineering at QUT for
providing the necessary facilities and technical support. Many thanks to the
Structural Laboratory staff members for assistance with fabricating, preparing and
testing the cold-formed steel specimens.
It is my pleasure to thank my post-graduate colleges, friends and church members
for their support and contribution to this research.
Finally, I like to express my sincere appreciation to my wife, daughter and parents
for their endless support and encouragement throughout the duration of this project.
Looking back, God’s hands were clearly on me. Glory and praise be to the Lord.
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
ii
KEYWORDS
Cold-formed steel compression members, light gauge high strength steels, elevated
temperatures, axial compression strength, reduced yield strength, reduced elasticity
modulus, stress-strain model, unstiffened element, stiffened element, local buckling
interaction, fire safety design, local buckling, effective width, local buckling stress,
buckling coefficient, simulated fire test, finite element analysis.
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
iii
ABSTRACT The importance of fire safety design has been realised due to the ever increasing loss
of properties and lives caused by structural failures during fires. In recognition of the
importance of fire safety design, extensive research has been undertaken in the field
of fire safety of buildings and structures especially over the last couple of decades. In
the same period, the development of fire safety engineering principles has brought
significant reduction to the cost of fire protection. However the past fire research on
steel structures has been limited to heavier, hot-rolled structural steel members and
thus the structural behaviour of light gauge cold-formed steel members under fire
conditions is not well understood. Since cold-formed steel structures have been
commonly used for numerous applications and their use has increased rapidly in the
last decade, the fire safety of cold-formed steel structural members has become an
important issue. The current design standards for steel structures have simply
included a list of reduction factors for the yield strength and elasticity modulus of
hot-rolled steels without any detailed design procedures. It is not known whether
these reduction factors are applicable to the commonly used thin, high strength steels
in Australia. Further, the local buckling effects dominate the structural behaviour of
light gauge cold-formed steel members. Therefore an extensive research program
was undertaken at the Queensland University of Technology to investigate the local
buckling behaviour of light gauge cold-formed steel compression members under
simulated fire conditions.
The first phase of this research program included 189 tensile coupon tests including
three steel grades and six thicknesses to obtain the accurate yield strength and
elasticity modulus values at elevated temperatures because the deterioration of the
mechanical properties is one of the major parameters in the structural design under
fire conditions. The results obtained from the tensile tests were used to predict the
ultimate strength of cold-formed steel members. An appropriate stress-strain model
was also developed by considering the inelastic characteristics.
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
iv
The second phase of this research was based on a series of more than 200 laboratory
experiments and corresponding numerical analyses on cold-formed steel
compression members to investigate the local bucking behaviour of the unstiffened
flange elements, stiffened web elements and stiffened web and flange elements at
elevated temperatures up to 800°C. The conventional effective width design rules
were first simply modified considering the reduced mechanical properties obtained
from the tensile coupon tests and their adequacy was studied using the experimental
and numerical results. It was found that the simply modified effective width design
rules were adequate for low strength steel members and yet were inadequate for high
strength cold-formed steel members due to the severe reduction of the ultimate
strength in the post buckling strength range and the severe reduction ratio of the
elasticity modulus to the yield strength at elevated temperatures. Due to the
inadequacy of the current design rules, the theoretical, semi-empirical and empirical
effective width design rules were developed to accurately predict the ultimate
strength of cold-formed steel compression members subject to local buckling effects
at elevated temperatures. The accuracy of these new design methods was verified by
comparing their predictions with a variety of experimental and numerical results.
This thesis presents the details of extensive experimental and numerical studies
undertaken in this research program and the results including comparison with
simply modified effective width design rules. It describes the laboratory experiments
at elevated temperatures undertaken using an electric furnace. It also describes the
advanced finite element models of cold-formed steel compression members
developed in this research including the appropriate mechanical properties, initial
imperfections, residual stresses and other significant factors. Finally, it presents the
details of the new design methods proposed for the cold-formed steel compression
members subject to local buckling effects at elevated temperatures.
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
v
PUBLICATIONS
(a) International Refereed Journal Papers
1. Jung Hoon Lee, Mahen Mahendran and Pentti Makelainen (2003), Prediction of
Mechanical Properties of Light Gauge Steels at Elevated Temperatures, Journal
of Constructional Steel Research, Vol 59 Iss12 pp. 1517-1532.
2. Jung Hoon Lee and Mahen Mahendran (2004), Local Buckling Behaviour of
High Strength Cold-formed Steel Unstiffened Elements subject to Axial
Compression, Journal of Constructional Steel Research (In preparation)
3. Jung Hoon Lee and Mahen Mahendran (2004), Local Buckling Behaviour of
Unstiffened Elements at Elevated Temperatures, Thin-Walled Structures (In
preparation)
4. Jung Hoon Lee and Mahen Mahendran (2004), Local Buckling Behaviour of
Stiffened Elements at Elevated Temperatures, Thin-Walled Structures (In
preparation)
(b) International Refereed Conference Papers
5. Jung Hoon Lee, Mahen Mahendran and Pentti Makelainen (2001), Buckling
Behaviour of Thin-walled Compression Members at Elevated Temperatures,
Sixth Pacific Structural Steel Conference, Beijing, China, pp. 634-641.
6. Jung Hoon Lee and Mahen Mahendran (2002), Effect of Fire on the Local
Buckling Behaviour of Cold-formed Steel Compression Members, ACMSM
2002, Gold Coast, Australia, pp. 491-494.
7. Jung Hoon Lee and Mahen Mahendran (2002), Mechanical Properties of Light
Gauge Steels at Elevated Temperatures, Second International Conference on
Advances in Structural Engineering and Mechanics, Busan, Korea.
8. Jung Hoon Lee and Mahen Mahendran (2003), Local Buckling Behaviour and
Design of High Strength Unstiffened Steel Plate, Ninth Asia-Pacific Conference
on Structural Engineering and Construction, Bali, Indonesia.
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
vi
9. Jung Hoon Lee, Mahen Mahendran and Pentti Makelainen (2003), Local
Buckling Behaviour and Design of Unstiffened Steel Plate Elements at Elevated
Temperatures, Ninth Asia-Pacific Conference on Structural Engineering and
Construction, Bali, Indonesia.
10. Jung Hoon Lee and Mahen Mahendran (2004), Local Buckling Behaviour and
Design of Cold-formed Steel Compression Members at Elevated Temperatures,
Fourth International Conference on Thin-Walled Structures, Loughborough,
England (In print).
11. Jung Hoon Lee and Mahen Mahendran (2004), Behaviour and Design of Lipped
Channel Members Subject to Local Buckling Effects at Elevated Temperatures,
Eighteenth Australasian Conference on the Mechanics of Structures and
Materials, Perth, Australia (In preparation)
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
vii
TABLE OF CONTENTS Acknowledgements i Keywords ii Abstract iii Publications v Table of Contents vii List of Figures xi List of Tables xv Notations xvii Statement of Original Authorship xix Chapter 1. Introduction 1-1 1.1 Overview 1-1 1.2 Definition of Fire Resistance 1-4 1.3 Problem Definition 1-5 1.4 Objectives, Scope and Research Plan 1-6 1.5 Contents of the Thesis 1-9 Chapter 2. Literature Review 2-1 2.1 General 2-1 2.1.1 Historical Background 2-1 2.1.2 Elements in Fire Safety Engineering 2-3 2.1.3 Behaviour of Fire 2-6 2.2 Local Buckling 2-6 2.2.1 Background Theory of Local Buckling 2-6
2.2.2 Local Buckling Effects and Effective Width Concept at Ambient Temperature 2-9
2.3 Design Methods of Steel Compression Members at Elevated Temperatures 2-12
2.3.1 AS 4100 2-12 2.3.2 BS 5950 2-15 2.3.3 Eurocode 3 2-16 2.3.4 Previous Research 2-18 2.4 Method of Investigations 2-34 2.4.1 Simulated Fire Tests 2-35 2.4.2 Finite Element Analysis 2-39 2.5 Literature Review Findings 2-40 Chapter 3. Mechanical Properties of Light Gauge Steels at
Elevated Temperatures 3-1
3.1 Experimental Investigations 3-1 3.1.1 Test Method 3-1 3.1.2 Test Specimens 3-2
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
viii
3.1.3 Test Rig and Procedure 3-4 3.2 Deterioration of Mechanical Properties with Increasing Temperatures 3-6
3.2.1 Determination of Yield Strength and Elasticity Modulus 3-6 3.2.2 Yield Strength 3-8 3.2.3 Elasticity Modulus 3-13 3.3 Comparison of Yield Strength and Elasticity Modulus Predictions 3-15 3.4 Stress-Strain Model 3-19 3.5 Ductility Increase of High Strength Steels with Temperatures 3-23 3.6 Summary 3-25 Chapter 4. Local Buckling Behaviour of Unstiffened Elements at Ambient Temperature
4-1
4.1 General 4-1 4.2 Effective Width Concept 4-2 4.3 Experimental Investigations 4-6 4.3.1 Test Specimens 4-6 4.3.2 Test Set-up 4-8 4.3.3 Determination of Local Bucking Stress and Effective Width 4-9 4.4 Numerical Analyses 4-10 4.4.1 Finite Strip Analysis 4-11 4.4.2 Finite Element Analysis 4-11 4.4.2.1 Elements and Finite Element Meshes 4-11 4.4.2.2 Material Properties 4-12 4.4.2.3 Loading and Boundary Conditions 4-13 4.4.2.4 Initial Geometric Imperfections and Residual Stresses 4-14 4.4.2.5 Analysis 4-19 4.4.2.6 Verification of Finite Element Model 4-19 4.5 Results and Discussions 4-27 4.5.1 Local Bucking Behaviour 4-27 4.5.2 Ultimate Strength Behaviour 4-28 4.5.3 Ultimate Strengths and Effective Widths 4-29
4.5.4 Design of High Strength Unlipped Channel Members Subject to Local Buckling Effects 4-34
4.5.5 Effects of Strain Hardening 4-38 4.5.6 Imperfection Sensitivity Analyses 4-39 4.6 Direct Strength Method 4-41 4.7 Summary 4-42 Chapter 5. Local Buckling Behaviour of Unstiffened Elements at Elevated Temperatures
5-1
5.1 General 5-1 5.2 Experimental Investigations 5-2 5.2.1 Testing Facilities 5-2 5.2.2 Test Specimens 5-2 5.2.3 Test Method 5-5
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
ix
5.2.4 Determination of the Local Buckling Stress and Ultimate
Strength at Elevated Temperatures 5-8
5.3 Finite Element Analysis 5-10 5.3.1 Model Description 5-10 5.3.1.1 Finite Elements and Meshes 5-10 5.3.1.2 Material Behaviour at Elevated Temperatures 5-11 5.3.1.3 Load and Boundary Conditions 5-14 5.3.1.4 Geometric Imperfections and Residual Stresses 5-15 5.3.1.5 Thermal Factors 5-17 5.3.2 Analysis 5-18 5.3.2.1 Elastic Eigenvalue Analyses 5-18 5.3.2.2 Non-linear Analyses 5-18 5.3.3 Verification Study 5-19 5.4 Results and Discussions 5-29 5.4.1 Local Buckling Behaviour 5-29 5.4.2 Variations of Effective Width at Elevated Temperatures 5-33 5.4.3 Effective Width Rules for Local Buckling Behaviour
at Elevated Temperatures 5-43
5.4.3.1 Theoretical Design Method 5-43 5.4.3.2 Semi-empirical Design Method 5-48 5.4.3.3 Simplified Design Method 5-56 5.4.4 Ductility of High Strength Steels at Elevated Temperatures 5-62 5.5 Summary 5-63 Chapter 6. Local Buckling Behaviour of Stiffened Elements at Elevated Temperatures
6-1
6.1 General 6-1 6.2 Modification of the Design Rules of Local Buckling Stress and Effective Width at Elevated Temperatures 6-2
6.3 Experimental Investigations 6-3 6.3.1 Test Specimens 6-3 6.3.2 Test Set-up and Procedure 6-6 6.4 Finite Element Analysis 6-8 6.4.1 Elements 6-8 6.4.2 Material Properties at Elevated Temperatures 6-8 6.4.3 Geometric Imperfections and Residual Stresses 6-9 6.4.4 Methods of Analysis 6-11 6.4.5 Half Length Model 6-11 6.4.6 Quarter-wave Buckling Length Model 6-18 6.5 Results and Discussions 6-24 6.5.1 Local Buckling Behaviour 6-24 6.5.2 Effective Width at Elevated Temperatures 6-27 6.6 Summary 6-37
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
x
Chapter 7. Interactive Local Buckling Behaviour of Stiffened Web and Flange Elements at Elevated Temperatures
7-1
7.1 General 7-1 7.2 Experimental Investigations 7-1 7.2.1 Test Specimens 7-1 7.2.2 Test Set-up and Procedure 7-4 7.3 Finite Element Analysis 7-4 7.3.1 Model Descriptions 7-5 7.3.2 Analysis 7-6 7.3.3 Half Length Model 7-6 7.3.4 Quarter-wave Buckling Length Model 7-13 7.4 Results and Discussions 7-18 7.4.1 Local Buckling Stress 7-18 7.4.2 Effective Width at Elevated Temperatures 7-20 7.5 Summary 7-30 Chapter 8. Conclusions and Recommendations 8-1 Appendix Appendix A Stress-Strain Curves of Tensile Coupon Tests at Elevated Temperatures A-1
Appendix B ABAQUS Residual Stress Subroutines A-4 B1. Abaqus Subroutine used for Unstiffened Elements A-4 B2. Abaqus Subroutine used for Stiffened Elements A-6 B3. Abaqus Subroutine used for Members subject to Local Buckling Interaction A-8
Appendix C Example Calculations A-10 C1. Direct Strength Method A-10 C2. Effective Width A-12 Appendix D Axial Compression Load versus Axial Shortening Curves at Ambient Temperatures (Chapter 4) A-14
Appendix E Axial Compression Load versus Axial Shortening Curves at Elevated Temperatures A-16
E1. Chapter 5 A-16 E2. Chapter 6 A-18 E3. Chapter 7 A-21 Appendix F Ultimate Strength of Stub Columns including Three Repeats A-26
References R-1
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
xi
LIST OF FIGURES
Figure 1-1 Fires and Fatalities (1983 - 1991) 1-1 Figure 1-2 Property Losses from Structural Damages 1-2 Figure 1-3 Application of Cold-formed Steel Structures 1-2 Figure 1-4 Buckling Failures of Steel Structures in Fire 1-3 Figure 2-1 Development of Fire and Related Phenomena 2-6 Figure 2-2 A Simply Supported Plate Uniformly Compressed in one Direction 2-7
Figure 2-3 k Factors as a Function of Ratio a/b 2-8 Figure 2-4 Effective Width and Stress Distribution of Uniformly Compressed Stiffened Elements 2-10
Figure 2-5 Variation of Yield Stress 2-14 Figure 2-6 Variation of Elasticity Modulus 2-14 Figure 2-7 Ramberg-Osgood Curve 2-19 Figure 2-8 Variation of Column Strength with Different Levels of Slenderness Ratio, Residual Stress and Load Eccentricity 2-20
Figure 2-9 Stress-Strain Curves determined by the Modified Ramberg- Osgood Model and Tests 2-21
Figure 2-10 Variation of TYTE ,, /φφ at Elevated Temperatures 2-29 Figure 2-11 Comparison of Effective Width Formulae at Elevated Temperatures 2-30
Figure 2-12 Specimen Dimensions 2-32 Figure 2-13 Tensile Test Set-up 2-32 Figure 2-14 Stress-Strain Curves obtained from Steady- and Transient- State Tests 2-34
Figure 2-15 Full Scale and Small Scale Tests 2-35 Figure 2-16 Temperature Effectiveness to Light and Heavy weight Materials 2-36
Figure 2-17 Converting Temperature Strain Curves to Stress-Strain Curves 2-38 Figure 2-18 Steel Temperature Corresponding to Standard 2-38 Figure 3-1 Dimensions of Tensile Test Specimens 3-2 Figure 3-2 Test Rig 3-5 Figure 3-3 Test Set-up and Data Acquisition 3-5 Figure 3-4 Determination of Mechanical Properties 3-7 Figure 3-5 Typical Stress-strain Curves at Low and High Temperatures and Yield Strength Determination (0.95mm-G550) 3-10
Figure 3-6 Comparison of Yield Strength Reduction Factor from Equations 3.1 and 3.2 with Test Results 3-11
Figure 3-7 Variation of Yield Strength at Different Strain Levels 3-12 Figure 3-8 Comparison of Elasticity Modulus Reduction Factors from Equations 3.3(a) to 3.3(c) with Test Results 3-14
Figure 3-9 Failure Modes of Tensile Specimens at Temperatures of 20 to 800°C 3-15
Figure 3-10 Comparison of the Variation of Yield Strength at Elevated Temperatures with Current Steel Design Rules 3-16
Figure 3-11 Comparison of the Variation of Elasticity Modulus at Elevated Temperatures with Current Steel Design Rules 3-18
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
xii
Figure 3-12 Effect of Parameter n on Stress-Strain Curve 3-21 Figure 3-13 Effect of Parameter β on Stress-Strain Curve 3-21 Figure 3-14 Stress-strain Curves from Equation 3.4 and Tests 3-22 Figure 3-15 Stress-Strain Curves at Elevated Temperatures 3-24 Figure 3-16 Failure Modes at Different Temperatures 3-25 Figure 4-1 Ultimate Strength of Plate Elements 4-4 Figure 4-2 Unlipped C-section 4-7 Figure 4-3 Test Set-up 4-9 Figure 4-4 Axial Compression Load versus Out-of-plane deflection2 Method 4-10
Figure 4-5 Labelling Method of Elements in FEA 4-12 Figure 4-6 Stress-strain Relationship used in FEA 4-13 Figure 4-7 FEA Model 4-14 Figure 4-8 Definition of Geometric Imperfections 4-16 Figure 4-9 Geometric Imperfections and Residual Stresses 4-18 Figure 4-10 Local Buckling of Flanges in Experiments and FEA 4-20 Figure 4-11 Axial compression load versus axial shortening curves 4-22 Figure 4-12 Determination of Local Buckling Load 4-23 Figure 4-13 Comparison of Buckling Coefficients from Tests and FEA 4-27 Figure 4-14 Local Buckling Failure of Flanges 4-29 Figure 4-15 Comparison of Ultimate Loads of Unlipped Channel Sections from Test and FEA results with those from Current Design Rules
4-31
Figure 4-16 Comparison of Ultimate Loads of Flanges from Test and FEA Results with those from Current Design Rules 4-31
Figure 4-17 Comparison of Test Results with Those from Modified Design Rules using a Reduced Yield Stress of 0.9fy
4-35
Figure 4-18 Comparison of Test Results with Equation 4.10 Predictions 4-37 Figure 4-19 Modified Effective Width Curve for High Strength Unstiffened Elements 4-38
Figure 4-20 Comparison of Results from the Direct Strength Method, Tests and FEA 4-42
Figure 5-1 Test Set-up 5-3 Figure 5-2 Section Geometry – Unlipped Channel 5-4 Figure 5-3 Comparison of the Transient State (TST) and Steady State Test (SST) Methods 5-7
Figure 5-4 Modelling of Material Behaviour at Elevated Temperatures in FEA 5-12
Figure 5-5 Comparison of Actual Stress-Strain Curves with Idealised Curves used in FEA 5-13
Figure 5-6 Quarter-wave Buckling Length Model used in FEA 5-14 Figure 5-7 Assumed Residual Stress Distribution with Temperature Effects 5-17
Figure 5-8 Elastic Buckling Stress versus Buckling Half-wave Length 5-20 Figure 5-9 Axial Compression Load versus Axial Shortening Curves at Elevated Temperatures (Test Series G5-2) 5-22
Figure 5-10 Axial Compression Load versus Axial Shortening Curves at Elevated Temperatures (Test Series G2-2) 5-24
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
xiii
Figure 5-11 Axial Compression Load versus Out-of-plane Displacement Curves 5-26
Figure 5-12 Failure Modes Observed in Different FEA Models 5-27 Figure 5-13 Failure Mode Observed in Experiments 5-28 Figure 5-14 Buckling Coefficients at Elevated Temperatures 5-30 Figure 5-15 Local Buckling Failure Modes 5-32 Figure 5-16 Variations of Effective Width Ratio with Temperature 5-37 Figure 5-17 Effective Width Ratio versus Section Slenderness at Various Temperatures 5-39
Figure 5-18 Comparison of Reductions to Elasticity Modulus and Yield Strength at Elevated Temperatures 5-42
Figure 5-19 Comparisons of Design Curves with Test and FEA Results 5-47
Figure 5-20 Relationship between CT and γT at Temperature T 5-50 Figure 5-21 Comparisons of Effective Width Ratios based on Equation 5.27 with Test and FEA Results at Elevated Temperatures
5-51
Figure 5-22 Relationship between CT and γT using FEA Results 5-52 Figure 5-23 Comparisons of Effective Widths based on Equation 5.29 with FEA Results at Elevated Temperatures 5-53
Figure 5-24 Comparisons of Effective Widths based on Equation 5.29 with Test Results at Elevated Temperatures 5-54
Figure 5-25 Comparisons of Effective Widths based on Equation 5.29 with FEA and Test Results using a Reduced Yield Stress of 0.9fy
5-55
Figure 5-26 Effects of Mechanical Properties on Effective Width 5-56 Figure 5-27 Comparisons of Effective Width Ratios based on Equation 5.31 with Test and FEA Results at Elevated Temperatures
5-58
Figure 5-28 Comparisons of Effective Width Obtained from British Standard (BS 8818) with Test and FEA Results 5-61
Figure 5-29 Axial Compression Load versus Axial Shortening Curves of Test Series G5-3 5-62
Figure 6-1 Section Geometry – Lipped Channel 6-3 Figure 6-2 Typical Buckling Stress Plot of Chosen Lipped Channel Sections (G5-L2) 6-4
Figure 6-3 Test Set-up 6-7 Figure 6-4 Residual Stress Distributions and Geometric Imperfections 6-10 Figure 6-5 Residual Stress Output from FEA model 6-10 Figure 6-6 Half Length Model 6-12 Figure 6-7 Comparison of Failure Modes at Different Temperatures 6-15 Figure 6-8 Axial Load versus Axial Shortening Curves from Test and Half Length FEA Model 6-15
Figure 6-9 Axial Compression Load versus Out-of-plane Displacement Curves 6-16
Figure 6-10 Quarter-wave Buckling Length Model 6-18 Figure 6-11 Axial Load versus Axial Shortening Curves from Test and FEA 6-21
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
xiv
Figure 6-12 Support Constraint Conditions in FEA 6-23 Figure 6-13 Reduction of Local Buckling Stress at Elevated Temperatures 6-24 Figure 6-14 Variation of Buckling Coefficients with Temperatures 6-25 Figure 6-15 Local Buckling Failure Modes 6-26 Figure 6-16 Variation of Effective Width of Stiffened Web Element 6-28 Figure 6-17 Elasticity Modulus to Yield Strength Ratio versus Temperature 6-28
Figure 6-18 Comparisons of Theoretical Design Curves with FEA Results of Stiffened Elements 6-29
Figure 6-19 Comparisons of Effective Widths from Equations 5.27 and 5.29 with FEA Results of Stiffened Web Elements at Elevated Temperatures
6-33
Figure 6-20 Comparisons of Effective Widths from Equation 5.31 with FEA Results of Stiffened Web Elements at Elevated Temperatures
6-34
Figure 7-1 Section Geometry – Lipped Channel 7-2 Figure 7-2 Typical Buckling Stress Plot of the Chosen Lipped Channel Sections (G5-I2) 7-3
Figure 7-3 Test Set-up 7-4 Figure 7-4 Initial Geometric Local Imperfection 7-5 Figure 7-5 Half Length Model 7-6 Figure 7-6 Comparison of Local Buckling Failure Modes at 600°C 7-10 Figure 7-7 Axial Load versus Axial Shortening Curves from Test and Half Length FEA Model 7-11
Figure 7-8 Axial Compression Load versus Out-of-plane Displacement Curves 7-12
Figure 7-9 Quarter-wave Buckling Length Model 7-13 Figure 7-10 Axial Load versus Axial Shortening Curves from Test and Quarter-wave Buckling Length Model 7-17
Figure 7-11 Variation of Buckling Coefficient at Elevated Temperatures 7-18 Figure 7-12 Local Buckling Failure Modes at Different Temperatures 7-20 Figure 7-13 Variation of Effective Width of Stiffened Web and Flange Elements 7-21
Figure 7-14 Comparisons of Theoretical Design Curves with FEA Results of Stiffened Web and Flange Elements at Elevated Temperatures
7-23
Figure 7.15 Comparisons of Effective Widths from Equations 5.27 and 5.29 with FEA Results of Stiffened Web and Flange Elements at Elevated Temperatures
7-26
Figure 7.16 Comparisons of Effective Widths from Equation 5.31 with FEA Results of Stiffened Web and Flange Element at Elevated Temperatures
7-27
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
xv
LIST OF TABLES
Table 1-1 Annual Fire Death Rate (1989 - 1994) 1-2 Table 2-1 Historical Events of Structural Fire Protection 2-2 Table 2-2 Active and Passive Measures of Fire Protection 2-5 Table 2-3 Local Plate Bucking Coefficients under Uniform Compression 2-9 Table 2-4 Variation of Yield Stress based on AS 4100 2-13 Table 2-5 Variation of Elasticity Modulus based on AS 4100 2-14 Table 2-6 Limiting Temperatures for Compression Members 2-16 Table 2-7 Reduction Factors for Cold-Formed Steel Members 2-16 Table 2-8 Mechanical Properties at Temperature T 2-18 Table 2-9 Factors for Stress-strain-temperature Relationships 2-19 Table 2-10 Reduction of Mechanical Properties at Elevated Temperature (S350GD+Z) 2-33
Table 3-1 Details of Test Specimens 3-3 Table 3-2 Reduction Factors of Yield Strength (fy,T/fy,20 ) 3-9 Table 3-3 Reduction Factors of Elasticity Modulus (ET/E20 ) 3-13 Table 3-4 Coefficients β for Equation 3.4 3-22 Table 4-1 Details of Test Specimens and FEA Models 4-7 Table 4-2 Additional FEA Models 4-8 Table 4-3 Results from Tests and FEA (G250 steel) 4-24 Table 4-4 Results from Tests and FEA (G550 steel) 4-25 Table 4-5 Comparison of Ultimate Strength Ratios 4-32 Table 4-6 Comparison of Test Ultimate Strength Ratios based on 0.9fy 4-35 Table 4-7 Comparison of Ultimate Strength Ratios based on Equation 4.10 4-37
Table 4-8 Comparison of Results from FE Models with and without Strain Hardening 4-39
Table 4-9 Effects of Imperfection on Ultimate Strength 4-40 Table 5-1 Test Specimens 5-4 Table 5-2 Reduction Factors for Elasticity Modulus (ET/E20) and Yield Strength (fy,T/fy,20)
5-9
Table 5-3 Residual Stresses Reduction Factor 5-17 Table 5-4 Comparison of Elastic Buckling Stresses and Ultimate Strengths at 20°C and 600°C 5-20
Table 5-5 Test and FEA Results of Buckling Stress and Ultimate Strength 5-21
Table 5-6 Experimental and FEA Effective Widths and Comparison 5-34 Table 6-1 Test Specimens 6-4 Table 6-2 Reduction Factors for Elasticity Modulus (ET/E20) and Yield Strength (fy,T/fy,20)
6-5
Table 6-3 Comparison of Local Buckling Stress and Buckling Coefficient from Test and Half Length FEA Model at Elevated Temperatures
6-13
Table 6-4 Comparison of Ultimate Strength and Effective Width from Test and Half Length FEA Model at Elevated Temperatures
6-14
Table 6-5 Comparison of Local Buckling Stress and Buckling Coefficient from Tests and Quarter-wave Buckling Length FEA Model at Elevated Temperatures
6-19
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
xvi
Table 6-6 Comparison of Ultimate Strength and Effective Width from Tests and Quarter-wave Buckling Length FEA Model at Elevated Temperatures
6-20
Table 6-7 Additional FEA Results of Effective Width for High Strength Stiffened Web Elements 6-32
Table 7-1 Test Specimens 7-2 Table 7-2 Reduction Factors for Elasticity Modulus (ET/E20) and Yield Strength (fy,T/fy,20)
7-3
Table 7-3 Comparison of Local Buckling Stress and Buckling Coefficient from Tests and Half Length FEA Model at Elevated Temperatures
7-8
Table 7-4 Comparison of Ultimate Strength and Effective Width from Tests and Half Length FEA Model at Elevated Temperatures 7-9
Table 7-5 Comparison of Local Buckling Stress and Buckling Coefficient From Tests and Quarter-wave Buckling Length FEA Model at Elevated Temperatures
7-15
Table 7-6 Comparison of Ultimate Strength and Effective Width from Tests and Quarter-wave Buckling Length FEA Model at Elevated Temperatures
7-16
Table 7-7 Additional FEA Results of Effective Width for High Strength Stiffened Web and Flange Elements 7-22
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
xvii
NOTATIONS A cross section area Ae effective cross section area a plate length b flange width (or plate width) be effective flange width be,T effective flange width at temperature T C non-dimensional parameter D flexural rigidity d web depth de effective web depth de,T effective web depth at temperature T E (E20) elasticity modulus at ambient temperature ET elasticity modulus at temperature T Et tangent modulus Et,T tangent modulus at temperature T EΦ ratio of elasticity modulus at temperature T to that at ambient
temperature e eccentricity ep non-dimensional proof stress f* design stress fcr local buckling stress fcr,T local buckling stress at temperature T f0.1,T 0.1% proof stress at temperature T fy (fy,20) yield strength at ambient temperature fy,T yield strength at temperature T fΦ ratio of yield strength at temperature T to that at ambient temperature H total length of column HL half-wave buckling length I second moment of area i radius of gyration of cross section k buckling coefficient kE,T reduction factor of elasticity modulus kT buckling coefficient at temperature T ky,T reduction factor of yield strength kλ(T) defined as function of k,y,T and k,y,T le effective length Nx critical value of compression force Nc,T critical load at temperature T Ne,T maximum elastic buckling load at temperature T Np,T plastic collapse load at temperature T Nu,T axial load at the fire limit state n non-dimensional parameter P applied axial force Pcr local bucking load Py crushing load
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
xviii
Pu ultimate compressive strength R load ratio r corner radius T temperature t thickness U strain energy W external force α residual stress reduction factor β constant for Ramberg-Osgood model βT constant for Ramberg-Osgood model at temperature T γ non-dimensional parameter γT non-dimensional parameter at temperature T δ out-of-plane displacement ε strain εs,low strain at which strain-hardening commences at low temperatures εs,high strain at which strain-hardening commences at high temperatures εT strain at temperature T εy strain at yield stress εe elastic strain εp plastic strain η plasticity reduction factor ηT plasticity reduction factor at temperature T λ slenderness ratio λT slenderness ratio at temperature T ν Poisson’s ratio ρ ratio of effective width to actual width of the plate σ stress σu,high ultimate stress at high temperatures σu,low ultimate stress at low temperatures σy,high yield stress at high temperatures σy,low yield stress at low temperatures χ constant
Local bucking behaviour and design of cold-formed steel compression members at elevated temperatures
xix
STATEMENT OF ORIGINAL AUTHORSHIP
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.
Signed:
Date:
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-1
Chapter 1. Introduction
1.1 Overview In recent times, the use of cold-formed steel structures has increased in residential,
industrial and commercial buildings. This has led to a realization of the importance
of fire safety design for cold-formed steel structures. However, most research into
the fire safety design of steel structures has been mainly focused on hot-rolled or
heavier steel assemblies.
The growing interest in fire safety design is proportional to the increased loss of
properties and lives each year. The number of fatalities and amount of damage
related to low-rise buildings and dwellings is much larger than damage in high-rise
buildings and other types of buildings such as public buildings. As shown in Figure
1.1, a comparison of the frequency of fires and fatalities including conflagration and
damage between high-rise and low-rise buildings clearly indicates that low-rise
buildings are more in need of improved fire safety design. Low-rise buildings in
Figure 1.1 are one or two storey residential buildings. Therefore if they include
office buildings up to four storeys, the significance of fire safety design in low-rise
buildings becomes even more important. Further, the death rate and the loss of
properties by fire in Australia have rapidly increased during the last decade (see
Table 1.1 and Figure 1.2).
(a) Number of Fires (b) Civilian Fatalities
Figure 1.1 Fires and Fatalities (Bennetts et al., 2000)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-2
Table 1.1 Annual Fire Death Rate (1989 - 1994)(Quintiere, 1998; SCRCSSP, 2000)
Country Annual Deaths per 105 persons Country Annual Deaths
per 105 persons Russia 10.60 Czech republic 1.21
Hungary 3.31 Germany 1.17 India 2.20 0.93
Finland 2.18 3.3(1998)* South Africa 2.00
Australia 4.7(1999)*
USA 1.95 New Zealand 0.92 Denmark 1.64 Spain 0.86 Norway 1.60 Poland 0.80 Canada 1.58 Austria 0.74 Japan 1.52 Netherlands 0.63
United Kingdom 1.49 Switzerland 0.53 * refers to the collected data from Steering Committee for the Review of Commonwealth State Service Provision (SCRCSSP, 2000)
Figure 1.2 Property Losses from Structural Damages (SCRCSSP, 2000)
(a) Studs
(c) Floor deck
(b) Residential use
(d) Framing with joists
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-3
Cold-formed steel structures are mainly used in low-rise buildings. Figures 1.3(a) to
(d) show various usages of cold-formed steel members. Thus, these data show the
significance of fire safety design of cold-formed steel structures. This is important
not only in relation to saving buildings, but also to saving lives. Structures should
last until evacuation can be completed because saving lives is very important.
When a steel structure is exposed to fire, the steel temperature increases and hence
its strength and rigidity are significantly reduced, leading to various buckling
deformations and failures (Figures 1.4(a) and (b)). Damage to steel structures in fires
tends to be higher than damage to reinforced concrete or heavy timber structures due
to a higher thermal conductivity and thinner material. Fire safety design of steel
structures will therefore greatly reduce the cost of fire protection and loss of lives.
Figure 1.4 Buckling Failures of Steel Structures in Fire
(a) Global buckling
(b) Local buckling
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-4
Many design solutions for structural steel members under fire conditions have been
proposed by international and domestic standards such as Eurocode 3-1.2 (ECS,
1995), British Standard 5950 (BSI, 1995) and Australian Standard 4100 (SA, 1998),
and by a few researchers (Olawale and Plank, 1988; Janss, 1982; Burgess and Najjar,
1994; Talamona et al., 1997; Toh et al., 2000) using various methods. However,
these methods are considered appropriate only for hot-rolled steel structures. Since
there are many significant differences between hot-rolled and cold-formed steel
members due to their manufacturing processes, geometry and structural
characteristics, pertinent design rules are required for cold-formed steel members
under fire conditions.
1.2 Definition of Fire Resistance
A general definition of ‘fire resistance’ is immunity to the effects of fire up to a
required degree (Malhotra, 1982). Before the expression ‘fire resistance’ was coined,
the term ‘fire proof’ was used in the nineteenth century to describe the construction
withstanding the effect of fire (Shields, 1987). However, the term confused people in
various fields because it referred not only to structural design for fire resistance but
also non-combustible materials. Therefore, the term ‘fire proof’ was replaced by the
term ‘fire resistance’ to reflect the actual structural fire safety design process.
“From an engineering point of view, fire safety design can be defined as the
application of engineering principles to the effects of fire in order to reduce the loss
of structural adequacy by quantifying the risks and hazards involved and to provide
an optimal solution to the application of preventive or protective measures” (Purkiss,
1996). The definitions of structural fire resistance given by the British standard BS
5950 Part 8 (BSI, 1995), Parts 20 and 21 (BSI, 1987) are categorized in accordance
with the ability of components responsible for fire resistance such as limiting
temperature, time period and load capacity, which should be achieved without
causing failure under fire. Similar methods for the purpose of unprotected steel
structures are used in AS 4100 (SA, 1998) and Eurocode 3 Part 1-2 (ECS, 1995).
Building Code of Australia (ABCB, 2000) also provides the fire resistance level for
building elements which is determined as the grading periods in minutes for three
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-5
criteria i.e., structural adequacy, integrity and insulation. Summaries of the
definitions recommended in the standards above are as follows.
Load bearing capacity – The ability of a structure or a member to resist force or
moment caused by structural load and by the effect of fire load including material
deterioration.
Limiting temperature (Critical temperature) – The temperature at which failure is
expected in a structural member under fire conditions.
Period of fire resistance – The required time to withstand the exposure to fire or to
sustain structural ability without failure under fire conditions.
This research focused on the load bearing capacity of cold-formed steel members
subject to local buckling effects without the use of fire retardant materials.
1.3 Problem Definition
Thin-walled steel structures made of high strength steel (yield stress fy≥550MPa) are
widely used as load bearing members in industrial, commercial and residential
buildings in Australia. They are mainly used in low-rise applications. Fire damage in
low-rise buildings is far greater than in high-rise buildings as shown in Figure 1.1,
and the corresponding losses from structural damages are exorbitant. The
significance of fire safety design of thin-walled steel structures has therefore become
very important. However their structural behaviour under fire conditions is not well
understood as, in the past, fire research has been limited to heavier, hot-rolled
structural steel assemblies. The following issues have influenced the proposed
research topic in recognition of the requirements of fire resistance of cold-formed
steel structures.
1. There are no proper design rules or guidelines in Australia for cold-formed steel
members subject to axial compression. Since the Australian standard does not
give any suitable design formula for cold-formed steel compression members
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-6
under fire conditions, the fire resistance design currently used is conservative or
very approximate.
2. Eurocode 3 (EC3, 1995) presents a design method for fire resistance based on
hot-rolled steel members. However, this standard does not take into account the
effect of local buckling. The influence of local buckling in cold-formed steel
members has to be considered because of the thin and slender cross-sections.
Thus, the design rule specified in Eurocode 3 is not applicable to cold-formed
steel members.
3. It is very important to improve the understanding of the local buckling behaviour
of cold-formed steel members subject to axial compression and the variation of
material properties at elevated temperatures. The current design rules provide
data on the degradation of material properties, elasticity modulus and yield stress
for hot-rolled steels and some limited cold-formed steel grades. No data is
available for Australian high strength steels such as G550 and G500.
4. Eurocode (EC, 1995) recommends a critical temperature equal to 350°C for cold-
formed steel members. This recommendation can be used safely as a
conservative limit. However, this limit is considered too conservative and there is
a need to improve the fire safety design of cold-formed steel members.
5. Since the current practice for fire safety design based on laboratory testing is
inadequate and very expensive, suitable advanced analysis methods must be
developed to achieve the accurate prediction of fire resistance of cold-formed
steel compression members.
1.4 Objectives, Scope and Research Plan
The overall purpose of this research is to develop adequate design rules for cold-
formed steel compression members commonly used in Australia with the aid of both
simulated fire tests at elevated temperatures and advanced numerical analyses.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-7
The specific objectives of this research are
• To develop the experimental data of mechanical properties for low (fy<450 MPa)
and high (fy≥450 MPa) strength light gauge steels at elevated temperatures so
that the reduced mechanical properties can be used in the design rules for the fire
safety of cold-formed steel structures.
• To investigate the local buckling behaviour of cold-formed steel members
subject to axial compression at elevated temperatures using both simulated fire
tests and advanced numerical analyses. Three governing cases are to be
considered in this research i.e., local bucking of unstiffened element using
unlipped channel members, local bucking of stiffened element using lipped
channel members and interactive local buckling of stiffened web and flange
elements using lipped channel members.
• To develop fire testing facilities for cold-formed steel compression members and
to undertake compression member tests (unlipped and lipped channel members)
at elevated temperatures using both steady state and transient state test methods.
• To develop accurate finite element models that are capable of simulating the
local buckling behaviour of cold-formed steel compression members (unlipped
and lipped channel members) at elevated temperatures.
• To validate the finite element model of cold-formed steel compression members
at elevated temperatures by comparison with experimental results.
• To investigate the applicability of conventional effective width principles for the
unstiffened and stiffened elements of cold-formed steel compression members
(unlipped and lipped channel members) at elevated temperatures.
• To develop new design methods for cold-formed steel compression members
subject to the local bucking of unstiffened and stiffened elements at elevated
temperatures.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-8
• To validate the new design methods through comparison with test and finite
element analysis results. The design methods can be then included in the cold-
formed steel structures standard AS/NZS 4600 (SA, 1996).
The following limits applied to the simulated fire tests, advanced numerical analyses,
test materials, member classification and section types.
• Materials used for tests at elevated temperatures and finite element analyses in
this research were cold-formed light gauge steels with three steel grades
commonly used in Australia (G250, G500 and G550).
• This research is focused on the cold-formed steel column members in cold-
formed steel wall frames, which are mainly governed by local buckling.
Therefore it was limited to single column tests and analyses. Thus, this research
did not include the behaviour of connections between beams and columns. With
further research, it could be extended to long members, connections and cold-
formed steel wall frame systems.
• Section geometries were limited to lipped and unlipped channel sections to
investigate the local buckling behaviour of unstiffened flanges elements and
stiffened web elements and the local buckling interaction.
• The steady state test method was mainly used to investigate the structural
behaviour at elevated temperatures. However, the transient state test method was
also used to validate the results from the steady state tests.
• A finite element analysis program, ABAQUS, and a finite strip analysis program,
THINWALL, were used to obtain the numerical results of the elastic local
buckling stress of cold-formed steel members at varying temperatures, whereas
the former was used to determine their non-linear ultimate strengths.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-9
1.5 Contents of the Thesis
The layout of this thesis is as follows:
Chapter 1 presents an introduction to fire safety design including fire damage
statistics and the definition of fire resistance. It defines the problem
from current research and the objectives of this research.
Chapter 2 presents the literature review describing the findings pertinent to this
research project based on various design approaches by international
standards and past research, fire test methods, general solution methods,
mechanical properties at elevated temperatures and previous numerical
investigations.
Chapter 3 presents the mechanical properties and a modified model for light gauge
steels at elevated temperatures including the wide range of thicknesses
and steel grades used in this research.
Chapter 4 presents the experimental and numerical local buckling investigation of
unstiffened flange elements made of high strength steels at ambient
temperature since the current design rules for the effective width are
based on experiments of low grade steels.
Chapter 5 presents the simulated fire tests and finite element analyses of cold-
formed steel compression members subject to local buckling effects of
unstiffened flange elements at elevated temperatures. It describes the
details of test procedures and numerical analyses. Theoretical, semi-
empirical and simplified empirical design methods were proposed and
validated using unlipped channel members.
Chapter 6 presents the simulated fire tests and finite element analyses of cold-
formed steel compression members subject to local buckling effects of
stiffened flange elements at elevated temperatures. The adequacy of the
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 1-10
design methods developed in Chapter 5 was assessed for stiffened web
elements using lipped channel sections.
Chapter 7 presents the interactive local buckling behaviour of stiffened web and
flange elements at elevated temperatures based on simulated fire tests
and finite element analyses and verifies the proposed design formulae
developed in Chapter 5.
Chapter 8 presents the significant findings from this research and recommendations
for further research.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-1
Chapter 2. Literature Review
This chapter presents a review of available literature in the areas of fire safety
design, research, test and design methods used in fire engineering and behaviour of
structures under fire conditions.
2.1 General
2.1.1 Historical background
The commencement of positive steps in fire safety design was taken as a result of the
Great Fire of London in 1666, which destroyed four-fifths of the city before being
brought under control (Boring et al., 1981). In 1667, the first regulations were
drafted, and in 1774, further consolidated and broadened.
The considerable advances in the knowledge of fire safety design and fire testing
were made by the British Fire Protection Committee (BFPC), which was established
in 1897. The interest in fire protection has been spreading throughout the world since
that time and the International Fire Prevention Congress in London was launched in
1903. In the United States, the American Society for Testing and Materials (ASTM)
set up a committee to produce fire test standards and issued a standard of time-
temperature curve in 1917. The ISO committee presented the fire test specifications
in 1961. In 1981, European Commission on Constructional Steelwork Steel
Construction issued provisions for fire safety design of steel structures (Malhotra,
1982). The Australian standard (SA, 1998) presented fire resistance requirements
which include the mechanical properties of hot-rolled steels at elevated temperatures
and the time based design. The definition of the time based design is given in
Section 2.1.2. Major historical events relevant to fire safety design are shown in
Table 2.1.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-2
Table 2.1 Historical Events of Structural Fire Protection
1600 1700 1800 1900 1940 1980
1666 The Great fire of London
1667 The Rebuilding Act drafted 1774 The Act of 1774 embodied in fire safety legislation
1790 First fire test by the ‘Associated Architects’ 1844 Metropolitan Building Act
1897 British Fire Prevention Committee(BFPC) established 1899 BS 476 Parts 1 and 8 : Development of standard fire curve
1902 Fire test facility in the USA 1903 First international fire prevention congress in London
1910 Fire test facilities in the USA and Germany 1917 ASTM-C19 issued with the standard curve 1921 Royal commission appointed
1932 BS 476 issued provisions on fire resistance tests 1935 Fire offices’ committee testing laboratory at Borehamwood
1961 ISO committee formed on fire test specifications 1962 Building regulations for Scotland, England and Wales issued
1972 ASTM-CP110 issued on fire resistance 1975 Joint committee of the institute of structural engineers and
concrete society issued guidance on concrete design for fire
1981 ECCS issued recommendations for the design of steel structures
1998 AS 4100 issued provisions for the mechanical
properties of hot-rolled structural steels
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-3
2.1.2 Elements in fire safety engineering
In order to design against fire in buildings, various objectives must be considered
based on aesthetic, functional, technological and economic issues. Fire safety
engineering includes any area from building structures to controlling of fire using
ventilation, escape, the risks and the potential hazards of fire. The main purposes of
fire safety engineering are to ensure the safety of people and property, and to prevent
the structure of a building from becoming unstable.
In fire safety engineering, four strategies can be identified. The brief explanation of
each strategy is as follows:
1. Detection and warning system
Smoke detection and early warning systems are generally provided for occupants to
be warned and evacuate safely (ABCB, 1996). They have important roles that inform
the occupants about fires and trigger active fire systems. There are three kinds of
tactics in this area. Firstly, fire detection can be used to inform people about a fire
and to provide them enough time to evacuate. Secondly, smoke-controlling system
can be used to eliminate problems caused by burning with any toxic material.
Thirdly, fire-fighting systems can be installed to extinguish fire, i.e. sprinkler
system.
2. Evacuation
This can be achieved by well-designed escape routes and also by educating the
occupants of the building. The length of escape routes should be considered by
architects. Therefore Building Code of Australia (ABCB, 1996) regulates the number
of exits required and exit travel distances in accordance with building classes.
3. Controlling of ignition and fire
This system is to control the growth of fire and flammability that should be limited
as much as possible. A conventional method to prevent a building from rapid fire
spread is to install vertical or horizontal fire compartments.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-4
4. Structural protection
Structural elements in a building can be categorised as frame, wall, partition, floor,
roof, beam, column, and so on. Each element under fire conditions which affects the
structural stability of a building needs to be designed against structural collapse in
addition to the need to prevent total collapse of a building during a fire. The
structural design philosophy for fire resistance has changed in recent times. In the
past, most buildings were designed based on an ambient temperature of 20°C and
were then protected by placing fire retardant materials on the sides of members.
However, the buildings today are designed to be able to withstand fire. Thus, the
building fire design calculation can be based on higher temperatures such as 600°C
or 800°C. In order to achieve this the difference in the use of standard fire curves and
real fire condition should be properly investigated for the adequate design.
Three different methods are used to provide structural protection against fire, which
are
1) Time determination
The time based design is used to determine the time to have a period of structural
adequacy. Structural members should have adequate time to resist structural
deterioration during which their load capacity is not rapidly lost under fire. AS 4100
(SA, 1998) provides a method to determine the time at which a limiting temperature
is attained. Some numerical expressions were derived to estimate the collapse time
of hot-rolled steel columns under fire conditions (Skowronski, 1993, Chandrasekaran
and Mulcahy, 1997).
2) Critical temperature
The critical temperature method can be used to determine the limiting temperature of
a structural member under fire. The critical temperature is defined as the temperature
at which failure is expected to occur in a structural steel member. The criteria for
calculating the critical temperature are provided in AS 4100 (SA, 1998), EC 3 Part
1.2 (ECS, 1995) and BS 5950 Part 8 (BSI, 1995) based on hot-rolled structural steel
members.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-5
3) Member capacity design
A strength based solution can be used for endurance of a building at elevated
temperatures. This is a method to sustain member ability to fulfil its required
functions and to keep its required strength under fire conditions. Eurocode 3 (ECS,
1995) deals with the fire resistance of hot-rolled structural steel members. Olawale
(1988) and Burgess (1992) presented a finite strip method for analysing the
behaviour of pin-ended columns under fire. However, the specification of Eurocode
3 is not applicable to cold-formed structural steel members because it does not
include the effects of the local buckling behaviour. Moreover, most previous
research on time determination and critical temperature were also conducted based
on hot-rolled and heavier structural steel members. Therefore, the importance of fire
research on cold-formed steel members is obvious, and the studies of structural
behaviour of cold-formed structural steel members at elevated temperatures are
being performed actively in many countries such as the UK, USA, European
countries and Queensland University of Technology in Australia.
The fire protection can be largely categorised into two provisions of active and
passive measures (Malhotra, 1982). Elements within active measures are installation
of methods or functions and fire controlling system on the occurrence of a fire which
generally includes fire detection, warning systems and fire fighting facilities at the
early stage. Passive measures mainly involve built systems such as building
construction and structural protection. Details of each item are illustrated in Table
2.2. The structural design for fire protection is a part of passive measures.
Table 2.2 Active and Passive Measures of Fire Protection
Active provisions Passive provisions Alarm system Smoke control system Detection Fire-fighting and fire control system Sprinklers Fire safety management system In-built fire fighting system
Compartments Escape provisions Prevention of structural collapse (or Adequate structural performance) Contents and linings Control of flammability
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-6
2.1.3 Behaviour of Fire
Based on a time versus temperature curve, Figure 2.1 describes the development of
fire and the related phenomena during a fire. The period including ignition by a heat
source and starting a fire within a compartment can be called the initiation period.
Temperatures in this period remain relatively low and the possibility for escape is
quite high. In the growth period, temperatures start increasing substantially with the
aid of sufficiently flammable materials. It can be seen that the transition from the
growth period to fully developed period is made in this period. Flame and smoke are
also produced. The maximum burning activity is observed in the steady period. Flash
over is an important phenomenon which can happen during growth period and result
a sudden increase of burning and temperature. All combustibles within the enclosure
are burnt and the temperature begins to fall as well. During the decay period, the risk
of propagation by penetration of constructional components remains.
Figure 2.1 Development of Fire and Related Phenomena
(Shields and Silcock, 1987)
2.2 Local Buckling
2.2.1 Background theory of local buckling
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-7
The elastic critical local bucking stress at ambient temperatures has been thoroughly
experimented and investigated by many researchers (Timoshenko and Gere., 1961;
Thompson and Hunt, 1973). Local buckling stress is determined using the plate
slenderness and local buckling coefficient with the corresponding half-wave
buckling length. Assuming that a rectangular plate is uniformly compressed in one
direction along the sides x = 0 and x = a, and the boundary conditions of all four
edges of the plate are simply supported as shown in Figure 2.2, the out-of plane
deflection of the buckled plate and the work of the forces acting in the plate can be
represented by the double trigonometric series.
∑∑∞
=
∞
=
=1 1
sinsinm n
mn byn
axmaw ππ
∫ ∫
∂∂
=∆a b
x dxdywNW0 0
2
21
χ (2.1a)
Substituting for w into Equation 2.1(a), it becomes
∑∑∞
=
∞
=
=∆1 1
2
222
8m n
mnxa
maNabW π (2.1b)
Figure 2.2 A Simply Supported Plate Uniformly Compressed in one Direction
The strain energy is given by
∑∑∞
=
∞
=
+=∆
1 1
2
2
2
2
22
4
8 m nmn b
namaDabU π (2.2)
Thus, adopting a concept of the energy method that the work of external forces is
equal to the strain energy of bending, the critical value of compressive forces, Nx
becomes
S.SS.S
S.SS.S
a
b
x
y
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-8
UW ∆=∆
∑∑ ∑∑∞
=
∞
=
∞
=
∞
=
+=
1 1 1 1
2
2
2
2
22
422
2
88 m n m nmnmnx b
namaDabamN
ab ππ
2
2
2
2
2
2
22
+=
bn
am
mDaN x
π
It is obvious that the smallest value of Nx is obtained by taking n equal to 1. For
instance, the critical load, when m is equal to 1, can be represented as Equation
2.3(a). If the width of a plate is constant and the length, a, gradually changes, the
numerical factors change with the ratio a/b.
2
2
2
+=
ba
ab
bDN x
π (2.3a)
It can be seen that the critical load acquires its minimum value when a=b (Figure
2.3).
2
24b
DN xπ
= (2.3b)
For other proportions of plates, Equation 2.3(b) can be represented in the following
form.
2
2
bDkNx
π= (2.3c)
Figure 2.3 k Factors as a Function of Ratio a/b
k
0123456789
10
a/b
m=1
2 4 531 2 6 12 20
2 3 4 5
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-9
In Equation 2.3(a), D is the flexural rigidity of the plate.
)1(12 2
3
ν−=
EtD
Therefore, the critical buckling stress of a plate element in compression is given by
2
2
2
)1(12
−
=btEkfcr
υπ
(2.4)
where k and b/t are the local buckling coefficient and the plate slenderness ratio,
respectively. The values of the local buckling coefficient differ according to
supported edges and loading conditions. The local buckling coefficient k in the case
of uniform compression loading is shown in Table 2.3.
Table 2.3 Local Plate Bucking Coefficients under Uniform Compression
Boundary Condition Local Buckling Coefficient, k Half-Wavelength
4.0
b
6.97
0.66b
0.425 0.675
∞ 2b
1.247
1.636b
2.2.2 Local buckling effects and effective width concept
at ambient temperature
A major advantage of cold-formed steel structures is their high strength to weight
ratio. On the other hand, since they are often thin-walled members, a phenomenon of
S.S
S.SS.S S.S
S.SS.SBuilt-in
Built-in
FreeS.SS.S
S.S
Built-inS.SS.S
Free
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-10
local buckling becomes substantially important in their design. The local buckling
behaviour of singly symmetric cold-formed steel columns was experimentally
studied at ambient temperature by Young and Rasmussen (1998, 1999). The local
buckling inducing overall bending reduces the strength of a column because of the
shift in the effective centroid of the section, particularly in pin-ended columns. In
fixed-ended columns, local buckling does not significantly induce overall bending
because of the balance by a shift in the line of action of the external force. However,
for stub column tests end boundary conditions do not influence the ultimate strength.
The effective width concept is used to determine the maximum strength of cold-
formed steel members in compression. For elements under uniform compression
with simply supported edges, the ratio of the effective width to the actual width of
an element was proposed as a function of fcr/fy by earlier researchers (von Karman,
1932; Winter, 1947; Chilver, 1954; Bambach and Rasmussen, 2002). To avoid
complex analysis in design, the stress distribution at both sides of an element is
generally replaced by a uniform stress f* along an effective width, be. Figure 2.4(b)
shows the effective width and stress distribution of a stiffened element with uniform
compression. Von Karman (1932) introduced a concept of effective width and
stated that f*bet equals the total load on the element. Most international and
domestic design rules accept this concept for the design of cold-formed steel
members under compression.
(a) Actual stress distribution (b) Effective width of element Figure 2.4 Effective Width and Stress Distribution of Uniformly
Compressed Stiffened Elements
The effective width formula in AS/NZS 4600 (SA, 1996) and EC 3 Part 1.3 (ECS,
1997) is based on the Winter’s formula (1968) that takes into account the reduction
in strength resulting from geometric imperfections and residual stresses from the
b
f*
be/2b
f*
2b /e
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-11
cold-forming process. In both these design rules, a non-dimensional plate
slenderness ratio λ at the design stress is used, based on the local buckling stress
formula.
eb b= for λ ≤ 0.673
eb bρ= for λ > 0.673
where ρ , the effective width factor, is given by
0.1
22.01≤
−
=λλρ (2.5)
Ef
tb
kEkf
tb
ff
cr
*
2
*2* 052.1)1(12
=−
==πνλ (2.6)
where *f is the design stress in the compression element calculated on the basis of
the effective design width. In the case when nominal section capacity in compression
is based on yielding, *f can be taken as the yield stress yf . In other cases, *f is
determined in accordance with the governing buckling modes of members.
The ratio of the effective width to full flat width of an element under compression in
BS 5950-5 (BSI, 1998) is also determined by a function of the ratio of the
compressive stress fc on the effective element to the local buckling stress fcr. The
compressive stress fc can be taken as the yield stress for stub columns. The effective
width equations in BS 5950-5 (BSI, 1998) give slightly conservative values, but the
difference is very little.
eb b= for λ ≤ 0.123
eb bρ= for λ > 0.123
( )435.0/141
1
−+=
crc ffρ (2.7)
2
904.0
=btEkfcr (2.8)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-12
Alternatively, the effective width equations proposed by Weng and Pekoz (1986) can
be used for stiffened compression elements subject to local buckling effects.
for λ < 0.673, ρ = 1 (2.9a)
for 0.673 <λ < cλ , λ
λρ
461.0358.1 −==
bbe (2.9b)
for λ ≥ λc, λλρ
−
==
22.01
bbe
(2.9c)
where cr
y
ff
=λ and 0.256 0.328 yc
fbt E
λ = +
A recent study into the effective width for unstiffened elements of cold-formed steel
compression members was conducted by Bambach and Rasmussen (2002). Their
study based on experiments on mild steel plates with 5 mm nominal thickness and
average yield strength of 295 MPa demonstrated the conservatism of conventional
effective width rules. The modified effective width rule developed by Bambach and
Rasmussen (2002) is as follows.
43
8.0−
= λbbe (2.10)
where λ is the section slenderness ratio as given in Equation 2.6.
2.3 Design Methods for Steel Compression Members
at Elevated Temperatures
2.3.1 AS 4100
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-13
Three methods of fire safety design are provided in AS 4100 (SA, 1998), namely the
deterioration of mechanical properties of steel, the limiting steel temperature and the
period of structural adequacy at elevated temperatures. Firstly, the variation of the
yield stress and elasticity modulus with temperatures is stipulated for hot-rolled steel
members. Secondly, the limiting steel temperature criterion is stipulated by a
function of the ratio of the fire design load to the design capacity at room
temperature. Thirdly, the period of structural adequacy is determined by determining
the limiting temperature of a steel member. Since this research is focused on the
strength based design, the temperature and time based design methods stipulated in
AS 4100 (SA, 1998) are not used in this study.
The variation of the yield stress at elevated temperatures is categorised into two
ranges of temperature lower than 215°C and higher than 215°C. The reduction
factors for the yield stress of steel member are given by the following equations and
Table 2.4.
( )
1.0(20)
y
y
f Tf
= 0°C<T≤ 215°C (2.11a)
( ) 905(20) 690
y
y
f T Tf
−= 215°C<T≤ 905°C (2.11b)
Table 2.4 Variation of Yield Stress based on AS 4100 (SA, 1998)
Reduction factor at temperature (°C)
The influence of temperature on the modulus of elasticity of hot-rolled steel is given
by the following equations and Table 2.5.
( ) 1.0(20) 2000 ln
1100
E T TE T
= +
0°C<T≤ 600°C (2.12a)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-14
690 1
( ) 1000(20) 53.5
TE TE T
− =−
600°C<T≤ 1000°C (2.12b)
Table 2.5 Variation of Elasticity Modulus based on AS 4100 (SA, 1998)
Reduction factor at temperature (°C) 20 100 200 300 400 500 600 700 800 900
1.00 0.98 0.94 0.88 0.80 0.68 0.5 0.32 0.19 0.01
Figures 2.5 and 2.6 present the variation of mechanical properties of steel as
recommended by AS 4100 (SA, 1998) and other international standards such as the
Eurocode (ECS, 1995) and British Standard (BSI, 1995). All the values are based on
hot-rolled steels except for the yield stress variation in the British Standard.
Figure 2.5 Variation of Yield Stress
Figure 2.6 Variation of Elasticity Modulus
Eurocode 3. 1-2
2000 12001000800600400
AS 4100
k T
oC
0.2
0.4
Tem perature
Reduction Factor
0.6
0.8
11.0 kE,T
A S 4 1 0 0
0 2 0 0 4 0 0 6 0 0 1 0 0 0 1 2 0 08 0 0
E u ro c o d e 3 . 1 -2
B S 5 9 5 0 . 8
k T
B S 5 9 5 0 . 8 0 .5 % S tra in( )
( 0 .5 % )S tra in,C o ld - fo rm e d s te e l
oC
R e d u c t io n F a c to r
T e m p e ra tu re
1
0 .8
0 .6
0 .4
0 .2
ky,T 1.0
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-15
There are no specific design rules for the strength prediction of steel structures under
fire conditions in AS 4100 (SA, 1998). Without an understanding of structural
behaviour under fire conditions, the resulting structural design against fire can be
inaccurate. The investigation into the structural behaviour under fire conditions is
therefore essential.
2.3.2 BS 5950
The strength of a column under fire conditions stipulated in BS 5950 (BSI, 1995) is
based on a load ratio of the column exposed on four sides. A hot-rolled steel member
which has a load ratio, R≤0.6, is assumed to have an inherent fire resistance of 30
minutes without fire protection. The maximum section factor, Hp/A (ratio of
perimeter to the cross section area), for an unprotected steel column is limited to 50
m-1 in BS 5950 Part 8 (BSI, 1995). The load ratio, R, for a column subject to pure
compression is given by
c
Tu
fAN
R⋅
= , (2.13)
fc : compressive strength
Nu,T : ultimate strength at the fire limit state
The design standard tabulates limiting temperatures for a given load ratio. The
limiting temperatures for hot-rolled steel compression members are given in Table
2.6 but not for cold-formed steel members. The design method in BS 5950 (BSI,
1995) is therefore based on the temperature based design rather than strength based
design. However, the British Standard provides strength reduction factors for hot
rolled steel and cold-formed steel members separately. Three kinds of strain levels of
0.5%, 1.5% and 2.0% are used in accordance with member classification that should
be considered in the structural performance under fire. The reduction factors at a
strain of 2.0% are considered for composite members in bending, protected with fire
protection materials. A strain level of 1.5% is used for non-composite members in
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-16
bending which are unprotected or protected with fire protection materials, which
demonstrates their ability to remain intact. The other strain level of 0.5% can be used
for members not covered in the strain levels of 1.5 and 2.0%. Thus, the reduction
factors at a strain level of 0.5% should be adopted for non-composite members in
compression. Table 2.7 gives the strength reduction factors for cold-formed steel
members at elevated temperatures in accordance with different strain levels. For the
temperatures higher than the values given, linear extrapolation may be used.
Intermediate values may be obtained by linear interpolation.
Table 2.6 Limiting Temperatures for Compression Members (BSI, 1995)
Limiting temperature at a load ratio of :
Table 2.7 Reduction Factors for Cold-Formed Steel Members (BSI, 1995)
Strength reduction factor at temperature (°C)
2.3.3 Eurocode 3
The design resistance of a compression member with a class 1, class 2 or class 3
cross section at temperature T is determined from
[ ] AfkTT yTy ,u 2.1/)()(N χ= (2.14a)
Nu(T) : ultimate strength of axially loaded column at elevated temperature
A : cross sectional area
yf : yield strength at room temperature
)()()(
1)(22 TTT
Tλϕϕ
χ−+
=
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-17
λλ λ )()( TkT =
TETy kkTK ,, /)( =λ , A
A
fEeff
y/πλλ =
λ : slenderness ratio for the relevant buckling mode
( )[ ]2)(2.0)(121)( TTT λλαϕ +−+= (2.14b)
α : imperfection factor (0.34 for cold-formed steel channel sections)
ky,T, kE,T : reduction factors for yield strength and modulus of elasticity
Eurocode classifies four classes of cross sections to take into account plastic or
elastic analysis, and the possible limits on the resistance of cross sections.
Class 1 cross sections are those which can form a plastic hinge with the rotation
capacity required for plastic analysis.
Class 2 cross sections are those which can develop their plastic moment resistance,
but have limited rotation capacity.
Class 3 cross sections are those in which the calculated stress in the extreme
compression fibre of the steel member can reach its yield strength, but local buckling
is likely to prevent development of the plastic moment resistance.
Class 4 cross sections are those in which it is necessary to make explicit allowances
for the effects of local buckling.
For the temperatures below 400°C, the strain-hardening is allowed for increasing the
yield strength of members, so that the reduction factor for effective yield strength is
constant up to 400°C (see Figure 2.5).
The influence of local buckling on a thin-walled steel compression member must be
taken into account to determine the compressive strength since the interaction effect
of local and overall buckling results in a reduction of the overall column strength.
Although the parameter Kλ(T) includes the effective area Aeff, the design rule in EC
3 Part 1.2 (ECS, 1995) does not fully include the effects of local buckling. Cold-
formed steel structures are included in Class 4 cross sections. Therefore, Eurocode
(ESC, 1995) has a provision limiting the critical temperature to 350°C for cold-
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-18
formed steel members. However, this provision is considered too conservative
(Ranby, 1998: Wang and Davies, 2000: Feng et al., 2003a and b).
The Eurocode reduction factors for the effective yield strength and modulus of
elasticity at elevated temperatures, relative to the appropriate value at 20°C, ky,T and
kE,T, are given in Table 2.8 and plotted in Figures 2.5 and 2.6. Linear interpolation
can be used for intermediate values of steel temperature. These factors are
recommended on the basis of hot-rolled steel members. Thus, they may be
overestimated for cold-formed steel members due to the special characteristics of
light gauge steels including their manufacturing process and chemical contents
associated with thin materials.
Table 2.8 Mechanical Properties at Temperature T (ECS, 1995)
2.3.4 Previous research A number of researchers have developed and proposed various recommendations in
fire safety issues of structural stability. Some of their research and their solutions in
relation to material modeling, fire resistance design of structural steel members
subject to axial compression and the local buckling behaviour at elevated
temperatures are presented in the following sections.
Ramberg-Osgood model
The behaviour of steel columns at increasing temperature was investigated based on
Ramberg-Osgood equations by Olawale and Plank (1988). The degradation in
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-19
structural properties of steel is included by adopting the stress-strain-temperature
relationship at temperature T. The modified form of Ramberg-Osgood equation is
represented as
Tn
Ty
T
T
Ty
T
TT EE
+
=
,
,
73
σσσσ
ε (2.15)
where nT is a constant that describes the shape of the knee of the stress-strain
relationship, typically taken as 30 for hot-rolled steel and 25 for cold-formed steel at
ambient temperature. Figure 2.7 shows an idealized stress-strain curve at ambient
temperature based on Ramberg-Osgood equation. Table 2.9 shows the important
parameters σy,T, ET and nT developed by Olawale and Plank (1988) for use with the
modified Ramberg-Osgood model based on tensile tests at varying temperatures.
Figure 2.7 Ramberg-Osgood Curve Table 2.9 Factors for Stress-strain-temperature Relationships
Temperature range Yield stress (σy,T) Elastic modulus (ET) nT 4600/T+α #20°C<T≤ 400°C yσ (0.978-9.74×10-5T) E(1-1.27×10-6γ 2) 2650/T+α ∗
400°C<T≤ 500°C yσ (1.553-1.55×10-3T) E(1-1.27×10-6γ 2 ) 2400/T+α 500°C<T≤ 600°C yσ (2.34-3.143×10-2T) E(1-1.402×10-6γ 2) 3900/T+α 600°C<T≤ 690°C yσ (1.374-1.56×10-3T) E(1-1.402×10-6γ 2) 3600/T+α 690°C<T≤ 800°C yσ (1.12-1.28×10-3T) E(1-1.402×10-6γ 2) 4600/T+α # Temperature range of 20°C<T≤ 250°C ∗ Temperature range of 250°C<T≤ 400°C γ = T-20°C α = T/[500ln(T/1750)]
Strain εi
n
y
iyii EE
+=
σσσσ
ε73
Stre
ss σ i
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-20
Using the Ramberg-Osgood equation (Equation 2.15), the variation of the critical
stresses at elevated temperatures was given to show the influence of slenderness,
residual stress and eccentricity of loading as plotted in Figure 2.8 (Olawale and
Plank, 1988). The ultimate load of columns varied as a function of both temperature
and slenderness ratio as shown in Figure 2.8(a). This indicates that columns with
different slenderness ratios behave differently, hence this should be considered in
determining the limiting temperature or the critical strength at a temperature.
Assuming that the residual stress distribution across a section remains the same at
elevated temperatures, the effect of residual stresses on the strength loss of columns
under fire conditions is shown in Figure 2.8(b).
(a) Slenderness Ratio (b) Residual Stress
(c) Load Eccentricity Figure 2.8 Variation of Column Strength with Different Levels of Slenderness Ratio, Residual Stress and Load Eccentricity (Olawale and Plank, 1988)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-21
The curves in Figure 2.8(b) show similar decreasing rates. Since the residual stresses
tend to decrease as the temperature increases, the differences shown in Figure 2.8(b)
may become even smaller. Small eccentric loading provides almost identical curves,
but the curve for large eccentricity shows a substantial improvement in the
temperature range of 200°C to 600°C because of the initial low capacity at 20°C
compared with that for small eccentricities as seen in Figure 2.8(c).
Since the degradation in material properties obtained using the Ramberg-Osgood
equation was based on hot-rolled structural steels, an investigation into the
pertinency of this model for thin-walled steel members was conducted by Outinen et
al. (1997). The results derived by the modified Ramberg-Osgood equation were
compared with the stress-strain relationship of thin-walled light gauge steel S355
widely used in Finland. The stress-strain curves determined for structural steel S355
were based on the transient state tensile tests. Outinen et al. (1997) found that the
Ramberg-Osgood model modified by Olawale and Plank (1988) was applicable to
low-grade thin-walled steels only up to a non-proportional strain of 0.2%. Thus a
constant parameter of 3/7 instead of 6/7 was used in Equation 2.15. However their
study did not include high strength light gauge steels commonly used in Australia.
Ali et al. (2004) indicated that use of different steel grades could cause scatter in the
data of mechanical properties. Therefore more studies are required to verify the
accuracy of the modified Ramberg-Osgood model for thin-walled light gauge steels
including high strength steels.
Figure 2.9 Stress-Strain Curves determined by the Modified Ramberg-Osgood Model and Tests (Outinen et al., 1999)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-22
Janss’s (1982) method
On the basis of the European Recommendations for Steel Construction at ambient
temperature, an analytical method for the design of steel columns in fire conditions
was proposed (Janss, 1982). He adopted an analytical expression for non-
dimensional buckling curves in the design of axially loaded bare metal sections.
Based on the relationship between the column slenderness ratio and the critical stress
of a column and the reduction of material properties, the following expression is
derived for each case of buckling curve for steel members at elevated temperatures.
( ) ( )[ ]
−+−+×−
+−+= 222
22
2,
, 42.012
12
2.01TTT
TT
TT
cr
TcrTcN λλλα
λλλλα
σσ
(2.16)
Equation 2.16 can be simplified as follows.
cTc NkN 1, = (2.17)
λλ2
1kk
T =
Where
α = imperfection factor
λ = the relative slenderness for the relevant buckling mode
1750ln767
1,1 T
Tkcr
Tcr +==σσ
(20°C ≤ T < 600°C) (2.18a)
4401000
1108,
1 −
−
==T
T
kcr
Tcr
σσ
(600°C ≤ T ≤ 1000°C) (2.18b)
1109.15105.34108.11102.17 527394122 +×+×−×+×−== −−−− TTTT
EEk T (2.18c)
Nc in Equation 2.17 is the column strength at ambient temperature.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-23
In the past, available fire testing methodology and failure criterion of axially loaded
columns at elevated temperature were not adequate to determine the fire resistance of
steel compression members. Thus, conservative results were obtained when the
numerical values derived from Janss’ proposal were compared with the experimental
results. The time at which the thermal elongation was cancelled out by the shrinkage
if a column was considered as the time of failure, and the temperature at the time of
failure was measured as the critical temperature. Moreover, it appears that the
reduction factor of elasticity modulus becomes zero at about 625°C (Equation
2.18(c)). This design approach can be used to safely predict the strength of hot-rolled
steel compression members.
Burgess and Najjar (1994)
Burgess and Najjar (1994) developed an analytical approach using finite element
analysis and the well-known Perry-Robertson analysis to investigate the behaviour
of steel columns at elevated temperatures. A column is generally subject to uniform
heating. However, their modelling considers a specific circumstance subject to the
local fire that causes thermal gradients within its section. It is more complicated to
analyse the column behaviour under temperature gradients because many variables
such as the degree of fire protection on a face exposed to fire or the degree of
deterioration of yield stress and modulus of elasticity across its section and so on,
need to be considered. In order to simplify the variation under temperature gradients,
the linear reduction of the mechanical properties across the section was assumed.
The capability of the simplified analysis was assessed by comparing the results from
nonlinear finite element analyses with those from experiments. The comparison
showed that the assumption of the linear stress distribution under temperature
gradients was found to be adequate.
Franssen et al.(1995)
A proposal for steel columns subjecte to axial compressive forces under fire
conditions was attempted by means of both experimental and numerical studies
(Franssen et al, 1995, 1998, Talamona et al, 1997). Since the design rule in Eurocode
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-24
(ECS, 1995) can produces values of low safety level because the safety level varies
with different slenderness ratios, a modification accounting for the imperfection
factor and appropriate safety level was conducted, based on the material properties
of steel at elevated temperatures as defined in the Eurocode (ECS, 1995). The
procedure of this proposal is as follows.
AfkTT yTy ,u )()(N χ= (2.19a)
)()()(
1)(22 TTT
Tλϕϕ
χ−+
=
λλ λ )()( TkT =
TETy kkTK ,, /)( =λ
lλλλ = ,
yl f
EiH πλλ == ,
++= )()(1
21)(
2TTT λλαϕ (2.19b)
α βε= , 235 / yfε =
α : imperfection factor β : severity factor ky,T, kE,T : reduction factors of yield strength and elasticity modulus (EC3-1.2)
In fact, parameters in Equations 2.19(a) and (b) are the same as those defined in
Eurocode 3 (ECS, 1997), except that the safety factor 1.2 in Equation 2.19(a) and the
initial constant 0.2 in Equation 2.19(b) do not exist in Eurocode 3 (ECS, 1997).
The numerical analyses were conducted to verify this optimised proposal using two
finite element programs, SAFIR and LENAS developed in Belgium and France,
respectively (Franssen et al, 1995; Talamona et al, 1997). The ideal model using one
quarter cross-section of hot-rolled I-section with nominal yield stresses of 235 MPa
and 355 MPa was analysed. The thermal elongation and material properties at
elevated temperatures were taken from Eurocode 3. The main hypotheses of the
modelling were that end boundary conditions were simply supported and column
imperfection was assumed as a sinusoidal member imperfection of H/1000 without
local imperfection. Triangular residual stress distribution of maximum values was
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-25
given by 0.3×fy if h/b >1.2 and 0.5×fy if h/b ≤1.2, where h and b are depth and width
of the cross sections. The constant values across the thickness of the web and the
flanges were assumed, and uniform temperature distribution around the section and
across the thickness was considered. The sections modelled in these programs did
not allow for local buckling effects.
Since the proposed Equations 2.19(a) and (b) were formulated based on the
observations from the numerical analysis in the range 400°C to 800°C, the results
drawn from the formula can be conservative for temperatures higher than 800°C and
lower than 400°C (Talamona et al, 1997). In the case of temperatures lower than
400°C, the test results would not be compatible with the results from the above
formulae. One of the reasons could be that the reduction factor of yield stress is
constant up to 400°C. Therefore Talamona et al. (1997) recommended that it was
preferable and more logical to interpolate reduction factors for yield stress in
temperature range of 100°C to 400°C.
Rankine approach
A simple analytical proposal based on the Rankine principle was derived to
determine the compressive strength of steel columns under fire by Toh et al (2000).
The Rankine formula is based on the interaction of strength and stability. The plastic
critical load Np that is based on the squashing or yield load in this equation is to
determine the strength of a column with an assumption of a pure plastic behaviour.
The elastic-buckling load Ne is to consider column stability based on the Euler curve.
This background theory was applied to columns under fire conditions and the
modified equation is as follows.
TeTpTc NNN ,,,
111+= (2.20)
where Nc,T = critical load at temperature T
Np,T = plastic collapse load at temperature T
Ne,T = maximum elastic-buckling load at temperature T
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-26
In this formula, the load eccentricity was considered to take the beam-column effect
into account. The temperature distribution within a member was assumed to be
uniform across the section and along the member height. Thermal restraint in the
longitudinal axis, initial imperfection, the effect of thermal gradient, residual stresses
and local buckling were simply ignored in this analysis. Three different support
conditions, namely, pinned-pinned, pinned-fixed and fixed-fixed, were investigated
because the elastic-buckling load is influenced by boundary conditions. The
maximum elastic-buckling load at temperature T, Ne,T, is obtained by the following
equation with an assumption that a column is perfectly straight.
2
2
,e
TTe l
IEN π=
where ET is the elasticity modulus of material at elevated temperatures
The plastic collapse load Np,T of a concentrically loaded column is simply yield load,
and for an eccentrically loaded column, Np,T is reduced by the bending moments
caused by eccentricity. However, Toh et al (2000) did not consider the local buckling
and distortional buckling effects in developing their design formula because the aim
of the design was based on hot-rolled steels. Therefore, this modification based on
the Rankine formula will provide overestimated results for cold-formed steel
compression members.
Critical temperature of steel members
Wong and Szafranski (2004) developed an elastic method for calculating the critical
temperatures of steel members under fire conditions considering the effects of both
static and thermal loadings. For the design of steel members in fire, the following
load ratio must be satisfied.
1,,
, ≤
∑
tdfi
dfi
RE
(2.21)
where Efi,d is the design effect of fire actions and Rfi,d,t is the corresponding design
resistance of steel members.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-27
Equation 2.21 was used to determine the critical temperature of steel members
subject to bending and deflection limit. Equation 5.21 can be rewritten to satisfy the
bending moment ratio (MT/M) at temperature T.
Tyy
TyTf
fM
M,
, Φ=≤ (2.22)
where fy,T is the yield strength at temperature T.
In Equation 5.22, MT can be divided into two sections i.e. static loading and thermal
loading as given in Equation 5.23.
MT = M*L + M*
T (2.23)
where M*L is determined from the static loading and M*
T from the thermal loading.
In Equation 5.23, M*T can be represented by the load vector PT generated from the
contribution of the thermal load. For each member with cross section area A and
coefficient of thermal expansion ψ, the thermal load vector PT is determined by
PT = EAψФT (2.24)
where ФT = ФE,T (T – T0), EET
TE =Φ , and T0 = ambient temperature
Hence for each member,
M*T = ФT M*
e (2.25)
where M*e is obtained from the results of EAψ.
By substituting Equations 5.23 and 5.25 into 5.22, the critical temperature of each
member is obtained as given in Equation 5.26.
ФT M*e + M*
L - Фy,T = 0 (2.26)
In a similar way, the critical temperature of steel members subject to deflection limit
can be obtained.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-28
Equations 5.27 presents the critical temperature of each member subject to deflection
limit.
∆=−+Φ
= )( 0,
TTeTE
L δδδ (2.27)
where δL is the deflection due to static loading at ambient temperature, δe is the
deflection due to the thermal loading, EAψ, and ∆ is the deflection limit.
Local buckling of steel plates at elevated temperatures
An investigation into the local buckling characteristics of flat hot-rolled steel plate
elements at elevated temperatures was carried out by Wong and Tan (1999). Using
the concept of effective width for plate design, formulation for the calculation of
effective width of plate elements at elevated temperatures was presented and its
variations using different codes were compared. A common practice to estimate the
maximum compressive strength is by the use of the effective cross-section through
the concept of effective width. For comparison purposes, three expressions of the
effective width ratio expressed as a function of the ratio of local buckling stress to
yield stress were adopted.
Ty
Tcr
T
Te
ff
bb
,
,, = (Von Karman) (2.28a)
−=
Ty
Tcr
Ty
Tcr
T
Te
ff
ff
bb
,
,
,
,, 22.01 (AISI) (2.28b)
Ty
Tcr
T
Te
ff
bb
,
,, 85.015.0 += (TNO Holland) (2.28c)
It is common to use the critical buckling stress fcr of a plate element from plate
bending theory to monitor the transition between buckling and yielding. Using the
values of k = 4, E20 = 200000 MPa and ν = 0.3, Equations 2.28(a) to (c) become
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-29
Ty
TE
T
Te
tbbb
,
,,
/54
φφ
= (VonKarman) (2.29a)
−=
Ty
TE
Ty
TE
T
Te
tbtbbb
,
,
,
,,
/5422.01
/54
φφ
φφ
(AISI) (2.29b)
Ty
TE
T
Te
bb
,
,, 135.015.0φφ
+= (TNO Holland) (2.29c)
TET
EE
,20
φ= , Tyy
Ty
ff
,20,
, φ=
The variation of the ratio TYTE ,, /φφ with temperature in accordance with different design codes is plotted in Figure 2.10.
Figure 2.10 Variation of TYTE ,, /φφ at Elevated Temperatures (Wong and Tan, 1999)
To examine the variation of the effective width ratio at elevated temperatures using
reduction factors regulated in the design codes of AS 4100 (SA, 1998), ASCE (Lie,
1992) and BS 5950 (BSI, 1995), Equations 2.29(a) to (c) have been plotted as shown
in Figure 2.11. Figure 2.11 shows that there is a significant difference in calculating
the effective width of a plate element when different effective width formulae are
used. However, Wong and Tan (1999) showed that a general trend of effective width
was to increase at elevated temperatures. Hence, the assumption that the local
buckling characteristics applied to plate at room temperature can also be adopted for
plates at elevated temperatures will give conservative results in design. However,
cold-formed steels may have different variation ratios of the elasticity modulus to the
yield strength at elevated temperatures when compared with hot-rolled steels. Due to
the different mechanical behaviour of cold-formed steels at elevated temperature, the
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-30
assumption that the local buckling characteristics applied to cold-formed steel
elements at ambient temperature can be adopted safely at elevated temperatures may
not be correct.
Figure 2.11 Comparison of Effective Width Formulae at Elevated
Temperatures (Wong and Tan, 1999)
Feng et al. (2003a and b) recently conducted an experimental and numerical study to
investigate the local buckling and distortional buckling behaviour of cold-formed
steel compression members with and without service holes. Their tests were carried
out based on the steady state test method. During the heating process, the specimens
were allowed for free thermal expansion without axial restraint. In their steady state
tests, temperature was increased up to pre-set temperature and then held the
temperature for a period of time to allow the furnace and specimen temperature to
reach equilibrium. They found that the furnace temperature was transferred to the
specimens after about 15 min even though there was an initial temperature difference
between the furnace and specimens.
(a) AS 4100 (b) ASCE
(c) BS5950
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-31
In their experimental study, only the final failure mode could be observed at elevated
temperatures because the tests were conducted in the furnace. From the failure
modes, it was found that the buckling mode of channel sections subject to axial
compression was similar to that at ambient temperature. However, in the case of
specimens with service holes, they observed that the specimens which were
predominantly governed by local buckling at lower temperatures (≤400°C) changed
to distortional buckling mode at higher temperatures (≥550°C). From the
observation, they concluded that regardless of the temperature increase, perforation
had a significant effect on the strength of cold-formed steel columns depending on
the thickness of the material and the location of the service hole.
Feng et al. (2003b) also conducted the finite element analyses to simulate the tested
specimens used by Feng et al. (2003a) and validated the numerical study by
comparing with the test results. In the numerical analyses, the mechanical properties
are an important parameter. Therefore, the gradual yielding behaviour was simulated
by using several stress-strain curve steps with different tangent modulus instead of
using the perfect elastic-plastic model commonly used at ambient temperature. From
the case study for the effect of initial imperfection undertaken by Feng et al. (2003b),
i.e., t, 0.1t and 0.01t, it was found that the effect of initial imperfection on the
ultimate strength was very small (less than 5%) and the numerical results agreed well
with the experimental results.
From the experimental and numerical results, Feng et al. (2003a and b) showed that
the use of current design rules considering the reduced yield strength and elasticity
modulus was adequate to predict the ultimate strength of cold-formed steel
compression members subject to local and distortional buckling effects. However
their study was limited to thicker cold-formed steels (1.2 mm to 2.0 mm) and low
strength steel (minimum yield strength of 350 MPa). Since thin (≤0.95 mm) and high
(minimum yield strength of 550 MPa) strength cold-formed steels are commonly
used with various applications in Australia and thus their structural behaviour may
be different, further research is required to prove the accuracy and reliability of
current design rules at elevated temperatures.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-32
Mechanical properties of cold-formed steels
As the mechanical properties of the yield strength and elasticity modulus at elevated
temperatures are the most important factor for fire safety design of steel structures,
an experimental research on mechanical properties under fire conditions was carried
out in Finland during the years 1994 to 1999 (Outinen, 1999; Outinen and
Makelainen, 2000).
Two types of test methods, namely, transient state and steady state test methods were
used in the tensile tests of cold-form steels with minimum yield strength of 350 MPa
and hot-rolled steel with minimum yield strength of 355 MPa and 420 MPa. The test
specimens were decided in accordance with the European Standard EN 10 002-5 (see
Figure 2.12), and the tests were conducted under stress rate control. The stress rate of
loading was 0.52 N/mm2 which caused a strain rate of 0.003/min to the specimens.
Figure 2.12 Specimen Dimensions
Figure 2.13 Tensile Test Set-up (Outinen, 1999)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-33
A test equipment consisting of heating oven, extensometer, temperature detecting
elements and temperature controller was used as seen in Figure 2.13. The gauge
lengths of the extensometer for hot-rolled and cold-formed structural steels were 25
mm and 50 mm, respectively. The air temperature and the specimen temperature
were measured separately during the test using temperature detecting elements.
The test results of mechanical properties for cold-formed structural steel S350GD+Z
are illustrated and compared with the reduction values in EC3-1.2 (ECS, 1995) and
BS5950 (BSI, 1995) in Table 2.10. The results showed that the reduction factors in
EC3-1.2 (ECS, 1995) and BS5950 (BSI, 1995) were slightly unconservative
compared with test results. It should also be noted that the values given in EC3-1.2
(ECS, 1995) as reduction factors of the elasticity modulus are based on hot-rolled
structural steel and the reduction values in BS5950 are based on cold-formed steel
with minimum yield strength of 275 MPa. Cold-formed steels commonly used in
Australia have higher minimum yield strength of 550 MPa than the steels tested and
used in European countries and the UK. It is therefore necessary to validate the
mechanical properties regulated in the current design standards for thin high strength
cold-formed steels.
Table 2.10 Reduction of Mechanical Properties at Elevated Temperature (S350GD+Z)
ET/E20 fy,T/fy,20 Test BS5950 Temperature
(°C) Test EC3-1.2 0.5% 1.5% 2.0% 0.5% 1.5% 2.0% 20 1.00 1.00 1.000 1.000 1.000 1.000 1.000 1.000100 0.90 1.00 - - - 0.973 1.000 1.000200 0.87 0.90 - - - 0.945 1.000 1.000300 0.62 0.80 0.840 0.952 0.996 0.834 0.949 1.000400 0.40 0.70 0.722 0.815 0.866 0.680 0.815 0.867500 0.36 0.60 0.470 0.532 0.555 0.471 0.556 0.590600 0.23 0.31 0.271 0.338 0.352 0.269 0.349 0.390700 0.10 0.13 - - - - - - 800 - 0.09 - - - - - -
The difference caused by the two test methods was also studied observing the results
from the transient state and steady state tensile tests by Outinen (1999). The stress-
strain curves of cold-formed steel S350GD+Z from steady and transient state tests at
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-34
temperatures 300°C, 400°C, 500°C and 600°C are plotted in Figure 2.14. Since the
transient state test method appears to be closer to realistic fire conditions, Outinen
preferred the use of the transient state test method to the steady state test method for
single element fire tests. However, as shown in Figure 2.14, the results obtained
from the transient state test agree reasonably well with those from the steady state
tests. It is still necessary to verify the accuracy of the steady state test method for
cold-formed steel compression members due to the different loading action causing
restraint of thermal expansion and buckling behaviour.
Figure 2.14 Stress-Strain Curves obtained from Steady and Transient State Tests (Outinen, 1999)
2.4 Method of Investigations
In general, two ways of investigations, tests and computing analysis, are used to
study the effects of fire on structural behaviour. The techniques of fire testing have
been developed over many years and are subject to vary as each test has different
focus to be analyzed. The new materials, building components and methods of
construction are being developed. The full-scale and small-scale fire tests under a
scenario are sometimes used. Figure 2.15 shows several kinds of full-scale and
small-scale fire tests with different purposes.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-35
(a) Fire spread test (b) Smoke in stairs shaft (1/7 scale)
(c) Full-scale office fire test
Since the full-scale fire tests are expensive and provide limited test data of
parametric studies required for research purposes, the simulated fire tests are useful
to investigate the behaviour of structural elements or the material properties with
different temperatures or load ratios. Therefore, this research deals with simulated
fire tests and computing numerical analyses.
2.4.1 Simulated fire tests
There are two types of testing methods commonly used for studying fire resistance
of steel structural members and material properties of steel; steady state test and
transient state test.
Figure 2.15 Full Scale and Small Scale Tests (Bennetts et al., 2000)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-36
Steady state test
The steady-state heat flow occurs when the temperature gradient within the material
settles down to a constant value. The rate of heat energy flowing through material
under steady-state condition can be expressed as follows.
Heat flow rate, tAW θξ ∆⋅⋅
= (2.30)
ζ = thermal conductivity of material
A = cross-sectional area
t = thickness
∆θ = thermal difference across the faces
For a composite material comprising several layers which have different thermal
conductivity and thickness, Equation 2.30 should be modified to take into account
these factors of the ratios. This problem is not considered in this report due to single
element investigation.
Figure 2.16 Temperature Effectiveness to Light and Heavy weight Materials
(Shields, 1987)
Both the steady and transient temperature variations are affected by the material
weight components. Figure 2.16 illustrates the effectiveness of temperature
variations to light and heavy weight materials. Light weight material reaches the
temperature plateau region earlier than heavy weight material does. Thus, the
temperature gradient of light weight material is steeper than that of heavy weight
material.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-37
The overall process of the steady state test method is that a specimen is first heated
up to a pre-selected temperature with a heating rate on a specimen of 10 Co /min to
20 Co /min. For cold-formed steels, a heating rate of 20 Co /min should be used due to
higher temperature effectiveness as shown in Figure 2.16. However, the heating rate
may not influence test results from the steady state test method due to constant
thermal diffusivity. After reaching the pre-selected temperature, the specimen is
loaded with a strain rate or a loading rate until it fails. The axial load versus axial
shortening curve can be directly obtained from test records.
Transient steady test
The fire process in ignition and growth phases is a transient process producing a
varying heat energy output. A general form of the transient heat flow process
occurred by the temperature variation is expressed as follows.
tx ∂
∂=
∂∂ θ
αθ 12
2
(2.31)
The numerical solution of Equation 2.31 can be achieved by finite differences
method with a suitable interval time. This method is used to solve a simple transient
heat flow problem.
In the transient state test method, a specimen is first loaded up to a planned load ratio
and subject to a constant load. Then the specimen is heated until the load approaches
zero under a constant temperature rate with a range of 10°C/min to 20°C/min. The
temperature versus axial shortening curves are plotted under the different loading
ratios during the tests. These results are then converted into load versus axial
shortening or stress-strain curves. One example of converting temperature-strain
curves to stress-strain curves is shown in Figure 2.17. The failure temperature is
considered as the critical temperature at the loading ratio. The transient state test is
also called the failure temperature test.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-38
Figure 2.17 Converting Temperature Strain Curves to Stress-Strain Curves (Outinen, 1999)
ISO 834 (1999) and ISO/TR 13387 (1999) provide the standard time-temperature
curve for fire tests based on the transient state test method. The time-temperature
curve is given by
( )18log345 1020 ++= tTT (2.32)
Figure 2.18 Steel Temperature Corresponding to Standard
The international standards for fire resistance tests do not specify the construction of
a furnace in detail. This causes some potential problems when making comparison
between tests from different furnaces especially for small scale individual member
testing where the temperature on the surface of a specimen is highly related to the
mass in a furnace. Therefore, it is necessary that the specimen temperature be
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-39
measured during the simulated fire test. The heating rates are chosen by Aasen
(1985) to give a nominal steel temperature rise rate of about 20°C/min for most of
the steel tests. Twilt et al. (1994) and Lawson (2001) also showed a similar
increasing trend of steel temperature corresponding to the standard temperature rate.
2.4.2 Finite element analysis
The purposes of using a computer-aided analysis are to reduce the cost and to save
time for the structural behaviour investigation. Moreover, extensive parametric
studies with various factors can be effectively conducted with the aid of computing
packages. Most computer-aided analyses are today based on the finite element
method that is conceptually straightforward in that complex structures are idealised
by a network or meshes of simple interlocking structures. The use of the finite
element method to analyze structures requires great care due to various factors
affecting numerical solutions. There are a number of available finite element analysis
programs allowing thermal loading, buckling and non-linear analysis of shell
element structures. Some of the analysis and design tools are as follows:
ABAQUS (USA), ANSYS (USA), ASAS (Canada), BERSAFE (UK), ACASTEM
(France), CASTOR (France), FEMFAM (Germany), PUCK-2 (Italy) etc.
The use of the finite elements analysis (FEA) was recently introduced into the
investigation of the structural behaviour of cold-formed steel structures at elevated
temperatures. Ranby (1998, 1999) showed the applicability of FEA to the behaviour
of thin-walled steel members, verifying that FEA could closely predict experimental
results. Following his work, Kaitila (2002) further validated the use of FEA for the
prediction of the ultimate strength for low grade (minimum yield strength of 350
MPa) cold-formed steel columns integrated a light-framed wall with gypsum boards
attached on both sides. Feng et al. (2003b) also showed the suitability of FEA to
determine the capacity of cold-formed steel members at elevated temperatures,
considering proper material behaviour, element size and initial imperfection. In order
to obtain accurate results, Feng et al. (2003b and c) simulated mechanical behaviour
based on the measured yield strength and elasticity modulus in FEA models. This
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-40
method should be used because the mechanical properties have a significant role to
the ultimate strength of cold-formed steel members at elevated temperatures. The
details of mechanical behaviour in FEA are to be demonstrated in Section 5.3.1.
2.5 Literature Review Findings
Extensive literature reviews as described in the earlier sections have enabled the
accumulations of the required knowledge in the following topics: Fire test methods
for structural steel members, Current international and domestic standards,
Analytical design approaches based on Ramberg-Osgood model, Rankine approach,
Mechanical properties of steel under fire conditions, Fire safety design concept and
Computing analyses using finite element modelling.
Since the use of cold-formed steel sections in residential, industrial and commercial
buildings is rapidly increasing, the research of cold-formed steel structural members
under fire is highly required due to lack of knowledge of the behaviour of cold-
formed steel members. Therefore, many research projects on the behaviour of cold-
formed structural steel members are being carried out in many countries to
understand the structural responses of cold-formed steel structures against fire. The
development of design rules for the cold-formed steel structures would bring about
considerable reductions to the cost of fire protection and in saving of lives.
Following is a summary of the literature review reported in this chapter.
• Load bearing capacity under fire conditions can be attained by time based,
limiting (critical) temperature based and strength based designs. Since the
critical temperature varies in accordance with load ratio, the time based design is
rarely used. The limiting temperature specified in Eurocode (ECS, 1995) and
Australian standards (SA, 1996) is determined mainly based on the load ratio
expressed as a function of the strength capacity at elevated temperatures.
Therefore, it is essential that the behaviour of steel members under fire exposure
be understood and developed for the determination of load bearing capacity
based on strength based design.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-41
• There are two kinds of fire test methods currently used for structural members,
i.e. Steady state test and transient state test. ISO provides the time-temperature
curve for fire test. However, the effects of temperature on a specimen can be
varying because the international and domestic standards do not clearly define
the characteristics of fire test equipment. The transient state test method might be
closer to realistic temperature variation than the steady state test, but the results
obtained from the transient state test method are close to those from the steady
state test for tensile tests and the former test is more complicated to carry out and
obtained load displacement curves. The research on the simulated fire test should
be conducted to study the effect of using the different test methods on
compression tests at elevated temperatures.
• During fire conditions, the mechanical properties of steel, elasticity modulus and
yield strength drop rapidly above 400 °C due to high thermal conductivity and
diffusivity. These thermal characteristics affect hot-rolled and cold-formed steels
with different levels of degradation in the elasticity modulus and yield strength.
At present, no appropriate data of the mechanical properties is available for
commonly used thin, high strength cold-formed steels.
• A few international and domestic standards are available for fire safety design:
EC3 Part 1.2 (ECS, 1995), BS 5950 (BSI, 1995) and AS 4100 (SA, 1998). The
current standards mainly specify the reduction factors of the elasticity modulus
and yield strength for structural design under fire conditions. Strain levels of
0.5%, 1.5% and 2.0% for the yield stress are used in the British standards in
accordance with member classification. This should be investigated to determine
their effects on the structural performance under fire conditions since the typical
yield plateau in the stress-strain curve does not exist at high temperatures.
• The structural design specification for fire resistance is regulated only in
Eurocode (ECS, 1995). However, the design rules for the fire resistance of steel
structures do not include the local buckling effects. Since the local buckling on
cold-formed steel members results in a significant reduction of compressive
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 2-42
strength, this design rule is not applicable to cold-formed steel structural
members.
• An analytical formula for steel columns at elevated temperatures using EC3-1.2
(ECS, 1995) has been proposed by Talamona et al. (1997), and compressive
resistance of steel columns in fire using Rankine method has been recommended
by Toh et al. (2000). Further studies into the behaviour of steel members under
axial compressive force have been undertaken by other researchers (Janss, 1982,
Burgess, 1992, Poh and Bennetts, 1995, Neves, 1995). However, none of these
researchers considered the local buckling effects due to little effect of the local
buckling in hot-rolled steel structures.
• The use of finite element analysis is very effective to investigate the structural
behaviour of cold-formed steel members under fire conditions. The applicability
of FEA into cold-formed steel members at elevated temperatures was shown with
verification study. However, the effects of some factors such as true material
behaviour, residual stress and the strain-hardening at elevated temperatures
which may influence the column behaviour are unknown and those were simply
ignored or too simplified in the past research.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-1
Chapter 3. Mechanical Properties of Light Gauge Steels at Elevated Temperatures
The deterioration of the mechanical properties of yield strength and modulus of
elasticity is considered as the primary element affecting the performance of steel
structures under fire. However, the variation of mechanical properties for light gauge
and high strength steels at elevated temperatures is not known. Light gauge steels
with thicknesses less than 1 mm and yield strengths greater than 550 MPa are
commonly used in Australia. Therefore an extensive experimental study into the
deterioration of mechanical properties for both low and high strength light gauge
steels at elevated temperatures was conducted to derive accurate reduction factors of
mechanical properties and set the database as fundamental information for fire safety
design. This experimental study includes light gauge cold-formed steels with three
steel grades and six thicknesses from 0.4 mm to 1.2 mm. This chapter presents the
details of the experimental study, the results, the empirical equations for the
reduction factors and a stress-strain model at elevated temperatures.
3.1 Experimental Investigations
3.1.1 Test method
The simulated fire test currently used for studying the fire resistance of a structural
member can be categorized into three types of test methods; steady-state test,
transient-state test and ISO test (ISO/TR 13387-2, 1999) as discussed in Section
2.4.1. The transient state and ISO test methods are based on temperature variations
under a constant load. Due to the difficult process of plotting the stress-strain curves
from temperature-strain curves, approximate values of mechanical properties can not
be avoided. The steady state test method is based on a constant temperature under
increasing static loading. In this study, the steady state test method was therefore
used due to its simplicity and accurate data acquisition. Other researchers have also
used the steady state test method for the same reasons (Feng et al, 2003).
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-2
Some researchers consider the transient state tests to be capable of implicitly
including the creep effects which can occur in building fires. The creep effects are
time dependant and are influenced by both the applied load and temperature. Further,
both the steady state and transient state tests are usually completed within an hour
and thus include only a limited amount of creep behaviour. In the case of such a
short period of exposure of steel members to elevated temperatures, the effects of
creep strain may be negligible (Wang, 2002). Outinen (1999) conducted both the
steady state and transient state tests using zinc-coated low strength light gauge steels.
His results showed that the difference was very small. The tensile test results
obtained by Outinen (1999) are given in Section 3.3. Therefore it can be stated that
there is little difference between the two types of tests in relation to the creep effects.
In the case of light gauge steels tested in this study, the specimen temperature can be
considered the same as air temperature since the steels are very thin (≤ 1.2mm) and
have a high thermal conductivity. However, adequate time was allowed in the tests
so that the specimens reached the air temperature.
3.1.2 Test specimens
The dimensions of tensile test specimens were determined in accordance with AS
1391 (SA, 1991) as shown in Figure 3.1. There were two holes at both ends of each
specimen to enable fixing to the loading shafts located at the top and bottom ends of
a furnace. These holes were designed so that they did not affect the specimen
fracture that occurred in the middle of the specimen. Preliminary trial tests were
conducted at ambient temperature to confirm that the holes did not affect the
specimen behaviour and to achieve deformation and fracture in the required zone.
Figure 3.1 Dimensions of Tensile Test Specimens
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-3
AS1391 (SA, 1991) specified that for tensile testing test pieces shall be cut so that
longitudinal axes are either at right angle or parallel to the direction of final rolling
or extrusion. Using tensile coupon tests, Rogers and Hancock (1998) reported that
the yield and ultimate strengths of G550 steels obtained from specimens cut along
the longitudinal direction were significantly lower than the values from specimens
cut along the transverse direction. However, the test results obtained from G250
steels showed that the cutting direction did not appear to have any significant
influence on the material properties. Therefore all the specimens tested in this study
were cut along the longitudinal direction.
Table 3.1 Details of Test Specimens
Note: * Mean value based on three specimens Temperatures are 20, 100 to 800°C at 100°C intervals (9 cases)
Table 3.1 presents the details of test specimens including the key parameters of steel
grade, thickness and temperature. The specimens were made of three steel grades,
G550, G500 and G250 (minimum yield stress of 550, 500 and 250 N/mm2 (MPa),
respectively), with nominal thicknesses of 0.4, 0.6, 1.0 and 1.2 mm. This resulted in
seven cases (see Table 3.1) with nine different temperatures and three tests per each
case. A total of 189 tests was therefore undertaken in this study.
All the specimens were tested first at ambient temperature to set the data base of the
materials used in this study and the values of the yield strength, the modulus of
elasticity, coating and base metal thicknesses were measured. The measured yield
stresses of the steels obtained on the basis of the 0.2% offset proof stress method and
the elasticity moduli at ambient temperature are given in Table 3.1. The coating and
SteelGrade
Minimum yield
strength (MPa)
Nominal thickness
(mm)
Measured coating
thickness (mm)
Measured base metal thickness
(mm)
Measured yield
strength* (MPa)
Measured elasticity modulus*
(MPa) 0.4 0.013 0.404 345 210200 0.6 0.013 0.564 335 220000 G250 250 1.0 0.010 0.964 317 219000
0.42 0.027 0.409 722 223800 0.6 0.027 0.592 715 224500 G550 550
0.95 0.024 0.936 636 220500 G500 500 1.2 0.024 1.148 585 220900
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-4
base metal thicknesses are also included in Table 3.1. The total thickness of each
specimen was measured accurately using a micrometer. A special coating thickness
gauge was then used to measure the thickness of the zinc/aluminium coated surface.
The base metal thickness was finally obtained from the above measurement and used
to determine the yield strength and elasticity modulus at varying temperatures.
Cold reducing and cold forming processes are used to increase the strength of the
steels used in this study. Steel properties are controlled by heat treatment. G250
sheet steels are fully recrystallised whereas G500 and G550 steels are stress relief
annealed and further processed through a tension levelling mill to improve the finish
quality (Rogers and Hancock, 1996). Chemical properties of these steel grades as
specified by the manufacturer are as follows: carbon 0.12%, phosphorus 0.03%,
manganese 0.5% and sulphur 0.035% for G250 steels, and carbon 0.2%, phosphorus
0.04%, manganese 1.2% and sulphur 0.03% for G500 and G550 grade steels.
3.1.3 Test rig and procedure
All the tensile tests at both ambient temperature and elevated temperatures were
conducted in the Structures Laboratory in the School of Civil Engineering,
Queensland University of Technology. A specially made electrical furnace heated by
glow bars was built for the simulated fire tests to determine the mechanical
properties of light gauge steels (see Figure 3.2). A hydraulic actuator was used to
apply the tension load to the specimens. A load cell was used to measure the applied
tension load whereas the elongation of the specimen was measured in the middle of
the specimen using a specially modified extensometer with a gauge length of 15 mm
as shown in Figure 3.2. The two thermocouples located at the top and bottom of test
specimens showed that the temperatures were evenly distributed throughout the
specimens because of shorter specimen length and associated rapid transfer of heat
within the thin specimens.
The simulated fire tests were carried out based on the steady state test method in
which the specimen was electrically heated up to a pre-selected temperature and then
loaded until it failed while maintaining the temperature. The elevated temperature
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-5
range selected in this study was 100°C to 800°C at intervals of 100°C with a heating
rate of 15~20°C/min from ambient temperature.
Figure 3.2 Test Rig
Figure 3.3 Test Set-up and Data Acquisition
Hydraulic Actuator
Load Cell
Furnace
Glow Bars
Specimen
Extensometer
Thermo- couples
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-6
As the yield strength of steels dropped to less than 10% at temperatures above
800°C, the tests were not conducted beyond 800°C. The heating rate was initially set
to 20°C/min, but was automatically reduced to about 15°C/min to minimise the
overshooting past the pre-selected temperature. In this study, thermal expansion of
specimens was allowed by maintaining a zero tension load during the heating
process. After reaching the pre-selected elevated temperature, the temperature was
maintained while applying the tension load to each specimen until loading failure.
Figure 3.3 shows the arrangement of the overall test set-up including the data
acquisition system.
3.2 Deterioration of Mechanical Properties with Increasing
Temperatures
3.2.1 Determination of yield strength and elasticity modulus
Although the 0.2% proof stress method or the 0.5% total strain method are
commonly used to determine the yield strength at ambient temperature, different
strain levels of 0.2, 0.5, 1.0, 1.5 and 2.0% are also adopted for determining the yield
strength at elevated temperatures. In the 0.2% proof stress method, the yield strength
is the stress value at the intersection of the stress-strain curve and the 0.2% strain
proportional line (see Figure 3.4(a)). The same process is used to obtain the yield
strength at 0.2% strain level at elevated temperatures. Meanwhile, the yield strengths
at other strain levels are those corresponding to the stress values at the intersection of
the stress-strain curve and the non-proportional line specified by the strain levels as
shown in Figure 3.4(a).
The primary reason for adopting 0.5, 1.5 and 2.0% strain levels at elevated
temperatures is that unlike the stress-strain behaviour at ambient temperature, the
distinct plateau zone is not present at high temperatures. Due to large strain and
material weakening, the stress-strain curve is gradually rounded out at elevated
temperatures (see Figure 3.4(a)). Therefore the use of the 0.2% proof stress might
lead to conservative reduction factors at high temperatures. However, the yield
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-7
strength at high strain levels have to be used with great care due to structural
instability problems under fire conditions. The British Standard BS5950 (BSI, 1990)
provides the yield strength reduction factors at three strain levels of 0.5, 1.5 and
2.0%. It allows the use of 1.5 and 2.0% strain levels in a limited way when structural
stability remains intact at these levels of stain.
(a) Yield strength (b) Elasticity modulus
Figure 3.4 Determination of Mechanical Properties
In this study, the yield strengths at four strain levels of 0.2, 0.5, 1.5 and 2.0% were
obtained for the purpose of comparison. However the values determined by the 0.2%
proof stress method were used in deriving the empirical equations for the reduction
factors of the yield strength at elevated temperatures since the practice of using other
strain levels has not been clearly defined and accepted widely.
The modulus of elasticity was determined from the stress-strain curve based on the
tangent modulus of the initial elastic linear curve as used at ambient temperature (see
Figure 3.4(b)).
In this experimental study, the test specimens were under load for a slightly smaller
duration compared with the load duration if a transient state test method was used.
Hence this study using steady state tests might have given slightly higher strengths.
However, Outinen (1999) performed both steady state and transient state tests and
his results showed that the difference between the results from these two methods is
σ1.5
σ2.0
σ0.2 σ0.5
Stre
ss, σ
Strain, % 0.2 0.5 1.5 2.0
Stre
ss, σ
Strain, ε
E= tan α= εσ∆∆
α
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-8
small for light gauge cold-formed steels. On average, it was about 5% in the
temperature range from 300 to 600°C with hardly any difference at high
temperatures.
3.2.2 Yield strength
Table 3.2 presents the reduction factors showing the deterioration of yield strength of
light gauge steels which are commonly used in the construction industry at elevated
temperatures. The reduction factors (fy,T/fy,20) were determined from the ratios of
yield strength at elevated temperatures to that at ambient temperature (20°C)
corresponding to four strain levels 0.2, 0.5, 1.5 and 2.0%. As discussed in the earlier
section, the 0.2% values in Table 3.2 were based on yield strengths determined using
the 0.2% proof stress method, however, the other yield strength values based on 0.5,
1.5 and 2.0% strains were obtained non-proportionally at 0.5, 1.5 and 2.0% strains.
Figures 3.5(a) and (b) show a typical set of stress-strain curves at low and high
temperatures and the determination of the yield stress corresponding to different
strain levels. As clearly seen in Figure 3.5(b), cold-formed steels contain different
stress-strain behaviour at high temperatures including large strain and less well-
defined yielding point. The yield stresses corresponding to 1.5 and 2.0% strain levels
are close to the ultimate stress for cold-formed steels. Therefore, the use of 0.2%
proof stress is recommended at not only ambient temperature, but also at elevated
temperatures.
Based on the yield strength results obtained from 189 tensile tests, empirical
equations were generated to determine the yield stresses of the large range of cold-
formed light gauge steels at elevated temperatures. Equations 3.1(a) to 3.1(d) show
the simplified empirical formulae for the reduction factors of the yield strength at
temperature T, i.e. fy,T/fy,20 where fy,T and fy,20 are the 0.2% proof stresses at elevated
temperature T and ambient temperature 20°C. Figure 3.6 compares the predictions
from Equations 3.1(a) to 3.1(d) with the test results and a good agreement can be
seen.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-9
0.42 mm G550 steel 0.6 mm G550 steel Temp. (°C) 0.2 % 0.5 % 1.5 % 2.0 % 0.2 % 0.5 % 1.5 % 2.0 % 20 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
100 0.993 0.993 0.979 0.981 0.986 0.986 0.986 0.986 200 0.977 0.907 0.965 0.981 0.924 0.894 0.947 0.975 300 0.872 0.866 0.885 0.893 0.873 0.845 0.944 0.962 400 0.821 0.781 0.861 0.861 0.774 0.701 0.833 0.888 500 0.652 0.601 0.699 0.736 0.578 0.549 0.662 0.726 600 0.491 0.411 0.531 0.597 0.464 0.394 0.493 0.569 700 0.297 0.163 0.279 0.355 0.239 0.187 0.261 0.305 800 0.127 0.099 0.139 0.167 0.141 0.056 0.131 0.167
0.95 mm G550 steel 1.2 mm G500 steel Temp. (°C) 0.2 % 0.5 % 1.5 % 2.0 % 0.2 % 0.5 % 1.5 % 2.0 % 20 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
100 0.988 0.988 0.984 0.984 0.982 0.982 0.976 0.983 200 0.969 0.940 0.953 0.953 0.941 0.905 0.949 0.949 300 0.893 0.861 0.931 0.931 0.877 0.877 0.898 0.915 400 0.797 0.711 0.828 0.890 0.842 0.771 0.864 0.905 500 0.622 0.590 0.656 0.711 0.719 0.676 0.749 0.762 600 0.478 0.452 0.500 0.563 0.491 0.386 0.532 0.559 700 0.267 0.197 0.312 0.342 0.313 0.253 0.345 0.362 800 0.127 0.095 0.135 0.164 0.122 0.071 0.118 0.169
0.4 mm G250 steel 0.6 mm G250 steel 1.0 mm G250 steel Temp. (°C) 0.2 % 0.5 % 1.5 % 2.0 % 0.2 % 0.5 % 1.5 % 2.0 % 0.2 % 0.5 % 1.5 % 2.0 % 20 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
100 0.971 0.971 0.945 0.972 0.971 0.971 0.985 0.985 0.987 0.987 0.984 0.984 200 0.915 0.911 0.917 0.931 0.961 0.952 0.953 0.953 0.935 0.944 0.953 0.953 300 0.898 0.898 0.897 0.923 0.889 0.889 0.862 0.892 0.879 0.879 0.885 0.901 400 0.754 0.783 0.833 0.861 0.794 0.794 0.802 0.826 0.788 0.801 0.824 0.849 500 0.600 0.642 0.667 0.694 0.581 0.594 0.615 0.646 0.608 0.615 0.634 0.644 600 0.435 0.435 0.472 0.527 0.413 0.413 0.447 0.492 0.427 0.411 0.439 0.482 700 0.202 0.236 0.305 0.333 0.253 0.253 0.309 0.323 0.212 0.223 0.264 0.301 800 0.104 0.115 0.167 0.202 0.205 0.186 0.228 0.246 0.144 0.153 0.183 0.217
Table 3.2 Reduction Factors of Yield Strength (fy,T/fy,20 )
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-10
(a) 20 °C
(b) 800 °C
Figure 3.5 Typical Stress-strain Curves at Low and High Temperatures
and Yield Strength Determination (0.95mm-G550)
0.120,
, =y
Ty
ff
20°C ≤ T < 100°C (3.1a)
3926
20,
, 10969.11008.300045.0964.0 TTTff
y
Ty −− ⋅+⋅−+= 100°C ≤ T ≤ 350°C (3.1b)
Strain, %
MPa
0
50
100
150
200
0 0.5 1 1.5 2 2.5 3
Stre
ss
0.2
Strain, %
MPa
0
100
200
300
400
500
600
700
800
0 0.5 1 1.5 2 2.5 3
Stre
ss
0.2
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-11
72.40144.0514.1 5/1
20,20,
,
+×
−=yy
Ty
fT
ff
400°C ≤ T ≤ 750°C (3.1c)
1.020,
, =y
Ty
ff
T = 800°C (3.1d)
Note: Use linear interpolation for 350°C < T < 400°C and 750°C < T < 800°C.
fy,20 in Equation 3.1(c) is in MPa.
Figure 3.6 Comparison of Yield Strength Reduction Factor from Equations 3.1 and 3.2 with Test Results
Alternatively, the following equation can be safely used for both low and high
strength light gauge steels in the temperature range from 20°C to 800°C. As seen in
Figure 3.6, Equation 3.2 provides a lower bound to most of the test data.
4123826
20,
, 109.7101020004.00065.1 TTTTff
y
Ty −−− ⋅+−⋅+−=
20°C ≤ T ≤ 800°C (3.2)
0
0.2
0.4
0.6
0.8
1
1.2
0 100 200 300 400 500 600 700 800 900
Eqs. 3.1(a)-(d)
Eq. 3.2
0.42(G550)
0.6(G550)
0.95(G550)
1.2(G500)
0.4(G250)
0.6(G250)
1.0(G250)
fy,T/fy,20
°C
G550
G250
Temperature
Red
uctio
n fa
ctor
s of y
ield
stre
ngth
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-12
0.00.20.40.60.81.01.21.4
0.2% 0.5% 1.5% 2.0%
0.42-G550
0.6-G5500.95-G550
1.2-G5000.4-G250
0.6-G2501.0-G250
f y (0
.5,1
.5, 2
.0%
)
f y
(0.2
%)
The yield strength reduction factors were in general determined as a function of
temperature (Olawale and Plank, 1988; Outinen et al., 1997; SA, 1998). Temperature
is the major element causing the degradation of yield strength of steels. However, it
was found that there are some differences of about 10% in the yield strength between
the low and high steel grades in the temperature range of 400°C to 750°C. The G500
and G550 steels gave better efficiencies than the G250 steels (see Figure 3.6). This
improved result might have been caused by the possible recovery of ductility for the
G500 and G550 high strength steels at elevated temperatures. Therefore, Equations
3.1(b) and (c) include the yield strength of steel (fy,20) in addition to the temperature
(T) in order to improve the accuracy in predicting the reduction factors. The test
results showed that the deterioration of yield strength in relation to various
thicknesses is reasonably consistent. Thus, the influence of thickness (0.4 ~ 1.2mm)
on the deterioration of yield strength can be considered negligible (< 5%).
Figure 3.7 Variation of Yield Strength at Different Strain Levels
Figure 3.7 shows the effect of using different strain levels of 0.5, 1.5 and 2.0%. The
ratios of reduction factors for yield strength at the strain levels of 0.5, 1.5 and 2.0%
to that using conventional 0.2% proof stress method were calculated and plotted in
Figure 3.7. As seen in Figure 3.7, the 0.5% strain based yield strengths are on
average 5.7% lower than those based on the 0.2% proof stress method whereas the
1.5 and 2.0% strain level based yield strengths are on average 11.2 and 16.5% higher
than the 0.2% values, respectively.
Ave. 1.000 0.943 1.112 1.165
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-13
3.2.3 Elasticity Modulus
Elevated temperature also leads to the deterioration of elasticity modulus. As for the
yield strength, the modulus of elasticity is also an important factor determining the
buckling behaviour of thin-walled steel members. As for the yield stress, elevated
temperatures rapidly transfer materials to the inelastic deformation associated with
large strain and significantly reduce the elasticity modulus. Table 3.3 presents the
reduction factors (ET/E20) for elasticity modulus at elevated temperatures. They were
determined as a ratio of elasticity modulus at elevated temperatures (E20) to that at
the ambient temperature of 20°C (E20).
Table 3.3 Reduction Factors of Elasticity Modulus (ET/E20 )
Evaluations of test results showed that the reduction factors obtained from the G550
and G500 steels are slightly lower than those from the G250 steels as shown in
Figure 3.8. This result might have been due to the variation of carbon content in the
steels which affects the structure and properties of steels at high temperatures
(Jackson and Dhir, 1988). However, the difference of carbon content between G250
and G550 steels is quite small. The carbon content guaranteed by the manufacturer is
0.12% and 0.2% for G250 and G550 (or G500) steels, respectively. The effect of
chemical components was not investigated in this study. However, the discrepancy
in the reduction factors amongst the steel grades and thicknesses is less than 5%
except at 500°C. Therefore, the influence of steel grade and thickness on the
degradation of elasticity modulus was not taken into account in the equations
determining the reduction factors at elevated temperatures. The calculated values
(°C)
0.42-G550
0.6-G550
0.95-G550
1.2-G500
0.4-G250
0.6-G250
1.0-G250
20 1.000 1.000 1.000 1.000 1.000 1.000 1.000 100 1.000 0.982 1.000 0.967 0.98 0.99 0.977 200 0.894 0.882 0.902 0.878 0.891 0.9 0.887 300 0.702 0.675 0.682 0.661 0.721 0.754 0.734 400 0.583 0.541 0.564 0.521 0.608 0.599 0.624 500 0.473 0.469 0.411 0.414 0.481 0.521 0.529 600 0.238 0.243 0.275 0.208 0.247 0.293 0.29 700 0.114 0.119 0.123 0.154 0.172 0.172 0.17 800 0.072 0.042 0.068 0.049 0.092 0.085 0.098
TempSteel
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-14
from Equations 3.3(a) to (c) are compared with the test results in Figure 3.8, and a
good agreement is seen.
0.120
=EET 20°C ≤ T ≤ 100°C (3.3a)
)100(0014.0120
−−= TEET 100°C < T ≤ 500°C (3.3b)
203.03.000122.0
12001
20
−+
−=
T
T
EET 500°C < T ≤ 800°C (3.3c)
Figure 3.8 Comparison of Elasticity Modulus Reduction Factors from Equations 3.3(a) to (c) with Test Results
Figure 3.9 shows the typical failure modes of tensile test specimens at different
temperatures from 20 to 800°C. The failure modes changed from a brittle fracture at
45° to ductile fracture across the specimen width. At the temperature of 800°C the
base metal tended to be slightly melted and seriously softened, and hence the
specimens yielded rather than fracturing at 800°C. Inspection of failed test
specimens showed that the zinc and aluminium coating started to scale off at the
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 100 200 300 400 500 600 700 800 900
Eqs. 3.3(a)-(c)
0.42(G550)
0.6(G550)
0.95(G550)
1.2(G500)
0.4(G250)
0.6(G250)
1.0(G250)
°C
Red
uctio
n fa
ctor
s of e
last
icity
mod
ulus
Temperature
ET/E20
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-15
temperature of 400°C and completely scaled off at the temperature of 700°C as seen
in Figure 3.9.
Figure 3.9 Failure Modes of Tensile Specimens at Temperatures of 20 to 800°C
3.3 Comparison of Yield Strength and Elasticity Modulus
Predictions
The British Standard BS 5950 (BSI, 1995) provides the reduction factors for the
yield strength of light gauge steels at three strain levels of 0.5, 1.5 and 2.0% up to
600°C, but the reduction factors for the elasticity modulus are not stipulated. BS
5950 allows the use of reduction factors based on 1.5 and 2.0% strain levels with the
requirement that the structural stability must remain intact at elevated temperatures.
The European Standard (ECS, 1995) and Australian Standard (SA, 1998) also define
the reduction factors for the yield strength and elasticity modulus. The reduction
factors of yield strength and elasticity modulus obtained from this research are
compared with those given in the above mentioned design standards. Even though
the reduction factors specified in EC 3.1-2 (ECS, 1995) and AS 4100 (SA, 1998) are
based on hot-rolled steels, they are also plotted in Figures 3.10 and 3.11.
Outinen (1999) carried out simulated fire tests based on both the steady state test
(temperature range of 300°C to 600°C) and transient state test methods to investigate
800 700 600 500 400 300 200 100 20°C
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-16
the deterioration of light gauge steel of S350GD+Z (minimum yield strength of 350
MPa). The test conducted with S350GD+Z showed that the steady state test method
gave larger yield strength reduction factors in the temperature range of 300°C to
600°C (see Figure 3.10(a)). Since the strength reduction factors at 0.2% strain level
from the transient state test are too conservative, he recommended the use of
reduction factors based on 2.0% strain level.
(a) Reduction Factors at 0.2% Strain Level
*SS and TS indicate the steady state and transient state test, respectively
(b) Reduction Factors at 0.5% Strain Level
0
0.2
0.4
0.6
0.8
1
1.2
0 100 200 300 400 500 600 700 800 900
BS 5950
0.42(G550)
0.6(G550)
0.95(G500)
1.2(G500)
0.4(G250)
0.6(G250)
1.0(G250)
Temp.(°C)
Red
uctio
n fa
ctor
s of y
ield
stre
ngth
fy,T/fy,20
Figure 3.10 Comparison of the Variation of Yield Strength at Elevated Temperatures with Current Steel Design Rules
0
0.2
0.4
0.6
0.8
1
1.2
0 100 200 300 400 500 600 700 800 900
Eqs. 3.1(a)-(d)
AS 4100
Outinen (TS,2001)Outinen (SS,2001)0.42(G550)
0.6(G550)
0.95(G550)
1.2(G500)
0.4(G250)
0.6(G250)
1.0(G250)
Red
uctio
n fa
ctor
s of y
ield
stre
ngth
fy,T/fy,20
Temp.(°C)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-17
(c) Reduction Factors at 1.5% Strain Level
(d) Reduction Factors at 2.0% Strain Level Figure 3.10 Comparison of the Variation of Yield Strength at Elevated Temperatures with Current Steel Design Rules
0
0.2
0.4
0.6
0.8
1
1.2
0 100 200 300 400 500 600 700 800 900
BS 5950
0.42(G550)
0.6(G550)
0.95(G500)
1.2(G500)
0.4(G250)
0.6(G250)
1.0(G250)
Temp.(°C)
Red
uctio
n fa
ctor
s of y
ield
stre
ngth
fy,T/fy,20
Temp.(°C) 0
0.2
0.4
0.6
0.8
1
1.2
0 100 200 300 400 500 600 700 800 900
BS 5950
EC3
0.42(G550)
0.6(G550)
0.95(G500)
1.2(G500)
0.4(G250)
0.6(G250)
1.0(G250)
Red
uctio
n fa
ctor
s of y
ield
stre
ngth
fy,T/fy,20
Temp.(°C)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-18
As seen in Figures 3.10(b) to (d), the comparison of test results shows that the
reduction factors of yield strength in BS 5950 (BSI, 1995) are conservative at
temperatures above 400°C. The yield strength reduction factors given in EC 3.1-2
(ECS, 1995) appear to be overestimated as expected (see Figure 3.10(d)). However,
the factors at temperatures above 600°C are lower than those determined for the light
gauge steels in this study. Unlike BS 5950 and EC 3.1-2, the reduction factors of
yield strength in AS 4100 are based on the 0.2% proof stress method (SA, 1998;
Proe et al., 1986). There is no reduction to yield strength up to 215°C and beyond
this temperature the yield strength decreases linearly up to 905°C according to AS
4100 rules (see Figure 3.10(a)). The comparison of the test results with the reduction
factors of yield strength in AS 4100 (SA, 1998) shows that AS 4100 provides
slightly conservative values at the middle temperatures from 400°C to 500°C, but
unconservative values at high temperatures above 700°C for light gauge steels.
Figure 3.11 Comparison of the Variation of Elasticity Modulus at Elevated
Temperatures with Current Steel Design Rules
The moduli of elasticity at elevated temperatures specified in AS 4100, EC 3.1-2 and
the experimental values obtained by Outinen (1999) are compared with the test
results in Figure 3.11. The comparison indicates that the reduction factors in AS
4100 and EC 3.1-2 are too high for light gauge steels. The reduction factors obtained
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 100 200 300 400 500 600 700 800 900
Eq. 3.3
AS4100
EC3
Outinen
0.42(G550)
0.6(G550)
0.95(G550)
1.2(G500)
0.4(G250)
0.6(G250)
1.0(G250)Red
uctio
n fa
ctor
s of e
last
icity
mod
ulus
Temp.(°C)
ET/E20
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-19
by Outinen (1999) are slightly conservative when compared with the test results.
This is due to the lower bound elasticity modulus values from his transient state
tests. Outinen (1999) therefore proposed the same reduction factors as those in EC
3.1-2 for the elasticity modulus of structural steel S350GD+Z.
In summary, the reduction factors given in the current steel design standards for
thicker hot rolled steels are unconservative for light gauge cold-formed steels. The
strength degradation also commences at lower temperatures for these light gauge
steels. These observations and results may be due to the special characteristics of
light gauge steels including their manufacturing process and chemical contents
associated with thin materials. High shape factor expressed as t he ratio of perimeter
to the cross-sectional area also contributes to the degradation of the mechanical
properties. The cold-forming process enhances the yield strength at ambient
temperature, but this gain in strength is lost at elevated temperatures.
3.4 Stress-strain Model
From the test results, a stress-strain model showing the non-linear elastic-plastic
relationship of light gauge steels with temperature was developed based on a
Ramberg-Osgood model (Ramberg and Osgood, 1943). In the past, a stress-strain
relationship at elevated temperatures was developed by Olawale and Plank (1988)
for hot-rolled steels and by Outinen et al. (1999, 2000) for a cold-formed steel S355
steel (minimum yield strength of 355 MPa). Olawale and Plank (1988) modified the
Ramberg-Osgood law by replacing the normally used constant 0.002 with β(fy,T/ET)
as seen in Equation 3.4. This allowed them to include the variation of mechanical
properties with increasing temperature. However the parameter β in Equation 3.4
was taken as a constant value of 3/7 and 6/7 by Olawale and Plank (1988) and
Outinen et al. (1999, 2000), respectively. The other parameter n which decides the
slope of the inelastic zone and the plastic stress in the stress-strain curve was
determined as a function of the temperature by both researchers. Their n values
varied from 4 to 14 for a temperature range 300 to 800°C.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-20
n
Ty
T
T
Ty
T
TT f
fEf
Ef
+=
,
,βε (3.4)
where εT is the strain at temperature T
fT is the stress at temperature T
fy,T is 0.2% proof stress at temperature T
ET is the elasticity modulus at temperature T
In this study, a different approach was used to determine an accurate stress-strain
model for light gauge steels at elevated temperatures. Experimental results showed
that the nonlinear inelastic region of stress-strain curves displayed a similar slope in
the temperature range of 300°C to 800°C. This observation is different to other
metals that had varying slopes in the inelastic region. For such cases, the parameter n
was varied to accurately simulate the stress-strain curves. Therefore in this case a
constant n value of 15 was used although the original Ramberg and Osgood
definition of n (=
T
Ty
ff
,1.0
,ln
2ln , where f0.1,T is the 0.1% proof stress at temperature T)
gave a value in the range of 3 to 15 for the experimental stress-strain curves obtained
here.
To demonstrate the effects of parameters n and β on the stress-strain curves, four
values of n and β were used in this study. Figures 3.12 and 3.13 show the effect of
parameters n and β on the stress-strain curves and compare with an experimental
stress-strain curve for 0.42mm G550 at 500°C. It is seen in Figure 3.12 that as the
value of n decreases the slope in the inelastic region increases. The stress-strain
curve for n = 15 provides a reasonably accurate slope and a lower bound to the
stress.
In Figure 3.13, the effect of β on the stress-strain curve is demonstrated. As seen in
Figure 3.13, the parameter β has no effect on the slope of the stress-strain curve but
affects the stress in the inelastic region. Therefore in order to model the varying
inelastic stress levels with temperature, suitable values of β were chosen. These
values are given in Table 3.4. The stress-strain relationship in the elastic region at
any temperature T is decided by the function of fT/ET in Equation 3.4. The yield
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-21
stress and the elasticity modulus at temperature T (fy,T, ET) to be used in Equation
3.4 can be obtained from Equations 3.1 to 3.3.
Figure 3.12 Effect of Parameter n on Stress-Strain Curve
Figure 3.13 Effect of Parameter β on Stress-Strain Curve
0
100
200
300
400
500
600
700
800
0 0.5 1 1.5 2 2.5 3
n = 5 n = 10 n = 15
n = 25
β = 0.45
0.42-G550 (500°C)
Stre
ss (M
Pa)
Strain (%)
0
100
200
300
400
500
600
700
800
0 0.5 1 1.5 2 2.5 3
n = 15
β = 0.45
0.42-G550 (500°C)
β = 1.5 β = 3.0
β = 0.1
Stre
ss (M
Pa)
Strain (%)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-22
Table 3.4 Coefficients β for Equation 3.4
Temp. (°C) 20 ~ 300 400 500 600 700 800 β 3.5 0.8 0.45 0.1 0.02 0.001
Figure 3.14 Stress-strain Curves from Equation 3.4 and Tests
0
100
200
300
400
500
600
700
800
0 0.5 1 1.5 2 2.5 3
Stre
ss (M
Pa)
Strain (%)
G550-0.42 (700°C)
G550-0.42 (500°C)
G550-0.42 (200°C) Test Eq. 3.4
(a) G550 steels
0
100
200
300
400
500
0 0.5 1 1.5 2 2.5 3
Stre
ss (M
Pa)
Strain (%)
G250-0.6 (700°C)
G250-0.6 (500°C)
G250-0.6 (200°C)
Test Eq. 3.4
(b) G250 steels
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-23
The comparison of the curves from the stress-strain model with test results shows a
good agreement for both low and high strength steels as shown in Figure 3.14.
Therefore, the modified stress-strain model given by Equation 3.4 can be used in the
temperature range of 20°C to 800°C for light gauge steels. However, it must be
noted that this model does not give the correct corner region for lower temperatures
(≤ 200°C). Even the high strength steels used in this study have almost a perfect
bilinear stress-strain relationship at lower temperatures, which can be modelled with
a very high n value (n > 200). Unless the n value is varied, this shortcoming cannot
be eliminated. Since assuming a constant n value of 15 affects only the corner region
of stress-strain curves at lower temperatures, it was considered acceptable.
3.5 Ductility Increase of High Strength Steels with Temperature
One of the advantages of using steel in building construction is its ability to
withstand limited deformation without fracture. In this regard, high strength light
gauge steels have lower failure strains when compared with low strength and mild
steels. This is a disadvantage of using high strength light gauge steels. Lower
ductility of high strength steels is caused by heat treatment associated with
recrystallisation to achieve high strength. Thus this can lead to a larger reduction
factor for the yield strength of thin high strength steels.
In the tensile coupon tests conduced at elevated temperatures in this study, an
interesting result was observed that high strength steels with lower ductility at
ambient temperature gained greater ductility at high temperatures as seen in Figure
3.15(a). Unfortunately, the extensometer used in this study could only measure
strains up to about 3.5%. Therefore although the failure strain at high temperatures
was considerably larger than that at low temperatures, all the failure strains in Figure
3.15(b) appeared to be the same. It is clearly seen from the stress-strain curves that
the ductility of high strength steels has significantly improved at elevated
temperatures. Therefore the lack of ductility may not be a disadvantage for light
gauge high strength steel structures at elevated temperatures and thus fire safety.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-24
Figure 3.15 Stress-strain Curves at Elevated Temperatures
(20, 100, 200, 300, 400, 500, 600, 700, 800 °C)
The improvement of ductility with increasing temperature cannot be seen in Figure
3.15(b) due to the limitation of strain measuring device. However Figure 3.16 shows
difference in the failure modes of high strength steels at ambient temperature and
high temperature of 800°C. Test results showed that a brittle shear failure occurred
0
100
200
300
400
500
600
700
800
0 0.5 1 1.5 2 2.5 3 3.5
Stre
ss (M
Pa)
Strain (%)
800°C
600°C
400°C
200°C
20°C
(a) 0.42mm – G550 steel
0
100
200
300
400
500
600
700
0 0.5 1 1.5 2 2.5 3 3.5
(b) 0.95mm – G550 steel
Stre
ss (M
Pa)
Strain (%)
800°C
600°C
400°C
200°C 20°C
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-25
for high strength steels at ambient temperature whereas the specimens yielded
without fracture at high temperatures as shown in Figure 3.16(b). From this
investigation, it can be stated that the ductility of high strength steels has
significantly improved at high temperatures. Other stress-strain curves obtained from
tensile tests are included in Appendix A.
Figure 3.16 Failure Modes at Different Temperatures
(brittle failure versus ductile failure)
3.6 Summary
This chapter has described an experimental investigation on the mechanical
properties of light gauge steels undertaken to determine such properties and to
develop suitable guidelines to predict the yield strength and elasticity modulus at
elevated temperatures. The investigation included three grades and six thicknesses of
light gauge steels. The reduction factors for yield strength and elasticity modulus
which are the most important parameters for the structural design under fire
conditions were determined from the tests. It was found that the yield strength in the
temperature range of 400°C to 750°C was influenced by steel grade, but no such
observation was found for the modulus of elasticity. Based on extensive
experimental results, suitable predictive equations were developed to determine the
yield strength and the elasticity modulus at temperatures from 20°C to 800°C by
taking into account the effects of steel grade and temperature. These equations can
(a) 20°C (brittle shear failure) (b) 800°C (ductile yielding failure)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 3-26
be used for the large range of cold-formed light gauge steels considered in this
investigation.
A stress-strain model based on a Ramberg-Osgood curve was developed for light
gauge steels at elevated temperatures. The comparison of the stress-strain curves
from the developed model with test results showed that the new stress-strain model
can be adopted for light gauge steels at elevated temperatures.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-1
Chapter 4. Local Buckling Behaviour of Unstiffened Elements at Ambient Temperature
The current design method (AISI, 1996; ECS, 1997; SA, 1996) for stiffened and
unstiffened elements of cold-formed steel members subject to local buckling effects
under axial compression is based on the same effective width rules regardless of
steel grades. Prior to investigating the local buckling behaviour of cold-formed steel
compression members at elevated temperatures, experimental and numerical studies
were conducted to investigate whether the conventional effective width approach is
adequate for unstiffened elements made of thin high strength steels (yield stress ≥
550 MPa) with low ductility. This chapter presents the details of the experimental
and numerical studies, a new proposal for the accurate prediction of effective widths
of unstiffened elements and the ultimate capacity of high strength cold-formed steel
compression members subject to local buckling effects. It also reviews the direct
strength method recently developed for cold-formed steel members.
4.1 General
Local and distortional buckling are the most common failure modes of short cold-
formed steel compression members. Since the elements of cold-formed steel
members are very thin, elastic local buckling occurs first. However, these elements
have considerable post buckling strength. The well known effective width concept
first developed by von Karman (1932) and experimentally modified by Winter
(1947) is commonly adopted in the American, Australian and European design codes
(AISI, 1996; ECS, 1997; SA, 1996) for predicting the ultimate strength of cold-
formed steel members subject to local buckling effects.
Past researches into the development of effective width design rules were based on
low strength and thicker steels (yield stress fy ≤ 350 MPa) and the same rules are
currently used in computing the strength of cold-formed steel members made of thin
high strength steels. Very thin (0.42 ≤ t ≤ 1.2 mm) and high strength steels are
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-2
commonly used in Australia with many applications in residential, industrial and
commercial buildings. However, it is not known whether the current effective width
design rules are adequate for thin high strength cold-formed steel members.
Therefore a series of experimental and numerical studies was conducted to
investigate the adequacy of the current effective width rules and other relevant
design rules using stub column tests and numerical analyses of unlipped channel
members with slender flanges. Effect of strain hardening, which is in general
ignored, was also investigated using numerical studies in this research. Moreover,
the direct strength method recently developed by Schafer and Pekoz (1998) and
Schafer (2002) to eliminate the complexity of the design procedures based on
effective width calculations of individual elements was also investigated to study its
accuracy for unlipped channel members subject to local buckling of unstiffened
flange elements under axial compression.
4.2 Effective Width Concept
The effective width concept based on the stress redistribution over a reduced plate
width was first derived by von Karman (1932). Assuming a uniform stress
distribution on a reduced width of stiffened plates, von Karman introduced the
effective width concept as given in Equation 4.1 to eliminate the complexity of using
the differential equations of large deflection theory. After the concept was introduced
a number of researchers (Heimerl, 1947; Schuette, 1947; Winter, 1947; Chilver,
1953; Bambach and Rasmussen, 2002) proposed improved effective width rules
expressed in terms of the ratio of the maximum stress at the supports to the elastic
buckling stress.
y
creff
bb
= (4.1)
where be/b = effective width/total flat width.
Winter (1947) developed effective width equations for stiffened and unstiffened
elements on the basis of experimental investigations by modifying the von Karman’s
formula. The effective width equations currently used in the American, Australian
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-3
and European design codes are based on the Winter’s proposal for stiffened
elements. His equations are as follows.
1=bbe for λ ≤ 0.673 (4.2a)
( ) 0.1/22.01≤
−=
λλ
bbe for λ > 0.673 (4.2b)
crf
fmax=λ (4.2c)
where fcr is the local buckling stress and fmax is the maximum design stress in the
compression element. fmax can be replaced by the yield stress fy for stub columns used
in this study as they are not subject to distortional, torsional or flexural buckling.
The theoretical elastic local buckling stress used in Equation 4.2(c) is given by
22
2
)/)(1(12 tbEkfcr
υπ
−= (4.3)
The buckling coefficient k is conservatively recommended as 0.43 for unstiffened
elements in the design codes (AISI, 1996; ECS, 1997; SA, 1996). In order to include
the rotational rigidity along the longitudinal edges of high strength unstiffened
elements provided by the adjoining web, actual buckling coefficients from a rational
elastic buckling analysis are used in this study instead of simply adopting a buckling
coefficient k of 0.43.
Equations 4.2(a) to (c) were initially proposed for stiffened elements by Winter
(1947). Winter also proposed an empirical effective width equation for unstiffened
elements based on test results. This equation provides slightly unconservative
predictions when compared with Equations 4.2(a) to (c). Winter’s equation for
unstiffened elements is as follows.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-4
−=
λλ1298.00.1119.1
bbe (4.4)
The Winter’s effective width equation for unstiffened elements was not included in
the design codes because of concerns regarding undesirable distortion of unstiffened
elements in the post-buckling range and lack of extensive experimental verification
(Kalyanaraman et al., 1977; Yu, 2000). The effective width equation for stiffened
elements is therefore used for both stiffened and unstiffened compression elements.
Winter originally used a constant 0.3 instead of 0.298 as in Equation 4.4.
Kalyanaraman (1976) proposed a constant 0.298 on the basis of his test results and
an analytical study of post-buckling behaviour. The equation using a constant of
0.298 in Equation 4.4 is also known as the Winter’s formula.
Von Karman’s formula was also modified by a few researchers (Lind et al. 1976;
Mulligan and Pekoz, 1984; Bambach and Rasmussen, 2002) using the term
crffmax=λ , where fmax and fcr are the maximum stress at the supported edge and the
critical buckling stress, respectively. Herzog (1987) developed a simplified average
ultimate strength concept in which the capacity of a column is determined based on a
percentage of the yield strength or squash load. Figure 4.1(a) describes the
conventional effective width concept first introduced by von Karman whereas Figure
4.1(b) shows Herzog’s average ultimate strength concept in which the ultimate
strength is determined by a parameter a governed by the slenderness of an element.
The hatched area in Figures 4.1(a) and (b) represents the ultimate load of a plate
element.
(a) Effective width concept (b) Herzog’s average ultimate strength
Figure 4.1 Ultimate Strength of Plate Elements
be/2 be/2 b b
Pu/2 Pu/2
Pu=αPyActual stress distribution
Actual stress distribution
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-5
A recent study into the effective width of unstiffened elements of cold-formed steel
compression members was conducted by Bambach and Rasmussen (2002). Their
experimental study of mild steel plates with 5 mm nominal thickness and average
yield strength of 295 MPa demonstrated the conservatism of conventional effective
width rules. The modified effective width rule developed by Bambach and
Rasmussen (2002) is as follows.
43
8.0−
= λbbe (4.5)
where λ is the section slenderness ratio as given in Equation 4.2(c).
Due to the uncertainty of the behaviour of thin and high strength cold-formed steels,
Section 1.5 of AS/NZS 4600 (SA, 1996) limits the design yield strength to the lesser
of 75 percent of their yield strength and 450 MPa for G550 steels with a thickness
less than 0.9 mm. Yang and Hancock (2002) conducted stub column tests using
rectangular boxes and hexagon sections composed of stiffened elements and verified
that the limitation of 75 percent yield strength is too conservative. Therefore, based
on experimental results, they recommended the use of 90 percent yield strength. It
was also concluded that the lack of strain hardening on high strength steels affects
stocky sections more than slender sections. Furthermore, most of their test results did
not reach the computed values based on the effective width design rules for stiffened
elements. This implies that the conventional effective width rules overestimate the
strength of stiffened elements made of thin high strength steels unless a reduced
yield strength (90%) is used as recommended by Yang and Hancock (2002).
Past researches on the effective width of unstiffened elements have been limited to
low strength and thicker steels with a yield stress less than 350 MPa. The structural
behaviour of unstiffened elements made of thin high strength steels is therefore
unknown. This investigation therefore included unstiffened elements with high
slenderness ratios and yield stresses subject to local buckling effects under pure axial
compression.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-6
4.3 Experimental Investigations
The aim of this experimental study was to investigate the local buckling behaviour of
unstiffened elements of high strength cold-formed steel compression members.
Therefore a series of stub column tests was conducted on unlipped C-sections (plain
channel sections) with slender flange elements. The channel sections were made of
G250 (minimum fy = 250 MPa) and G550 grade steels (minimum fy = 550 MPa)
representing low and high strength cold-formed steel members. Both low and high
strength steel columns were chosen so that the effects of the higher steel grade could
be investigated. Two different thicknesses were used for low strength steel while
there were four different thicknesses for high strength steels. All the test specimens
were designed so that the column behaviour would be governed by local buckling of
unstiffened flange elements. The finite strip analysis program THINWALL
(Papangelis and Hancock, 1998) was used to achieve this.
4.3.1 Test Specimens Fifteen G550 steel columns and eleven G250 steel columns were chosen with three
repeats each. This resulted in a total of 78 tests with a b/t ratio of 18 to 94 (see Table
4.1). The corresponding non-dimensionalised slenderness ratio λ was 0.72 to 4.20
(see Tables 4.3 and 4.4). The width to thickness ratio (b/t) of unstiffened elements is
limited to 60 in AS/NZS 4600 (SA, 1996). However, in this research, it was
extended to about 90 for G250 steels. Some specimens were of unusual geometry to
prevent any occurrence of web local buckling. This was verified by preliminary tests
and finite strip analyses. However, all the specimen made of G550 steels were within
the b/t ratio limit of 60. The profile and dimensions of specimens chosen are shown
in Figure 4.2 and Tables 4.1 and 4.2.
The total and coating thicknesses of each specimen were measured using a
micrometer gauge and a special coating thickness gauge, respectively. The base
metal thicknesses were then obtained from the above measurements and were used to
determine the local buckling stresses and effective widths of flanges. The 0.2% proof
stress was measured using tensile coupon tests except for 0.8 mm thickness steel
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-7
made of G550 steel. The yield stress of 0.8 mm thickness steel measured by
Mahendran and Jeevaharan (1999) was used in this study as the steel sheets used
were from the same batch.
Figure 4.2 Unlipped C-section
Table 4.1 Details of Test Specimens and FEA Models
Test Grade t (BMT) (mm) b (mm) d (mm) HL
(mm) fy
(MPa) b/t
G5-C1 0.936 18.5 18.0 45 636 20 G5-C2 0.936 24.5 22.7 55 636 26 G5-C3 0.936 34.0 32.0 85 636 36 G5-C4 0.936 38.5 36.7 95 636 41 G5-C5 0.936 48.7 47.0 110 636 52 G5-C6 0.936 56.0 53.0 135 636 60 G5-C7 0.790 19.5 12.8 40 656 25 G5-C8 0.790 29.0 18.5 65 656 37 G5-C9 0.790 38.5 28.5 80 656 49 G5-C10 0.415 9.3 8.3 20 722 22 G5-C11 0.415 15.0 13.5 35 722 36 G5-C12 0.415 20.0 18.3 45 722 48 G5-C13 0.415 22.0 19.2 50 722 53 G5-C14 0.590 19.2 17.9 45 715 33 G5-C15
550
0.590 23.5 22.0 55 715 40 G2-C1 1.560 27.4 25.0 65 298 18 G2-C2 1.560 37.5 32.5 80 298 24 G2-C3 1.560 47.3 44.5 105 298 30 G2-C4 1.560 63.0 57.0 140 298 40 G2-C5 1.560 77.0 75.5 160 298 49 G2-C6 1.560 88.5 83.0 200 298 57 G2-C7 1.150 49.0 38.5 105 312 43 G2-C8 1.150 29.0 23.0 65 312 25 G2-C9 1.150 74.0 46.0 180 312 64 G2-C10 1.150 88.5 46.0 200 312 77 G2-C11
250
1.150 108.5 47.5 215 312 94 * BMT and HL refer to base metal thickness and half-wave buckling length, respectively. Corner radius r = 1.2 mm for all specimens.
d
b
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-8
Table 4.2 Additional FEA Models
Test Grade t
(BMT, mm)
b (mm) d (mm) HL (mm)
fy
(MPa) b/t
G5-A1 0.936 15.0 10.0 30 636 16 G5-A2 0.936 30.0 25.0 60 636 32 G5-A3 0.936 40.0 35.0 90 636 43 G5-A4 0.790 30.0 20.0 60 656 38 G5-A5 0.790 65.0 20.0 120 656 82 G5-A6 0.790 50.0 20.0 100 656 63 G5-A7 0.790 47.5 20.0 95 656 60 G5-A8 0.790 40.0 20.0 80 656 51 G5-A9
550
0.790 27.0 15.0 55 656 34 G2-A1 1.560 20.0 15.0 40 298 13 G2-A2 1.560 50.0 45.0 110 298 32 G2-A3 1.560 70.0 65.0 165 298 45 G2-A4 1.150 75.0 50.0 160 312 65 G2-A5 1.150 90.0 50.0 185 312 78 G2-A6
250
1.150 110.0 50.0 220 312 96 * BMT and HL refer to base metal thickness and half-wave buckling length, respectively. Corner radius r = 1.2 mm for all models.
The measured values were used for predicting the ultimate column strengths based
on the current design codes and for obtaining the effective widths from test results.
The specimen dimensions and measured yield stresses are given in Tables 4.1 and
4.2. Since the variation in the measured values was small, Tables 4.1 and 4.2 present
the average value in each case.
4.3.2 Test Set-up
Fixed-end support conditions were considered adequate for the stub column tests.
The length of test specimens was decided based on three times the buckling half-
wave length. Additional lengths of 20 mm were added at the top and bottom ends so
that any restraint at the fixed ends did not cause reductions to the buckling half-wave
lengths and stresses. The test specimens were located between the large cross heads
of a universal testing machine and loaded in axial compression until the specimens
collapsed. No eccentricity was therefore allowed on specimens.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-9
Figure 4.3 shows the overall test set-up including the locations of displacement
transducers. The out-of-plane deflection of both flanges was measured at mid-height,
and an additional displacement transducer was located on the web to monitor any
occurrence of web deformation. The axial compression load versus axial shortening
curves were monitored well beyond the ultimate load using an in-built displacement
transducer.
Figure 4.3 Test Set-up
4.3.3 Determination of Local Buckling Stress and Effective Width
The experimental local buckling load can be obtained visually during the test.
However, it is difficult to decide this for imperfect plates. Therefore, determining the
experimental local buckling load inevitably depends on observers’ experience and
judgement. To avoid this problem, the out-of-plane displacements of flanges were
monitored at mid-height during the tests in order to determine the experimental local
buckling stress. Local buckling stress was then obtained from the experimental
compression load versus out-of-plane deflection2 curve as shown in Figure 4.4
(Venkataramaiah and Roorda, 1982).
DT (lateral displacement)
Fixed end support DT (axial
shortening)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-10
The effective width of flanges was directly obtained from the test results as the web
was fully effective and no distortional or flexural torsional buckling occurred. All the
specimens failed by local buckling of flanges.
Figure 4.4 Axial Compression Load versus Out-of-plane deflection2 Method
(Venkataramaiah and Roorda, 1982)
Single plate elements were sometimes used in experimental studies to simulate the
ideal boundary conditions that are present between the web and flanges (Pokharel,
and Mahendran, 2001; Bambach and Rasmussen, 2002). However, in this research
the more practical C-section members were considered with simple test conditions,
but appropriate allowances were made for the rotational rigidity along the
longitudinal edge between the web and flanges in associated strength calculations by
using the relevant elastic buckling coefficients.
4.4 Numerical Analyses
In the numerical analyses, the test specimens were modelled first, ie. a total of 26
models with a range of section slenderness λ from 0.72 to 4.20. Following the
validation of numerical models, finite element analyses were extended to include a
larger λ range of 0.5 to 4.6 (15 models) and the effects of strain hardening (4 models)
as shown in Tables 4.3 and 4.4. This resulted in a total of 45 numerical models.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-11
4.4.1 Finite Strip Analysis
In this study, a finite element analysis (FEA) program ABAQUS (HKS, 2002) and a
finite strip analysis (FSA) program THINWALL (Papangelis and Hancock, 1998)
were used. Prior to using finite element analyses, THINWALL was used to obtain
the theoretical elastic local buckling stresses, and to determine the appropriate
section sizes for FEA and experiments on the basis of section buckling behaviour.
Local bucking stresses from THINWALL were re-examined after analysing each
model using FEA. Buckling stresses from THINWALL are compared with those
from experiments and FEA in Section 4.4.2.6.
4.4.2 Finite Element Analysis
4.4.2.1 Elements and finite element meshes
Element type needs to be defined to provide proper degrees of freedom to simulate a
model’s deformation. Many shell modelling options are available in ABAQUS. The
fully integrated S4 shell element is an element type for general purpose which allows
for transverse shear and large strains. Therefore the S4 element will be more accurate
than S4R5 element, but the processor time, disk usage and memory required for S4
elements are far greater than S4R5 elements. The difference in ultimate strength
results between using the S4 and S4R5 elements was found to be negligible (less
than two percent) for models of structural steel members with negligible transverse
shear effect (Avery, 1998). Therefore, the element type S4R5 which allows
arbitrarily large rotations, but small strains, was selected for the analyses in this
research. The S4R5 element is a quadrilateral with four nodes and five degrees of
freedom per node with the use of reduced integration (see Figure 4.5).
Preliminary analyses were required to decide the most suitable finite element mesh.
The finite element mesh should be fine enough to analyse a model accurately, but
excessively finer mesh is not economical in terms of processor time and disk usage.
The convergence studies undertaken showed that a 3.5 mm mesh was adequate to
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-12
analyse the local buckling and nonlinear ultimate strength behaviour of unlipped
channel section stub columns considered in this study.
S 4 R 5
Figure 4.5 Labelling Method of Elements in FEA
4.4.2.2 Material properties
The material properties required for elastic and nonlinear analyses are yield strength
(fy), elasticity modulus (E) and Poisson’s ratio. All models were assumed to
implement either perfect plasticity or isotropic strain-hardening behaviour. Perfect
plastic behaviour assumes no strain hardening and no change of yield strength with
increasing plastic strain, and was considered adequate for G550 steels. In fact, for the
G550 zinc/aluminium alloy-coated light gauge steels used here, low ductile
characteristics with about 2% strain at failure (based on 50 mm gauge length) and
negligible strain hardening were observed. A few material models that simulated the
experimentally observed stress-strain relationship with a reduced fracture strain of
2% were attempted. However, it appeared that ABAQUS did not recognize the
reduced fracture strain. Therefore, a perfect plasticity model was used for all G550
steels (see Figure 4.6(a)).
To consider the strain-hardening behaviour of G250 steels, a field option defining
solution dependent variables of plastic strain values was used in the material models.
The stress-strain relationships used for G550 and G250 steels are shown in Figures
4.6(a) and (b). Measured yield and ultimate strengths were used and εs in Figure
4.6(b) was taken as 4.2% on the basis of tensile test results of G250 steels. The
procedure and details of tensile coupon tests were discussed in detail in Chapter 3.
Degree of freedom Reduced integration
Number of nodes
Stress/displacement shell
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-13
The strain-hardening modulus was taken as 0.04 times the elasticity modulus
obtained experimentally.
(a) G550 steel (b) G250 steel
Figure 4.6 Stress-strain Relationship used in FEA
The elasticity modulus and Poisson’s ratio were taken as 210 GPa and 0.3,
respectively. The elasticity modulus obtained from tensile tests was in the range of
about 210 to 223 GPa. It was found from a few case studies that the effect of such
variations in the elasticity modulus on member strength was about 1.8%. Therefore
the elasticity modulus of 210 GPa was used in all the bifurcation bucking and
nonlinear inelastic analyses.
4.4.2.3 Loading and boundary conditions
Axial compression load was represented as a concentrated nodal force. It was
distributed to the edge nodes at the end along the loading axis by the rigid surface.
The R3D4 rigid surface element was used to create pinned end restraints. This
element is a quadrilateral with four nodes and three translational degrees of freedom
per node. As the element has no rotational degrees of freedom, perpendicular shell
elements attached by common nodes to a rigid surface comprising R3D4 elements
are free to rotate about the attached edge. Local buckling rotations are therefore
unconstrained. Single point constraints were used along the plane of symmetry to
provide the ideal conditions of quarter-wave length model.
σu = σy
εy Strain
Stress
perfect plasticity model
E
Stress
Strain εy εs
strain hardening
E
0.04E σy
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-14
All the models used in this study were half of half-wave length models. This
modelling approach was considered adequate to investigate the experimental and
ideal local buckling behaviour because of the symmetric nature of the local buckling
waveform. A half-length model was considered unnecessary to simulate the
behaviour of test specimen as the difference in results from this and the half of half-
wave length model was minimal. The boundary condition of single point constraints
was applied to restrain the rotation about the axes perpendicular to the loading axis
so that the buckling mode can be symmetrical as required. An elastic strip was also
used to eliminate any stress concentrations in the elements in contact with the rigid
surface. Details of the mesh and boundary conditions are shown in Figure 4.7.
Figure 4.7 FEA Model
4.4.2.4 Initial geometric imperfections and residual stresses
Prior to undertaking nonlinear analyses, it was necessary to include initial geometric
imperfections and residual stresses. Initial geometric imperfections can be included
as local and global imperfections. In this study, the governing failure mode was local
buckling, and global buckling did not occur. Therefore only the local bucking
imperfections were included.
A number of researchers used the measured local imperfections in the FEA of cold-
formed steel members and studied the sensitivity of these imperfections
(Sivakumaran and Abdel-Rahman, 1998; Schafer and Pekoz, 1998; Young and Yan,
2000; Dubina and Ungureanu, 2002). Walker (1975) observed the lack of agreement
between a real plate and a perfect flat plate prior to loading and thus derived
R3D4 elements
Compression Load (SPC,12356)
Elastic strip S4R5 elements
Plane of symmetry (SPC345)
B 1/4 wave length
Flange
Web
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-15
theoretically specified values of the plate initial imperfection on the basis of
Robertson’s (1925) deflection analysis. The Walker’s proposal to predict the
magnitude of initial imperfections is given by the following equation.
∆ = β t √Py/Pcr (4.6)
where ∆ is the magnitude of initial imperfection;
β is a constant that can be adjusted to fit experimental results;
Py is the crushing load;
Pcr is the buckling load.
Due to the variations of edge restraint for various geometries and different
imperfections, Walker (1975) recommended a value of 0.3 for β. Because of the
many variables involved in the buckling and crushing loads, Equation 4.6 needs to
be revised in order to specify major functions affecting the initial imperfections.
Substituting the theoretical local buckling stress Equation 4.3 into Equation 4.6 leads
to the following.
Ek
fb y
2
2 )1(123.0
π
ν−=∆ (4.7)
As seen in Equation 4.7, the magnitude of initial imperfections in Equation 4.6 is
therefore determined by the plate width for a given steel grade and section geometry
irrespective of the type of plate element, i.e. unstiffened or stiffened element.
However, the steel thickness t might also play an important role in the initial
imperfections due to cold-forming processes used. Therefore, there is a need to
consider observed initial imperfections in cold-formed steel sections.
Schafer and Pekoz (1996, 1998) collected the data on measured geometric
imperfections of cold-formed channel sections from existing experimental data and
recommended realistic magnitudes of geometric imperfections using statistical and
probabilistic analyses. Figures 4.8(a) and (b) show the two types of geometric
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-16
imperfections (Types 1 and 2 observed by Schafer and Pekoz (1998)). Maximum
magnitudes for these types of imperfections are given by the following equations.
∆1 = 6te-2t or 0.006×plate width (4.8)
∆2 = t (4.9)
where ∆1, ∆2, t and plate width are in mm.
(a) Type 1 (b) Type 2
Figure 4.8 Definition of Geometric Imperfections
For the prediction of Type 1 imperfections, either a plate width or thickness was
taken into account based on certain trends in experimental data as shown in Equation
4.8. The imperfection magnitude for Type 2 is given by a function of thickness as
seen in Equation 4.9, ignoring the effect of plate width. Based on the plots of
measured imperfection magnitude versus slenderness, Schafer and Pekoz (1996)
found that no trend existed for the Type 2 imperfections. Thus, in order to simplify
the determination of imperfections, the maximum deviation from the straightness for
Type 2 was conservatively proposed to be equivalent to the plate thickness. In the
reality of the imperfection magnitudes, large scatters exist for both Type 1 and Type
2 imperfections not only depending on the plate slenderness or thickness, but also on
manufacturers. Therefore, researchers have used slightly different magnitudes in
their analyses. To validate the effect of imperfection magnitudes on unlipped channel
members subject to flange local buckling, further sensitivity studies have been
conducted with varying imperfection magnitudes and discussed in Section 4.5.6.
Local imperfections are in general taken on the basis of either material thickness or
the width of plate elements. Despite the drawback in determining the maximum
geometric imperfections, the theoretical and probabilistic studies provide a
∆1
∆2
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-17
reasonable understanding in the treatment of imperfections for finite element
analyses. In this study on unlipped channel sections governed by flange local
buckling, the local imperfection magnitude of t (thickness) as proposed by Schafer
and Pekoz (1998) was used on the critical elastic eigenbuckling mode as shown in
Figure 4.9(a). The local imperfections were introduced in all of the possible locations
where local buckling occurred.
Due to the cold-forming process, residual stresses are present with varying
magnitudes, and can cause members to fail prematurely. It is therefore necessary to
include appropriate residual stresses in the FEA. Schafer and Pekoz (1998) gives the
details of flexural residual stress distributions in cold-formed steel sections based on
their measurements on lipped channels. The residual stresses could be considered to
be the same for unlipped channel sections because the maximum flexural residual
stress always occurs at the corners, and the residual stress in the flanges and lips is
comparatively small. For both lipped and unlipped channel members, the major
influence of flexural residual stress occurs at the corners. Membrane residual stresses
generated during the cold-working process were considered reasonably small to be
ignored in the FEA (Schafer and Pekoz, 1998; Young and Rasmussen, 1999).
Therefore, the flexural residual stress distribution for press braked unlipped channel
sections was assumed as shown in Figure 4.9(b) based on Schafer and Pekoz (1998).
Figure 4.9(c) shows the residual stress output from FEA. Five interpolation points
were used to vary the residual stress through the thickness. As seen in Appendix A
and Figure 4.9(b), the residual stress is linearly changed from tension residual stress
to compression residual stress with five interpolation points.
The user defined residual stress was modelled using the SIGINI Fortran user
subroutine in ABAQUS. It was necessary to ensure that the coordinate system for
stress components was correctly defined to produce the residual stresses on the
required axis. This was achieved by defining the variables of SIGNI for each axis.
For this purpose, the global coordinate system was used defining the stress values on
each element. The subroutine including the user defined residual stress distribution
is given in Appendix A. This subroutine defines the variation of the residual stress
thought the thickness. It was ensured that tension residual stress on the outside of the
fold and compression on the inside were correctly defined as seen in Figure 4.9(b)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-18
(a) Local geometric imperfections
(b) Assumed flexural residual stress distribution
(c) FEA output of residual stresses on unlipped channel members
Figure 4.9 Geometric Imperfections and Residual Stresses
t
t
b
0.08fy 0.33fy
0.17fy
+σr
-σr
outside
inside
Residual stress through thickness
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-19
4.4.2.5 Analysis
The quarter-wave buckling length was determined based on the results obtained from
THINWALL (see Tables 4.1 and 4.2). Prior to using the results from THINWALL,
the quarter-wave buckling length was validated by comparing with the FEA results
of the variation of buckling load with different half-wave buckling lengths. Two
methods of elastic bifurcation buckling and nonlinear static analyses were used for
the 45 unlipped C-section models. Elastic bifurcation buckling analysis was used
first to obtain the critical eigenvalue and the associated buckling mode which was
required to include the initial local imperfections. The residual stress in channel
members is not self equilibrating and thus causes undesirable load vectors including
moment or reaction force. To cope with this problem, two separate steps in the
nonlinear analysis are required. Residual stresses containing no applied loads and all
degrees of freedom constrained were defined on a model with initial imperfections.
Nonlinear static analysis was run with one increment, requesting reaction force and
moment. This step was to detect and eliminate the reaction force and moment caused
by the residual stress load case. Two nonlinear static analysis steps, the first step for
residual stress equilibrium and the second step for an applied compression load, were
then finally submitted. Applied load magnitudes were set higher than the expected
ultimate capacity.
4.4.2.6 Verification of finite element model
The finite element analysis is useful in saving time and resources when compared
with experiments, and is therefore increasingly used in research. However, it is
necessary to verify the accuracy of FEA results considering all the relevant
assumptions before undertaking detailed parametric studies. For the purpose of
validating the finite element model used in this investigation, all test specimens from
both steel grades were selected and the buckling stress (fcr) and ultimate strength (Pu)
from FEA were compared with those from experiments.
Tables 4.3 and 4.4 show the comparisons of FEA results with experimental and finite
strip analysis (FSA) results. The local buckling stresses (fcr) obtained from FEA
agree well with those from FSA and experiments with an overall average Test/FEA
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-20
and FSA/FEA ratio of 1.040 and 1.035, respectively. Likewise, the comparison also
shows good agreement in the prediction of ultimate strengths with an overall average
Test/FEA ratio of 0.952 and the coefficient of variation of 0.045. However, the
ultimate strength appears to be slightly overestimated by FEA for high strength
steels. This might have been due to the difference between the true and assumed
mechanical properties. Further discussions on experimental and numerical results are
given in Section 4.5.
The failures of all the specimens were predominantly governed by flange local
buckling with three half waves as the web element was compact. The FEA showed
similar behaviour in a quarter wave length model as seen in Figures 4.10(a) and (b).
Therefore, a quarter-wave length model can be successfully used to simulate three
half-wave stub column tests of unlipped channel sections.
1/4 wave length
Figure 4.10 Local Buckling of Flanges in Experiments and FEA
(a) Three half waves - Tests
(b) Quarter-wave buckling length - FEA
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-21
Figures 4.11(a) and (b) compare typical axial compression load versus axial
shortening curves from FEA and experiments for low and high strength steels,
respectively. In these figures, G2 and G5 refer to the steel grades of G250 and G550.
The FEA curves agree reasonably well with the experimental curves despite the
possible experimental errors associated with measurements, loading eccentricity and
imperfections of specimens. Appendix D provides more axial compression load
versus axial shortening curves obtained for test and FEA.
The axial compression load versus out-of-plane displacement curves were used to
obtain the experimental local bucking load as shown in Figures 4.12(a) and (b). Ideal
bifurcation curve was seen in the local buckling of perfect plates. However, initial
geometric imperfection was present in most plates. This lead to gradual out-of-plane
displacement even at lower axial compression loads. Therefore, the axial
compression load versus out-of-plane displacement2 (P-δ2) curve was used due to the
presence of plate imperfections (Venkataramaiah and Roorda, 1982). Figure 4.12(b)
shows the typical transformation from the axial compression load versus out-of-
plane displacement curve. The local buckling load was determined from the point of
intersection of the two straight lines shown in these figures. The FEA results
corresponding to the experimental results are included in Figures 4.12(a) and (b),
which show a good agreement in the prediction of local buckling strength for both
low and high strength steels. The FEA always demonstrated imperfect plate
behaviour in the axial load versus out-of-plane displacement curves due to the initial
local imperfection included in FEA. In experiments, most results showed imperfect
plate behaviour in the axial load versus out-of-plane displacement curves, but test
specimen G2-C7 showed perfect plate behaviour. Hence for the purpose of
comparison, the P-δ and P-δ2 curves are plotted for test and FEA results in Figure
4.12(a).
From the comparisons of elastic buckling and ultimate strengths as shown in Tables
4.3 and 4.4 and Figures 4.10 to 4.12, the FEA including element types, boundary
conditions, residual stresses and local imperfections is considered suitable in
predicting the strength of unlipped channel members governed by flange local
buckling. The comparison also indicates that the use of a quarter-wave length model
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-22
enables accurate prediction of the local buckling stress and ultimate strength of
unlipped channel sections.
(a) G250 steels
(b) G550 steels Figure 4.11 Axial Compression Load versus Axial Shortening Curves
0
10
20
30
40
50
60
70
80
0 0.5 1 1.5 2
G2-C3(FEA)
G2-C5(FEA)
G2-C3(Test)
G2-C5(Test)
(mm)
(kN)
×
Axi
al c
ompr
essi
on lo
ad
Axial shortening
0
5
10
15
20
25
30
35
40
45
50
0 0.5 1 1.5
G5-C3(FEA)
G5-C6(FEA)
G5-C3(Test)
G5-C6(Test)
×
(kN)
(mm)
Axi
al c
ompr
essi
on lo
ad
Axial shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-23
(a) Test series G2-C7 (perfect and imperfect plate behaviour, P-δ and P- δ2 curves) (b) Test series G5-C6 (imperfect plate behaviour, P-δ2 curve)
Figure 4.12 Determination of Local Buckling Load
0
5
10
15
20
25
30
35
0 1 2 3 4
Test
FEA
(kN)
Axi
al c
ompr
essi
on lo
ad
Out-of-plane displacement
P-δ curve
Pcr
P-δ2 curve
(mm)
0
4
8
12
16
20
0 1 2 3 4
Test
FEA
Pcr
(kN)
Axi
al c
ompr
essi
on lo
ad
Out-of-plane displacement2 (mm2)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-24
FEA fcr Pu (kN) be/b
Specimen λ k fcr (MPa)
FSA FEA
Test FEA Eq. 4.2 Test FEA
Pu (Test) Pu (FEA)
Test FEA
be (Test) be (Eq. 4.2)
be (FEA) be (Eq. 4.2)
G2-C1 0.719 0.99 NA NA NA 31.1 33.8 34.1 0.99 0.83 0.86 1.15 1.19 G2-C2 1.026 0.87 283 1.03 0.88 42.5 45.2 42.5 1.06 0.85 0.80 1.11 1.04 G2-C3 1.234 0.95 196 1.03 1.08 52.1 52.8 49.5 1.07 0.66 0.62 1.06 0.95 G2-C4 1.691 0.90 104 1.02 1.05 57.0 57.8 58.1 0.99 0.53 0.54 1.03 1.05 G2-C5 2.092 0.92 68 1.08 0.93 66.1 65.8 67.2 0.98 0.41 0.45 0.96 1.01 G2-C6 2.357 0.92 54 1.05 1.07 70.7 72.0 72.1 1.00 0.40 0.41 1.04 1.06 G2-C7 1.728 0.96 104 1.06 1.01 30.3 31.7 32.1 0.99 0.54 0.54 1.07 1.07
24.8 0.97 0.75 1.04 G2-C8 1.107 0.82 255 1.08 1.1 22.4 24.1 25.5* 0.95 0.81 0.79* 1.12 1.09 37.8 0.94 0.38 1.03 G2-C9 2.478 1.10 51 0.98 1.19 35.8 35.4 38.8* 0.91 0.36 0.40* 0.98 1.09 38.4 0.98 0.32 1.04 G2-C10 3.024 1.06 34 1.10 1.12 35.9 37.7 39.6* 0.95 0.36 0.35* 1.17 1.14 39.2 1.01 0.27 1.04 G2-C11 3.625 1.11 24 1.01 1.11 37.1 39.7 40.8* 0.97 0.31 0.29* 1.20 1.12
G2-A1 0.501 1.22 NA NA - 27.2 - 27.1 - - 0.99 - 0.88 G2-A2 1.315 0.99 172 0.92 - 52.2 - 51.2 - - 0.61 - 0.96 G2-A3 1.996 0.84 75 1.05 - 61.1 - 63.0 - - 0.47 - 1.05 G2-A4 2.654 0.99 44 1.06 - 37.6 - 40.25 - - 0.39 - 1.13 G2-A5 3.063 1.07 33 1.09 - 38.6 - 40.1 - - 0.33 - 1.09 G2-A6 3.752 1.06 22 0.95 - 38.8 - 39.9 - - 0.26 - 1.04
Mean 1.03 1.05 0.98 1.06 1.04 COV 0.05 0.09 0.04 0.09 0.06
* FEA results including strain-hardening. k refers to local buckling coefficient λ = √fy/fcr where fy is from measured yield stress and the critical buckling stress fcr is from FEA. In the case of NA, experimental fcr was used for λ.
Table 4.3 Results from Tests and FEA (G250 steel)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-25
Table 4.4 Results from Tests and FEA (G550 steel)
k refers to local buckling coefficient λ = √fy/fcr where fy is from measured yield stress and the critical buckling stress fcr is from FEA.
FEA fcr Pu (kN) be/b Specimen λ k fcr
(MPa) FSA FEA
Test FEA Eq. 4.2 Test FEA
Pu (Test) Pu (FEA)
Test FEA
be (Test) be (Eq. 4.2)
be (FEA) be (Eq. 4.2)
G5-C1 1.20 0.93 441 0.98 1.03 25.7 25.4 25.6 0.99 0.68 0.69 1.00 1.01 G5-C2 1.66 0.86 241 1.03 0.88 28.8 26.5 27.4 0.97 0.47 0.49 0.90 0.94 G5-C3 2.33 0.83 121 1.09 0.98 34.8 30.7 32.8 0.94 0.29 0.36 0.75 0.93 G5-C4 2.74 0.78 88 1.01 0.95 37.2 33.4 35.2 0.95 0.26 0.30 0.77 0.90 G5-C5 3.19 0.91 64 1.00 0.94 44.9 38.5 41.9 0.92 0.22 0.24 0.75 0.86 G5-C6 4.20 0.74 39 1.05 0.86 46.6 40.3 43.3 0.93 0.17 0.21 0.75 0.93 G5-C7 1.81 0.74 224 1.07 1.03 16.3 16.7 18.2 0.92 0.51 0.55 1.05 1.16 G5-C8 2.51 0.73 101 1.04 0.94 20.3 19.1 21.0 0.91 0.32 0.36 0.89 1.05 G5-C9 3.54 0.83 66 1.03 0.98 25.1 23.5 24.9 0.94 0.22 0.25 0.83 0.95 G5-C10 1.62 0.72 282 1.04 1.07 5.4 4.8 5.3 0.91 0.38 0.48 0.78 0.96 G5-C11 2.11 1.11 162 0.98 1.02 7.9 7.1 7.9 0.89 0.30 0.42 0.75 0.98 G5-C12 3.16 0.88 72 1.02 1.05 9.9 8.8 9.6 0.92 0.27 0.42 0.75 0.95 G5-C13 3.29 0.99 67 1.03 1.05 10.3 8.9 9.3 0.96 0.23 0.36 0.68 0.94 G5-C14 2.11 0.84 146 0.97 0.98 14.6 13.2 14.8 0.89 0.37 0.44 0.88 1.03 G5-C15 2.87 0.66 78 1.01 1.05 15.5 14.1 15.0 0.94 0.23 0.31 0.79 0.91
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-26
Table 4.4 Results from Tests and FEA (G550 steel) (continued)
k refers to local buckling coefficient λ = √fy/fcr where fy is from measured yield stress and the critical buckling stress fcr is from FEA.
FEA fcr Pu (kN) be/b Specimen λ k fcr
(MPa) FSA FEA
Test FEA Eq. 4.2 Test FEA
Pu (Test) Pu (FEA)
Test FEA
be (Test) be (Eq. 4.2)
be (FEA) be (Eq. 4.2)
G5-A1 1.003 0.86 632 0.95 - 21.7 - 21.8 - - 0.81 - 1.00 G5-A2 1.895 0.96 177 1.02 - 32.6 - 32.2 - - 0.45 - 0.96 G5-A3 2.55 0.94 98 1.05 - 38.8 - 37.8 - - 0.33 - 0.92 G5-A4 2.322 0.92 121 1.03 - 24.1 - 25.0 - - 0.41 - 1.05 G5-A5 4.654 1.07 30 1.09 - 25.9 - 26.5 - - 0.21 - 1.03 G5-A6 3.932 0.89 42 1.11 - 24.5 - 25.5 - - 0.26 - 1.08 G5-A7 3.757 0.88 46 1.06 - 24.4 - 25.1 - - 0.26 - 1.04 G5-A8 3.290 0.81 60 0.98 - 23.9 - 25.1 - - 0.31 - 1.09 G5-A9 2.235 0.80 131 1.05 - 20.8 - 21.6 - - 0.43 - 1.07
Mean 1.04 1.03 0.93 0.82 0.99 COV 0.04 0.11 0.05 0.18 0.07
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-27
4.5 Results and Discussions
4.5.1 Local Buckling Behaviour
The bucking coefficients which demonstrate the effect of web restraint to flanges are
compared in Figure 4.13. All the values of the bucking coefficient k in Figure 4.13
were obtained from the experimental and numerical buckling stresses using Equation
4.3. In general, a buckling coefficient of 0.425 is used for an element with one
longitudinally free edge and the others simply supported under uniform compression.
Most of the buckling coefficients obtained from both tests and FEA were in the
range from 0.7 to 1.1 which demonstrated the presence of restraint conditions that
can be considered to be between pinned and fixed ends. The scatter of the
coefficients is reasonably constant (0.7 to 1.1) up to a section slenderness ratio λ of
4.6. The buckling coefficients of cold-formed steel members under uniform
compression are determined by the restraint provided by adjacent elements (Rhodes,
1991). Thus, the ratio of the length of flanges to that of web (b/d) is the critical factor
for unlipped channel members under the same support conditions. The buckling
coefficients in a shorter range as seen in Figure 4.13 might be therefore due to
similar b/d ratios of the test specimens and FEA models.
Figure 4.13 Comparison of Buckling Coefficients from Tests and FEA
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 1 2 3 4 5
G250 (Test)
G250 (FEA)
G550 (Test)
G550 (FEA)
k
λ
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-28
Interestingly, when the slenderness ratio exceeds 2.5, a slightly bigger scatter of
buckling coefficients was seen for high strength steels. The difference in the
buckling coefficients between G250 and G550 steels was about 0.3 when
experimental and numerical results were considered. However, the results of G550
steel members did not show a clear trend of buckling coefficient variation. Further,
it might have been caused by slightly higher b/d ratios of G250 steel members than
those of G550 steel members in the case of slender sections. There was no clear
effect of steel grade on the rotational rigidity of unstiffened elements constrained by
the web in unlipped channel sections.
4.5.2 Ultimate Strength Behaviour
During the tests, local flange buckling occurred first, which then influenced the post
buckling and ultimate strength behaviour of all the specimens. Therefore it can be
stated that the overall behaviour was governed by local flange buckling.
Experimental study showed that each specimen had three stages of behaviour,
namely, local buckling stage, post buckling stage and collapse stage. The web and
flange elements did not deform until reaching the local buckling load even though
the out of plane straightness appeared in imperfect plate elements. Once the axial
load reached the local buckling load, the axial stiffness decreased significantly while
further axial loading increased, which is the so called post buckling stage. The local
buckling mode involved the rotation of flanges with the web in its original position
as expected. The three half waves of local flange buckling were seen in the post
buckling range as shown in Figure 4.10(a) and channel members continued to
deform while maintaining the three half waves until reaching the ultimate strength.
This behaviour appeared on both low and high strength steel specimens. After
reaching the ultimate strength, the specimens collapsed with decreasing axial load.
Figures 4.14(a) and (b) show the typical failure modes of low and high strength steel
members governed by local flange buckling. As seen in Figures 4.14(a) and (b),
three local buckling waveforms of flange did not exist any longer due to the collapse
of unlipped channel members at the weakest location. Similar buckling and collapse
pattern was seen in all the tests.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-29
It was also observed from the stress-strain curves that a considerable post-buckling
strength was present for high strength steels and high b/t ratios. On the other hand, a
few compression members with low b/t ratios failed at a high load closer to material
yielding with a high local buckling strength but reduced post buckling strength. This
is in good agreement with theoretical predictions.
Finite element analyses confirmed these observations from tests. However, the axial
compression load sometimes dropped rapidly after reaching the ultimate strength
during tests whereas the FEA results showed gradual unloading (see Figure 4.11
(b)). This might be due to the assumption of ideal perfect-plastic material behaviour
in FEA.
(a) G250 steel (b) G550 steel
Figure 4.14 Local Buckling Failure of Flanges
4.5.3 Ultimate Strengths and Effective Widths
Tables 4.3 and 4.4 present the ultimate failure strengths of the unlipped channel
members (Pu) and the corresponding effective widths (be) from AS/NZS 4600
(Equation 4.2), tests and FEA for unlipped channel sections. The effective widths of
flange elements were calculated based on the assumption that the web is fully
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-30
effective. They were computed based on the ultimate strengths from FEA and tests.
To predict the effective widths based on AS/NZS 4600 (SA, 1996), Equations 4.2 (a)
to (c) were used with the slenderness ratio λ based on elastic local buckling stress fcr
from FEA. This is because the difference in fcr from experiments and FEA was small
and the FEA values were more consistent. Appendix C presents the example
calculations to obtain the effective widths presented in Tables 4.3 and 4.4.
Figure 4.15 plots the non-dimensionalised test and FEA ultimate strength Pu with
respect to section capacity Ns calculated using the current design rules (AISI, 1996;
ECS, 1997; SA, 1996) against the plate slenderness ratio b/t. The comparisons in
Figure 4.15 appear to show a reasonably good agreement with the theoretical
calculations presenting the approximate range of the column strength ratios from
0.87 to 1.11 even though most of the test results of high strength steels are placed in
the range of 0.87 to 1.0. However, in evaluating the effective widths of unlipped
channel sections subject to local flange buckling effects, it may not be appropriate to
use the total axial strength in the theoretical calculations and comparisons. Although
the web element of unlipped channel sections considered in this study are fully
effective, the local buckling effects of unstiffened flange elements are not evident in
the comparisons. Therefore, the strengths of flange elements were determined by
subtracting the capacity of the fully effective web (Pu-Pu,web and Ns-Ns,web) and were
plotted in Figure 4.16 against the plate slender ratios (b/t).
This procedure resulted in a larger scatter of ratios (0.68 to 1.19) of experimental and
numerical strengths to design strengths when compared with those in Figure 4.15.
This was considered a more accurate and realistic comparison in the investigation
into the local buckling behaviour of unstiffened flange elements. As shown in Figure
4.16, the predictions from the current effective width rules agree well with test and
FEA results of low strength steels because the current design rules were developed
on the basis of low strength steels. The FEA results are also in reasonably good
agreement with the test results for low strength steels.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-31
Figure 4.15 Comparison of Ultimate Loads of Unlipped Channel Sections from
Test and FEA results with those from Current Design Rules
Figure 4.16 Comparison of Ultimate Loads of Flanges from Test and FEA Results with those from Current Design Rules
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80 100
G2-1.6(Test)G2-1.6(FEA)
G2-1.2(Test)G2-1.2(FEA)
G5-0.95(Test)
G5-0.95(FEA)G5-0.8(Test)
G5-0.8(FEA)G5-0.6(Test)
G5-0.6(FEA)
G5-0.42(Test)G5-0.42(FEA)
s
uNP
b/t
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 20 40 60 80 100
G2-1.6(Test)
G2-1.6(FEA)
G2-1.2(Test)
G2-1.2(FEA)
G5-0.95(Test)
G5-0.95(FEA)
G5-0.8(Test)
G5-0.8(FEA)
G5-0.6(Test)
G5-0.6(FEA)
G5-0.42(Test)
G5-0.42(FEA)
webss
webuu
NNPP
,
,
−
−
b/t
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-32
Table 4.5 Comparison of Ultimate Strength Ratios
Test FEA Series t(BMT)
(mm) Pu-Pu,web Ns-Ns,web Range Mean Pu-Pu,web
Ns-Ns,web Range Mean
G2-C1 1.15 1.19 G2-C2 1.11 1.04 G2-C3 1.06 0.95 G2-C4 1.03 1.05 G2-C5 0.96 1.01 G2-C6
1.560
1.04
0.96 -
1.15
1.06 (0.062)
1.06
0.95 -
1.19
1.05 (0.075)
G2-C7 1.07 1.07 G2-C8 1.12 1.04 G2-C9 0.98 1.03 G2-C10 1.17 1.04 G2-C11
1.150
1.20
0.98 -
1.20
1.11 (0.078)
1.04
1.03 -
1.07
1.04 (0.014)
G2-A1 - 0.88 G2-A2 - 0.96 G2-A3
1.560 -
- - 1.05
0.88 -
1.05
0.96 (0.088)
G2-A4 - 1.13 G2-A5 - 1.09 G2-A6
1.120 -
- - 1.04
1.04 -
1.13
1.09 (0.041)
G5-C1 1.00 1.01 G5-C2 0.90 0.95 G5-C3 0.76 0.93 G5-C4 0.79 0.91 G5-C5 0.75 0.84 G5-C6
0.936
0.75
0.75 -
1.00
0.83 (0.124)
0.93
0.84 -
1.01
0.94 (0.053)
G5-C7 1.05 1.16 G5-C8 0.89 1.05 G5-C9
0.790 0.83
0.83 -
1.05
0.92 (0.123)
0.95
0.95 -
1.16
1.05 (0.100)
G5-C10 0.78 0.96 G5-C11 0.75 0.98 G5-C12 0.75 0.95 G5-C13
0.415
0.68
0.68 -
0.78
0.74 (0.057)
0.94
0.94 -
0.98
0.96 (0.017)
G5-C14 0.88 1.03 G5-C15 0.590 0.79
0.79-0.88
0.84 (0.075) 0.91
0.91- 1.03
0.97 (0.087)
G5-A1 - 1.00 G5-A2 - 0.96 G5-A3
0.936 -
- - 0.92
0.92 -
1.00
0.96 (0.041)
G5-A4 - 1.05 G5-A5 - 1.03 G5-A6 - 1.08 G5-A7 - 1.04 G5-A8 - 1.09 G5-A9
0.790
-
- -
1.07
1.03 -
1.09
1.06 (0.022)
Note: Numbers in parenthesis are COV values
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-33
In contrast to a good agreement for low strength steels, the current design rules
overestimated the compressive strength of cold-formed steel members made of high
strength steels. Most of the specimens made of high strength steels did not reach the
strengths predicted by the design rules as seen in Figure 4.16 and Table 4.5.
However, FEA provided results that agreed with predicted design strengths, but not
with test results for high strength steels as seen in Figure 4.16. In particular, the
ultimate strengths for 0.42 mm high strength steels are consistently lower than those
for other thickness steels.
The reasons for the observed reduced test strengths of high strength steel sections
may be due to the absence of perfect plasticity as assumed in the FEA, lack of strain
hardening and in general the lack of ductility of high strength steels. Yang and
Hancock (2002) also observed reduced test strengths for 0.42 mm G550 steel
specimens and attributed this to lack of strain hardening. The presence of the so-
called Bauschinger effect in high strength steels could have also contributed to the
reduced test strengths, i.e., the high strength steels may have a smaller compressive
yield stress than the assumed tensile yield stress. Macadam et al. (1988) showed that
the yield strength in compression was about 85% of that in tension for high strength
cold-formed steels even though their study was limited to thicker cold-formed steel
plates exceeding 1.35 mm. Therefore, the different yield stresses in tension and
compression should be considered in the FEA and design strength calculations.
However, replacing the tension yield stress by the compression yield stress may not
be accurate as there are both tension and compression zones even in members
subject to axial compression load. Further research is required to investigate this
issue. Unlike high strength cold-formed steels, low strength steels (mild steels) are
considered as isotropic materials. Low strength steels are annealed to a greater extent
in comparison with high strength steels and thus this procedure results in near
isotropic material properties for low strength steels (Rogers and Hancock, 1998).
This resulted in a good agreement between experiments and FEA for low strength
steels.
Overall, the results of G250 steels agreed well with the design curves currently used
in the Australian Standards. This is confirmed by the mean values of 1.06 and 1.04
for be(Test)/be(Eq. 4.2) and be(FEA)/be(Eq. 4.2) in Tables 4.3 and 4.5. On the
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-34
contrary, Figure 4.16 and Tables 4.4 and 4.5 show that Equations 4.2(a) and (b)
overestimate the predictions when compared with G550 steel test results. Table 4.4
shows a mean value of 0.82 for be(Test)/be(Eq. 4.2). The FEA provided slightly
higher predictions than test results and therefore agree reasonably well with the
design curve.
Since the effective width rules in AS/NZS 4600 (SA, 1996) and AISI (1996)
specification are based on experimental studies of low strength cold-formed steel
members, they were unable to predict the strengths of cold-formed steel members
made of high strength steels (≥ 550 MPa) with low ductility and anisotropy
characteristics.
4.5.4 Design of High Strength Unlipped Channel Members Subject to
Local Buckling Effects
Effective width rules used in the strength prediction in cold-formed steel design were
reviewed in Section 4.5.3 for both low and high strength steel unstiffened elements
using unlipped channel stub column tests and FEA. Experimental and numerical
results have confirmed the adequacy of current effective width formula in AS/NZS
4600 (SA, 1996) for unstiffened elements made of low strength steels. However, the
design rules overestimate the effective width of high strength (G550) cold-formed
steel unstiffened elements. Similarly, it was found from the recent research by Yang
and Hancock (2002) that experimental results were predicted unconservatively using
the current design rules for stiffened elements and the use of 75% reduction in the
yield stress is too conservative. Therefore, they recommended a reduction factor of
0.9 on the basis of their experimental results, i.e., 0.9fy. In this study, this
recommendation was used for high strength steels and the revised design strength
predictions are compared with test results.
Table 4.6 and Figure 4.17 present the ratios of experimental results to design
strengths based on 90% yield strength. 0.42 mm thickness steels are still placed in
the lower band despite the mean of 0.94. This might be due to greater effects
discussed earlier for those high strength steels.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-35
Figure 4.17 Comparison of Test Results with Those from Modified
Design Rules using a Reduced Yield Stress of 0.9fy (G550 steel)
Table 4.6 Comparison of Test Ultimate Strength Ratios based on 0.9fy
Test t (BMT) (mm)
Pu-Pu,web Ns-Ns,web
Range Mean COV
G5-C1 1.13 G5-C2 1.01 G5-C3 0.98 G5-C4 1.05 G5-C5 1.06 G5-C6
0.936
1.08
0.98-1.13 1.05 0.050
G5-C7 1.17 G5-C8 1.07 G5-C9
0.790 1.08
1.07-1.17 1.11 0.050
G5-C10 0.97 G5-C11 0.93 G5-C12 0.96 G5-C13
0.415
0.90
0.90-0.97 0.94 0.034
G5-C14 1.07 G5-C15 0.560 1.04
1.04-1.07 1.05 0.021
Yang and Hancock’s (2002) recommendation of using 0.9fy for high strength steels
with a thickness less than 0.90 mm appears to eliminate the problem as the strength
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 20 40 60 80
0.95
0.8
0.6
0.42
webss
webuu
NNPP
,
,
−
−
b/t
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-36
ratios are now greater than 1.05 for all the steel thicknesses except for 0.42 mm
G550 steel (see Table 4.6). Therefore for 0.42 mm G550 steels, a reduction factor
less than 0.9 may be needed. Overall, it can be concluded that a design approach
based on a reduced yield stress may be adequate to address the reduced ultimate
strength of high strength steels observed in this study.
Alternatively, a suitable modification of effective width rules was attempted for
unstiffened elements of high strength cold-formed steel members on the basis of
experimental results. A simplified effective width equation is given with respect to
the ratio of the buckling stress to the yield stress expressed as the inverse plate
slenderness. The simplified equation is as follows.
max
745.0ff
bb cre = ≤ 1.0 (4.10)
Equation 4.10 can also be used for low strength steels but will result in conservative
effective widths. From Equation 4.10 the slenderness limit is defined as 0.745 that is
slightly greater than the current slenderness limit of 0.673. Substituting the critical
elastic buckling stress formula, Equation 4.3 into Equation 4.2 (c), the theoretical
limiting width to thickness ratio can be derived. The following equations for width to
thickness ratio limit are derived corresponding to slenderness ratios of 0.673 and
0.745.
yf
kEtb
2lim 1
038.0υ
π−
=
for λ = 0.673 (4.11a)
yf
kEtb
2lim 1
046.0υ
π−
=
for λ = 0.745 (4.11b)
Using Equations 4.11(a) and (b), limiting values of (b/t)lim are obtained as 7.7 and
8.5 for slenderness ratios of 0.673 and 0.745, respectively when the buckling
coefficient is conservatively taken as 0.425. When the lowest experimental buckling
coefficient of 0.64 is used, (b/t)lim becomes 9.5 for the slenderness limit of 0.673.
This implies that the use of 0.745 as the slenderness limit is reasonable. Figure 4.18
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-37
and Table 4.7 show the accuracy of Equation 4.10 for high strength cold-formed
steel compression members.
Figure 4.18 Comparison of Test Results with Equation 4.10 Predictions
(G550 steel)
Table 4.7 Comparison of Ultimate Strength Ratios based on Equation 4.10
Test t(BMT) (mm)
Pu-Pu,web Ns-Ns,web
Range Mean COV
G5-C1 1.05 G5-C2 1.02 G5-C3 0.98 G5-C4 1.03 G5-C5 1.04 G5-C6
0.936
1.02
0.98-1.05 1.02 0.024
G5-C7 1.09 G5-C8 1.02 G5-C9
0.790 1.04
1.02-1.09 1.05 0.034
G5-C10 0.99 G5-C11 0.94 G5-C12 0.97 G5-C13
0.415
0.93
0.93-0.99 0.96 0.029
G5-C14 1.06 G5-C15 0.590 1.01 1.01-1.06 1.04 0.034
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80
0.95
0.8
0.6
0.42
webss
webuu
NNPP
,
,
−
−
b/t
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-38
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Test
FEA
The difference between the proposed effective width formula given in Equation 4.10
and the current effective width design rule is seen in Figure 4.19. The effective width
of unstiffened elements was derived directly from axial compression strengths due to
the fully effective web. Figure 4.19 confirms the inadequacy of the current effective
width design rules for high strength steels as shown in Figure 4.16. The adequacy of
the modified effective width rule is proven for unstiffened elements by plotting
experimental and numerical effective widths against the section slenderness ratio as
shown in these figures.
Figure 4.19 Modified Effective Width Curve for High Strength
Unstiffened Elements
4.5.5 Effect of Strain Hardening
Effect of strain-hardening was investigated using the FEA. In general, strain-
hardening is not considered in material modelling because it requires additional
modelling efforts and material test data including strain hardening parameters. It
should be therefore useful to determine whether the effect of strain hardening can be
ignored.
y
creff
bb
745.0=
−=
y
cr
y
creff
ff
bb
22.01
be/b
λ
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-39
Eight models with and without strain-hardening were analysed and their results are
compared in Table 4.8. Analysis results showed that strain-hardening included in
FEA modelling was not considerably effective as shown in Table 4.8. The mean
difference of effective width and ultimate strength between the models with and
without strain-hardening was 5.5% and 3.3%, respectively. A possible reason for this
is that the models reached their ultimate strength at lower strains that are less than
those corresponding to the strain hardening region. Therefore, the ultimate tensile
strength of low strength steels at the end of its strain-hardening zone did not
determine the ultimate strength of unlipped channel sections governed by local
flange buckling. From the observation of the effect of strain hardening, it can be
concluded that strain hardening can be omitted in the FEA models used in parametric
studies. However, it will provide slightly conservative results.
Table 4.8 Comparison of Results from FE Models with and without Strain Hardening
Perfect elastic-plastic model
Strain-hardening model Pu,e-p be,e-p Flange
(mm) Web (mm)
BMT (mm) Pu
(kN) be b
Pu (kN)
be b
Pu,s-h be,s-h
29 23.0 1.15 24.8 0.75 25.5 0.79 0.97 0.96 74 46.0 1.15 37.8 0.38 38.8 0.40 0.97 0.95
88.5 46.0 1.15 38.4 0.32 39.6 0.35 0.97 0.94 108.5 47.5 1.15 39.2 0.27 40.8 0.29 0.96 0.93
Mean 0.968 0.945 COV 0.005 0.014
Note: Pu,e-p is the ultimate strength without strain hardening (perfect elastic-plastic
model) and Pu,s-h is the ultimate strength including strain hardening effects.
4.5.6 Imperfection Sensitivity Analyses
In most cold-formed steel members, initial geometric imperfections are present as
the magnitude of out-of-straightness due to the manufacturing process. The
magnitude and pattern of the initial imperfection existing in members are different
from one to another. Therefore, researchers generally use different expressions for
the modelling of imperfections in the finite element analysis as discussed in Section
4.4.2.4 because there are no specified rules for imperfections. Therefore, in this
study, a sensitivity study on initial geometric imperfections was undertaken to assess
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-40
the effects of imperfection magnitudes on unlipped channel members subject to local
buckling effects. For this study, theoretical and practical recommendations by other
researchers were considered.
Walker (1975) attempted to theoretically specify the initial deflections in the same
manner as Robertson’s (1925) analysis. The magnitude of the imperfections
proposed by Walker (1975) has already been introduced in Equation 4.6.
Schafer and Pekoz (1996, 1998) used a practical approach based on the accumulated
experimental data from a number of researchers. Experimental data were sorted into
two types of geometric imperfections as illustrated in Figure 4.8 and analysed in
detail. As described in Equations 4.8 and 4.9, Schafer and Pekoz (1996, 1998)
recommended simple and practically alternative rules in accordance with the
imperfection types. Further, all the data were statistically estimated as cumulative
distribution function (CDF). Therefore, in this imperfection sensitivity study, each
case was additionally analysed with 25%, 50% and 75% exceedance imperfections.
This imperfection sensitivity study was limited to Type 2 imperfection because
unlipped channel members used in this study are governed by Type 2 imperfection.
The CDF values for the exceedance imperfections are given in Table 4.9.
Table 4.9 Effects of Imperfection on Ultimate Strength
Ratio of ultimate strength / ultimate strength based on Eq.4.9 Specimen Eq. 4.9 (t) 25%(0.64t) 50%(0.94t) 75%(1.55t) Walker’s
(Eq. 4.6) G2-C10 1.000 1.009 1.003 0.986 1.005
G5-C3 1.000 1.014 1.005 0.981 1.007
Table 4.9 shows the comparison of the ultimate strengths for different imperfection
magnitudes. The magnitudes of initial imperfections for G2-C10 and G5-C3
specimens obtained from Walker’s recommendation were 1.03 mm and 0.65 mm,
respectively. The largest difference was 1.9% when the magnitude based on 75%
exceedance imperfection was used and the average difference is 1.0% as seen in
Table 4.9. This sensitivity study has therefore proved that the local bucking
behaviour studied here is not sensitive to imperfections, and that the use of thickness
for the local imperfection magnitude is rational for unlipped channel members
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-41
subject to flange local buckling. The numerical results obtained by Kaitila (2002)
also showed that the strength difference was small when different local
imperfections were considered.
4.6 Direct Strength Method
Due to the complexity of the current design procedures for cold-formed steel
structures, Schafer and Pekoz (1998) developed an alternative design method called
the direct strength method. The new design proposal avoids the necessity of effective
width calculations. Since the unlipped channel section is not in the list of pre-
qualified column sections that includes lipped channel, hat section and rack section
(Schafer, 2002), this research reviewed the applicability of direct strength method to
unlipped channel sections subject to local buckling. In the direct strength method, the
ultimate axial strength Pn of columns subject to local buckling is given by the
following equations.
1=y
nPP
for λ ≤ 0.776 (4.11a)
−
=
4.04.0
15.01y
cr
y
cr
y
nPP
PP
pP
for λ > 0.776 (4.11b)
where Pcr is the elastic local buckling load while Py is the squash load as determined
in the current standards.
Figure 4.20 compares the test and FEA results with a strength curve computed using
the direct strength method. Appendix C provides example calculations used to obtain
Figure 4.20. The direct strength method gives a slightly conservative prediction as
the slenderness ratio (λ=√Py/Pcr) increases. Interestingly, a lower scatter was shown
for G550 steels, particularly for slender sections. From the test and FEA results
obtained in this investigation, it can be seen that the direct strength method can be
safely used for unlipped channel sections subject to local buckling effects, but will
provide slightly conservative predictions for slender sections.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-42
Figure 4.20 Comparison of Results from the Direct Strength
Method, Tests and FEA
4.7 Summary
Experimental study and finite element analyses were conducted at ambient
temperature to investigate the adequacy of the current effective width rules for
unstiffened flange elements of high strength cold-formed steel members subject to
local buckling effects under axial compression. The test and numerical results have
shown that the currently used effective width rule is adequate in predicting the
effective width of unstiffened elements in low strength cold-formed steel members,
but is inadequate for high strength steel members due to the different mechanical
properties compared with low grade steels. Therefore, a new design rule has been
recommended for high strength steels based on test and FEA results.
There are some discrepancies in the ultimate strength results between experiments
and FEA for high strength steel members. This may be due to the perfect plasticity
assumption in FEA, low ductility characteristics including lack of strain hardening,
and possible Bauschinger effects in thin high strength steels. However, these effects
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Schafer and Pekoz
G550-Test
G250-Test
G550-FEA
G250-FEA
Pn/Py
√Py/Pcr
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 4-43
are not present in low strength steels and hence the current design rules and FEA
were able to predict the ultimate strength adequately.
Effects of strain hardening are in general ignored in the numerical analyses due to
the complexity of modelling. Eight models with and without strain hardening were
analysed to investigate the influence of strain hardening for unlipped channel
members. The FEA results showed that strain hardening does not significantly
increase the member strength due to the failure of models prior to reaching the
ultimate strength of materials and thus is negligible for unlipped channel members
subject to local buckling. A sensitivity study of initial imperfections showed that the
use of thickness for the local imperfection magnitude is rational for unlipped channel
members subject to flange local buckling.
A new design approach, the direct strength method, was also reviewed in this
chapter. The predictions made using the direct strength method agreed reasonably
well with test and FEA results of both low and high strength cold-formed steel
members. This method can therefore be safely used for unlipped channel sections
subject to flange local buckling effects under axial compression, but will provide
slightly conservative predictions for slender sections.
Chapter 4
Underdetermined Blind Source
Separation
4.1 Introduction
BSS has its root in array signal processing. Signals from some particular sources
first pass through an intermediate medium (with possibly noise), which modifies
the original source signals, then arrive at an array of sensors. The observed output
of each sensor is a mixture of all the source signals. It is desired to recover the
unobserved source signals from the observed mixtures; this problem is known as
source/signal separation. If neither the structure of the medium transfer nor the
source signals are known, we are said to be in a “blind” context. One often assumes
in such a context that the source signals are mutually independent in order to
facilitate the separation [1]. BSS is important when modeling the transfer from the
sources to the sensors is difficult or when no a priori information is available about
83
84 Chapter 4. Underdetermined Blind Source Separation
the mixtures.
BSS is also known as: blind array processing, signal copy, Independent Com-
ponent Analysis (ICA), and waveform preserving estimation. It has emerged over
the past decade to become an important area of signal processing, being signified
by an ongoing series of dedicate conferences [2] and appearing as special sessions in
many signal processing conferences. Useful reviews of BSS theories and algorithms
can be found in [1, 3–7].
BSS has many applications in areas that involve the processing of signals from a
sensor array, which offers spatial diversity. Typical examples of BSS are seen in: (i)
radar and sonar applications (separation and recognition of sources from antenna
arrays, robust source localization from ill–calibrated arrays [8]), (ii) communications
(multiuser detection in communication systems [9]), (iii) speech processing (speaker
separation, also called the “cocktail party” problem; speech recorded in the presence
of background noise and/or competing speakers, automatic voice recognition in
noisy acoustic environments [10]), and (iv) biomedical signal processing (separation
of Electroencephalogram (EEG) signals [11,12]).
BSS can be categorized into different classes according to the way the signal
structures are “forced”/conditioned using some particular criteria so that we can
restore the original structure of the source signals. These different classes are [7]:
probability structure forcing, spectral/time–coherence structure forcing, and TF
structure forcing.
When signals are nonstationary, the TF structure forcing approach was intro-
duced to achieve the separation, by Belouchrani and Amin [13, 14]. This approach
defines a STFD that combines both TF diversity and spatial diversity. The benefit
of using STFDs in an environment of nonstationary signals is the direct exploita-
tion of the information it offers due to the signal nonstationarity. In contrast to
BSS approaches using SOS and HOS (see [6] and references therein), this approach
allows the separation of Gaussian sources with identical spectral shape but with
different TF localization properties. Moreover, the effects of spreading the noise
4.1. Introduction 85
power, while localizing the source energy in the TF domain, amounts to increasing
the SNR [15]. Subsequent works have been carried out by Belouchrani, Amin and
their co–workers on the further development of this approach and its applications
to communications [16–18].
A drawback of most BSS algorithms is that they fail to separate sources in
situations where there are more sources than sensors [19]. Mathematically, the
invertability of mixing matrix, that is often used for separation, is no longer satis-
fied [20]. This challenging problem, known as the Underdetermined Blind Source
Separation (UBSS), has recently been studied in [20,21] where the discrete sources
were treated, in [19,22–24] where a priori knowledge of the probability density func-
tions of the sources was needed; and in [10] where disjoint orthogonality displayed
by STFT was exploited.
UBSS for nonstationary signals is investigated in this chapter. The TF structure
forcing approach above is chosen in order to take advantage of TF signal processing,
over the classical time–only and frequency–only signal processing. We will propose
a TF–UBSS algorithm that uses the main assumption of TF orthogonality. In
particular, TF orthogonality facilitates the selection of TF points lying on the TF
supports (signatures) of all source signals, hence a clustering process allows the
identification/separation of the TF signatures. We make a distinction here regarding
the previous approach using TF information [13, 14, 16–18] that we are now in the
underdetermined situation in which the previous approach is inapplicable due to
the non–invertability of the mixing matrix.
The chapter is organized as follows. Section 4.2 presents the data model and
assumptions, especially the notion of TF orthogonality. Section 4.3 recalls the
definition and properties of STFD matrices. Section 4.4 proposes the TF–UBSS al-
gorithm. Section 4.5 provides an illustrative demonstration of the usefulness of the
algorithm by some simulated experiments. Section 4.6 presents an enhanced version
of the algorithm using MWVD to achieve better selection of TF points. Section 4.7
provides another method to enhance the selection of TF points using image com-
86 Chapter 4. Underdetermined Blind Source Separation
ponent extraction. Several measurements for numerical performance evaluation are
given in Section 4.8. The last section is for concluding remarks and perspectives.
4.2 Signal model and assumptions
Assume that an n–dimensional vector s(t) = [s1(t), s2(t), . . . , sn(t)]T ∈ C(n×1)
corresponds to n nonstationary complex source signals si(t), i = 1, . . . , n. The
source signals are transmitted through a medium so that an array of m sensors
picks up a set of mixed signals represented by an m–dimensional vector x(t) =
[x1(t), x2(t), . . . , xm(t)]T ∈ C(m×1). Each observed signal xj(t), j = 1, . . . ,m, at
each time instance t has been mixed by the transmission medium which may have
also been corrupted by AWGN η(t) = [η1(t), η2(t), . . . , ηm(t)]T ∈ C(m×1). Consid-
ering the instantaneous linear mixture case, the observed signals can be modeled
as:
x(t) = As(t) + η(t), (4.1)
where A ∈ C(m×n) is called the mixing matrix. The instantaneity means that A
does not depend on t. The signal model is illustrated in Figure 4.1.
In the underdetermined situation, i.e. the UBSS problem, we have n > m. The
mixing matrix A is no longer invertible [20], thus any previous approaches in the
determined BSS problem (i.e. n ≤ m) is generally no longer applicable. Note that,
as m approaches infinity, the quantity (n−m) approaches zero (since n > m), thus,
UBSS becomes determined BSS. Therefore, one may approximately use the usual
methods in the determined BSS case to achieve the separation; in other words, this
happens when n−m is small compared to m.
We made the following two assumptions. The first assumption is usually made in
the context of BSS, and the second assumption is the main feature which facilitates
the proposal of our TF–UBSS algorithm.
As1) The column vectors of matrix A = [a1, a2, . . . , an] are assumed to be pairwise
4.2. Signal model and assumptions 87
SEPARATION
SOURCE
BLIND
!#" $ %& ' (
)*+,&!- . )/0
$ %& ' 1 2)*0
354687 9 :3;9 :<9=0>
?A@
?B?C
?D
E C F CF @EG@
F<HE H
?B
?A@?C
?D
Figure 4.1: UBSS: Schematic diagram.There are n unobserved source signals si(t), i = 1, . . . , n, to be separated fromm observed mixed signals xj(t), j = 1, . . . ,m, corrupted by AWGN. The under-determined case corresponds to n > m.
linearly independent, i.e., for any i, j ∈ 1, 2, . . . , n and i 6= j, ai and aj are
linearly independent. Obviously, if two sources, for example s1(t) and s2(t),
have linearly dependent vectors, i.e. a2 = αa1, their separation is, then,
inherently impossible since we can write
x(t) = As(t) + η(t), (4.2)
where A = [a1, a3, . . . , an] and s(t) = [s1(t) + αs2(t), s3(t), . . . , sn(t)]T . It
is also known that BSS is only possible up to an unknown scaling and an
unknown permutation [25]. We take the advantage of this indeterminacy to
assume, without loss of generality, that the column vectors of A have a unit–
norm, that is ‖ai‖ = 1 for all i.
As2) The sources are assumed to have different structures and localization proper-
ties in the TF domain. More precisely, we assume the sources to be orthogonal
in the TF domain (Figure 4.2) as stated in the following definition:
Definition 4.1 (TF orthogonality).
88 Chapter 4. Underdetermined Blind Source Separation
Let S1(t, f) and S2(t, f) be TFDs of two source signals s1(t) and s2(t), respec-
tively. Let Ω1 and Ω2 be the corresponding TF supports of S1 and S2, that
is S1(t, f) 6= 0 if and only if (t, f) ∈ Ω1,
S2(t, f) 6= 0 if and only if (t, f) ∈ Ω2.(4.3)
The sources s1(t) and s2(t) are said to be orthogonal in the TF domain
if the following satisfies:
Ω1 ∩ Ω2 = ∅. (4.4)
time
tfrequen y f
1 2Time-frequen y orthogonality
Figure 4.2: TF orthogonality.The TF supports of two sources are disjoint in the TF domain.
The above definition can be applied to any TFDs. It is clear that the TF
orthogonality is too restrictive and will almost never be satisfied exactly in practice.
Fortunately, only approximate orthogonality, said quasi–orthogonality, is needed to
achieve source separation, as will be shown in Section 4.5.2. Note that the source TF
orthogonality can be considered as a particular type of source sparse decomposition,
which can be used to achieve source separation [22,26,27].
A physical example of TF (quasi)–orthogonality is observed in a musical per-
formance; several musical instruments, e.g. a base–guitar and a lead–guitar, play
simultaneously but create musical sounds with different instantaneous frequency
laws. These laws can have some overlapping in the TF domain, representing the
4.3. Spatial time–frequency distributions 89
quasi situation. This happens when the very high (frequency) notes produced by
the base–guitar coincide in frequency with the very low (frequency) notes produced
by the lead–guitar, and these notes are played at the same duration of time.
Two arguments may be given here regarding the assumption of TF orthogonal-
ity, before going further introducing the TF–UBSS algorithm. Firstly, one would
think that if the sources are TF orthogonal then simple TF masking (if the source
TF signatures are known) and TF synthesis in the TF domain would be sufficient
to recover the source signals without using any sophisticated algorithm. However,
in the context of blind separation, source TF signatures are unknown. The pro-
posed algorithm will allow us to extract the source TF signatures from the spatial
information offered by the sensors; hence, the source signals. Secondly, one may
use an image processing technique to achieve a classification of different source TF
components (as distinguished to source TF signatures) as has been done in [28].
With the obtained TF components, this classification still fails to obtain the source
TF signatures because it is well possible that a source TF signature can have mul-
tiple TF components. Our algorithm provides necessary information to allow for
the determination of which TF components belong to one particular source, thus
allowing blind separation of multicomponent source signals.
4.3 Spatial time–frequency distributions
We provide here some definitions that will be used throughout the chapter.
Definition 4.2 (Spatial TFD [13]).
Let z(t) be a vector containing n signals z1(t), . . ., zn(t); z(t) = [z1(t), . . . , zn(t)]T ∈C(n×1). The Spatial Time–Frequency Distribution (STFD) matrix is math-
ematically defined as
Dzz(t, f)∆=
∞∑l=−∞
∞∑k=−∞
φ(k, l)z(t + k + l)zH(t + k − l)e−j4πfl, (4.5)
90 Chapter 4. Underdetermined Blind Source Separation
where t and f represent the time index and the frequency index, respectively, the
superscript (H) denotes the complex conjugate transpose operator, and φ(m, l) is a
TFD time–lag kernel. The matrix Dzz(t, f) ∈ C(n×n) varies with respect to t and f .
Its (t–f) elements are obtained from the TFD as:
[Dzz(t, f)]ij = Dzizj(t, f)
=∞∑
l=−∞
∞∑k=−∞
φ(k, l)zi(t + k + l)z∗j (t + k − l)e−j4πfl, i, j = 1, 2, . . . , n (4.6)
with z∗j being the complex conjugate of zj.
Note that Dzz(t, f) is a matrix; when evaluated at a TF point (to, fo), its ele-
ments are the values of Dzizj(to, fo) using (4.6).
Next, we will define the notion of cross– and auto–source STFDs, which are
slightly modified from those defined in [16] for more clarity. Before doing so, let
us recall the notions of “auto–term” and “cross–term” in the literature of TFSP.
Given a signal with multiple IF components, an auto–term TF point in the TF
representation of this signal represents the “true” energy concentration of the signal
at that point in time and frequency. A cross–term TF point, on the other hand,
represents a “ghost” energy concentration of the signal though the concentration
may visually appear high at this point the TF representation. This “ghost” effect
comes from the bilinearity of the TFD that applies on the signal among its IF
components [29].
Above, the TFD is applied on only one signal. In our context, we consider
several source signals, and each of which may have multiple IF components.
Definition 4.3 (Cross– and auto–source STFD).
Let z1(t) and z2(t) be two different source signals with possibly multiple IF compo-
nents, and that they be displayed on the TF representation by a TFD ρ(t, f) through
the computation of the STFD Dzz(t, f) where z(t) = [z1(t)z2(t)]T .
(a) An auto–source TF point (ta, fa) of a source zi(t), i = 1, 2, is a point in
4.3. Spatial time–frequency distributions 91
the TF representation where the energy concentration of zi(t) is evaluated by
the auto–TFD ρzizi(ta, fa).
(b) A cross–source TF point (tc, fc) between source z1(t) and z2(t) is a point
in the TF representation where the energy concentration is evaluated by the
cross–TFD ρz1z2(tc, fc)1.
(c) For an auto–source TF point (ta, fa), the STFD matrix computed at that point
is called an auto–source STFD matrix, denoted by Dzz(ta, fa) .
(d) For a cross–source TF point (tc, fc), the STFD matrix computed at that point
is called a cross–source STFD matrix, denoted by Dzz(tc, fc).
A few remarks can be made according to the above definition. For simplicity,
hereafter, we use “point” to mean “TF point”.
• The energy concentration at an auto–source point can be “true” if zi(t) is
monocomponent but that can also be “ghost” if multicomponent. The lat-
ter means that the auto–source point coincides with the cross-term point if
the source is multicomponent. This will be illustrated in Experiment 4.5.2
(Figure 4.8.l).
• Since the diagonal elements of the matrix Dzz(t, f) are evaluated by the auto–
TFD, this STFD matrix at an auto–source point, Dzz(ta, fa), becomes an
auto–source STFD matrix and that it is quasi-diagonal (i.e. its diagonal
entries are close to one).
• Since the off–diagonal elements of the matrix Dzz(t, f) are evaluated by the
cross–TFD, this STFD matrix at a cross–source point, Dzz(tc, fc), becomes a
1The cross–TFD is defined, similar to the auto–TFD as in (2.13), as below:
ρz1z2(t, f) ∆=∫∫∫ ∞
−∞ej2πν(u−t) Γ (τ, ν) z1(u +
τ
2)z∗2(u− τ
2) e−j2πfτ dν du dτ (4.7)
When z1(t) = z2(t), the cross–TFD becomes the auto–TFD.
92 Chapter 4. Underdetermined Blind Source Separation
cross–source STFD matrix and that it is quasi–off–diagonal. (i.e. its diagonal
entries are close to zero).
Applying (4.5) to the linear data model (4.1), assumed a noise–free environment,
leads to the following expression:
Dxx(t, f) = ADss(t, f)AH , (4.8)
where Dss(t, f) and Dxx(t, f) are the source and mixture STFD matrices, respec-
tively. Further from the above remarks, since the sources are assumed to be TF
orthogonal, the diagonal entries of Dss(t, f) are:
• all equal to zero except for one value, if the STFD matrix Dss(t, f) is evaluated
at an auto–source point since only one source active at this point.
• all equal to zero, if the STFD matrix Dss(t, f) is evaluated at a point other
than an auto–source point.
Therefore, if Ωi is the TF support of source signal si(t), the following is achieved:
Dxx(t, f) = Dsisi(t, f)aia
Hi , ∀ (t, f) ∈ Ωi. (4.9)
It is the particular structure in (4.9) that will be used for our TF–UBSS.
We also note that, if, on the other hand, the sources do not satisfy the TF
orthogonality assumption such that at an auto–source point there are k sources
active (i.e. there is an overlap, on the TF representation, of the TF signatures of
these sources), then among the diagonal entries of Dss(t, f) there will have exactly
k values different from zero if k ≤ m, or at maximum m values different from zero
if k > m. This observation may be used to provide a test on TF orthogonality,
and further to analyse TF–nonorthogonality. However, detailed treatments of TF
non–orthogonality, e.g. the degree of acceptable non–orthogonality for successfully
achieving UBSS, is not carried out in this thesis and is subject to future research;
this issue is of importance when dealing with speech signals rather LFM signals.
4.4. TF-UBSS algorithm 93
4.4 TF-UBSS algorithm
Thanks to the structure in (4.9), the following observation is deduced for two auto–
source (t1, f1) and (t2, f2) corresponding to the same source si(t):Dxx(t1, f1) = Dsisi(t1, f1)aia
Hi ,
Dxx(t2, f2) = Dsisi(t2, f2)aia
Hi .
(4.10)
The above observation implies that Dxx(t1, f1) and Dxx(t2, f2) have the same prin-
cipal eigenvector ai. Therefore, all the auto–source points associated with the same
principal eigenvector belong to the TF support of one particular source signal.
This leads to the principal idea of our TF–UBSS algorithm as follows. We
first obtain only auto–source points from the TF representation. If we are able to
cluster auto–source points on the TF domain into different sets, each associating
with one principal eigenvector, then these sets represent different TF signatures
corresponding to different underlying source signals. And if each source signal can be
recovered from its set, we are able to achieve the UBSS. In particular, corresponding
to a principal eigenvector of one particular source, the estimated TFD values of
this source at its auto–source points are obtained as the principal eigenvalues of
the STFD matrices at those points. Hence, we can use a TF synthesis method to
recover the source waveform.
4.4.1 Separation algorithm
The proposed TF–UBSS algorithm includes four main procedures as shown in Fig-
ure 4.3 and its schematic diagram is illustrated in Figure 4.4. Details of these
procedures are given next.
94 Chapter 4. Underdetermined Blind Source Separation
TF–UBSS algorithm
Procedure 1: STFD computation and noise thresholding
Procedure 2: Auto–source TF point selection
Procedure 3: Clustering and source TFD estimation
Procedure 4: Source signal synthesis
Figure 4.3: TF-UBSS algorithm: Procedures.
Sele tionAuto-termDxx(t; f)x(t) = As(t)
ClassierWVDSTFD Dsnsn(t; f)C2 C1
Cn
f(ta; fa)gDs1s1(t; f) s1(t)s2(t)sn(t)Ds2s2(t; f) TF-SynTF-Synget TFD TF-Synget TFD
get TFDFigure 4.4: TF-UBSS algorithm: Schematic diagram.
4.4.1.1 STFD computation and noise thresholding
Given L observation vectors x(1), . . . ,x(L), the STFD matrices Dxx(t, f) defined
according to (4.5), can be estimated using time–lag domain discrete implementa-
tion [30] as below:
Dxx(l, k) =M∑
p=−M
M∑q=−M
g(q − l, p)x(q + p)xH(q − p) e−j4πpk/L, (4.11)
4.4. TF-UBSS algorithm 95
where g(l, p) is a discrete time–lag kernel, M = (L − 1)/2, and l = 1, . . . , L. The
elements of Dxx(l, k) are obtained from the TFD as:
[Dxx(l, k)
]ij
= Dxixj(l, k)
=M∑
p=−M
M∑q=−M
g(q − l, p) xi(q + p)x∗j(q − p) e−j4πpk/L, i, j = 1, . . . ,m. (4.12)
In the later simulations (Experiment 1 and Experiment 2), we will use the WVD for
computing the STFD matrices. The WVD of an analytic signal x(t) is defined as
in (2.11). Its discrete implementation is of the form in (4.12) without the time–lag
kernel g(l, p).
These STFD matrices are next processed to extract the source signals. In order
to reduce the computational complexity, by processing only “significant” STFD
matrices, a noise thresholding step is then carried out for removing those points
with negligible energy. More precisely, a threshold ε1 (typically, ε1 = 0.05 of the
point with maximum energy) is used to keep only the points (ts, fs) with sufficient
energy:
If: ‖Dxx(ts, fs)‖ > ε1
then: keep (ts, fs)(4.13)
4.4.1.2 Auto–source TF point selection
The second procedure of the algorithm consists of separating the auto–source points
from cross–source points using an appropriate testing criterion.
In the determined case, where the number of sensors is greater than or equal to
the number of sources and the mixing matrix A is of full–column rank, a selection
procedure that exploits the off–diagonal structure of the cross–source STFD matri-
ces has been proposed in [16]. This selection procedure proceeds through two steps
as follows:
96 Chapter 4. Underdetermined Blind Source Separation
– Data whitening : Let W denote an m × n matrix such that (WA)(WA)H =
UUH = I, i.e. WA is an n × n unitary matrix. Matrix W is referred to
as the whitening matrix since it whitens the signal part of the observations.
Pre– and post–multiplying the STFD matrices Dxx(t, f) by W lead to the
whitened STFD matrices:
Dxx(t, f) = WDxx(t, f)WH = UDss(t, f)UH (4.14)
In practice, W is often computed as an inverse squared root of the sample
estimate covariance matrix of the observation.
– Testing : Given a whitened cross–source STFD matrix Dxx(tc, fc), we have:
trace Dxx(tc, fc) = traceUDss(tc, fc)U
H
= trace Dss(tc, fc)
≈ 0 (4.15)
Based on this observation, the following test is given:
If:trace
Dxx(tc, fc)
norm
Dxx(tc, fc)
< ε2
then: (tc, fc) is a cross–source point
(4.16)
where the threshold ε2 is a positive scalar no greater than 1 (typically ε2 =
0.8).
Contrary to the determined case explained above, the matrix U in the under-
determined case is non–square with more columns than rows, and consequently
UHU 6= I represents the projection matrix onto the row space of U. Therefore,
Eq. (4.15) becomes only an approximation; a good one if (m − n) is “small” as
observed in our simulation results (see Figure 4.7 of Experiment 1 and Figure 4.8
of Experiment 2 in Section 4.5).
4.4. TF-UBSS algorithm 97
Another method, alternative to the above approximation projection method,
consists of exploiting the sources TF orthogonality. Under this assumption, each
auto–source STFD matrix is of rank one, or at least has one “large” eigenvalue com-
pared to its other eigenvalues. Therefore, one can use rank selection criteria, such as
Minimum Description Length (MDL) or Akaike Information Criterion (AIC) [31],
to select auto–source points as those corresponding to STFD matrices of selected
rank equal to one. For simplicity, we use the following criterion (see Figure 4.11 of
Experiment 3 in Section 4.6):
If:
∣∣∣∣∣∣λmax
Dxx(t, f)
norm
Dxx(t, f)
− 1
∣∣∣∣∣∣ > ε2
then: (t, f) is a cross–source point
(4.17)
where ε2 is a small positive scalar (typically, ε2 = 0.3), and λmax · represents the
largest eigenvalue of the matrix in the bracket.
Comparing the above two methods for auto–source point selection based on
approximation projection and TF orthogonality for the underdetermined case shows
a similar performance (see Figure 4.11).
4.4.1.3 Clustering and source TFD estimation
Once the auto–source points have been selected, a clustering procedure based on the
sources spatial directions/signatures is performed. This clustering is based on the
observation that two STFD matrices corresponding to the same source signal have
the same principal eigenvector. Moreover, the corresponding principal eigenvalues
are given by the desired source TFD. This implies that if we apply an appropriate
clustering procedure on the set auto–source points, we will be able to obtain the
separate TF signatures of the source signals. Specifically, we consider the following
steps:
– For each auto–source point, (ta, fa), compute the main eigenvector, a(ta, fa),
98 Chapter 4. Underdetermined Blind Source Separation
and its corresponding eigenvalue, λ(ta, fa), of Dxx(ta, fa).
– As the vectors a(ta, fa) are estimated up to a random phase ejφ, φ ∈ [0, 2π),
we force them to have, without loss of generality, their first entries real and
positive. These vectors are then clustered into different classes Ci. Mathe-
matically, a(ti, fi) and a(tj, fj) belong to the same class if:
d(a(ti, fi), a(tj, fj)) < ε3 (4.18)
where ε3 is a properly chosen positive scalar and d is a distance measure
(different strategies for choosing the threshold ε3 and the distance d or even
the clustering method can be found in [32]). As an example, we use a dis-
tance measure, in the simulated experiments in Section 4.5, according to their
angles:
d(a1, a2) = arccos(aTi aj), i 6= j (4.19)
where a = [Re(a)T ; Im(a)T ]T and ‖a‖ = 1.
– Set the number of sources equal to the number of classes and, for each source
si (i.e. each class Ci), estimate its TFD as:
Dsisi(t, f) =
λ(ta, fa), if (t, f) = (ta, fa) ∈ Ci
0, otherwise. (4.20)
4.4.1.4 Source signal synthesis
Having obtained the source TFD estimates Dsisi, we then use an adequate source
synthesis procedure to estimate the source signals si(t) (i = 1, . . . , n). The recovery
of the waveform (in time) of a signal from its TFD is made possible thanks to the
following inversion property of the WVD [15]
x(t) =1
x∗(0)
∫ ∞
−∞ρwvd
x (t
2, f) ej2πft df , (4.21)
4.4. TF-UBSS algorithm 99
which implies that the signal can be reconstructed to within a complex exponential
constant ejα = x∗(0)/|x(0)| given |x(0)| 6= 0.
Some TF synthesis algorithms can be found in [15,33–35]. Among them, [33] pro-
vides a well–known synthesis algorithm recovering a signal from its WVD estimate.
Since we use WVD to compute our STFD matrices, we opt to use this synthesis
algorithm for recovering our original sources. Below, this algorithm is summarized
to assist the understanding of our UBSS algorithm (without any contribution of the
author of this thesis).
Given the TFD estimate of source s(t), denoted by Dss(t, f), find the signal
s(t) that its WVD, denoted by ρwvds (t, f), best approximates Dss(t, f) in the least
square sense, i.e. minimizing the following:
J(s) =
∫ ∞
−∞
∣∣∣Dss(t, f)− ρwvds (t, f)
∣∣∣2 df. (4.22)
The above minimization leads to the discrete computation of the synthesized signal
s(l), l = 0, . . . , L− 1, as below:s(2k) = se(k), for k = 0, . . . , Le − 1; Le = b(L + 1)/2c
s(2k − 1) = so(k), for k = 1, . . . , Lo; Lo = bL/2c, (4.23)
where se = [se(0)se(1) · · · se(Le − 1)]T and so = [so(1)so(2) · · · se(Lo)]T are the nor-
malized principal eigenvectors of the matrices Ce and Co, representing the even and
odd samples of s(k). The elements of these matrices are computed as:
ce(q + 1, p + 1) = y(q + p, q − p) + y∗(q + p, p− q),
for q, p = 0, . . . , Le − 1
co(q, p) = y(q + p + 1, q − p) + y∗(q + p + 1, p− q),
for q, p = 1, . . . , Lo
, (4.24)
where y(l, p) is the discrete inverse Fourier transform of Dss(t, f). If the phase of
100 Chapter 4. Underdetermined Blind Source Separation
the recovered signal is important, the phase can be corrected using the original
signal s(t) by computing:αe = tan−1[<∑Le−1
k=0 s(2k)s∗e(k)
/=∑Le−1
k=0 s(2k)s∗e(k)]
αo = tan−1[<∑Lo
k=1 s(2k − 1)s∗o(k)
/=∑Lo
k=1 s(2k − 1)s∗o(k)] (4.25)
then replacing se(k) and so(k) in (4.23) by se(k)ejαe and so(k)ejαo respectively.
Above, <· and =· denote the real part and imaginary part, respectively.
4.4.2 Discussion
It is essential to address the following issues regarding the above proposed algorithm
for UBSS.
4.4.2.1 Underdeterminacy
In the above description of the procedures involved in the proposed algorithm, we
do not use the information of the number of signals (n) and the number of sensors
(m). Therefore, our TF–UBSS algorithm is general in the sense that it is not only
specific to UBSS but it can also be used for determined BSS. However, in this work,
we only provide results for UBSS since it imposes a challenge in the area of BSS as
we have explained in the introductory section.
In the simulated experiments that will be shown later, we choose m = 2. Obvi-
ously, to exploit the spatial diversity offered by a sensor array, the minimum value
for m is two. This, however, is the most difficult case given a fixed number of
signals, contrast to an intuition that m = 2 is the simplest. This is due to the
fact that more sensors will provide more spatial diversity, hence more information.
On the other hand, we only use n = 3 source signals, as is the simplest case for
UBSS given that m = 2, in our simulation. This selection serves our purpose as
to illustrate the new approach, rather than to provide a very detailed performance
4.4. TF-UBSS algorithm 101
analysis on the approach.
4.4.2.2 TF orthogonality
It is important to have orthogonal sources in the TF domain in order to achieve
the blind separation of the sources. It is clear that this is too restrictive and
will almost never be satisfied exactly in practice. Nonetheless, as shown in the
simulation Section 4.5, it suffices that the source signals may need only to satisfy
a TF quasi–orthogonality condition for the signal separation to be achieved. The
term “quasi” implies that most of the energy of one source is localized in the TF
region disjoint from the TF regions of all other sources, as illustrated on Figure 4.5.
time
tfrequen y f1 2
Time-frequen y quasi-orthogonality
Figure 4.5: TF quasi–orthogonality.Small overlapping of the two TF supports is allowable (Ω1 ∩ Ω2 ≈ ∅); i.e. mostof the energy of one source is localized in the TF region disjoint from the TFsupport of all other sources.
4.4.2.3 Choice of the TFD
We have chosen the WVD to compute the STFD matrices for our simulation. The
reason stems for, first, the fact that it is an invertible TFD up to a constant
phase [15]; and second, the WVD is the optimal TFD for LFM signals (used in
the simulations). In general, the choice of the TFD should be made according to
the nature of the application of interest and the properties desired in the TFD, as
102 Chapter 4. Underdetermined Blind Source Separation
explained in [15].
4.4.2.4 Noise thresholding
The threshold used for removing the noisy points can be chosen based on the SNR
and the possible structure of the mixed signals. The noise thresholding, however,
is used mainly for the benefit of reducing the computational complexity, and so is
not a critical factor in the proposed algorithm.
4.4.2.5 Auto–source point selection
We have proposed three selection criteria to separate the auto–source points from
the cross–source points in the TF plane. These criteria require a good choice of
the thresholding parameter as well as the signal TFD (a good choice of the TFD is
proposed in Section 4.6).
4.4.2.6 Vector clustering
A simple algorithm for vector clustering was used in the simulations in order to illus-
trate the feasibility of UBSS. More sophisticated algorithms (see [32] and references
therein) should be applied to achieve robust separation.
4.4.2.7 Number of sources
We have observed in the experiments that the number of classes, obtained from
the clustering procedure, was greater than the actual number of sources. Simple
thresholding scheme, based on energy leveling, was used to eliminate the classes with
insignificant energy compared to others. These classes may or may not be considered
as noise, depending on the nature of the sources in the particular application of
interest. At this stage, problems may arise if one or more sources have much
4.4. TF-UBSS algorithm 103
higher energy than others, in which the proposed UBSS algorithm may be used in
conjunction with a deflation technique [36].
4.4.2.8 TF synthesis
The source signatures, after a proper classification procedure, can be reconstructed
to obtain their original waveforms through the use of TF synthesis. We have applied
in our simulations a classical but seminal algorithm (without any TF masking),
proposed by Boudreaux–Bartels et al. [33]. Other synthesis algorithms can be found
in [15, 34, 35]. The successful recovery of original signal waveforms depends on the
signal type, choice of TFD, the robustness of vector clustering procedure, and the
performance of the TF synthesis algorithm itself.
On the other hand, instead of using TF synthesis, we may apply the time–
varying notched filter approach as sketched in Figure 4.6 in which selection block is
composed of all the steps from Procedure 4.4.1.1 to Procedure 4.4.1.3. Information
of notched filter design can be found in [37]. This approach is useful when the TF
synthesis algorithm corresponding to the TFD in used is not yet available.
4.4.2.9 Computational complexity
The total cost of computation is broken down into separate costs corresponding to
different procedures in the proposed algorithm. Major contributions to the total
cost Ctotal come from the computations of (i) STFD matrices (C1), (ii) the Singular
Value Decomposition (SVD) of the STFD matrices for separating auto–source points
from cross–source points (C2), (iii) clustering (C3), and of source synthesis (C4).
Note that, we use the values of SVD already obtained for the estimation of source
TFDs.
Ctotal ≈ C1 + C2 + C3 + C4 (4.26)
Denote n, m, L, Na, Nc the number of source signals, sensors, signal samples,
auto–source points, and cross–source points, respectively. CL is the cost for the
104 Chapter 4. Underdetermined Blind Source Separation
Time-varyingNotched filter
Time-varyingNotched filter
Time-varyingNotched filter
InterpolationFilter
InterpolationFilter
InterpolationFilter
WVDSTFD C2 C1Cn
f(ta; fa)g
Dsnsn(t; f)Ds2s2(t; f)get TFDget TFDget TFD
Tra eTest Classierx(t) Dxx(t; f) Ds1s1(t; f)
x1x2xm ~s1 + ~s2 + + ~sn
fn(t)f2(t)f1(t) sn~sn~s1 + + ~sn1
s2~s2~s1 + ~s3 + + ~sns1~s1~s2 + + ~sn
SELECTIONBLOCK
x(t) = As(t)
Figure 4.6: TF-UBSS algorithm using notch filters: Schematic diagram.
TFD computation of a signal of length L. Following are the associated costs:
C1 = CL ×m(m + 1)
2(4.27)
C2 = (Na + Nc)×O(m3) (4.28)
C3 ≈Na(Na + 1)
2(4.29)
C4 ≈ n×O(L3) (4.30)
Note that the computation of CL depends on the TFD method, signal length and
the number of FFT points used. If a sophisticated clustering method, then C3 is
expected to increases. Overall, C2 and C3 are the most expensive computation due
to the high numbers of auto–source points and cross–source points present in the
TF representation; obviously, these numbers are dependent on the number of source
signals to be separated.
4.5. Experiments 105
4.5 Experiments
The algorithm in Section 4.4 is experimentally tested for the following two situa-
tions: (i) with TF orthogonal sources, and (ii) with TF almost orthogonal sources.
In both experiments, a uniform linear array of m = 2 sensors, having half wave-
length spacing, is used. It receives signals from n = 3 independent source signals,
each of length L = 128, in the presence of AWGN with SNR level of 20 dB. The
source signals arrive at different angles, 30, 45 and 60, respectively. The WVD
was used to compute the STFD matrices.
4.5.1 Experiment 1: TF orthogonal sources
The sources are chosen to be all monocomponent LFM signals (Figure 4.7.a–c) and
are well separated in the TF domain (Figure 4.7.d–f). The choice of LFM signals is
motivated, but not limited, from the practical application of such signals in radar
application [38] and communications [39]. The ‘noisy’ points appearing in the data
mixture (Figure 4.7.g) are first removed using energy thresholding (Figure 4.7.h;
there seemed to be no difference due to a visual effect, however, a significant num-
ber of points were indeed removed). The cross-source points are removed using
threshold ε3 (Figure 4.7.i). After the vector classification procedure with ε2, three
classes containing three TF signatures representing the three original source sig-
nals are separated (Figure 4.7.j–l). Finally, estimates of the three original source
waveforms are obtained (Figure 4.7.m–o), and their corresponding TF representa-
tions (Figure 4.7.p–r), resembling the original sources (Figure 4.7.a–c) and their TF
representations (Figure 4.7.d–f), respectively.
By comparing the original with the estimates of source waveforms, it is concluded
that the proposed UBSS algorithm is successful. However, an amplitude fading at
the two ends of the recovered signals is due to the poor TFD energy concentration
in the vicinity of the TF support boundaries. In addition, though significant cross–
106 Chapter 4. Underdetermined Blind Source Separation
source points have been removed, there remain a number of them in the classified
TF signatures.
4.5.2 Experiment 2: TF quasi–orthogonal sources
As stated previously, we show here an example where the separation is achievable
in a quasi–orthogonal condition. A combination of 2 monocomponent LFM signals,
s1(t) and s2(t), and 1 multicomponent LFM signal, s3(t), are used for the testing
source signals (Figure 4.8.a–c). Similar to Experiment 1, the auto–source points
are obtained, so are the separation of TF signatures, and finally the estimates of
source signal waveforms (Figure 4.8.d–r).
4.5. Experiments 107
0 20 40 60 80 100 120 140−1.5
−1
−0.5
0
0.5
1
1.5
signal s1(t)
(a) s1(t)0 20 40 60 80 100 120 140
−1.5
−1
−0.5
0
0.5
1
1.5
signal s2(t)
(b) s2(t)0 20 40 60 80 100 120 140
−1.5
−1
−0.5
0
0.5
1
1.5
signal s3(t)
(c) s3(t)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
20
40
60
80
100
120
WVD of signal s1
frequency (Hz)
time (
sec)
(d) WVD of s1(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
WVD of signal s2
frequency (Hz)
time (
sec)
(e) WVD of s2(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
WVD of signal s3
frequency (Hz)
time (
sec)
(f) WVD of s3(t)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
20
40
60
80
100
120
WVD of mixture from sensor 1
frequency (Hz)
time (
sec)
(g) WVD of x1(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
TFD of auto−terms and cross−terms only
frequency (Hz)
time (
sec)
(h) auto & cross points0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
TFD of auto−terms only
frequency (Hz)
time (
sec)
(i) auto–source points
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
20
40
60
80
100
120
TF signature of class 3 / 3
frequency (Hz)
time (
sec)
(j) TF signature of s1(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
TF signature of class 1 / 3
frequency (Hz)
time (
sec)
(k) TF signature of s2(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
TF signature of class 2 / 3
frequency (Hz)
time (
sec)
(l) TF signature of s3(t)
0 20 40 60 80 100 120 140−1.5
−1
−0.5
0
0.5
1
1.5Synthesized signal from TF signature 3 / 3
(m) s1(t)0 20 40 60 80 100 120 140
−1.5
−1
−0.5
0
0.5
1
1.5Synthesized signal from TF signature 1 / 3
(n) s2(t)0 20 40 60 80 100 120 140
−1.5
−1
−0.5
0
0.5
1
1.5Synthesized signal from TF signature 2 / 3
(o) s3(t)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
20
40
60
80
100
120
WVD of synthesized signal from TF signature 3 / 3
frequency (Hz)
time (
sec)
(p) WVD of s1(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
WVD of synthesized signal from TF signature 1 / 3
frequency (Hz)
time (
sec)
(q) WVD of s2(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
WVD of synthesized signal from TF signature 2 / 3
frequency (Hz)
time (
sec)
(r) WVD of s3(t)
Figure 4.7: Experiment 1: TF-UBSS algorithm with TF orthogonality.Three monocomponent LFM signals s1(t), s2(t) and s3(t) (a–c), being the sourcesignals, were tested. The recovered source signals shown in (m–o) indicated thesuccess of the UBSS.
108 Chapter 4. Underdetermined Blind Source Separation
0 20 40 60 80 100 120 140−1.5
−1
−0.5
0
0.5
1
1.5
signal s1(t)
(a) s1(t)0 20 40 60 80 100 120 140
−1.5
−1
−0.5
0
0.5
1
1.5
signal s2(t)
(b) s2(t)0 20 40 60 80 100 120 140
−1.5
−1
−0.5
0
0.5
1
1.5
signal s3(t)
(c) s3(t)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
20
40
60
80
100
120
WVD of signal s1
frequency (Hz)
time (
sec)
(d) WVD of s1(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
WVD of signal s2
frequency (Hz)
time (
sec)
(e) WVD of s1(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
WVD of signal s3
frequency (Hz)
time (
sec)
(f) WVD of s1(t)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
20
40
60
80
100
120
WVD of mixture from sensor 1
frequency (Hz)
time (
sec)
(g) WVD of x1(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
TFD of auto−terms and cross−terms only
frequency (Hz)
time (
sec)
(h) auto & cross points0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
TFD of auto−terms only
frequency (Hz)tim
e (se
c)
(i) auto–source points
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
20
40
60
80
100
120
TF signature of class 2 / 3
frequency (Hz)
time (
sec)
(j) TF signature of s1(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
TF signature of class 1 / 3
frequency (Hz)
time (
sec)
(k) TF signature of s2(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
TF signature of class 3 / 3
frequency (Hz)
time (
sec)
(l) TF signature of s3(t)
0 20 40 60 80 100 120 140−1.5
−1
−0.5
0
0.5
1
1.5Synthesized signal from TF signature 2 / 3
(m) s1(t)0 20 40 60 80 100 120 140
−1.5
−1
−0.5
0
0.5
1
1.5Synthesized signal from TF signature 1 / 3
(n) s2(t)0 20 40 60 80 100 120 140
−1.5
−1
−0.5
0
0.5
1
1.5Synthesized signal from TF signature 3 / 3
(o) s3(t)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
20
40
60
80
100
120
WVD of synthesized signal from TF signature 2 / 3
frequency (Hz)
time (
sec)
(p) WVD of s1(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
WVD of synthesized signal from TF signature 1 / 3
frequency (Hz)
time (
sec)
(q) WVD of s2(t)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
20
40
60
80
100
120
WVD of synthesized signal from TF signature 3 / 3
frequency (Hz)
time (
sec)
(r) WVD of s3(t)
Figure 4.8: Experiment 2: TF-UBSS algorithm with TF quasi–orthogonality.A mixture of two monocomponent and one multicomponent LFM signals s1(t),s2(t) and s3(t) (a–c), being the source signals, were tested. s1(t) and s2(t) overlapin TF domain. Source s3(t) was not falsely separated into two monocomponentsources.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-1
Chapter 5. Local Buckling Behaviour of Unstiffened Elements at Elevated Temperatures
5.1 General
The major advantage in the use of cold-formed steel members is their high strength
to weight ratio. Due to the recent development of advanced technologies, thin (0.42
mm ≤ t ≤ 1.2 mm) and high strength (yield strength ≥ 550 MPa) cold-formed steels
are commonly used in many applications. Thus, the local buckling associated with
thin steels is a significant factor to be considered in the design of cold-formed steel
members as reflected by extensive research undertaken at ambient temperature.
However, the local buckling behaviour of cold-formed steel members at elevated
temperatures is not well understood even though the significance of fire safety
design is increasingly being recognised due to large losses of lives and properties.
Therefore, detailed experimental and numerical studies on the local buckling
behaviour of cold-formed steel members subject to uniform axial compression at
elevated temperatures were undertaken in this research project. Applicability of the
effective width design method commonly used at ambient temperature to predict the
ultimate strength was investigated for elevated temperatures.
As the first phase of this research project, the local buckling behaviour of unstiffened
elements in a series of unlipped channel members made of low and high strength
steels was investigated including the variation of the effective width at elevated
temperatures. For these purposes, experimental and finite element studies were
undertaken with great attention to testing procedures and relevant numerical factors.
The current design rules were also simply modified taking into consideration of the
reduction factors of yield strength and elasticity modulus, and the suitability of this
modification was investigated by comparing with experimental and numerical
results. This chapter presents the details of the experimental and finite element
analyses including accurate material behaviour and all other significant factors.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-2
5.2 Experimental Investigations
5.2.1 Test Facilities
A specially designed electrical furnace with a maximum temperature of 1200ºC was
built to conduct experimental investigations simulating fire conditions. Temperatures
in the furnace were controlled by a microcomputer based temperature indicating
system, and the temperature increasing rate was adjustable. Two thermo-couples
were installed in the furnace for the purposes of monitoring air temperature and
safety lock-up while another two more thermo-couples were set up at the top and
bottom of test specimens to monitor specimen temperatures. Four glow bars were
used to emit heat evenly, and the heating system was found to be very efficient as
shown by the measured air temperature inside the furnace and specimen temperature.
A small hole on the front door of the furnace allowed limited observation of the
buckling behaviour during testing. Unfortunately, the behaviour of columns was
difficult to see during the tests due to the darkness inside the furnace.
A 300 kN capacity Tinius Olsen universal testing machine (UTM) was used to apply
the axial compression load to the test specimens with fixed ends. An in-built
extensometer was used to measure axial shortening while a displacement transducer
was used to measure the out-of-plane displacement at the mid-height of specimens.
The overall test set-up is shown in Figures 5.1(a) and (b).
5.2.2 Test Specimens
A total of 30 tests including 6 types of geometry and 5 different temperatures were
conducted to investigate the local buckling behaviour of unstiffened flange elements
of unlipped channel members subject to axial compression at elevated temperatures
(see Figure 5.2). All the specimens were carefully designed using a finite strip
analysis program THINWALL so that their behaviour was governed by the local
buckling of unstiffened flange elements. They were made of G550 (minimum yield
strength of 550 MPa) and G250 steels (minimum yield strength of 250 MPa) with
base metal thicknesses of 0.936 and 1.56 mm, respectively.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-3
(a) Schematic diagram
(b) Overall test set-up
Figure 5.1 Test Set-up
Insulation
Furnace
Glow bars
Displacement transducer
Test Specimen
Fixed end support
UTM base
Thermo-couples
Glow bars
Displacement transducer
Test Specimen
Fixed end support
Compression load application
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-4
Figure 5.2 Section Geometry – Unlipped Channel
The base metal thicknesses were obtained by measuring total and coating thicknesses
using a micrometer gauge and a special coating thickness gauge, respectively. These
measured values were then used in all the strength calculations. The specimen length
was decided on the basis of three times the half-wave buckling length that was
obtained from THINWALL. Figure 5.8 located later in this chapter shows typical
buckling stress plots from THINWALL. An additional 40 mm was considered to
allow for the end effects so that three buckling waves appeared along the specimens.
Preliminary tests at ambient temperature showed that this additional length allowed
the development of three buckling waves. Major variables in this study were
temperature T, section slenderness ratio λ (=√fy/fcr) and steel yield strength fy.
Details of test specimens are shown in Figure 5.2 and Table 5.1.
Table 5.1 Test Specimens
THINWALL Test series Grade b
(mm) d
(mm)
t (BMT, mm)
Measured fy (MPa) fcr
(MPa) λ
HL (mm)
G5-1 18.7 17.8 0.936 636 392 1.274 40 G5-2 33.7 32.1 0.936 636 122 2.283 80 G5-3
G550 48.5 47.3 0.936 636 61 3.229 100
G2-1 27.5 25.5 1.560 298 485 0.784 60 G2-2 47.5 45.1 1.560 298 173 1.312 100 G2-3
G250 77.2 75.1 1.560 298 67 2.109 200
*BMT and HL refer to base metal thickness and half wave buckling length, respectively. Corner radius r = 1.2mm for all specimens Temperatures considered were 20, 200, 400, 600 and 800°C.
b
d
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-5
In the design standards (AISI, 1996; ECS, 1997; SA, 1996), a buckling coefficient of
0.43 is conservatively used for unstiffened elements to consider the effect of
rotational rigidity at ambient temperature. However, it is not known whether the
rotational rigidity of unstiffened elements remains the same at elevated temperatures.
Therefore, the buckling coefficient was measured to investigate any variation of
rotational rigidity at elevated temperatures using the conventional local buckling
stress formula and the reduced elasticity moduli at elevated temperatures. The
measured values were used to predict the ultimate strength.
5.2.3 Test Method
The steady state test method was chosen for the simulated fire tests. Using this
method, the specimen was first heated up to a pre-selected temperature. It was then
loaded until it failed while maintaining the pre-selected temperature. The
temperature range chosen in this study was from ambient temperature to 800ºC at
200ºC intervals with a heating rate of 20ºC/min (see Table 5.1). When the
temperature reached the pre-selected value, it exceeded the pre-selected value by a
small margin, but the difference was less than 2% even at the higher temperatures of
600 and 800ºC.
The creep effects associated with high temperature and loading occur in building
fires. In an experimental study, the creep behaviour may influence the test results
depending on the test methods used. The transient state test method in which
temperature increases under a constant load may more closely simulate the creep
effects than the steady state test method. However, it was found that the steady state
test is also capable of simulating the creep effects for the following reasons, and
more accurate results of axial ultimate strengths can be produced by the steady state
test method.
Firstly, a typical creep test at elevated temperatures is conducted under a constant
temperature and a constant load since creep is a time-dependant behaviour (Callister,
2000). Therefore, both test methods consider one constant factor for the creep
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-6
behaviour, either temperature or load. It is unknown which factor is more critical.
Therefore it can be stated that a specimen tested by either the steady state or transient
state test method is considered to be subject to creep behaviour.
Secondly, the time duration of steel members exposed to elevated temperatures is so
short that the effects of creep strain may be negligible for practical considerations of
steel structures under fire conditions (Wang, 2002). Tests conducted by either the
steady state or transient state test method are terminated approximately within an
hour. Therefore, it can be stated that there is very little effect of creep and thus little
difference between the test methods. When the steady test method was used in this
study, a significant time of about 20 minutes was allowed between reaching the pre-
selected temperature and loading commencement. Hence the test duration was about
the same in both steady state and transient state test methods.
Thirdly, tensile test results of zinc-coated light gauge steels showed that the
difference between the steady state and transient state tests was very small (Outinen,
1999). This has already been discussed with experimental results obtained by
Outinen (1999) as shown in Section 3.3.
Fourthly, additional experiments were conducted to validate the steady state test
method using both the transient state and steady state test methods. Two test series of
G5-2 and G2-2 were selected and the results are shown in Figures 5.3(a) and (b).
These results showed that the difference in the ultimate strengths between the two
test methods was quite small.
It was also found from this verification study that the accurate ultimate strength was
difficult to obtain from the transient state test method due to limited loading rates in
the test procedure unless the initial loading rate was the same as the ultimate load.
Thermal expansion at high temperature appeared to have been restrained by axial
loading, particularly with high loading rates. It resulted in low axial shortening in the
transient state test as seen in Figures 5.3(a) and (b).
Therefore, the experimental data produced by the steady state test method can be
considered to be relatively practical and reliable.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-7
Figure 5.3 Comparison of the Transient State Test (TST) and Steady State
Test (SST) Methods
0
5
10
15
20
25
30
0 0.5 1 1.5 2
SST
SST
TST
TST
(a) Test series G5-2
600ºC
400ºC
(kN)
Axi
al c
ompr
essi
on lo
ad
Axial shortening mm
mm 0
5
10
15
20
25
30
35
40
0 1 2 3 4
SST
SST
TST
TST
(b) Test series G2-2
400ºC
600ºC
(kN)
Axi
al c
ompr
essi
on lo
ad
Axial shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-8
5.2.4 Determination of the Local Buckling Stress and Ultimate Strength
at Elevated Temperatures
The ultimate strength at elevated temperatures was directly obtained from the data
record of the universal testing machine. It was not possible to use visual observation
to determine the local buckling stress during fire tests. Therefore the local buckling
load was determined from the axial load versus out-of-plane displacement curve at
elevated temperatures. For a perfect plate, the bifurcation of the axial compression
versus out-of-plane displacement clearly appears, and thus the local buckling load
can be easily obtained from the axial compression load versus out-of-plane
displacement curves. However, for an imperfect plate, it is difficult to determine the
local buckling load from the axial compression load versus out-of-place
displacement curves. Venkatarimaiah and Roorda (1982) therefore developed a
method capable of determining the local buckling load from the axial compression
load versus out-of-plane displacement2 (P-δ2) curves. In this study, the experimental
local buckling load was obtained using this P-δ2 method. The illustration of the
method is shown in Figure 4.4 (Chapter 4).
Using the theoretical elastic local buckling stress formula (see Equation 4.2) and
assuming that the Poisson’s ratio ν is unchanged at elevated temperatures, the local
buckling stress equation at elevated temperatures (fcr,T) can be simply modified by
considering the reduced elasticity modulus ET. The modified equation is given by
( )( )22
2
,/112 tb
Ekf TT
Tcrυ
π
−= (5.1)
It is useful to investigate the effect of rotational restraint by an adjacent web element
on the local buckling behaviour of unstiffened flange elements at elevated
temperatures. Using Equation 5.1, the variation of the buckling coefficient kT that
represents the effect of web restraint to flange elements is investigated. Substituting
the experimental local buckling stress into Equation 5.1, the buckling coefficient can
be obtained at elevated temperatures as follows.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-9
( )
T
TcrT E
tbfk
2, /11.1
= (5.2)
The reduced elasticity modulus to be used in Equation 5.2 is based on tensile test
results (see Chapter 3). The reduction factors of mechanical properties for G250
(1.56 mm) and G550 (0.936 mm) steels are given in Table 5.2. The reduction factors
of G550 steels were based on the measured values whereas the factors of G250 steels
were obtained based on the proposals of mechanical properties in Chapter 3
(Equations 3.1 and 3.3) because 1.56 mm G250 steels were not tested.
Table 5.2 Reduction Factors for Elasticity Modulus (ET/ E20) and
Yield Strength (fy,T/ fy,20)
Temp. (ºC) 20 200 400 600 800 G550 1.000 0.902 0.564 0.275 0.068 ET
E20 G250 1.000 0.860 0.580 0.281 0.058 G550 1.000 0.969 0.797 0.478 0.127 fy,T
fy,20 G250 1.000 0.937 0.779 0.425 0.140 Note: fy,20 = 636, 298 MPa for G550 and G250 steels as in Table 5.1.
The elasticity modulus measured from tensile tests was 211 and 220 GPa for G250
and G550 steels, respectively. As discussed in Section 4.4.2.2, the effect of such
variation in the elasticity modulus on member strength was minimal. Therefore the
elasticity modulus E20 of 210 GPa was used for calculation of the buckling
coefficient and the finite element analyses as used in Chapter 4. Poisson’s ratio υ is,
in general, assumed as an independent factor for structural steels at elevated
temperature for numerical solutions because it is unknown and is very difficult to
define under fire conditions (Kaitila, 2002; Zha, 2003). In this research, the
sensitivity study of Poisson’s ratio was therefore conducted at the temperature of
600ºC using a finite element analysis program ABAQUS. For the analytical study, a
test series of G5-2 was used including three cases of Poisson’s ratios of 0.1, 0.3 and
0.49 (υ must be less than 0.5). The analytical results showed the discrepancies of
0.6% and 4.5% for Poisson’s ratios of 0.1 and 0.49, respectively, when compared
with numerical results using Poisson’s ratio of 0.3. The difference is therefore
relatively small. It is also commonly assumed that Poisson’s ratio at elevated
temperatures remains the same (Kaitila, 2002; Zha, 2003). Therefore, in this study, it
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-10
was assumed that Poisson’s ratio did not vary with increasing temperatures and was
taken as 0.3.
5.3 Finite Element Analysis
Several commercial finite element analysis (FEA) programs are available for
research into the behaviour of steel structures. For this research, a finite element
analysis program ABAQUS standard version 6.3 (HKS, 2002) was used due to its
accuracy and efficiency in modelling buckling and non-linear effects including that
of residual stresses and geometric imperfections. A pre-processor MSC/PATRAN
was used to generate basic geometries, boundary conditions and loading system.
Using these two tools, which allowed modelling and performing the elastic buckling
and non-linear analyses, numerical simulation of the experiments at elevated
temperatures was achieved. QUT’s high performance computer centre provided the
facilities required for all the analyses. This section presents the details of finite
element analyses including accurate material behaviour, residual stresses and
geometric imperfections, analysis methods, procedures and verification of their
accuracy through comparisons with experimental results.
5.3.1 Model description
5.3.1.1 Finite elements and meshes
In order to decide the mesh size, convergence studies were required as preliminary
analyses so that computing processor time and disk usage are not excessive. The
finer the mesh, the more accurate the results are. Therefore, it is necessary to
optimize the mesh density of models since the difference is negligible after a certain
mesh size. From the results of preliminary analyses, approximately 3.5 mm square
meshes were found adequate for the bifurcation local buckling analysis and post-
buckling non-linear analysis.
There are a number of modelling options available for shell elements in ABAQUS.
In order to provide proper degrees of freedom to a model’s deformation, the S4R5
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-11
thin shell element type which allows arbitrarily large rotations with four nodes and
five degrees of freedom per node was considered suitable for three-dimensional
analysis. The definition of each label of the element type is illustrated in Figure 4.5
in Chapter 4.
The S4 shell element is fully integrated and is an element type for general purpose
which allows for transverse shear and large strains (HKS, 2002). Therefore, the S4
element will provide better results than the S4R5 element whereas the S4 element
requires far greater disk usage and memory when compared with the S4R5 element.
Avery (1998) conducted analytical studies to investigate the influences of the
element types and found that the difference is so small that either the S4 or S4R5
element can be used for structural steel members. The same approach was used in
the models at elevated temperatures.
5.3.1.2 Material Behaviour at Elevated Temperatures
One of the factors significantly influencing the local buckling and ultimate strength
behaviour at elevated temperatures is the mechanical properties. Therefore the
mechanical behaviour should be explicitly considered in FEA. The reduced
mechanical properties based on the reduction factors reported in Chapter 3 were
directly applied in the initial step of analyses with an assumption that Poisson’s ratio
remains unchanged at elevated temperatures. Due to the steady state analysis
associated with the thermal load, the time scale of conductivity according to
temperature increase was not required for the material modelling as temperature
increases.
Two important mechanical properties for the elastic and nonlinear analyses are the
yield strength and elasticity modulus. The finite element models for G550 and G250
steels were assumed with perfect plasticity and isotropic strain-hardening behaviour
up to 200ºC, respectively. Temperatures up to 200ºC are considered as low
temperatures in Figures 5.4(a) and (b). It was observed from tensile test results that
the G550 zinc/aluminium alloy-coated light gauge steels contained no strain-
hardening and low ductility at ambient temperature. Therefore, a few variations of
material models including low ductility using a descending stress beyond the
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-12
ultimate strength were attempted, but ABAQUS did not recognize the reduction in
material strength after reaching the ultimate strength. Perfect plasticity material
modelling was considered adequate for G550 steels at low temperatures.
A field option defining variables of plastic strain was used to consider the strain-
hardening effect of G250 steels. The measured ultimate and yield strengths were
used and the strain at which the strain-hardening (εs) commences was taken as 4.2%
at low temperatures and 8.5% at high temperatures based on the tensile test results of
G250 steels. The strain-hardening stiffness at low temperatures was taken as 4
percent of the elasticity modulus, 0.04Elow (see Figure 5.4(a)).
(b) Yield strength and elasticity modulus models for G550 Steels
Figure 5.4 Modelling of Material Behaviour at Elevated Temperatures in FEA
Stress
σu,low= σy,low low temperatures
Strain
0.16 σy,high σy,high
high temperatures
ET 0.2% proof stress
σu,high
εy
Strain
σy,low
εy εs,low εs,high
0.16 σy,high
0.04Elow
0.02Ehigh
ET
Stress
0.2% proof stress
low temperatures
high temperatures
σy,high
(a) Yield strength and elasticity modulus models for G250 Steels
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-13
For the temperatures of 400, 600 and 800ºC, several time steps using the field option
were used to model as accurately as possible the actual material behaviour at high
temperatures based on experimental stress-strain curves. The strain-hardening
stiffness for low strength steels was taken as 2 percent at high temperatures, 0.02Ehigh
(see Figure 5.4(a)). Even though the strain-hardening was found to be small at
800ºC, it was conservatively assumed as 2 percent of the elasticity modulus.
However, the strain-hardening did not affect the results much as member failure
occurred before reaching the strain level at the ultimate strength in tensile tests. It
was also found from tensile coupon tests that the ultimate strengths of G550 steel
were about 16% higher than the yield strength based on the 0.2% proof stress. In the
case of G250 steels, tensile strength gained 16% of yield strength based on the 0.2%
proof stress before strain-hardening commenced (see Figure 5.4(a)). Details of
tensile test results and comparisons at different strain levels are given in Chapter 3
and Appendix A. 0.2% strain was used for εy for both G250 and G550 steels. Details
of idealised stress-strain curves used in the material modelling in FEA are illustrated
in Figures 5.4(a) and (b). ‘Low’ and ‘High’ terms used in Figures 5.4(a) and (b)
indicate the low and high temperature ranges, respectively, while σy and σu refer to
the yield strength based on 0.2% proof stress and the ultimate strength at elevated
temperatures. Some actual stress-strain curves are compared with idealised curves to
demonstrate the accuracy of the latter in Figure 5.5.
0
100
200
300
400
500
600
700
0 0.5 1 1.5 2 2.5 3 3.5
Stress (MPa)
Strain(%)
Experimental stress-strain curves
stress-strain curves used in FEA
400 °C
600 °C
Figure 5.5 Comparison of Actual Stress-Strain Curves with Idealised Curves
used in FEA (G550 steel)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-14
5.3.1.3 Load and boundary conditions
The magnitude of load must be defined in the initial analysis step as a total value,
thus the applied load magnitude was set higher than the expected ultimate load. The
loading and boundary conditions used in the experiments were simulated. Axial
compression load was represented as a concentrated nodal force. The concentrated
load was distributed to edge nodes by the rigid surface. The R3D4 element was used
as the rigid surface to create the free rotated edge. The R3D4 element is a rigid
quadrilateral with four nodes and three translational degrees of freedom per node. As
the element has no rotational degrees of freedom, perpendicular shell elements
attached by common nodes to a rigid surface comprising R3D4 elements are free to
rotate about the attached edge. Local buckling rotations are therefore unconstrained.
Figure 5.6 Quarter-wave Buckling Length Model used in FEA
R3D4 elements (rigid surface)
P (SPC, 12456)
Elastic strip
S4R5
Plane of symmetry (SPC345)
Local buckling wave
:
Full
leng
th
Hal
f len
gth
Qua
rter-
wav
e
buck
ling
leng
th
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-15
As illustrated in Figure 5.6, three different models can be used to simulate the
symmetric buckling waveform, namely, a full length, a half length and a quarter-
wave buckling length model. However an idealised quarter-wave buckling length
model was used in this study and the model was validated as discussed in Section
5.3.3.
Single point constraints (SPC) were used at the plane of symmetry to provide the
ideal condition of the half of half-wave length model. The boundary condition of
each node to the loading direction was pinned due to the use of R3D4 elements on
the rigid surface. The boundary condition of single point constraints on the plane of
symmetry was applied to restrain the rotation about axes perpendicular to the loading
axis so that the buckling mode can be symmetrical as required. An elastic strip was
used to eliminate any stress concentrations at the elements contacting with the rigid
surface. The details of the quarter-wave buckling length model are shown in Figure
5.6.
5.3.1.4 Geometric imperfections and residual stresses
Geometric imperfections are included in FEA to allow for the errors in
manufacturing processes. Initial local deformation causes reduction in the ultimate
strength even though it is often small. Therefore it is necessary to define geometric
imperfections in each model. The geometric imperfection can be generated by
specifying nodal coordinates modified from previous buckling eigenmodes. In the
process, nodal coordinates are input as scale factors at the location of maximum
displacements. Local and global imperfection magnitudes are in general included in
the non-linear analyses of cold-formed steel members. All the models in this
research project were governed by local buckling only. Therefore, only the local
imperfections were included in the finite element models as deviation values from
the perfect geometry.
Many researchers have measured the local imperfections of cold-formed steel
members and studied their sensitivity on strength at ambient temperature
(Sivakumaran and Abdel-Rahman, 1998; Schafer and Pekoz, 1998; Young and Yan,
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-16
2000; Dubina and Ungureanu, 2002). The initial imperfection distribution is more
complicated under fire conditions and no relevant data is available for cold-formed
steel members at elevated temperatures. Kaitila (2002a) and Feng et al. (2003a,b)
showed from their experimental and numerical studies that the local imperfections
lead to a smaller effect on the ultimate strength of cold-formed steel members at high
temperatures than at ambient temperature. In real fire conditions, their effects are not
yet known and are very complicated due to the variations of increasing temperatures
and the change of residual stresses and material properties. Therefore, in current
research, the magnitude of local buckling imperfection is generally assumed to be
the same at elevated temperatures (Feng et al.; 2003, Zha; 2003; Kaitila, 2002b). The
same assumption was therefore applied in this study at elevated temperatures. Based
on the local imperfections used at ambient temperature (see Section 4.4.2.4), the
magnitude of t (thickness) was used as the initial imperfection on the critical
buckling mode obtained from elastic buckling analysis (see Figures 4.8(b) and
4.9(a)). The local imperfections were introduced in all of the possible locations
where local buckling occurred.
Residual stresses should also be taken into account in the FEA. Flexural and
membrane residual stresses occur with different magnitudes due to the folding
process of cold-forming with variation in accordance with elements and thicknesses.
These stresses can cause members to fail prematurely. It is therefore necessary to
include appropriate residual stresses in FEA models. The simplified variations in
flexural residual stresses recommended by Schafer and Pekoz (1998) were adopted
for models at ambient temperature. Membrane residual stresses generated during
cold-working process are too small and are ignored in the FEA (Schafer and Pekoz,
1998; Young and Rasmussen, 1999).
Residual stresses are relieved by exposing a member to elevated temperatures.
Therefore, a suitable reduction of residual stresses should be considered at elevated
temperatures. An annealing process to remove residual stresses is dependant upon
the amount of carbon contained in a material. Fully annealing temperature for steels
containing 0.2% carbon is approximately 830ºC (Callister, 2000). Based on the fully
stress relieving temperature, assumed to be 800ºC, the stress relieving factors for
residual stresses were determined by considering a linearly reducing rate at different
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-17
temperature levels. The residual stress reduction factor (a) is therefore given by
Equation 5.3 and the required values are given in Table 5.3.
a = 1.0181-0.00128·T (5.3)
where T (temperature) is in °C in the range of 20 ≤ T ≤ 800.
Table 5.3 Residual Stress Reduction Factor
Temp.( ºC) 20 200 400 600 800 α 1.00 0.76 0.51 0.25 0.00
Five interpolation points were used to vary the residual stress through the thickness.
The user defined residual stress was modelled using the SIGINI Fortran user
subroutine. Figure 5.7 shows the assumed residual stress distribution.
Figure 5.7 Assumed Residual Stress Distribution with Temperature Effects
5.3.1.5 Thermal factors
The thermal properties of conductivity, specific heat or thermal expansion can be
accounted by defining thermal coefficients or field variables. Due to the steady state
method used in this research, such thermal factors were not effective and were
omitted (HKS, 2002).
0.08αfy 0.33αfy
0.17αfy
+σr
-σr
outside
inside
Residual stress through thickness
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-18
5.3.2 Analysis
5.3.2.1 Elastic eigenvalue analyses
Eigenvalue buckling analyses were carried out to obtain the critical buckling load by
applying the Lanczos eigensolver method. It was essential to undertake buckling
analyses first to obtain the critical buckling mode to include the initial local
imperfections. It was found that the first buckling mode was always critical and the
most symmetrical. It might be worthwhile applying a combination of two or three
buckling modes due to the complexity of the distribution of the initial deformation.
However, the difference between the single critical mode and a combination of local
buckling modes was seen to be very little for the ultimate strength at both ambient
and elevated temperatures (Kaitila, 2002). Thus, the critical eigenvalue mode was
used with the local buckling imperfection.
5.3.2.2 Non-linear analyses
In the non-linear analyses, the length of quarter wave buckling length models was
based on the half-wave buckling length results from THINWALL (see Table 5.1)
whereas that of half length and full length models was based on the specimen length.
Prior to running post buckling analyses, two more steps were required to include
residual stresses. Residual stresses containing no applied loads and with all degrees
of freedom constrained were defined on a model with initial local imperfections.
Non-linear static analysis was then run with one increment, requesting reaction force
and moment. This step was to detect and eliminate the reaction force and moment
caused by the residual stress load case. Defining an applied load with the reaction
force and moment, two non-linear static analysis steps for residual stress equilibrium
and an applied compression load were finally submitted.
ABAQUS always increases loads with time steps. The increment was set to be
controlled by the initial time scale of 0.02 and automatic increment scheme, so that
no user intervention is required. For non-linear analyses, the modified Riks method
is used for the unstable static response of structures in load-displacement
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-19
relationship. At high temperatures, the material stiffness becomes considerably
weakened. The Riks method is also recommended for the unstable mechanical
behaviour and low stiffness. Thus, the modified Riks algorithm was included in all
the non-linear analyses.
5.3.3 Verification study
Based on the process introduced above, two of the test series G5-2 and G2-2 were
selected to verify the accuracy of the elastic buckling load and ultimate strength
results from FEA compared with experimental results. Three types of comparisons
were undertaken to verify the results of FEA modelling. Firstly, a finite strip analysis
(FSA) program THINWALL (Papangelis and Hancock, 1998) was used to obtain the
elastic critical buckling stress which was compared with FEA results. Secondly, the
experimental local buckling stress and ultimate strength results were used to validate
the FEA modelling for bifurcation and non-linear analyses. The last series of
comparison was attempted using full length, half length and half of half-wave
buckling length models.
To investigate the elastic buckling behaviour of cold-formed steel members,
numerical solutions based on the finite strip method such as THINWALL and
CUFSM are commonly used. These computational methods are quite reliable and are
currently used by many researchers (Schafer, 2002; Yang and Hancock, 2003). In
this research, the use of a finite strip analysis is considered efficient for validating
the elastic buckling solutions from FEA. Table 5.4 shows the results of the local
buckling stress (fcr) and ultimate strength (Pu) obtained from three different FEA
models, test and FSA for the test series of G5-2 and G2-2 at the temperatures of
20°C and 600°C. The local buckling stress from FSA is very close to the result
obtained from FEA. The test provided slightly higher results of the local buckling
stress. However, as seen in Tables 5.4 and 5.5, the results are in reasonably good
agreement for both low and high strength steel members. These results therefore
indicate that the FEA model based on the half of half-wave buckling length model
(quarter-wave buckling length) accurately predicts the elastic buckling stress and
ultimate strength.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-20
Table 5.4 Comparison of Elastic Buckling Stresses and Ultimate Strengths
at 20°C and 600°C
FEA Half of half-
wave length Half
length Full
length Test FSA
20°C 130 132 132 148 131 fcr (MPa) 600°C 37 35 35 47 37
20°C 35.9 35.2 35.2 35.2 - G5-2 Pu (kN) 600°C 13.8 13.5 13.5 14.5 -
20°C 180 182 182 190 183 fcr (MPa) 600°C 51 52 52 60 53
20°C 49.6 49.3 49.3 50.2 - G2-2 Pu (kN) 600°C 22.1 19.9 19.9 21.1 -
(a) Buckling Stress Curve at 20°C
(b) Buckling Stress Curve at 600°C
Figure 5.8 Elastic Buckling Stress versus Buckling Half-wave Length
Flange local buckling
70
Flange local buckling
70
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-21
Table 5.5 Test and FEA Results of Buckling Stress and Ultimate Strength
fcr,T (MPa) kT Pu (kN) Test series
Temp(°C)
fy,T
(MPa) Test FEA Test FEA FEA Exp Test FEA FEA
Exp 20 636 399 389 0.840 0.819 0.98 25.4 25.8 1.02 200 616 371 350 0.866 0.778 0.90 24.0 24.7 1.03 400 507 265 238 0.989 0.846 0.86 20.1 19.2 0.96 600 304 151 111 1.156 0.809 0.70 11.5 10.9 0.95
G5-1
800 81 30 26 0.929 0.766 0.82 2.6 2.7 1.04 20 636 148 130 1.012 0.889 0.88 35.2 35.9 1.02 200 616 135 123 1.023 0.888 0.87 33.2 33.5 1.01 400 507 84 77 1.018 0.889 0.87 24.0 24.1 1.00 600 304 47 37 1.168 0.876 0.75 14.5 13.8 0.95
G5-2
800 81 13 10 1.307 0.957 0.73 3.9 3.6 0.92 20 636 73 63 1.034 0.892 0.86 38.5 40.7 1.06 200 616 66 61 1.036 0.912 0.88 37.5 38.7 1.03 400 507 47 42 1.180 1.004 0.85 31.9 31.1 0.97 600 304 22 23 1.133 1.184 1.05 17.1 16.7 0.98
G5-3
800 81 7 6 1.395 1.249 0.90 4.8 4.4 0.92 Mean 0.86 0.99 COV 0.051 0.047
20 298 266 497 0.442 0.825 1.87 35.1 35.6 1.01 200 279 258 453 0.499 0.835 1.67 33.6 34.1 1.01 400 232 195 300 0.558 0.820 1.47 27.2 28.4 1.04 600 127 88 141 0.539 0.828 1.54 17.5 16.5 0.94
G2-1
800 42 23 31 0.650 0.819 1.26 4.2 3.8 0.90 20 298 190 180 0.945 0.892 0.94 50.2 49.6 0.99 200 279 164 165 0.942 0.907 0.96 48.0 46.8 0.98 400 232 127 110 1.085 0.897 0.83 36.3 38.9 1.07 600 127 60 51 1.101 0.893 0.81 21.1 22.1 1.05
G2-2
800 42 13 11 1.073 0.867 0.81 6.3 5.3 0.84 20 298 75 68 0.981 0.890 0.91 67.2 66.3 0.99 200 279 72 66 1.095 0.958 0.87 64.5 62.1 0.96 400 232 51 45 1.150 0.969 0.84 45.2 50.1 1.11 600 127 25 21 1.211 0.971 0.80 25.1 26.2 1.04
G2-3
800 42 6 5 1.243 1.112 0.89 7.5 6.5 0.87 Mean 1.09 0.97 COV 0.331 0.059
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-22
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5 3
Test
FEA
Axial shortening
Axi
al c
ompr
essi
on lo
ad
kN
mm
(b) 400 °C
Axial shortening
Axi
al c
ompr
essi
on lo
ad
kN
mm 0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2
Test
FEA
(a) 20 °C
Figure 5.9 Axial Compression Load versus Axial Shortening Curves
at Elevated Temperatures (Test Series G5-2)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-23
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5 3
Test
FEA
kN
Axi
al c
ompr
essi
on lo
ad
mm Axial shortening
(c) 600 °C
0
2
4
6
8
10
12
0 0.5 1 1.5 2 2.5 3
Test
FEA
Axial shortening
Axi
al c
ompr
essi
on lo
ad
kN
mm
(d) 800 °C
Figure 5.9 Axial Compression Load versus Axial Shortening Curves
at Elevated Temperatures (Test Series G5-2)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-24
Figure 5.10 Axial Compression Load versus Axial Shortening Curves
at Elevated Temperatures (Test Series G2-2)
0
10
20
30
40
50
0 0.5 1 1.5 2 2.5 3
Test
FEA
Axial shortening
Axi
al c
ompr
essi
on lo
ad
kN
mm
(b) 400 °C
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3
Test
FEA
Axial shortening
Axi
al c
ompr
essi
on lo
ad
kN
mm
(a) 20 °C
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-25
Figure 5.10 Axial Compression Load versus Axial Shortening Curves
at Elevated Temperatures (Test Series G2-2)
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5 3
Test
FEA
kN
Axi
al c
ompr
essi
on lo
ad
mm Axial shortening
(c) 600 °C
0
2
4
6
8
10
12
14
16
18
0 0.5 1 1.5 2 2.5 3
Test
FEA
Axial shortening
Axi
al c
ompr
essi
on lo
ad
kN
mm
(d) 800 °C
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-26
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6
Test
FEA
(a) 200°C
Out-of-plane displacement
mm
kN A
xial
com
pres
sion
load
0
5
10
15
20
25
0 1 2 3 4 5 6
Test
FEA
(b) 600°C
Out-of-plane displacement
mm
kN
Axi
al c
ompr
essi
on lo
ad
Figure 5.11 Axial Compression Load versus Out-of-plane Displacement Curves (Test Series G5-2)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-27
Larger buckling stresses of G2-1 series in FEA are shown in Table 5.5. This is
because the G2-1 section was close to the section slenderness limit and the buckling
stresses in FEA were obtained based on the linear elastic analysis. This resulted into
the buckling stresses greater than the yield strength.
Figures 5.8(a) and (b) show the elastic buckling curves corresponding to buckling
half-wave lengths of test series G5-2. As shown in Figure 5.8(b), the elastic buckling
curve was also obtained at a high temperature of 600°C using a reduced elasticity
modulus based on the reduction factors given in Table 5.2. Interestingly, these two
curves in Figures 5.8(a) and (b) show that the temperature does not influence the
buckling half-wave length and buckling mode. Similar observations were made in
FEA.
Typical axial compression load versus axial shortening curves predicted by FEA are
also compared with the measured experimental curves at the temperatures of 20, 400,
600 and 800°C in Figures 5.9 and 5.10. The ultimate strengths at low and high
temperatures obtained from FEA are in reasonably good agreement with test results.
Although Figure 5.9(a) shows a very good agreement of the ultimate strength at low
temperature, the ultimate strength from FEA is slightly lower than the test result at
high temperatures in some cases. However Table 5.5 results show that experimental
ultimate loads are overall in good agreement with FEA ultimate loads for both low
and high strength steel sections (mean FEA/Exp = 0.97 and 0.99). At low
temperatures, FEA appear to predict slightly higher strengths than experiments for
high strength steel sections (see Table 5.5). This might be due to the inability of FEA
to simulate the effect of any lack of ductility in high strength steel. Since high
strength steels gained greater ductility at elevated temperatures, the effects of lack of
ductility were expected to be minimal and Table 5.5 confirms this.
Figures 5.11(a) and (b) show the typical axial compression versus out-of-plane
displacement curves obtained from tests and FEA. The experimental out-of-plane
displacement was measured up to about 1.3 mm due to the limit of the measurement.
The axial load versus out-of-plane displacement curve was used to obtain the local
buckling load. It was found that the limited measurement of 1.3 mm was enough to
obtain the local buckling load from the P-δ2 curve. All these comparisons confirm
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-28
that the finite element model can be successfully used to simulate the experimental
local buckling behaviour of cold-formed steel compression members subject to
unstiffened flange element local buckling effects. Further comparisons of the results
of the local buckling stress and effective width from tests and FEA are also provided
in Section 5.4.
(a) Full-length (b) Half-length (c) Half of half-
wave length
Figure 5.12 Failure Modes observed in different FEA Models
Three different FEA models, that is, full length model, half length model and half of
half-wave buckling length (quarter-wave buckling length) model, were used for
stocky columns used in this study. As already shown in Table 5.4, there is very little
difference in the ultimate strength and buckling stress results. Moreover, the failure
modes from these three models and experiments also agree with each other as seen in
Figures 5.12 and 5.13. This again confirms the adequacy of half of half-wave
buckling length (quarter wave buckling length) models used in this study, even
though there are some unavoidable differences between the idealised FEA models
and experiments.
Figure 5.13 Failure Mode observed in Experiments
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-29
In this section, the results obtained from tests, FEA and FSA were compared with
each other. This comparison indicates that a quarter-wave length FEA model
including the appropriate factors associated with temperature and manufacturing
process can be considered adequate to investigate the local buckling behaviour and
provides suitable and reliable results of the elastic local buckling stress and ultimate
strength of cold-formed steel columns at elevated temperatures.
5.4 Results and Discussions
5.4.1 Local buckling behaviour
One of the most important behaviours associated with thin-walled steel members
subject to axial compression is local buckling. In most cases, cold-formed steel
sections are non-compact and hence local buckling effects influence the effective
area in current design standards (AISI, 1996; SA 4600, 1996; ECS, 1997). Likewise,
the phenomenon is also significant at elevated temperatures. The local buckling
behaviour of unstiffened flanges element made of high and low strength steels is
investigated at elevated temperatures up to 800ºC in this chapter.
A simplified relationship to obtain the local buckling stress in the elastic range at
temperature T, fcr,T, was proposed by Uy and Bradford (1995). The simplified
equation is given by
20
20,, EEff T
crTcr = for TyTcr ff ,, ≤ (5.4)
where fcr,20 and ET refer to local buckling stress at ambient temperature and the
elasticity modulus at temperature T, respectively.
According to Equation 5.4, the local buckling stress is controlled by the reduced
elasticity modulus when the local buckling stress is lower than the yield stress. The
buckling coefficient kT therefore remains the same at temperature T. This
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-30
0.0
0.5
1.0
1.5
2.0
0 200 400 600 800 1000
Test G2-1
Test G2-2
Test G2-3
FEA G2-2
FEA G2-3
`
ºC
kT
relationship leads to a conclusion that the rotational restraint provided by the web is
unchanged at elevated temperature.
The local buckling stresses were obtained from tests and FEA. Using the local
buckling stresses, the experimental and numerical buckling coefficients were derived
using Equation 5.2. The results of the local buckling coefficients are given in Table
5.5. Moreover, the comparisons of FEA results with test results are included in Table
5.5. The variation of buckling coefficients obtained from tests and FEA at elevated
temperatures are shown in Figures 5.14(a) and (b) for G250 and G550 steel sections,
respectively.
(a) G250 steel sections
(b) G550 steel sections
Figure 5.14 Buckling Coefficients at Elevated Temperatures
0.0
0.5
1.0
1.5
2.0
0 200 400 600 800 1000
Test G5-1
Test G5-2
Test G5-3
FEA G5-1
FEA G5-2
FEA G5-3
ºC
kT
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-31
The numerical buckling coefficients of the G2-1 series are not plotted in Figure
5.14(a) as the buckling stresses were greater than the yield strength. Although all the
sections used in this study were designed to be non-compact, the G2-1 section was
close to the section slenderness limit. It resulted in a greater buckling stress in FEA
whereas smaller experimental buckling coefficients were obtained when compared
with other sections as seen in Figure 5.14(a).
The variation of the buckling coefficient in Figures 5.14(a) and (b) presents the
effect of temperatures on the rotational restraint provided by the web for low and
high strength steel sections. Interestingly, regardless of the increase of temperature,
buckling coefficients are reasonably constant. For some cases at the temperature of
800ºC, the buckling coefficient slightly increased even though the reduction of the
elasticity modulus was considered. This might have been caused by the lower
reduction in the local buckling stress when compared with the reduction of elasticity
modulus at 800ºC. Further, even small experimental errors with low buckling loads
obtained at high temperature caused larger differences in the buckling coefficient
obtained from these loads. Despite the small discrepancy at 800ºC leading to
conservative results, it implies that the local buckling stress is reduced by a similar
ratio of the elasticity modulus at elevated temperatures. Similar behaviour was seen
in Figure 5.14(b) for high strength steel sections. Therefore, simply modified current
design rules considering the reduction of the elasticity modulus can be safely
recommended to predict the local buckling stress of unstiffened elements at elevated
temperatures.
Considering the reduction of elasticity modulus, most buckling coefficients are seen
constant at elevated temperatures even though temperature caused a slight increase
of the buckling coefficient in some cases. Therefore it can be stated that the
rotational restraint provided by the web to the flanges of unlipped channel members
remains unchanged at elevated temperatures when the reduced elasticity modulus is
considered. The above observation is relevant when the effects of temperature on the
boundary conditions of cold-formed steel structural members are considered under
fire conditions. No moment rotational constraint exists at pin ends. However, in most
structural compression members, end restraint effects are provided by adjacent
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-32
members or footings, causing various effective lengths. End conditions of a column
are determined by the rotational restraint on the basis of which an effective length
factor is provided. It is more complicated to consider the transition of end support
conditions under fire due to thermal effects such as material softening and thermal
expansion which cause reductions to the maximum load capacity.
Figure 5.15 Local Buckling Failure Modes
(a) Low strength steel sections
(b) High strength steel sections
20ºC 400ºC 800ºC
20ºC 400ºC 800ºC
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-33
Effects of column restraints under fire conditions were studied by a few researchers
(Ali et al., 1998; Valente and Neves, 1999; Neves et al., 2002). They found that the
failure temperature decreased with increasing axial restraint. In most research, end
supports were assumed unchanged, that is, restraining beams or plates were located
outside the furnace in the experimental condition or restraining factors were assumed
constant regardless of the temperature increase in the numerical study. This
assumption might be slightly different with end support conditions in real fires
because end supports are also subject to fire conditions. Therefore, the transition of
boundary conditions may exist during the fire event. Further, most research on the
effect of fire on boundary conditions has been limited to hot-rolled steel members.
In this investigation, only rotational restraint provided by the web was considered
because the web neither buckled nor translated and thus the effect of axial web
stiffness was not considered in this study. The results appeared to be predominantly
dependant upon the reduction of the elasticity modulus at elevated temperatures.
Therefore if the critical temperature is decreased as a result of different end
conditions, e.g. reduced critical temperature for fixed ends when compared with
pinned ends, it might have been caused by the axial restraint rather than the
rotational restraint. Further research is however required to confirm the effect of
temperatures on the boundary conditions of cold-formed steel structural members.
Figures 5.15(a) and (b) show the representative failure modes at low, middle and
high temperatures (20, 400 and 800°C). It was thought that some sections could have
failed by a combination of local buckling and distortional buckling at high
temperatures due to material softening and the reduced rotational restraint of flanges
provided by the web. However as seen in Figure 5.14, the effect of restraint at
elevated temperatures was reasonably constant. Thus all sections failed by flange
local buckling as seen in Figures 5.15(a) and (b). Therefore it could be stated that
there was no change of buckling modes at elevated temperatures for the unlipped
channel sections considered in this research, which were governed by flange local
buckling at ambient temperature.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-34
5.4.2 Variations of effective width at elevated temperatures
Determining the effective width is one of the most important factors in the design of
cold-formed steel members because of the local buckling associated with thin steels.
As discussed in Chapter 4, many researchers (von Karman et al., 1932; Schuette,
1947; Winter, 1947; Chilver, 1953; Kalyanaraman, 1976; Bambach and Rasmussen,
2002) proposed various methods to determine the effective width at ambient
temperature. However, no effective width rules exist for cold-formed steel
compression members at elevated temperatures. Nonetheless, Kaitila (2002) and
Feng et al. (2003a, b) found that the current effective width design rule for local
buckling can be extended to elevated temperatures by using the reduced yield
strength and elasticity modulus values. However, their research was limited to low
strength steels (≤ 350 MPa), and the accuracy of their conclusions to high strength
steels (> 550 MPa) commonly used in Australia is not known.
The use of stocky columns with a fully effective web enabled the experimental
effective widths of unstiffened flange elements to be directly obtained from the
ultimate strength. For this purpose, the reduction factors of mechanical properties in
Table 5.2 were used. The yield strength reduction factors in Table 5.2 based on the
0.2% proof stress were used in the strength calculations because the use of other
strain levels of 1.5% and 2.0% has neither been clearly defined nor widely accepted
for cold-formed steel compression members. British design standard (BSI, 1995)
also does not recommend the use of 1.5% and 2.0% strain levels for compression
members for the sake of structural ability and deformation. Further, in aluminium
structures in which similar stress-strain characteristics are seen, the yield strength
corresponding to a 0.2% strain is commonly used (Mazzolani, 1995). It is therefore
considered adequate that the 0.2% proof stress is used for the strength calculations at
elevated temperatures. Table 5.6 presents the experimental and numerical ultimate
strengths of unlipped channel members and the corresponding effective widths of
unstiffened flange elements at different temperatures. Example calculations of
effective widths are shown in Appendix C.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-35
Table 5.6 Experimental and FEA Effective Widths and Comparison
be/b λT Test series
Temp. (ºC) Test FEA Eq.5.5
Test Eq.5.5 FEA Test FEA
20 0.62 0.64 1.05 1.01 1.263 1.279 200 0.59 0.62 1.09 1.01 1.289 1.327 400 0.60 0.55 1.01 1.06 1.383 1.460 600 0.55 0.49 1.08 1.07 1.419 1.655
G5-1
800 0.39 0.40 1.35 1.24 1.643 1.765 20 0.36 0.38 1.20 1.07 2.073 2.212 200 0.35 0.36 1.20 1.12 2.136 2.238 400 0.24 0.25 1.54 1.43 2.457 2.566 600 0.25 0.21 1.44 1.53 2.543 2.866
G5-2
800 0.26 0.20 1.41 1.62 2.496 2.846 20 0.19 0.22 1.65 1.33 2.952 3.177 200 0.18 0.20 1.69 1.46 3.055 3.178 400 0.18 0.17 1.58 1.59 3.284 3.474 600 0.11 0.10 2.30 2.58 3.717 3.636
G5-3
800 0.14 0.10 1.96 2.56 3.402 3.674 Mean 1.43 1.44 COV 0.153 0.248
20 0.88 0.90 0.85 1.03 1.058 0.774 200 0.90 0.92 0.84 1.00 1.040 0.785 400 0.81 0.87 0.90 0.98 1.091 0.879 600 0.98 0.86 0.69 0.94 1.201 0.949
G2-1
800 1.05 0.96 0.59 0.73 1.351 1.164 20 0.65 0.63 1.01 1.02 1.252 1.287 200 0.66 0.63 0.97 1.01 1.304 1.300 400 0.54 0.59 1.15 0.99 1.352 1.452 600 0.53 0.58 1.10 0.94 1.455 1.578
G2-2
800 0.91 0.62 0.54 0.73 1.797 1.954 20 0.44 0.43 1.01 0.99 1.993 2.093 200 0.46 0.42 0.98 1.03 1.969 2.056 400 0.28 0.36 1.20 1.10 2.133 2.271 600 0.26 0.29 1.17 1.08 2.254 2.459
G2-3
800 0.55 0.41 0.63 0.78 2.646 2.898 Mean 0.91 0.96 COV 0.136 0.122
Adequate design rules are not available for the effective width of cold-formed steel
members at elevated temperatures. Therefore, a simple modification using current
design standards (AISI, 1996; SA, 1996; ECS, 1997) was first attempted to
investigate whether the conventional design rules can be used in the fire safety
design of cold-formed steel compression members.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-36
In order to modify the current design rules, the reduction factors of the yield strength
and elasticity modulus were used in the conventional effective width design rule
given by Equations 4.1(a) to (c). These effective width design rules at ambient
temperature can then be rewritten for an elevated temperature T as given in
Equations 5.5(a) to (c).
1=Tρ for 673.0≤Tλ (5.5a)
T
T
T
TeT b
bλλ
ρ
−
==
22.01, for 673.0>Tλ (5.5b)
Tcr
Ty
T
Ty
TT f
f
E
f
tb
k ,
,,052.1=
=λ (5.5c)
where kT is the buckling coefficient, ET is the elasticity modulus and fy,T and fcr,T are
the yield strength and critical buckling stress at temperature T, respectively.
In Equations 5.5(a) and (b), the slenderness limit of 0.673 used at ambient
temperature is assumed unchanged at elevated temperatures. The thickness
measurements before and after tests showed that the base metal thickness remained
the same at elevated temperatures. Therefore a constant thickness was used for the
prediction of effective width at elevated temperatures.
The review of current effective width rules at ambient temperature showed that the
conventional design rules were adequate for low strength steel sections whereas they
overestimated the ultimate strength of unstiffened elements made of high strength
steels at ambient temperature as discussed in Chapter 4. Therefore a simple
relationship considering the ratio of the local buckling stress to the yield stress was
proposed at ambient temperature on the basis of experimental and numerical results
(Equation 4.10 in Section 4.5.4). This proposal was also simply modified using the
reduced mechanical properties to predict the effective width and ultimate strength,
and the results were then compared with experimental and numerical results. The
modified equations for an elevated temperature T are given as follows.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-37
1, =b
b Te for 745.0≤Tλ (5.6a)
Ty
TcrTe
ff
bb
,
,, 745.0= for 745.0>Tλ (5.6b)
Tcr
TyT f
f
,
,=λ (5.6c)
In order to verify the accuracy of these design equations, the effective widths of
unstiffened flanges from experiments and finite element analyses were compared
with predictions from Equations 5.5 and 5.6. Table 5.6 gives the relevant parameters
be,T/b and λT, whereas Appendix C presents example calculations of these
parameters.
Figures 5.16(a) and (b) present the effective width variations of unstiffened flanges
made of G250 and G550 steels at varying temperatures. For the sections made of
both G250 and G550 steels, the effective width reduction was minimal at 200ºC as
the mechanical properties were not reduced much at 200ºC. The effective width of
low strength steel sections appeared to decrease slightly in the temperature range of
400ºC and 600ºC.
However, the comparison of experimental and numerical results with the predictions
from Equation 5.5 showed that the average ratios of the effective width from tests
and FEA to Equation 5.5 at 400ºC and 600ºC were 1.05 and 0.99 for low strength
steel sections, respectively. As seen in Figure 5.16(a), the modified design rules
provided conservative results at 20ºC and 800ºC. Therefore the use of current design
rules considering the reduced yield strength and elasticity modulus can be considered
to be reasonably adequate for low strength steel sections subject to local buckling
effects although the design rules overestimated the results slightly for the
temperature of 400ºC.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-38
(a) Low strength steel sections
(b) High strength steel sections
Figure 5.16 Variations of Effective Width Ratio with Temperature
Figures 5.17(a) to (e) compare the effective widths of unstiffened flanges from
experiments, FEA and proposed design Equations 5.5 and 5.6 in a different format,
i.e. effective width ratio be,T/b versus section slenderness λT(=√fy,T/fcr,T) for each
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 200 400 600 800 1000
Test
FEA
Eq.5.5
ºC
be,T/b
G2-1
G2-2
G2-3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 200 400 600 800 1000
Test
FEA
Eq.5.5
ºC
be,T/b
G5-3
G5-2
G5-1
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-39
temperature. Although the b/t ratios were the same for both test and FEA models, the
section slenderness (√fy,T/fcr,T) values were slightly different because of the
difference in local buckling stress (fcr,T) from test and FEA.
Figures 5.17(a) to (e) show that the effective width comparisons with design
equation are similar for low strength steel sections at the two extreme temperatures
of 20 and 800ºC. In fact, the design equation appears to be more conservative at
800ºC. The effective widths at the temperature of 800ºC considering the reduction in
mechanical properties are similar to those at ambient temperature, and even better
effective widths are seen in some cases. The reasons for this phenomenon can be
explained as follows. Firstly, the tensile yield strength reduction might be
conservative at 800ºC. Secondly, the effectiveness of the buckling coefficient
slightly increased at 800ºC as seen in Figure 5.14. The effective width is inversely
proportional to the section slenderness which is a function of the ratio of the yield
strength to the elasticity modulus (see Equation 5.5(c)). Due to these reasons, the
improvement of the effective width might be seen at 800ºC. Further, the ultimate
strength at 800ºC decreases to about 10% of that at ambient temperature. Therefore
small experimental and numerical errors cause a larger discrepancy in the cases with
lower ultimate strength and mechanical properties.
In contrast to the local buckling behaviour of low strength steel sections, a clear
trend of the effective width is seen for high strength steel sections in Figure 5.16(b)
and Figures 5.17(a) to (e). Simply modified current design rules clearly overestimate
the effective width of high strength steel sections. Equation 5.6 provides slightly
better prediction and yet consistently overestimates the effective width at high
temperatures. As temperature increased, the effective width gradually decreased
despite the fact that the local buckling coefficients of high strength steel sections
were reasonably constant at elevated temperatures as shown in Figure 5.14(b).
Therefore it implies that the deterioration of the ultimate strength of cold-formed
steel columns must have occurred in the post-buckling range.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-40
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 2 4 6
Eq.(5.5)
Eq.(5.6)
G2-20(Test)
G2-20(FEA)
G5-20(Test)
G5-20(FEA)
(a) 20ºC Tcr
Ty
ff
,
,
bb Te,
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 2 4 6
Eq.(5.5)
Eq.(5.6)
G2-200(Test)
G2-200(FEA)
G5-200(Test)
G5-200(FEA)
bb Te,
Tcr
Ty
ff
,
,
(b) 200ºC
Figure 5.17 Effective Width Ratio versus Section Slenderness at
Various Temperatures
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-41
Figure 5.17 Effective Width Ratio versus Section Slenderness at
Various Temperatures
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 2 4 6
Eq.(5.5)
Eq.(5.6)
G2-400(Test)
G2-400(FEA)
G5-400(Test)
G5-400(FEA)
(c) 400ºC
bb Te,
Tcr
Ty
ff
,
,
(d) 600ºCTcr
Ty
ff
,
,
bb Te,
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 2 4 6
Eq.(5.5)
Eq.(5.6)
G2-600(Test)
G2-600(FEA)
G5-600(Test)
G5-600(FEA)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-42
Figure 5.17 Effective Width Ratio versus Section Slenderness at
Various Temperatures
Possible reasons for the different local buckling behaviour between low and high
strength steel sections are considered here. The local buckling stress is theoretically
independent of the yield strength, but is considerably dependent on the width to
thickness ratio. Therefore it is well known that high strength steel sections include a
greater post buckling strength than low strength steel sections, which results in a
greater reduction in the ultimate strength of high strength steel sections.
The conventional local bucking stress formula can be therefore used for the
prediction of the local buckling stress at elevated temperatures for both low and high
strength steel sections considering the reduction of the elasticity modulus. However
the simply modified effective width design rules are not adequate for high strength
cold-formed steel compression members subject to local buckling effects despite the
inclusion of appropriately reduced yield strength and elasticity modulus.
(e) 800ºCTcr
Ty
ff
,
,
bb Te,
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 2 4 6
Eq.(5.5)
Eq.(5.6)
G2-800(Test)
G2-800(FEA)
G5-800(Test)
G5-800(FEA)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-43
Figure 5.18 Comparison of Reductions to Elasticity Modulus and Yield
Strength at Elevated Temperatures
In Equation 5.5(c), the ratio of the yield strength to the elasticity modulus is the
major parameter in the section slenderness influenced by elevated temperatures. It is
seen from Table 5.2 and Figure 5.18 that the elasticity modulus reduces at a faster
rate than yield strength for both steels, but this effect is more severe for high strength
steels. This also means that for higher strength steel sections the gap between
buckling and yield stresses has increased more at elevated temperatures than for low
strength steel sections, i.e. greater post buckling strength. The difference in the
reducing rates of the yield strength and elasticity modulus is expected to influence
the effective width. This is another important reason for the greater reduction in
effective widths observed for high strength steel sections at elevated temperatures
(see Figures 5.17(a) to (e)).
Rasmussen and Rondal (1997) investigated the behaviour and associated strength
curves of metal columns as a function of the material characteristics. Their research
showed that the strength curves were dependent on two important parameters (1)
Ramberg-Osgood parameter n defining the stress-strain characteristics (2) non-
dimensionalised proof stress ep = fy/E. Their study involved values of n from 3 to
100 and ep from 0.001 to 0.008 and clearly demonstrated the influence of both of
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 200 400 600 800 1000
G250
G550
Temperature T (°C)
20,
,
20 y
TyTff
EE
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-44
these parameters on column buckling strength. Rasmussen and Rondal’s results
confirm the many observations in this research. This research has first shown that
stress-strain characteristics are highly nonlinear at elevated temperatures and thus the
influence of parameters such as n on the strength results. More importantly, the
parameter ep increases as the temperature increases, in particular for high strength
steels as seen in Figure 5.18 and hence influences the local buckling strength results.
Therefore such changes to material characteristics with increasing temperatures, in
particular for high strength steels, must be taken into account in the development of
design rules.
5.4.3 Effective Width Rules for Local Buckling Behaviour at Elevated
Temperatures
5.4.3.1 Theoretical design method
The local buckling behaviour of cold-formed steel sections at elevated temperatures
is different to that at ambient temperatures as the former is characterised by highly
nonlinear stress-strain behaviour associated with gradual yielding. Therefore the
inelastic behaviour of cold-formed steel sections at elevated temperatures is of major
importance. The characteristics of such behaviour can be modelled using the
Ramberg-Osgood law as it is often used for elastic-perfect plastic materials, elastic-
hardening materials and inelastic materials (Mazzolani, 1995). This is the approach
used for stainless steel and aluminium structures in order to include the nonlinear
stress-strain relationships.
By employing the Ramberg-Osgood law, the stress-strain relationships at elevated
temperatures can be simulated well by considering the temperature effects.
n
Ty
T
T
TT f
fEf
+=
,χε (5.7)
Equation 5.7 can be divided into two sections, elastic strain єe and plastic strain єp, as
follows.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-45
n
Ty
Tp
T
Teff
Ef
==
,, χεε (5.8)
where
=
T
Ty
f
fn
,1.0
,ln
2ln (Mazzonlani, 1995), f0.1,T and fy,T = 0.1% and 0.2% proof
stresses at temperature T, respectively.
The elastic critical buckling stress at ambient temperature is in general defined as in
Equation 4.3. Many researchers (Stowell, 1948; Bleich, 1952; Buitendag and van
den Berg, 1994; Reyneke and van den Berg, 1996; Hopperstad et al., 1999) used this
elastic buckling stress equation to derive the critical local buckling stress in the
inelastic range. The inelastic buckling stress equation including a plasticity reduction
factor ηT at temperatures T is then given by
( )( )22
2
, /112 tb
Ekf TTp
Tcr υ
πη
−= (5.9)
The effective width concept introduced by von Karman et al. (1932) then becomes
Ty
TcrT
Te
f
f
bb
,
,, η= (5.10)
The plasticity reduction factor ηT for the determination of the elastic critical buckling
stress is modified including temperature effects based on the definition using the
tangent modulus as given in Equation 5.11 (Buitendag and van den Berg, 1994;
Reyneke and van den Berg, 1996).
T
TtT E
E ,=η (5.11)
The tangent modulus Et corresponding to stress values in the inelastic range was
developed based on Engesser’s original theory of buckling beyond the elastic limit.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-46
In the tangent modulus theory, the plasticity reduction factor is not affected by the
shape of the column section but dependent on the elastic-plastic properties of
materials (Bleich, 1952). Therefore the use of the tangent modulus was considered
adequate to predict the column strength at elevated temperatures. The tangent
modulus, Et, is defined as the slope of the stress-strain curve. For example, the rate
of the tangent modulus increases in proportion to dσ/dє. Therefore using the
Ramberg-Osgood law (Equation 5.7), the tangent modulus can be defined. Reyneke
and Van den Berg (1996) obtained the tangent modulus as the inverse of the first
derivative with respect to strain based on the Ramberg-Osgood law. The tangent
modulus derived by Reyneke and Van den Berg (1996) was used with consideration
of temperature effects on mechanical properties as given in Equation 5.12.
1
,,
,, −
+
=n
Ty
TTTy
TTyTt
ff
nEf
EfE
χ (5.12)
The elastic and plastic stresses can be defined by Hooke’s law and the Ramberg-
Osgood formula, respectively, in a similar manner to the elastic and plastic strains in
Equation 5.8.
eTT Ef ε= for elastic stress (5.13a)
np
TyT ff/1
,
=
χε
for plastic stress (5.13b)
( )
)1/(
,
/12nn
Ty
Tn
pf
En−
−=
χε (5.13c)
The plastic strain given in Equation 5.13(c) was defined by Hopperstad et al. (1999)
for the post-buckling region of unstiffened flange elements. For plates where the
ultimate strength is obtained in the post buckling region, the plastic strain should be
considered in the calculation of the ultimate strength. Therefore Hopperstad et al.
(1999) defined an expression for the plastic strain at which unstiffened elements
reached their ultimate strength in the post critical region. In Equation 5.13(c), it was
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-47
assumed that a parameter n is larger than 2. The tensile coupon test results showed
that the parameter n was larger than 3. Therefore Hopperstad et al.’s (1999) plastic
strain was used with simple modification including the reduced mechanical
properties. The plastic strain given in Equation 5.13(c) can be used for any material
by changing the two factors of n and χ. Therefore, it is considered adequate for the
mechanical property of cold-formed steels that varies with temperature.
Using the Ramberg-Osgood model at elevated temperatures defined in Equations
5.7, 5.8, 5.12 and 5.13, Equation 5.11 is expressed as follows.
n
Ty
T
TT
T
n
Ty
TTTy
TTy
T
TtT
ff
fnE
E
ffnEf
Ef
EE
+
=
+
==
−
,
1
,,
,
,
11
1
χ
χ
η (5.14)
Substituting the plastic stress Equation 5.13(b) into Equation 5.14, the plasticity
reduction factor is then given by
nn
Ty
Ty
Tn
T
T
f
fEn
nE1
,
1
,
/1)2(
1
1
−
−
−
+
=
χ
χ
χ
η (5.15)
Equation 5.15 can be simplified as follows.
Tn
Tyn
TyT
T
En
ff
nE /1,/1
, )2(11
1
χχ
χχ
η
−+
=
)1(22
21
1−−
=
−+
=n
n
nnTη (5.16)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-48
Figure 5.19 Comparison of Design Curves with Test and FEA Results
Based on the experimental tensile stress-strain curves obtained in this study (see
Chapter 3), a value of n in Equation 5.16 is taken as four constants (n = 3, 5, 7, 15).
Figure 5.19 compares theoretical design curves with experimental and numerical
results for high strength steel sections. This comparison shows reasonable adequacy
of the use of theoretical derivation considering temperature effects on mechanical
properties of cold-formed steels. It is seen in Figure 5.19 that the use of a constant n
of 3 is appropriate at high temperatures and the use of n = 5, 7 and 15 is adequate at
low temperatures.
The above observations in relation to Figure 5.19 comparisons correlate well with
the stress-strain characteristics and the developed Ramberg-Osgood model in
Chapter 3 for steels at elevated temperatures. In Chapter 3, it was found that the
nonlinearity of stress-strain curve increased at higher temperatures and hence smaller
n values less than 15 were needed to model exactly the stress-strain behaviour. This
agrees quite well with the need to use an ‘n’ value of 3 for high temperatures as
found in this chapter. Although Chapter 3 recommended a constant n value of 15 for
all the temperatures to simplify the stress-strain model, it may be necessary to vary n
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)Eq.(5.10)G5-20G5-200G5-400G5-600G5-800
be,T/b
λT
n=3
n=5
n=7 n=15
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-49
values as a function of temperature, particularly because the approach used in this
chapter does not have another factor such as the β factor used in Chapter 3.
5.4.3.2 Semi-empirical design method
Although the theoretical design method developed in Section 5.4.3.1 was adequate
for high strength steel sections at elevated temperatures, a semi-empirical design
approach used in steel structures was attempted in this section based on Winter’s
method because the theoretical design method was in general used for aluminium
alloy structures.
Effective width is the width of a steel plate element which buckles when the
compressive strength reaches the yielding point. Based on this concept, the effective
width is determined using the following steps (Yu, 2000).
22
2
)/)(1(12 tbEkf
ey
νπ
−= (5.17)
Equation 5.17 can be rewritten as for the effective width be.
yy
e fkE
tCfEktb =
−=
)1(12 2
2
νπ
(5.18)
kEf
tbC ye= (5.19)
Using the theoretical elastic local buckling stress formula (Equation 4.2), the
effective width ratio can be derived (von Karman, 1932) as given in Equation 4.1.
After this relationship was introduced, Sechler (1933) and Winter (1947) showed
that the term C in Equations 5.18 and 5.19 was dependant on the non-
dimensionalised parameter γ given next.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-50
=bt
fE
yγ (5.20)
Equations 5.19 and 5.20 can be rewritten considering the reduced yield strength and
elasticity modulus at elevated temperatures T as follows.
TT
TyTeT kE
ft
bC ,,= (5.21)
=
bt
fE
Ty
TT
,γ (5.22)
From experiments and FEA, effective widths of unstiffened flange elements were
obtained. Based on Equation 5.21, the term CT was evaluated for all the temperatures
considered in this study. In this evaluation, only the high strength steel sections were
considered. The corresponding non-dimensional parameter γT was also evaluated
using Equation 5.22. The non-linear relationship between CT and γT was established
as shown in Figure 5.20.
The following equation was derived based on the evaluation of CT and γT.
)71.112.657.71(152.0 32TTTTC γγγ +−+= (5.23)
Substituting Equations 5.21 and 5.22 into Equation 5.23, Equation 5.23 becomes
+
−
+=
2/3
,
3
,
22/1
,,, 71.112.657.71152.0
Ty
T
Ty
T
Ty
T
Ty
TTTe f
Ebt
fE
bt
fE
bt
fEk
tb
(5.24)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-51
Figure 5.20 Relationship between CT and γT at Temperature T
Equation 5.24 can be rearranged to express be/b as follows.
+
−
+=
2/3
,
3
,
22/1
,,
, 71.112.657.71152.0Ty
T
Ty
T
Ty
T
Ty
TTTe
fE
bt
fE
bt
fE
bt
fEk
bt
bb
(5.25)
Equation 5.25 can be rearranged in terms of the slenderness ratio λT defined by
Equation 5.5(c).
+−+=
332, 99.177.696.7116.0
TTTTTTT
Te
kkkbb
λλλλ (5.26)
Equation 5.26 is a semi-empirical effective width equation for unstiffened flange
elements subject to local buckling effects at elevated temperatures. Equation 5.26 is
finally simplified as given in Equation 5.27.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
G5-Test
G5-FEA
CT
γT
)71.112.657.71(152.0 32TTTTC γγγ +−+=
R=0.847
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-52
+−+=
32, 99.177.696.7116.0
TTTT
Te
bb
βββλ (5.27)
where
TT
Ty
Tcr
TyT kE
ftb
ff ,
,
, 052.1
==λ
TTT
TyT k
Ef
tb λβ =
= ,052.1
Figure 5.21 Comparisons of Effective Width Ratios based on Equation 5.27
with Test and FEA Results at Elevated Temperatures
For the prediction of effective width using Equation 5.27, the buckling coefficient kT
was assumed to remain the same at elevated temperature. In the calculation of βT, kT
was taken as 0.866 based on the FEA results at ambient temperature. Figure 5.21
compares the effective width predictions obtained from Equation 5.27 with
experimental and FEA results at elevated temperatures. It can be seen that Equation
5.27 provides a lower bound to the effective widths obtained from FEA and
experiments. Therefore Equation 5.27 can be safely used to predict the effective
width of unstiffened flange elements subject to local buckling effects at elevated
temperatures.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
G5-20
G5-200
G5-400
G5-600
G5-800
be,T/b
λT
Equation 5.27
Equation 5.5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-53
In order to reduce the scatter in the relationship between CT and γT at elevated
temperatures, only the FEA results were used in Figure 5.22 and an equation
expressing the relationship between CT and γT was derived as given by Equation
5.28.
)69.374.1021.131(102.0 32TTTTC γγ +−+= (5.28)
Figure 5.22 Relationship between CT and γT using FEA Results
The same derivation process was again used to derive a new effective width equation
as follows.
+−+=
32, 29.489.119.131107.0
TTTT
Te
bb
βββλ (5.29)
In Figure 5.23, the effective width predictions based on Equation 5.29 were
compared with FEA results at elevated temperatures. The prediction provides a
lower bound to the effective widths obtained from FEA. Therefore Equation 5.29 can
be successfully used to predict the effective width. However, as seen in Figure 5.22,
even the FEA data points do not follow a regular pattern. This is due to the variations
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
CT
γT
)69.374.1021.131(102.0 32TTTTC γγ +−+=
R=0.902
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-54
of the important parameter ep (fy/E) for these data points. The value of epvaried from
about 0.002 to 0.006 for the data in Figures 5.22 and 5.23.
Figure 5.23 Comparisons of Effective Widths based on Equation 5.29 with
FEA Results at Elevated Temperatures
Figure 5.24 compares the effective widths based on Equation 5.29 with test results.
In some cases, the effective widths obtained from experiments were less than the
predictions of Equation 5.29 as seen in Figure 5.24. It was found in Chapter 4 that
the disadvantages of the use of high strength steels were lack of ductility including
strain hardening and the Bauschinger effect present in high strength steels at ambient
temperature even for 0.95 mm thickness high strength steel, i.e. greater than the
AS4600 limit of 0.9 mm. It was also found in Chapter 3 that the ductility of high
strength steels significantly improved at high temperatures. However, there was a
small discrepancy between the test and FEA results at elevated temperatures as
shown in Figures 5.23 and 5.24. The test results appeared to be slightly below the
design curve based on Equation 5.29, which was developed using the FEA results.
This implies that despite the improved ductility at elevated temperatures, test results
were lower than the expected values. Hence it is possible that the Bauschinger effect
may be still present at high temperatures. However, there was no clear trend of the
discrepancy in the temperature range used in this study and only 0.95 mm thickness
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
G5-20
G5-200
G5-400
G5-600
G5-800
be,T/b
λT
Equation 5.5
Equation 5.29
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-55
was used in this study. Therefore further research is required to investigate the
ductility and Bauschinger effects of high strength steels at elevated temperatures.
Figure 5.24 Comparisons of Effective Widths based on Equation 5.29 with
Test Results at Elevated Temperatures
In this study, as was done in the same section for ambient temperatures, an additional
yield strength reduction factor of 0.9 was used in the calculation of effective widths
from experimental results to determine whether it improves the comparison in Figure
5.24. Such modified experimental effective widths were then compared with
predictions from Equation 5.29 in Figure 5.25. This comparison shows that Equation
5.29 can be used safely to predict the effective widths of unstiffened flange elements
made of high strength steels at elevated temperatures.
This section has proposed two equations 5.27 and 5.29 to predict the effective widths
of high strength steel unstiffened flange elements at elevated temperatures. Equation
5.27 was developed based on both experimental and FEA results while Equation
5.29 was based on FEA results. Hence Equation 5.27 not only includes the
behavioural changes due to elevated temperatures, but also the specific problems of
high strength steels at ambient temperatures. Since the current cold-formed steel
design codes (SA, 1996; AISI, 1996) address the latter by using a reduced yield
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
G5-20
G5-200
G5-400
G5-600
G5-800
be,T/b
Equation 5.5
Equation 5.29
λT
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-56
strength (0.9fy), it is recommended that Equation 5.29 is used in the design of high
strength steel unstiffened flange elements as the use of a reduced yield strength
already takes into account the high strength steel problems at ambient temperature.
Figure 5.25 Comparisons of Effective Widths based on Equation 5.29 with
FEA and Test Results using a Reduced Yield Stress of 0.9fy
5.4.3.3 Simplified design method
The design process based on the Ramberg-Osgood law as described in the last
section may be considered too complicated. Therefore a simpler alternative was also
empirically developed based on experimental and numerical effective widths at
elevated temperatures in order to simplify practical designs.
From the variation of effective width of high strength cold-formed steel compression
members with temperature, an approximate linear relationship between the reduction
of effective width and temperature including the reduction ratio of the elasticity
modulus to the yield strength was derived using experimental and numerical results
as shown in Figures 5.26(a) and (b). As clearly seen in Figures 5.26(a) and (b), the
reduction of the effective width of high strength steel sections is much larger than
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
G5-20(Test)G5-20(FEA)G5-200(Test)G5-200(FEA)G5-400(Test)G5-400(FEA)G5-600(Test)G5-600(FEA)G5-800(Test)G5-800(FEA)
be,T/b
λT
Equation 5.5
Equation 5.29
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-57
that of low strength steels sections. Further the reduction ratio of the effective width
of low strength steel sections is less than the ratio of the elasticity modulus to the
yield strength for low strength steels at elevated temperatures (see Figure 5.18).
Therefore current design rules including the reduced yield strength and elasticity
modulus can be used for low strength steel sections. However, a further modification
is required for high strength steel sections.
Figure 5.26 Effects of Mechanical Properties on Effective Width
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 200 400 600 800
G2-1
G2-2
G2-3
(a) Low strength steel section
be,T/be,20
Φ
Φ
,yfET
Effe
ctiv
e w
idth
redu
ctio
n ra
tio
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 200 400 600 800
G5-1
G5-2
G5-3c
be,T/be,20
Φ
Φ−=,20,
, 00063.01ye
Te
fE
Tbb
Φ
Φ
,yfET
(b) High strength steel section
Effe
ctiv
e w
idth
redu
ctio
n ra
tio
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-58
From the investigation of the reduced effective width at elevated temperature, a
linear equation was derived. This linear equation is given in Equation 5.30.
Φ
Φ−=,20,
, 00063.01ye
Te
fE
Tbb
(5.30)
where 20E
EE T=Φ and
20,
,,
y
Tyy f
ff =
Φ
Combining the above equation (be,T/be,20) and Equation 4.10, that is, the simplified
effective width equation at ambient temperature (be,20/b), an empirical effective
width equation (be,T/b) including the temperature effects is derived as given next.
( )Ty
TcrTe
ff
Tb
b
,
,, 00063.01745.0 −= (5.31)
The effective widths obtained from Equation 5.31 were compared with experimental
and numerical results in Figures 5.27(a) to (e). Equation 5.31 provides reasonably
accurate predictions at elevated temperatures.
In addition to the design proposals described above, the British Standard method (BS
8818, 1992) was also attempted to investigate the applicability to cold-formed steel
sections at elevated temperatures. In fact, the standard is for aluminium alloy
structures. Since there are similarities between aluminium alloy steels at ambient
temperature and light gauge steels at elevated temperatures, the British Standard
method was used in this study.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-59
Figure 5.27 Comparisons of Effective Width Ratios based on Equation 5.31
with Test and FEA Results at Elevated Temperatures
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-200(Test)
G5-200(FEA)
be,T/b
Tλ
(b) 200 °C
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-20(Test)
G5-20(FEA)
be,T/b
(a) 20 °C
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
Tλ
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-60
Figure 5.27 Comparisons of Effective Width Ratios based on Equation 5.31
with Test and FEA Results at Elevated Temperatures
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-400(Test)
G5-400(FEA)
be,T/b
(c) 400 °C
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
Tλ
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-600(Test)
G5-600(FEA)
be,T/b
(d) 600 °C
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
Tλ
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-61
Figure 5.27 Comparisons of Effective Width Ratios based on Equation 5.31
with Test and FEA Results at Elevated Temperatures
The equation given in BS 8818 (BSI, 1992) is simply modified using the reduced
yield strength and elasticity modulus. The modified equation is as follows.
+−=
3221,,1111.1TTT
TyTc kkfλλλ
σ (5.32)
where k1= 0.22, k2= 0 and TcrTyT ff ,, /=λ for non-welded outstands.
In BS8818 (BSI, 1992), λT was taken as 1.59(b/t)√fy/E assuming a buckling
coefficient value (k) of 0.438. However, in this case, a suitable kT value is 0.866
based on FEA results at ambient temperature (see Table 5.5). Therefore λT in this
case was based on the original definition of √fy,T/fcr,T.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-800(Test)
G5-800(FEA)
be,T/b
(e) 800 °C
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
Tλ
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-62
Figure 5.28 Comparison of Effective Width obtained from British Standard
(BS 8818) with Test and FEA results
In Figure 5.28, the effective widths obtained from Equation 5.32 are compared with
test and FEA results at elevated temperatures. This comparison shows that the
British Standard method for aluminium alloy structures is reasonably adequate up to
400°C but overestimates the results at high temperatures above 600°C. Therefore
Equation 5.32 was modified as follows to provide the best lower bound fit for all the
data points.
( )
−=
2,,83.1
06.0)83.1(
1
TTTyTc f
λλσ (5.33)
5.4.4 Ductility of High Strength Steels at Elevated Temperatures
As discussed in Chapter 4, one of the shortcomings of using high strength cold-
formed steels at ambient temperature is low ductility. Effects of low ductility of high
be,T/b
λT 0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.32)Eq.(5.33)G5-20G5-200G5-400G5-600G5-800
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-63
strength steels on the ultimate strength were clearly seen in the investigation at
ambient temperature. The tensile tests at elevated temperatures showed that with
reducing yield strength at high temperatures, high strength steels gained greater
ductility, without any strain-hardening (see Figure 3.15 in Chapter 3). In this section,
the ductility of cold-formed steel members subject to axial compression was
investigated using axial compression load versus axial shortening curves at different
temperatures.
Figure 5.29 Axial Compression Load versus Axial Shortening Curves of
Test Series G5-3
The failure strain at high temperature is much higher than at ambient temperature as
seen in Figure 5.29 although the ultimate strength was reduced due to the
deterioration of mechanical properties at high temperatures. One of the shortcomings
using finite element analyses (FEA) of high strength steel sections at ambient
temperature was that FEA was not able to simulate the exact material stress-strain
curves including the reduction of material strength after reaching the failure strain.
Therefore the assumption of perfect plasticity in FEA resulted in slightly higher
prediction of the ultimate compression load than test results at ambient temperature.
By recovering the ductility at high temperatures, the FEA predictions were in
reasonably good agreement with experimental results at high temperatures. The axial
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2 2.5
Test
FEA
Axi
al C
ompr
essi
on L
oad
Axial Shortening
kN
mm
20 ºC
200 ºC
400 ºC
600 ºC
800 ºC
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-64
compression load versus axial shortening curves of other test series are given in
Appendix E1.
5.5 Summary
This chapter has presented the details of experimental and finite element analyses of
cold-formed steel compression members subject to local buckling effects. Both
experimental and numerical investigations have provided a better understanding of
the local buckling behaviour of unstiffened flange elements at elevated temperatures.
Simulating the nonlinear mechanical properties of cold-formed steels at high
temperature was successfully achieved in the finite element analyses using field
option variables based on true stress-strain curves.
From this investigation, it was found that the local buckling stresses at elevated
temperatures were reduced as a function of the deterioration of the elasticity
modulus. Therefore, the theoretical equation for the local buckling stress can be
simply modified by including the reduced elasticity modulus. Further, the rotational
rigidity provided by the web to the adjoining unstiffened flanges of unlipped channel
members can be considered relatively independent of temperature increase as for the
calculation of the local buckling stress.
The effective width design rule was simply modified using the reduction factors of
the yield stress and elasticity modulus to predict the ultimate strength of cold-formed
steel compression members at elevated temperatures. Comparing the experimental
and numerical results with the modified design rules, it was found that the modified
rules were adequate for the unstiffened flange elements made of low strength steel,
but not for high strength steel unstiffened elements. The reason for the inadequacy of
the effective width rule was that most reduction of the ultimate strength of cold-
formed steel compression members at elevated temperatures occurred in the post
buckling strength range. Therefore, the effective width rules had to be re-established
for high strength steel sections considering the relationship in the reduction of the
mechanical properties and post-buckling strength associated with elevated
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 5-65
temperatures. New improved design rules using four different approaches were
developed and their details are given in this chapter.
The Ramberg-Osgood law was used first to develop an effective width design rule at
elevated temperatures for cold-formed steel members because of the plastic material
behaviour and the varying yield strength at elevated temperatures. In deriving the
plasticity reduction factor, tangent modulus was used, and the results were compared
with experimental and numerical results. The comparison showed that the theoretical
predictions were in good agreement with test and FEA results. Semi-empirical
design methods were also proposed considering the reduced yield strength and
elasticity modulus based on Winter’s (1947) approach. Due to the complexity of the
use of theoretical design curves, two empirical alternatives were also proposed based
on experimental and numerical results. They also provided reasonably accurate
predictions at elevated temperatures. It is recommended that the semi-empirical
design method is used in the design of high strength steel unstiffened flange
elements as the use of a reduced yield strength already takes into account the high
strength steel problems at ambient temperature.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-1
Chapter 6. Local Buckling Behaviour of Stiffened Elements at Elevated Temperatures
6.1 General
The local buckling behaviour of unstiffened flange elements at elevated temperatures
was presented in Chapter 5. The conventional design rules of the effective width and
the local buckling stress were simply modified first and applied to unstiffened flange
elements at elevated temperatures to verify the suitability and accuracy of the
modifications. Due to the inadequacy of simply modified effective width rules for
high strength steel sections, theoretical, semi-empirical and empirical design rules
were developed in Chapter 5. However, the modification of the local buckling stress
considering the reduced elasticity modulus was found to be applicable to unstiffened
elements at elevated temperature.
In order to investigate the local buckling behaviour of stiffened elements and the
adequacy of the design rules developed in Chapter 5, tests and finite element
analyses (FEA) were carried out using lipped channel sections subject to axial
compression that were predominantly governed by the local buckling of stiffened
web elements at elevated temperatures up to 800°C. Deficiencies in the knowledge
and understanding of the local buckling behaviour of cold-formed steel compression
members at high temperatures were highlighted in Chapter 5 even though the cold-
formed steel structures have been widely used in various applications with increasing
recognition of the significance of fire safety design. This research was therefore
undertaken with the goal of eliminating such deficiencies in the knowledge of local
buckling behaviour of cold-formed steel compression members at elevated
temperatures. This chapter presents the details of the investigation into the local
buckling behaviour of stiffened web elements in the lipped channel members and the
results.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-2
6.2 Modification of the Design Rules of Local Buckling Stress and
Effective Width at Elevated Temperatures
The design rules for the local buckling stress and effective width for cold-formed
steel compression members at elevated temperatures are inadequate due to the lack
of experimental evidence and understanding regarding their structural behaviour.
Simple modifications were therefore attempted using the conventional design rules
in Chapter 5. The reduced elasticity modulus at temperature T (ET) was used in the
elastic buckling stress equation with an assumption that Poisson’s ratio of 0.3
remains unchanged throughout the temperature range from ambient temperature to
800ºC. The modification of the local buckling stress is given in Equation 5.1. The
local buckling coefficient kT representing the rotational restraint to the stiffened
elements is given in Equation 5.2, which was derived from Equation 5.1. A design
proposal developed by Uy and Bradford (1995) was also studied. In their proposal,
the local buckling stress is mainly controlled by the reduction of the elasticity
modulus, and thus the degree of rotational restraint remains the same irrespective of
higher temperatures. The proposal to predict the local buckling stress at elevated
temperatures is given in Equations 5.4. Their proposal was also investigated for
stiffened web elements.
The conventional effective width design rule currently used in the design standards
(AISI, 1996; SA, 4600, 1996; ECS, 1997) was simply modified in Chapter 5 by
using the appropriately reduced yield strength and elasticity modulus values. The
modified rules are given in Equations 5.5(a) to (c). However these equations were
found to be inadequate for high strength unstiffened elements, and thus theoretical,
semi-empirical and empirical equations were proposed. The simple modification was
first used for stiffened elements at elevated temperatures and its predictions were
compared with experimental and numerical results. Following this, the theoretical,
semi-empirical and empirical proposals given by Equations 5.10, 5.27, 5.29 and 5.31
were used for stiffened web elements.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-3
6.3 Experimental Investigations
The objectives of the experimental investigations are to obtain the elastic local
buckling stresses and experimental effective widths at elevated temperatures, and
then to investigate the effects of elevated temperatures on the local buckling
behaviour of stiffened web elements. Further, the failure modes of cold-formed steel
compression members predominantly governed by the stiffened web element were
also studied to ascertain whether any transition of the buckling mode existed due to
increasing temperatures up to 800°C.
6.3.1 Test Specimens
This experimental program included cold-formed steel lipped channel sections
subject to axial compression governed by the local buckling of stiffened web
elements. Forty tests of cold-formed steel stub columns including the variables of
temperature, steel grade and section slenderness ratio λ (=√fy/fcr) were conducted to
investigate the local buckling behaviour of stiffened elements at elevated
temperatures. All the specimens were made of G550 steel (minimum yield strength
of 550 MPa) and G250 steel (minimum yield strength of 250 MPa).
Figure 6.1 Section Geometry – Lipped Channel
The cross section geometry, measured geometrical dimensions and properties of
sections are shown in Figure 6.1 and Table 6.1. The specimens were designed so that
they failed by web local buckling without flange or lip buckling. For this purpose,
the well known finite strip analyses program THINWALL (Papangelis and Hancock,
b
d
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-4
1998) was used. Figure 6.2 shows the typical buckling plot including the critical
buckling mode of the chosen lipped channel sections. The local web buckling was
then considered as the local buckling of the stiffened web element that is influenced
by the rotational restraint provided by the adjoining flange elements.
Table 6.1 Test Specimens
THINWALL Test
series Grade b (mm)
d (mm)
L (mm)
t (BMT, mm)
fy (MPa) fcr
(MPa) λ HL (mm)
G2-L1 8.5 57.0 9.1 0.964 317 301 1.03 40 G2-L2 18.0 76.5 14.0 0.964 317 176 1.34 60 G2-L3 27.1 118.0 18.5 0.964 317 67 2.18 90 G2-L4
G250
22.5 98.2 13.5 0.564 335 39 2.93 75 G5-L1 9.0 38.5 9.1 0.936 636 605 1.03 30 G5-L2 14.1 57.5 9.5 0.936 636 282 1.50 45 G5-L3 19.2 88.5 14.0 0.936 636 119 2.30 70 G5-L4
G550
23.5 97.5 14.0 0.785 650 70 3.05 75 *BMT and HL refer to base metal thickness and half-wave buckling length, respectively. Corner radius r = 1.2 mm for all specimens. Temperatures considered were 20, 200, 400, 600, 800ºC.
The yield stresses fy are the measured values.
Figure 6.2 Typical Buckling Stress Plot of the Chosen Lipped
Channel Sections (G5-L2)
45
mm
MPa
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-5
Prior to conducting simulated fire tests, a series of preliminary tests was conducted
for each section at ambient temperature to ensure that the behaviour of all the
specimens are governed by the local web buckling. Base metal thicknesses were
measured using a micrometer gauge for total thickness and a coating thickness
gauge. It was found from the measurement of thicknesses before and after testing
that the base metal thickness remained the same during fire tests. The specimen
length was taken as three times the half-wave buckling length (HL in Table 6.1) plus
an additional 20 mm at the top and bottom ends to allow for the end support
conditions.
The yield strength and elasticity modulus for the steels used in this study were
measured at the ambient and elevated temperatures (see Chapter 3). The relevant
reduction factors for the yield strength and elasticity modulus for this study are given
in Table 6.2 and were used in the calculation of the local buckling stress and ultimate
strength at elevated temperatures.
Table 6.2 Reduction Factors for Elasticity Modulus (ET/E20) and
Yield Strength (fy,T/fy,20)
20 200 400 600 800
G250 0.964 1.000 0.887 0.624 0.300 0.100 G250 0.564 1.000 0.900 0.599 0.293 0.098 G550 0.936 1.000 0.902 0.564 0.273 0.068
ET E20
G550 0.785 1.000 0.902 0.564 0.275 0.070 G250 0.964 1.000 0.935 0.788 0.427 0.150 G250 0.564 1.000 0.961 0.794 0.409 0.205 G550 0.936 1.000 0.969 0.797 0.478 0.127
fy,T fy,20
G550 0.785 1.000 0.969 0.784 0.472 0.141
As discussed in Chapter 3, four different strain levels of 0.2, 0.5, 1.5 and 2.0% can
be used in the determination of the reduced yield strength. The reduction factors for
the yield strength based on the 0.2% proof stress method were used in this study and
are shown in Table 6.2, because adopting the other strain levels has not been clearly
defined or widely accepted. In particular, increased attention is required to the
structural stability issues when higher strain levels of 1.5 and 2.0% are used.
BMT (mm)
Temp (°C) Steel Grade
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-6
6.3.2 Test set-up and procedure
All the tests were conducted using a specially designed furnace heated by four
electronic glow bars and containing temperature monitoring systems. The furnace
was described in detail with a schematic diagram in Section 5.2.1. Figure 6.3 shows
the experimental test set-up for stiffened web elements including a test specimen
inside the furnace. The air and specimen temperatures were monitored using two in-
built thermo-couples and two thermo-couples located at the top and bottom of a
specimen, respectively. The temperature distribution inside the furnace was
relatively even and very effective due to the use of four glow bars and the small
space used. The furnace temperature was controlled by a microcomputer based
temperature indicating system. The axial shortening and lateral displacement of the
specimen associated with the vertical compression loading were measured using an
extensometer installed in a 300 kN capacity universal testing machine and a
displacement transducer, respectively. As considered in the local buckling tests of
unstiffened flange elements, fixed-end support conditions were considered adequate
for the same reasons.
As for the tests of unstiffened flange elements at elevated temperatures described in
Chapter 5, all the simulated fire tests of stiffened web elements were conducted
under a steady state based on specimen temperature, the so-called, steady state test
method. The details of the test procedure have already been discussed in Section
5.2.3, and thus it is briefly summarised. Prior to applying the axial load, the
specimen was heated up to a pre-determined temperature. The increased temperature
was then maintained while the specimen was loaded until failure. The adequacy and
reliability of the steady state test method has already been discussed in Section 5.2.3.
From the tests, the axial compression load versus axial shortening curve and the axial
compression load versus lateral displacement curve were obtained at various
temperature levels. The temperature range was 20ºC to 800ºC with an interval of
200ºC (20ºC, 200ºC, 400ºC, 600ºC and 800ºC).
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-7
Figure 6.3 Test Set-up
Temperature overshooting, in which the temperature continued beyond the pre-set
value, was found when the specimens were heated. However, it was reasonably
small to be neglected. In particular, the temperature overshoot was less than 2% at
high temperatures.
The local buckling and ultimate loads were obtained from each test. From the
experimental ultimate strength results, the effective width of the stiffened web
element was obtained assuming that flanges and lips were fully effective. The
experimental local buckling load was obtained based on the load versus out-of-plane
displacement2 curves as developed by Venkatarimaiah and Roorda (1982). This
method was used as described in Sections 4.4.2.6 for unstiffened flange elements.
The local buckling coefficient kT was then derived using Equation 5.2.
In-built Thermo -couples
Thermo-couples
Displacement transducer
Glow bars
Fixed End
Specimen
Compression load application
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-8
6.4 Finite Element Analysis
6.4.1 Elements
A finite element analysis program ABAQUS 6.3 (HKS, 2002) was used to simulate
the behaviour of all the test specimens at elevated temperatures. Similar principles
used for unlipped channel sections were considered adequate for lipped channel
sections (see Section 5.3). The S4R5 element type was considered suitable to provide
accurate results because the shell element allows large rotations and sufficient
degrees of freedom with four nodes and five degrees of freedom (HKS, 2002). It is
necessary to avoid too coarse meshes for the sake of storage usage and
computational time, and thus there needs to be an appropriate decision made
regarding element mesh density for the accuracy of a model. A 3.5 mm length mesh
was considered adequate for the lipped channel sections used in this study based on a
series of convergence studies.
6.4.2 Material properties at elevated temperatures
The Mises and Hill yield surfaces allowing the perfect plasticity hardening behaviour
were implemented for the material behaviour of classical metal plasticity. Using this
method the mechanical properties at elevated temperatures were achieved defining
the accurate material behaviour associated with gradual yielding.
It was observed from tensile tests that G550 zinc/aluminium alloy-coated light gauge
steels contained no strain hardening at ambient and elevated temperatures. Thus as
used in Section 5.3.1.2, the perfect elastic-plastic model was considered adequate for
G550 steels at low temperatures (20°C and 200°C). Several time steps using the field
dependant option were implemented to model the material softening zone (gradual
yielding zone after the elastic behaviour) based on experimental stress-strain curves
at high temperatures. For the material model of low strength steels (G250), the
strain-hardening stiffness at low and high temperatures was taken as 4 and 2
percentage of the elasticity modulus, respectively. Even though the strain-hardening
was hardly found at the temperature of 800ºC in the experimental study, it was
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-9
assumed as 2 percentage of the elasticity modulus. However, the strain-hardening
did not affect the results much due to the failure of members before reaching the
large strains at high temperatures. The details of idealised mechanical properties
used in the finite element analyses of this study are illustrated in Figures 5.4(a) and
(b) for high and low strength steels, respectively.
6.4.3 Geometric imperfections and residual stresses
For the members subject to local buckling effects, initial geometric imperfections are
required to compensate for the errors in the manufacturing processes. Global
buckling imperfection was excluded because all the lipped channel members were
predominantly governed by the local buckling of the stiffened web element. An
imperfection magnitude of d/167 as recommended by Schafer and Pekoz (1998) was
used on the critical eigen-buckling mode as deviation values from the perfect
geometry (see Figure 6.4(a)). The details of earlier research and the procedure to
include the geometric imperfection in FEA are given in Section 4.4.2.4.
Figure 6.4(b) shows the assumed residual stress distribution at temperature T. This
was based on the residual stress distribution recommended by Schafer and Pekoz
(1998) for press braked lipped channel sections at ambient temperature. Five
interpolation points were used to consider the varying residual stress through the
thickness. As the residual stress is relieved with increasing temperature, the residual
stress should not be assumed to be the same for all the temperatures. Therefore, the
same stress relieving or residual stress reduction factors, as used for unlipped
channel sections (see Equation 5.3 and Table 5.3 in Section 5.3.1.4), were used for
lipped channel sections at elevated temperatures. The residual stress reduction
factors (a) given in Table 5.3 are based on a fully annealing temperature assumed as
800ºC. The user defined residual stress was modelled using the SIGINI Fortran user
subroutine with the variation of five interpolation points. Membrane residual stresses
for press-braked sections are considered too small and hence ignored in the
numerical studies (Schafer and Pekoz, 1998; Young and Rasmussen, 1999). Figure
6.5 shows the residual stress output obtained from FEA.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-10
Figure 6.4 Residual Stress Distributions and Geometric Imperfection
Figure 6.5 Residual Stress Output from FEA model
0.08αfy
0.33αfy
0.17αfy
+σr
-σr
outside
inside
Residual stress through thickness
0.33αfy 0.08αfy
∆ =d/167 d
(a) Initial geometric imperfection
(b) Assumed residual stress with temperature effects
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-11
6.4.4 Methods of analysis
The elastic eigenvalue buckling analysis and non-linear post buckling analysis were
carried out to investigate the structural behaviour of stiffened web elements at
elevated temperatures. Elastic buckling analyses were first used to obtain the critical
buckling mode by applying the Lanczos eigensolver method. Suitable magnitudes of
local buckling imperfection based on Figure 6.4(a) were then applied to the critical
buckling mode.
Non-linear static analyses were then undertaken in the second stage. In this stage,
several steps are required to include residual stresses. Residual stresses with no loads
and all degrees of freedom restrained were first applied to a model. This step was
performed to determine the reaction forces and moments caused by residual stresses.
Including the reaction forces and moments, two non-linear analysis steps for the
residual stress equilibrium and an applied axial load were finally submitted to obtain
the ultimate strengths at elevated temperature.
6.4.5 Half length model
Two kinds of models can be used to investigate the local buckling behaviour of stub
columns. An experimental model with a half length or a full length is used to
simulate the behaviour of test specimens whereas an ideal model with half-wave
buckling length or half of half-wave buckling length is used to simulate ideal
conditions. In this numerical study of lipped channel sections, both the experimental
and ideal models were used and validated.
To simulate the test conditions in the laboratory, a half length model was used with
appropriate boundary conditions. Figure 6.6 shows the half length model including
the model geometry, meshes and the loading pattern with appropriate boundary
conditions using multi-point constraints (MPC).
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-12
Figure 6.6 Half Length Model
The local buckling stress and ultimate strength obtained from the half length FEA
model were validated using experimental results. Tables 6.3 and 6.4 show the results
and comparison of local buckling stress (fcr,T) and ultimate strength (Pu,T) at
temperature T. The comparison of the numerical local buckling stress with
experimental local bucking stress shows mean values of 0.96 and 1.01 for low and
high strength steel sections, respectively. Similarly the mean values of the ultimate
strength at elevated temperatures were 0.97 and 0.99 for low and high strength steel
sections, respectively as given in Table 6.4. As seen in these comparisons of local
buckling stress and ultimate strength, the results from the half length FEA model
agreed well with experimental results for both low and high strength steel sections.
It is seen in Table 6.4 that the ultimate strength and effective width obtained from
FEA are slightly higher than those from tests for high strength steel members at low
temperatures of 20°C and 200°C. In particular, thinner high strength steel members
showed larger discrepancies. This behaviour was similar to that described in Chapter
4. Therefore this confirms the investigation reported in Chapter 4 regarding the lack
of ductility and Bauschinger effect of high strength steel members. Due to larger
reduction of the effective width at high temperatures, large discrepancy with small
effective widths is shown in Table 6.4. Thus COV values of the effective width were
shown to be larger than those values of the ultimate strength.
P (MPC, 12456)
S4R5 elements
Plane of symmetry (SPC345)
Dependant nodes
Elastic strip
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-13
Table 6.3 Comparison of Local Buckling Stress and Buckling Coefficient
from Test and Half Length FEA Model at Elevated Temperatures
fcr,T (MPa) kT λT Test Series
Temp. (ºC) Test FEA FEA
Test Test FEA Test FEA
20 301 314 1.04 5.30 5.53 1.026 1.005 200 289 280 0.97 5.73 5.56 1.013 1.029 400 211 202 0.96 5.95 5.70 1.052 1.075 600 108 98 0.91 6.34 5.75 0.961 1.009
G2-L1
800 37 32 0.86 6.51 5.63 0.636 0.684 20 180 176 0.98 5.71 5.58 1.327 1.342
200 167 154 0.92 5.97 5.50 1.332 1.387 400 110 109 0.99 5.59 5.54 1.457 1.464 600 60 53 0.88 6.34 5.60 1.289 1.372
G2-L2
800 21 17 0.81 6.66 5.39 0.844 0.938 20 72 76 1.06 5.43 5.73 2.098 2.042
200 68 69 1.01 5.78 5.87 2.088 2.073 400 45 48 1.07 5.44 5.80 2.278 2.206 600 24 23 0.96 6.03 5.78 2.038 2.082
G2-L3
800 9 7 0.78 6.79 5.28 1.289 1.462 20 39 39 1.00 5.93 5.93 2.931 2.931
200 35 36 1.03 5.91 6.08 3.033 2.990 400 22 24 1.09 5.57 6.08 3.409 3.264 600 12 11 0.92 6.08 5.57 2.952 3.083
G2-L4
800 4 4 1.00 6.25 6.76 2.315 2.315 Mean 0.96 COV 0.087
20 600 641 1.07 5.11 5.46 1.030 0.996 200 541 570 1.05 5.11 5.38 1.067 1.040 400 380 360 0.95 5.74 5.44 1.137 1.168 600 180 183 1.02 5.58 5.67 1.142 1.133
G5-L1
800 45 48 1.07 5.48 5.84 0.814 0.788 20 280 291 1.04 5.32 5.53 1.507 1.478
200 251 258 1.03 5.27 5.43 1.567 1.546 400 170 165 0.97 5.73 5.56 1.700 1.725 600 80 81 1.01 5.53 5.60 1.713 1.703
G5-L2
800 23 21 0.91 6.24 5.70 1.139 1.192 20 123 124 1.01 5.54 5.58 2.274 2.265
200 110 109 0.99 5.49 5.44 2.367 2.378 400 70 69 0.99 5.59 5.51 2.649 2.668 600 35 34 0.97 5.73 5.56 2.590 2.628
G5-L3
800 9 9 1.00 5.79 5.79 1.820 1.820 20 72 74 1.03 5.59 5.75 3.005 2.964
200 66 68 1.03 5.68 5.85 3.089 3.043 400 40 42 1.05 5.51 5.78 3.514 3.429 600 20 21 1.05 5.65 5.93 3.414 3.331
G5-L4
800 6 5 0.83 6.43 5.55 2.340 2.564 Mean 1.01 COV 0.057
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-14
Table 6.4 Comparison of Ultimate Strength and Effective Width
from Test and Half Length FEA Model at Elevated Temperatures
Pu,T de,T/d Test Series
Temp. (ºC) Test
(kN) FEA (kN)
FEA Test Test FEA FEA
Test 20 28.3 27.2 0.96 0.89 0.82 0.92
200 27.5 25.9 0.94 0.95 0.86 0.90 400 23.1 21.7 0.94 0.96 0.86 0.90 600 11.2 12.1 1.08 0.87 1.00 1.15
G2-L1
800 3.8 3.7 0.97 0.94 0.89 0.95 20 37.0 36.3 0.98 0.65 0.62 0.95
200 35.5 33.7 0.95 0.70 0.62 0.88 400 28.4 27.3 0.96 0.63 0.57 0.90 600 15.1 14.0 0.93 0.69 0.58 0.84
G2-L2
800 5.3 5.1 0.96 0.82 0.75 0.92 20 45.5 44.6 0.98 0.43 0.40 0.94
200 42.5 43.8 1.03 0.43 0.46 1.08 400 34.2 35.6 1.04 0.39 0.43 1.11 600 18.5 17.0 0.92 0.45 0.35 0.77
G2-L3
800 6.5 6.0 0.92 0.56 0.45 0.81 20 21.0 21.4 1.02 0.35 0.38 1.08
200 20.4 19.6 0.96 0.36 0.31 0.87 400 12.1 12.5 1.03 0.30 0.34 1.12 600 6.5 6.0 0.92 0.38 0.29 0.76
G2-L4
800 2.2 2.2 0.98 0.37 0.34 0.93 Mean 0.97 0.94 COV 0.046 0.114
20 43.0 43.9 1.02 0.81 0.85 1.05 200 40.6 41.0 1.01 0.76 0.78 0.94 400 34.1 33.4 0.98 0.75 0.68 0.82 600 19.5 18.7 0.96 0.66 0.59 0.89
G5-L1
800 4.8 4.5 0.94 0.53 0.43 0.81 20 47.9 49.3 1.03 0.50 0.53 1.04
200 46.1 47.0 1.02 0.49 0.52 1.06 400 38.2 37.4 0.98 0.45 0.42 0.94 600 21.5 19.8 0.92 0.37 0.27 0.72
G5-L2
800 5.5 5.2 0.95 0.32 0.25 0.79 20 59.2 62.2 1.05 0.33 0.39 1.18
200 57.1 60.0 1.05 0.32 0.38 1.18 400 45.6 45.1 0.99 0.26 0.25 0.95 600 26.0 26.5 1.02 0.21 0.23 1.10
G5-L3
800 6.8 7.0 1.03 0.19 0.22 1.16 20 52.5 55.1 1.05 0.23 0.29 1.24
200 52.2 54.3 1.04 0.23 0.27 1.18 400 40.2 39.4 0.98 0.20 0.18 0.91 600 22.9 22.0 0.96 0.15 0.11 0.74
G5-L4
800 7.1 6.6 0.93 0.17 0.12 0.70 Mean 0.99 0.97 COV 0.042 0.172
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-15
FEA (quarter-wave
buckling length) (b) 800°C
Figure 6.7 Comparison of Failure Modes at Different Temperatures
FEA (half-length) Test (full length)
(a) 20°C
Test (full length) FEA (half-length) FEA (quarter-wave
buckling length)
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3
Test
FEA
(a) Test Series G2-L4 (20, 200, 400, 600, 800 °C) mm
Axi
al C
ompr
essi
on L
oad
kN
Figure 6.8 Axial Load versus Axial Shortening Curves from Test and Half Length FEA Model
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-16
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3
Test
FEA
(b) Test Series G5-L4 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial shortening
Figure 6.8 Axial Load versus Axial Shortening Curves from Test and Half Length FEA Model
0
10
20
30
40
50
0 1 2 3 4 5 6
Test
FEA
(a) 200°C
Out-of-plane displacement
mm
kN
Axi
al c
ompr
essi
on lo
ad
Figure 6.9 Axial Compression Load versus Out-of-plane Displacement Curves (G5-L2)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-17
Figures 6.7(a) and (b) compare the failure modes obtained from test and FEA at
temperatures of 20°C and 800°C. The experimental failure modes show a good
agreement with the failure modes obtained from the half-length and quarter-wave
buckling length models. Tests and FEA showed that the lipped channel sections used
in this study were governed by the local buckling of stiffened web elements.
Figures 6.8(a) and (b) present a comparison of typical axial compression load versus
axial shortening curves obtained from the experiments and the half length FEA
model. Figures 6.9(a) and (b) show the typical axial compression versus out-of-plane
displacement curves obtained from tests and FEA. The maximum experimental out-
of-plane displacement that could be measured was limited to about 1.3 mm due to
the limitation in the displacement measuring device. However, the limited
measurement of 1.3 mm was adequate to obtain the local buckling load from the P-δ2
curve. All these comparisons confirm that the half-length model can be successfully
used to simulate the experimental local buckling behaviour of cold-formed steel
compression members subject to the effects of stiffened web element local buckling.
0
5
10
15
20
25
0 1 2 3 4 5 6
Test
FEA
(b) 600°C
Out-of-plane displacement
mm
kN
Axi
al c
ompr
essi
on lo
ad
Figure 6.9 Axial Compression Load versus Out-of-plane Displacement Curves (G5-L2)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-18
6.4.6 Quarter-wave buckling length model
The validation of half length model for stiffened web elements provided the
confidence in using FEA model for developing and validating design rules. In this
section, a quarter-wave buckling length model was attempted because this model
simulated the ideal conditions used in local buckling studies. The quarter-wave
buckling length model with appropriate boundary conditions including other
significant factors such as residual stress and geometric imperfections was used to
assess the design rules developed in Chapter 5.
Figure 6.10 describes the elements and boundary conditions of the quarter-wave
buckling length model. An applied axial load was distributed using a rigid surface
providing pin-end boundary conditions to the attached edge nodes. In order to create
pinned end restraint on the rigid surface, R3D4 element which provided free rotation
about the attached edge was used. Single point constraints (SPC) with proper
boundary conditions were also used to create the symmetric half-wave buckling
length so that the desired local buckling mode was simulated.
Figure 6.10 Quarter-wave Buckling Length Model
P (SPC, 12456)
S4R5
Plane of symmetry (SPC345)
Elastic strip
R3D4 elemetns (rigid surface)
A quarter-wave buckling length
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-19
fcr,T (MPa) kT λT Test Series
Temp. (ºC) Test FEA FEA/Test Test FEA Test FEA 20 301 296 0.98 5.30 5.22 1.026 1.035
200 289 265 0.92 5.73 5.25 1.013 1.058 400 211 192 0.91 5.95 5.42 1.052 1.103 600 108 92 0.85 6.34 5.41 0.961 1.041
G2-L1
800 37 30 0.81 6.51 5.25 0.636 0.706 20 180 165 0.92 5.71 5.23 1.327 1.386
200 167 145 0.87 5.97 5.20 1.332 1.430 400 110 103 0.94 5.59 5.21 1.457 1.506 600 60 49 0.82 6.34 5.22 1.289 1.427
G2-L2
800 21 17 0.81 6.66 5.28 0.844 0.938 20 72 70 0.97 5.43 5.28 2.098 2.128
200 68 64 0.94 5.78 5.47 2.088 2.152 400 45 43 0.96 5.44 5.20 2.278 2.331 600 24 21 0.88 6.03 5.39 2.038 2.179
G2-L3
800 9 7 0.78 6.79 5.23 1.289 1.462 20 39 37 0.95 5.93 5.67 2.931 3.009
200 35 34 0.97 5.91 5.67 3.033 3.077 400 22 22 1.00 5.57 5.67 3.409 3.409 600 12 10 0.83 6.08 5.19 2.952 3.233
G2-L4
800 4 4 1.00 6.25 6.30 2.315 2.315 Mean 0.90 COV 0.075
20 600 595 0.99 5.11 5.07 1.030 1.034 200 541 536 0.99 5.11 5.07 1.067 1.072 400 380 335 0.88 5.74 5.06 1.137 1.211 600 180 172 0.95 5.58 5.32 1.142 1.168
G5-L1
800 45 45 0.99 5.48 5.42 0.814 0.814 20 280 274 0.98 5.32 5.20 1.507 1.524
200 251 241 0.96 5.27 5.08 1.567 1.599 400 170 155 0.91 5.73 5.22 1.700 1.780 600 80 75 0.94 5.53 5.19 1.713 1.769
G5-L2
800 23 19 0.81 6.24 5.04 1.139 1.253 20 123 114 0.93 5.54 5.13 2.274 2.362
200 110 102 0.93 5.49 5.09 2.367 2.458 400 70 64 0.91 5.59 5.11 2.649 2.770 600 35 32 0.90 5.73 5.16 2.590 2.709
G5-L3
800 9 8 0.93 5.79 5.37 1.820 1.931 20 72 70 0.97 5.59 5.41 3.005 3.047
200 66 64 0.97 5.68 5.51 3.089 3.137 400 40 39 0.97 5.51 5.37 3.514 3.558 600 20 19 0.97 5.65 5.50 3.414 3.502
G5-L4
800 6 5 0.82 6.43 5.44 2.340 2.564 Mean 0.94 COV 0.057
Table 6.5 Comparison of Local Buckling Stress and Buckling Coefficient from
Tests and Quarter-wave Buckling Length FEA Model at Elevated Temperatures
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-20
Pu,T de,T/d Test Series
Temp. (ºC) Test
(kN) FEA FEA Test Test FEA FEA
Test 20 28.3 26.0 0.92 0.89 0.77 0.86
200 27.5 24.8 0.90 0.95 0.80 0.84 400 23.1 20.8 0.90 0.96 0.81 0.84 600 11.2 11.6 1.04 0.87 0.94 1.08
G2-L1
800 3.8 3.5 0.93 0.94 0.84 0.89 20 37.0 34.8 0.94 0.65 0.57 0.87
200 35.5 32.3 0.91 0.70 0.56 0.80 400 28.4 26.1 0.92 0.63 0.52 0.82 600 15.1 13.4 0.89 0.69 0.52 0.76
G2-L2
800 5.3 4.9 0.92 0.82 0.69 0.84 20 45.5 42.8 0.94 0.43 0.36 0.84
200 42.5 42.1 0.99 0.43 0.42 0.98 400 34.2 34.2 1.00 0.39 0.39 1.00 600 18.5 16.5 0.89 0.45 0.31 0.68
G2-L3
800 6.5 5.8 0.89 0.56 0.40 0.71 20 21.0 20.6 0.98 0.35 0.33 0.94
200 20.4 18.8 0.92 0.36 0.27 0.76 400 12.1 12.0 0.99 0.30 0.29 0.98 600 6.5 5.8 0.89 0.38 0.25 0.65
G2-L4
800 2.2 2.1 0.94 0.37 0.34 0.91 Mean 0.94 0.87 COV 0.046 0.126
20 43.0 41.7 0.97 0.81 0.76 0.94 200 40.6 37.4 0.92 0.76 0.62 0.82 400 34.1 30.3 0.89 0.75 0.54 0.72 600 19.5 17.7 0.91 0.66 0.50 0.76
G5-L1
800 4.8 4.4 0.91 0.53 0.40 0.75 20 47.9 47.4 0.99 0.50 0.49 0.98
200 46.1 45.6 0.99 0.49 0.48 0.98 400 38.2 36.3 0.95 0.45 0.37 0.83 600 21.5 20.0 0.93 0.37 0.28 0.76
G5-L2
800 5.5 5.1 0.93 0.32 0.24 0.76 20 59.2 59.2 1.00 0.33 0.33 1.01
200 57.1 57.1 1.00 0.32 0.33 1.03 400 45.6 42.9 0.94 0.26 0.20 0.78 600 26.0 25.2 0.97 0.21 0.18 0.87
G5-L3
800 6.8 6.7 0.98 0.19 0.17 0.92 20 52.5 54.1 1.03 0.23 0.27 1.16
200 52.2 53.2 1.02 0.23 0.26 1.11 400 40.2 39.0 0.97 0.20 0.18 0.89 600 22.9 21.8 0.95 0.15 0.11 0.71
G5-L4
800 7.1 6.6 0.93 0.17 0.10 0.61 Mean 0.96 0.88 COV 0.041 0.163
Table 6.6 Comparison of Ultimate Strength and Effective Width from Tests
and Quarter-wave Buckling Length FEA Model at Elevated Temperatures
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-21
Figure 6.11 Axial Load versus Axial Shortening Curves from Test and FEA
(a) Test Series G2-L4 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3
Test
FEA
Axial shortening
(b) Test Series G5-L4 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3
Test
FEA
Axial shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-22
As used for the validation of the half length model, the quarter-wave buckling length
model was analysed first using an elastic buckling analysis and then by a non-linear
analysis. The mean of the ratio of buckling stresses from FEA to that from tests was
found to be 0.90 and 0.94 for low and high strength steel sections, respectively (see
Table 6.5). The comparison of the ultimate strength with test results gave mean
values of 0.94 and 0.96 for low and high strength steel sections, respectively (see
Table 6.6). Further, as shown in Figure 6.7, the comparison of the failure modes at
elevated temperatures showed a good agreement with experimental results.
Typical axial compression load versus axial shortening curves predicted by FEA are
also compared with experimental curves at elevated temperatures as shown in
Figures 6.11(a) and (b). The ultimate strength and axial stiffness obtained from FEA
are overall in good agreement with test results at both low and high temperatures.
However the FEA results showed the presence of higher ductility at low
temperatures, particularly for high strength steel sections. This might be due to the
mechanical characteristics assumed in FEA. However, the axial compression load
versus axial shortening curves obtained from FEA are very similar to the
experimental curves at high temperatures due to the gain in ductility of high strength
steels.
However there are small discrepancies between the results obtained from the half
length and half-wave buckling length models. This is demonstrated by the
comparisons with test results in Tables 6.3 to 6.6 and Figures 6.8 to 6.11. The
discrepancy in results from the half length and quarter-wave buckling length models
was due to the difference in generating the boundary condition using R3D4 elements
and the multi-point constraints (MPC) in the two models. As seen from these results,
in particular from the mean FEA/Test ratios in Tables 6.3 to 6.6, the half length
model simulates the experimental behaviour more closely than the quarter-wave
buckling length model. Although this was expected, the FEA reported in the last
chapter did not find a noticeable difference between the results from the two types of
FEA models for unlipped channel sections. The following paragraphs therefore
explain the reasons for this. The rigid body using R3D4 elements is required to
create the ideal pin conditions for the half of half-wave buckling length model. This
element type allows entirely free rotation condition at each node on the rigid body as
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-23
shown in Figure 6.12(a). Therefore, the quarter-wave buckling length model using
R3D4 elements is the idealised pin-end theoretical model for local buckling.
(a) Rigid body
(b) Multi-point constraints
Figure 6.12 Support Constraint Conditions in FEA
In the half length model, the MPC which allows constraints to be imposed with
different degrees of freedom is commonly used to create boundary conditions. Each
node on the edge of lipped channel members is considered a dependent node. These
dependent nodes are connected to one independent node as rigid beams and
controlled by the independent node. Therefore the rotation about the attached edge is
restrained by rigid beams. This provided slightly higher results. Figure 6.12(b)
shows the method of generating boundary conditions using multi-point constrains.
However, this effect was minimal for unlipped channel sections due to their lower
rotational stiffness. This is the reason why very close results were obtained from the
half length and quarter-wave buckling length models in the case of unstiffened
flange elements of unlipped channel sections (see Section 5.3.3).
Rigid body
Free rotation
Dependent nodes
Independent nodes (Node to be input boundary conditions)
Rigid beam
Rotation constrained
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-24
6.5 Results and Discussion
This section uses the test and FEA results reported in the previous sections to
evaluate the local buckling and ultimate strength behaviour of stiffened web
elements considered in this study. The results from the half length FEA model were
used in the discussion when it included test results whereas the results from the
quarter wave buckling length FEA model were used when it included predictions
from design rules and proposals reported in Chapter 5.
6.5.1 Local buckling behaviour
The effects of temperature on the stiffened elements restrained by flanges in cold-
formed steel lipped channel sections were investigated by obtaining the buckling
coefficients from local bucking stresses at elevated temperatures. The local bucking
stresses obtained from both test and FEA results were considerably deteriorated at
elevated temperatures as shown in Figure 6.13. Therefore, the local buckling stress
equation used at ambient temperature needed to be revised and thus the theoretical
elastic buckling stress equation was simply modified using the reduced elasticity
modulus ET as given in Equation 5.1.
Figure 6.13 Reduction of Local Buckling Stress at Elevated Temperatures
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 200 400 600 800 1000
G250
G550
fcr,T/fcr,20
Temperature (ºC)
Red
uctio
n of
buc
klin
g st
ress
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-25
In Equation 5.1, the b/t ratio and Poisson’s ratio were assumed to be constants, and
the local bucking stress was predominantly affected by the reduced elasticity
modulus. It was found in Chapter 5 that using the reduced elasticity modulus the
simple modification allowed the accurate calculation of local buckling stresses of
unstiffened elements at elevated temperatures. Therefore, the same principle was
used in order to assess the modification for stiffened elements comparing
experimental and numerical results. Further, the rotational restraint provided by
adjoining flanges to the stiffened web element was studied at elevated temperature.
(b) G550 steels
Figure 6.14 Variation of Buckling Coefficients with Temperatures
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0 200 400 600 800 1000
G5-L1(Test)
G5-L2(Test)
G5-L3(Test)
G5-L4(Test)
G5-L1(FEA)
G5-L2(FEA)
G2-L3(FEA)
G2-L4(FEA)
kT
Temperature (ºC)
Buc
klin
g co
effic
ient
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0 200 400 600 800 1000
G2-L1(Test)
G2-L2(Test)
G2-L3(Test)
G2-L4(Test)
G2-L1(FEA)
G2-L2(FEA)
G2-L3(FEA)
G2-L4(FEA)
kT
Buc
klin
g co
effic
ient
Temperature (ºC) (a) G250 steels
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-26
The experimental and numerical results of the local bucking stress and buckling
coefficient are given in Table 6.3. The variation of the local buckling coefficient at
elevated temperatures is plotted in Figures 6.14(a) and (b). This demonstrates a
similar behaviour to that of unstiffened elements. Therefore, these results
demonstrate that the local buckling stress of stiffened elements is largely governed
by the reduced elasticity modulus at elevated temperatures. The predictions using the
modified elastic buckling stress equation were compared with experimental and
numerical results in Table 6.3. The mean and COV values demonstrate the adequacy
of the current buckling design rules considering the reduced elasticity modulus.
20ºC 600ºC 800ºC
Figures 6.15(a) and (b) show the representative local buckling failure modes at low,
middle and high temperatures for both low and high strength steel sections. No
distortional or other relevant bucking modes including flange local buckling was
(b) G550 steels
Figure 6.15 Local Buckling Failure Modes
(a) G250 steels 20ºC 600ºC 800ºC
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-27
found in the temperature range used in this study. The degree of rotational restraint
provided by the flanges was between a simple and a fixed support as indicated by the
buckling coefficients reported in Figures 6.14(a) and (b). This is probably the most
common situation of stiffened web elements of cold-formed steel channel members.
From this study, it can be noted that the rotational restraint provided by adjacent
compact flanges remained the same at elevated temperatures.
6.5.2 Effective width at elevated temperatures
The effective widths of stiffened web elements at elevated temperatures were
directly derived from the ultimate strengths obtained from experimental and
numerical results considering the reduced yield strength at elevated temperatures
because flange and lip elements were compact. The base metal thickness of the
materials was measured before and after the test. It was found that the base metal
thickness remained the same in the temperature range up to 800°C, and thus no
reduction in the material thickness was required in the effective width calculations.
The variation of the effective width of web element for G250 and G550 steels at
elevated temperatures is given in Figures 6.16(a) and (b). It appeared that the
effective width ratio is reasonably constant at elevated temperatures for low strength
steel sections when the reduced yield strength is considered. On the other hand, the
variation of the effective width appeared to decrease for high strength steel sections
as temperature increases even when the reduced yield strength was included. The
FEA results show a similar trend for both low and high strength steel sections. This
effective width variation is similar to that of unstiffened flange elements reported in
the last chapter.
Figure 6.17 shows the effect of temperature on the ratio of the elasticity modulus to
the yield strength. It clearly shows the different reduction ratios between the low and
high strength cold-formed steels. As discussed in Section 5.4.2, the ratio the
elasticity modulus to the yield strength has an important role on the behaviour of
steel columns (Rasmussen and Rondal, 1997). This might have caused different
ultimate strength behaviour between the low and high strength steel sections.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-28
0
5
10
15
20
25
30
20 200 400 600 800
G2-0.6
G2-1.0
G5-0.95
G5-0.8
Ty
TfE
,
ºC
Figure 6.17 Elasticity Modulus to Yield Strength Ratio versus Temperature
be,T/b
Temperature (ºC)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 200 400 600 800 1000
G2-L1(Test)
G2-L2(Test)
G2-L3(Test)
G2-L4(Test)
G2-L1(FEA)
G2-L2(FEA)
G2-L3(FEA)
G2-L4(FEA)
Effe
ctiv
e w
idth
at t
empe
ratu
re T
(a) G250 steels be,T/b
Temperature (ºC)
Effe
ctiv
e w
idth
at t
empe
ratu
re T
0.00
0.20
0.40
0.60
0.80
1.00
0 200 400 600 800 1000
G5-L1(Test)
G5-L2(Test)
G5-L3(Test)
G5-L4(Test)
G5-L1(FEA)
G5-L2(FEA)
G2-L3(FEA)
G2-L4(FEA)
(b) G550 steels
Figure 6.16 Variation of Effective Width of Stiffened Web Element
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-29
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq.(5.6)
Eq.(5.10)
G2-L200
G5-L200
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq.(5.6)
Eq.(5.10)
G2-L20
G5-L20
(a) 20 ºC Slenderness ratio at temperature T
be,T/b
Tcr
Ty
f
f
,
,
n=3
n=7
n=5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
n=15
Slenderness ratio at temperature T
Figure 6.18 Comparisons of Theoretical Design Curves with FEA
Results of Stiffened Web Elements
(b) 200 ºC
be,T/b
Tfcr
f Ty
,,
n=3
n=7 n=5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
n=15
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-30
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq.(5.6)
Eq.(5.10)
G2-L400
G5-L400
(c) 400 ºC
be,T/b
Tcr
Ty
f
f
,
,
n=3
n=7 n=5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
n=15
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq.(5.6)
Eq.(5.10)
G2-L600
G5-L600
Figure 6.18 Comparisons of Theoretical Design Curves with FEA
Results of Stiffened Web Elements
(d) 600 ºC
be,T/b
Tcr
Ty
f
f
,
,
n=3
n=7 n=5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
n=15
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-31
Figure 6.18 Comparisons of Theoretical Design Curves with FEA
Results of Stiffened Web Elements
As seen in Figures 6.18(a) to (e), the simply modified design rules (Equations 5.5
and 5.6) are adequate for the prediction of the effective widths of low strength cold-
formed steel members subject to local buckling effects, but it overestimated the
effective widths for high strength steel members at elevated temperatures. In these
figures, the FEA results were based on the quarter wave buckling length model.
As discussed in Section 5.4.2 using the results of unstiffened flange elements, the
effects of greater post bucking strength and different reduction rate of the yield
strength and elasticity modulus might have affected to the results of stiffened web
elements. To assess the reliability and adequacy of the design rules developed in
Chapter 5, additional numerical analyses were conducted using a quarter wave
buckling length model to cover a larger range of web slenderness ratio (d/t) and to
obtain more results for high strength steel sections subject to local web bucking
effects and the results are given in Table 6.7. These additional results have been
included in Figures 6.18(a) to (e).
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq.(5.6)
Eq. (5.10)
G2-L800
G5-L800
(e) 800 ºC
be,T/b
Tcr
Ty
f
f
,
,
n=3
n=7
n=5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
n=15
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-32
Table 6.7 Additional FEA results of Effective Width for High Strength
Stiffened Web Elements
Web d/t ratio
Temp. (°C)
fy.T (MPa)
fcr,T (MPa) λT Pu,T
(kN) de,T
(mm) de,T/d
20 636 430 1.22 45.5 27.4 0.60 200 616 390 1.26 43.1 25.7 0.56 400 507 260 1.40 34.3 23.3 0.51 600 304 126 1.55 19.6 19.9 0.43
49 (t=0.936)
800 81 30 1.64 4.9 15.8 0.34 20 636 225 1.68 53.5 28.9 0.44 200 616 204 1.74 51.2 27.8 0.43 400 507 126 2.01 39.1 21.4 0.33 600 304 60 2.25 22.1 16.7 0.26
69 (t=0.936)
800 81 15 2.32 5.7 14.4 0.22 20 636 153 2.04 55.1 31.6 0.39 200 616 135 2.14 52.1 29.3 0.37 400 507 85 2.44 39.5 22.3 0.28 600 304 42 2.69 22.7 18.8 0.23
85 (t=0.936)
800 81 11 2.76 5.8 15.7 0.20 20 650 94 2.63 48.9 24.8 0.30 200 630 85 2.72 47.0 24.1 0.29 400 510 52 3.12 34.6 15.5 0.19 600 307 27 3.36 20.1 12.5 0.15
104 (t=0.785)
800 92 8 3.45 5.95 11.7 0.14 20 650 54 3.47 57.1 24.9 0.22 200 630 47 3.66 55.2 24.6 0.22 400 510 30 4.11 41.3 16.2 0.15 600 307 17 4.22 23.8 11.8 0.11
143 (t=0.785)
800 92 5 4.32 7.1 11.7 0.10 Note: t is the base metal thickness.
It is interesting to note that the local buckling coefficient of stiffened elements is
reasonably constant for both low and high strength steel sections when appropriately
reduced mechanical properties were used. However, the effective width of high
strength steel sections was severely reduced at elevated temperatures. Similar to the
local buckling behaviour of unstiffened flange elements, it seems that the major
reduction of the ultimate strength of stiffened elements occurs in the post buckling
strength range. The resulting variation in post-buckling strengths between the low
and high strength steel members explain why there is a severe reduction in the
effective width of high strength steel sections than in low strength steel members.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-33
A theoretical design method of cold-formed steel sections subject to local buckling
effects including the temperature effects was proposed for unstiffened flange
elements in Section 5.4.3.1. Due to the similar local buckling behaviour of stiffened
web elements compared with that of unstiffened flange elements, the design methods
used in Chapter 5 were also used for the stiffened web elements and compared with
the numerical results. In Figures 6.18(a) to (e), the effective widths obtained from
Equation 5.10 (theoretical design method) were compared with the numerical
effective widths of stiffened web elements obtained from the quarter wave buckling
length FEA models. G2 and G5 in Figures 6.18(a) to (e) refer to the low and high
strength steel sections, respectively. The comparison shows that the theoretical
design method is applicable to stiffened web elements at elevated temperatures. The
use of a parameter n of 3 is adequate at high temperatures greater than 600°C and the
use of n = 5, 7 and 15 is adequate at low temperatures. However, the use of a
constant n of 15 appeared to be inadequate at the temperature of 400°C as shown in
Figure 6.18(c).
Figure 6.19 presents the comparison of effective widths obtained from semi-
empirical design methods developed in Chapter 5 with numerical effective widths of
stiffened web elements obtained from the quarter wave buckling length FEA models.
The comparison shows that the predictions using the semi-empirical design methods
(Equations 5.27 and 5.29) provide a lower bound to the effective widths of stiffened
web elements at elevated temperatures.
Slenderness ratio at temperature T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
G5-L20
G5-L200
G5-L400
G5-L600
G5-L800
be,T/b
Tol
Ty
ff
,
,
Equation 5.5
Equation 5.27
Figure 6.19 Comparisons of Effective Widths from Equation 5.27 and 5.29 with FEA Results of Stiffened Web Elements at Elevated Temperatures
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Equation 5.29
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-34
Tol
Ty
ff
,
,
Figure 6.20 Comparisons of Effective Widths from Equation 5.31 with
FEA Results of Stiffened Web Elements at Elevated Temperatures
be,T/b
(a) 20C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-L20(FEA)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-L200(FEA)
be,T/b
Tol
Ty
ff
,
,
(b) 200C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-35
Figure 6.20 Comparisons of Effective Widths from Equation 5.31 with
FEA Results of Stiffened Web Elements at Elevated Temperatures
be,T/b
Tol
Ty
ff
,
,
(c) 400C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-L400(FEA)
be,T/b
Tol
Ty
ff
,
,
(d) 600C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-L600(FEA)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-36
Figures 6.20(a) to (e) also compare the effective widths obtained from the simplified
empirical design method (Equation 5.31) with the effective widths of stiffened web
elements obtained from the quarter wave buckling length FEA models. Due to the
similar local behaviour of stiffened and unstiffened elements, the use of Equation
5.31 is adequate for the prediction of effective widths of high strength stiffened web
elements at elevated temperatures.
In general, the local buckling behavioural characteristics of stiffened web elements
were similar to that of unstiffened flange elements at elevated temperatures.
Therefore the design rules developed for unstiffened flange elements in Chapter 5
were found to be applicable to stiffened web elements at elevated temperatures.
Figure 6.20 Comparisons of Effective Widths from Equation 5.31 with
FEA Results of Stiffened Web Elements at Elevated Temperatures
be,T/b
Tol
Ty
ff
,
,
(e) 800C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-L800(FEA)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 6-37
6.6 Summary
This chapter has presented the details of a series of experimental and numerical
studies conducted to investigate the local buckling behaviour of stiffened web
elements of lipped channel sections at elevated temperatures and the results. The
effective width design rule and the elastic buckling stress equation for ambient
temperature were simply modified for stiffened elements using the reduced
mechanical properties at elevated temperatures and their adequacy was assessed for
use at elevated temperatures.
The local buckling stress was predominantly governed by the reduction of the
elasticity modulus for both low and high strength steel sections at elevated
temperatures. It was also found that the rotational restraint provided by the adjoining
flange elements to the stiffened web element remained the same at elevated
temperatures. Therefore, the elastic buckling stress equation including the reduced
elasticity modulus was adequate for both low and high strength steel sections.
The predictions using the simply modified effective width rule for stiffened elements
were compared with the experimental and numerical results. It was found that the
simply modified effective width rule was adequate for low strength steel sections,
but was inadequate for high strength steel sections even though the reduction of the
mechanical properties was taken into account. This might have been caused by the
severe reduction in the post-buckling strength at elevated temperatures and the
different reduction ratio of the elasticity modulus to the yield strength. The
theoretical, semi-empirical and empirical design proposals for unstiffened elements
developed in Chapter 5 were used to predict the effective widths of stiffened
elements at elevated temperatures. The comparison of the effective width results
showed a good agreement for stiffened web elements made of high strength steels
and thus confirmed the adequacy of the design rules developed in Chapter 5.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-1
Chapter 7. Interactive Local Buckling Behaviour of Stiffened Web and Flange Elements at Elevated Temperatures
7.1 General
The effects of elevated temperatures on the local buckling behaviour of cold-formed
steel stiffened web and unstiffened flange elements have been investigated in
Chapters 5 and 6. Due to the inadequacy of the simply modified current design rules
for high strength steel members at elevated temperatures, theoretical, semi-empirical
and empirical effective width design rules were developed in Chapter 5. The new
design rules were then compared with experimental and numerical results and found
to be adequate for the stiffened web and unstiffened flange elements at elevated
temperatures in Chapters 5 and 6, respectively. In Chapters 5 and 6, the local
buckling behaviour of individual elements (unstiffened flange element and stiffened
web element) of cold-formed steel compression members was investigated. In this
chapter, the interactive local bucking behaviour of stiffened web and flange elements
of cold-formed steel compression members was investigated using experiments and
finite element analyses. The results were then compared with the predictions based
on the simply modified current design rules (Equation 5.5) and the new effective
width design rules developed in Chapter 5 (Equations 5.10, 5.27, 5.29 and 5.31).
7.2 Experimental Investigations
7.2.1 Test Specimens
A total of 50 tests were conducted to investigate the behaviour of lipped channel
members subject to the interactive local buckling of web and flange elements. All the
specimens were made of G250 (minimum yield stress of 250) and G550 (minimum
yield stress of 550) steels and were carefully designed so that their structural
behaviour was governed by the local buckling behaviour. The specimen length was
decided on the basis of three times the half-wave buckling length. An additional 40
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-2
mm was added to the specimen length to allow for the end effects so that three local
buckling waves appeared along the specimen length. Major variables in this study
were temperature, plate slenderness ratio and steel grade. The details of test
specimens are shown in Table 7.1.
Table 7.1 Test Specimens
THINWALL Test series Grade b
(mm) d
(mm) L
(mm)
t (BMT, mm)
fy (MPa) fcr
(MPa) λ HL
(mm) G2-I1 47.7 47.3 11.0 0.964 317 310 1.01 50 G2-I2 82.0 82.2 18.0 0.964 317 109 1.71 80 G2-I3 58.1 58.6 13.5 0.564 330 70 2.17 60 G2-I4 78.3 78.0 18.5 0.564 330 40 2.87 80 G2-I5
G250
97.0 97.5 23.5 0.564 330 26 3.56 100 G5-I1 38.5 38.2 9.5 0.936 636 443 1.20 40 G5-I2 58.7 58.6 14.0 0.936 636 195 1.81 60 G5-I3 78.1 78.2 18.5 0.936 636 110 2.40 80 G5-I4 38.3 38.1 9.5 0.409 722 83 2.95 40 G5-I5
G550
48.5 48.5 11.5 0.409 722 53 3.69 50 *BMT and HL refer to base metal thickness and half-wave buckling length, respectively. Corner radius r = 1.2 mm for all specimens. Temperatures considered were 20, 200, 400, 600, 800ºC.
The yield stresses fy are the measured values.
Figure 7.1 Section Geometry – Lipped Channel
The measured reduction factors of the yield strength and elasticity modulus are given
in Table 7.2. The measured mechanical properties at elevated temperatures were
used for the calculation of the local buckling coefficient and effective width. The
0.2% proof stress was used in this study for the same reasons given in Sections 5.4.2
and 6.3.1.
b
d
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-3
Table 7.2 Reduction Factors for Elasticity Modulus (ET/E20) and
Yield Strength (fy,T/fy,20)
20 200 400 600 800
G250 0.964 1.000 0.887 0.624 0.300 0.100 G250 0.564 1.000 0.900 0.599 0.293 0.098 G550 0.936 1.000 0.902 0.564 0.273 0.068
ET E20
G550 0.409 1.000 0.894 0.583 0.238 0.072 G250 0.964 1.000 0.935 0.788 0.427 0.150 G250 0.564 1.000 0.961 0.794 0.409 0.205 G550 0.936 1.000 0.969 0.797 0.478 0.127
fy,T fy,20
G550 0.409 1.000 0.977 0.821 0.491 0.127
Figure 7.2 shows the typical buckling stress versus half-wave buckling length curve
including the critical buckling mode of the chosen lipped channel sections subject to
interactive local buckling. The minimum local buckling stress for the G5-I2 section
was 195 MPa at the half-wave buckling length of 60 mm. Hence the test specimen
height for G5-I2 section was taken as 220 mm.
Figure 7.2 Typical Buckling Stress Plot of the Chosen Lipped
Channel Sections (G5-I2)
60
BMT (mm)
Temp (°C) Steel Grade
MPa
mm
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-4
7.2.2 Test Set-up and Procedure
The details of the testing facilities and procedure have already been discussed in
Sections 5.2.1, 5.2.3 and 6.2.2. The test set-up is shown in Figure 7.3.
Each test specimen was placed inside the furnace and the temperature was raised to
the pre-set value allowing free expansion of the specimen. Following this, an axial
compression load was applied to the test specimen until it failed, maintaining the
pre-set temperature (steady state test). The axial compression load versus out-of-
plane deflection2 curve was used to determine the local buckling load
(Venkatarimaiah and Roorda, 1982).
7.3 Finite Element Analysis
The details of FEA model at elevated temperatures were described in detail in
Sections 5.3 and 6.3. Therefore the model descriptions, relevant numerical factors
and analysis procedures are only briefly provided in this section.
Figure 7.3 Test Set-up
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-5
7.3.1 Model descriptions
The S4R5 element was used for the finite element analyses of the experimental and
ideal models, and 3.5 mm length meshes were found to be adequate for the accuracy
of the models at elevated temperatures. As used in Chapters 5 and 6, the elastic-
plastic model using the field dependent option was used to model the mechanical
characteristics at elevated temperatures (see Figures 5.4(a) and (b)).
Figure 7.4 Initial Geometric Local Imperfection
Based on Schafer and Pekoz’s proposal (1998), the maximum local imperfection
magnitude of plate width/167 was used with the critical local buckling mode. The
web was the governing element even though the buckling displacements on the web
and flanges were similar. Hence the maximum local imperfection was introduced to
the web and the corresponding imperfections were included to flanges and lips.
Figure 7.4 shows the magnitude of the local geometric imperfection taken as the
fabrication tolerance. The simplified variations in flexural residual stresses
recommended by Schafer and Pekoz (1998) were used in the nonlinear analyses at
ambient temperatures. However, the residual stresses are reduced when the members
are exposed to elevated temperatures. Therefore assuming the fully stress relieving
temperature of 800ºC (Callister, 2000) the residual stresses were reduced linearly for
temperatures ranging from 20 to 800ºC as given in Table 5.3 (see Figure 6.4(b)).
∆ =d/167 d
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-6
7.3.2 Analysis
The elastic bucking analysis was first carried out to obtain the local buckling stress
and the associated critical buckling mode. The local buckling imperfection was then
applied to the critical buckling mode. Following the elastic buckling analysis, the
non-linear static analysis was carried out after including the initial imperfection and
residual stresses.
7.3.3 Half length model
As in Sections 6.4.5 and 6.4.6, both half length model (experimental model) and
quarter-wave buckling length model (ideal model) were investigated in this study.
The half length model was first used to validate the finite element analyses at
elevated temperatures including appropriate boundary conditions by comparing with
the experimental results whereas the quarter-wave buckling length model was used
to verify the accuracy of design rules as the latter model simulates ideal conditions.
Figure 7.5 shows the half length model including the model geometry, meshes and
the loading pattern with appropriate boundary conditions using multi-point
constraints (MPC).
Figure 7.5 Half Length Model
The local buckling stresses and ultimate strengths obtained from the half length
model were validated by comparing them with corresponding experimental results.
Unlike the unstiffened flange and stiffened web elements considered in Chapters 5
P (MPC, 12456)
S4R5 elements
Plane of symmetry (SPC345) Dependant
nodes
Elastic strip
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-7
and 6, there were some uncertainties in deriving the experimental and numerical
effective widths. Although the lips were designed not to buckle using simple
theories, it was not clear whether they could be assumed to be compact in the
effective width calculations. Hence two extreme cases were considered in the
calculation of the effective width of web and flanges. Firstly, the lips were assumed
fully effective (compact), and secondly, the same rate of effective width reduction
was used for all the elements including the web, flanges and lips. Appendix C shows
example calculations of effective widths in these two cases.
Tables 7.3 and 7.4 show the numerical results and comparison of the critical local
buckling stress of the section (fcr,T), the buckling coefficient of web and flange
elements (kT), the ultimate strength of the member (Pu,T) and the effective width of
web and flange elements (be,T) at elevated temperature. It must be noted that the
buckling coefficients and effective widths are the same for both web and flange
elements as they have the same b/t ratio. The ratios of the numerical local buckling
stress to the experimental local bucking stress had the mean values of 0.94 and 0.95
for low strength and high strength steel sections, respectively. The mean and COV of
the ratio of the numerical and experimental ultimate strengths at elevated
temperatures was 0.97 and 0.079, respectively, for low strength steel sections and
1.03 and 0.057, respectively, for high strength steel sections as given in Tables 7.3
and 7.4. These comparisons of the local buckling stress and ultimate strength showed
a good agreement and confirmed that the half length model can be successfully used
to model the local buckling behaviour of cold-formed steel members subject to the
local buckling interaction at elevated temperatures.
At lower temperatures of 20°C and 200°C, it is seen in Table 7.4 that the ultimate
strength obtained from FEA are consistently higher than those from tests for high
strength steel members. Thinner steel members (t = 0.409 mm) showed larger
discrepancies. These observations were similar to those discussed in Chapters 4 to 6.
The lack of ductility and Bauschinger effect in high strength steel members
discussed in detail in Chapter 4 appeared to be present up to the temperature of
200°C and thus their effects were noticeable for thinner high strength steel members
at lower temperatures.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-8
fcr,T(MPa) kT λT Test
series Temp. (°C) Test FEA FEA/Test Test FEA Test FEA 20 298 305 1.02 3.61 3.70 1.031 1.019 200 280 274 0.98 3.83 3.74 1.028 1.039 400 190 192 1.01 3.69 3.73 1.147 1.141 600 97 93 0.96 3.92 3.76 1.180 1.205
G2-I1
800 40 32 0.80 4.85 3.88 1.095 1.225 20 105 109 1.04 3.84 3.99 1.738 1.705 200 91 96 1.05 3.69 3.96 1.804 1.756 400 71 69 0.97 4.61 4.05 1.876 1.903 600 37 34 0.92 4.92 4.15 1.910 1.993
G2-I2
800 10 11 1.10 5.23 4.03 2.191 2.089 20 76 73 0.96 4.13 3.97 2.084 2.126 200 72 69 0.96 4.35 4.17 2.098 2.143 400 49 45 0.92 4.44 4.08 2.312 2.413 600 26 22 0.88 4.71 3.99 2.279 2.477
G2-I3
800 8.5 7 0.86 5.13 4.23 2.828 3.117 20 44 43 0.98 4.24 4.14 2.739 2.770 200 40 38 0.95 4.28 4.07 2.815 2.888 400 29 25 0.86 4.57 4.01 3.006 3.237 600 14 13 0.86 4.49 4.17 3.105 3.223
G2-I4
800 5 4 0.83 5.35 4.28 3.688 4.123 20 29 28 0.97 4.36 4.21 3.373 3.433 200 26 25 0.96 4.35 4.18 3.492 3.561 400 18 17 0.94 4.51 4.26 3.815 3.926 600 9 8 0.89 4.51 4.01 3.873 4.108
G2-I5
800 3.2 3 0.95 5.35 5.01 4.610 4.761 Mean 0.94 COV 0.083
20 457 436 0.96 3.83 3.67 1.180 1.208 200 435 396 0.91 4.04 3.68 1.190 1.247 400 281 258 0.92 4.18 3.84 1.343 1.402 600 148 137 0.93 4.51 4.18 1.433 1.490
G5-I1
800 37 33 0.90 4.43 3.95 1.480 1.567 20 202 194 0.97 3.99 3.83 1.774 1.811 200 185 174 0.95 4.05 3.81 1.825 1.882 400 115 109 0.95 4.02 3.81 2.100 2.157 600 61 53 0.87 4.38 3.80 2.232 2.395
G5-I2
800 17 13 0.80 4.79 3.66 2.183 2.496 20 119 120 1.01 4.18 4.22 2.312 2.302 200 96 106 1.11 3.74 4.13 2.533 2.411 400 65 70 1.08 4.05 4.36 2.793 2.691 600 37 34 0.92 4.73 4.34 2.866 2.990
G5-I3
800 10 8 0.80 5.02 4.02 2.846 3.182 20 95 91 0.96 4.08 3.90 2.757 2.817 200 86 83 0.97 4.13 3.98 2.863 2.914 400 60 53 0.89 4.42 3.90 3.144 3.345 600 26 23 0.89 4.69 4.15 3.695 3.929
G5-I4
800 8 7 0.88 4.77 4.17 3.391 3.625 20 57 59 1.04 3.96 4.10 3.559 3.498 200 50 52 1.05 3.89 4.04 3.755 3.682 400 35 35 1.01 4.17 4.17 4.116 4.116 600 14 16 1.15 4.09 4.67 5.036 4.710
G5-I5
800 5 4 0.88 4.83 4.25 4.290 4.796 Mean 0.95 COV 0.087
Table 7.3 Comparison of Local Buckling Stress and Buckling Coefficient from Tests and Half Length FEA Model at Elevated Temperatures
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-9
Pu,T be,T/b Test
series Temp. (°C) Test
(kN) FEA (kN)
FEA Test Test FEA1 FEA2 FEA1
Test FEA1 FEA2
20 44.1 41.5 0.94 0.81 0.75 0.78 0.93 0.96 200 40.2 38.7 0.96 0.79 0.75 0.78 0.95 0.96 400 32 30.4 0.95 0.73 0.69 0.73 0.95 0.94 600 18.2 16.6 0.91 0.78 0.69 0.73 0.88 0.94
G2-I1
800 6.9 6.2 0.90 0.86 0.75 0.78 0.87 0.96 20 46.5 48.4 1.04 0.47 0.50 0.53 1.06 0.94 200 45.1 43.3 0.96 0.49 0.47 0.51 0.96 0.92 400 31.7 34.6 1.09 0.39 0.44 0.48 1.13 0.91 600 19.5 18.2 0.93 0.46 0.42 0.46 0.91 0.91
G2-I2
800 7.3 6.4 0.88 0.50 0.42 0.46 0.84 0.91 20 17.1 18.1 1.06 0.37 0.40 0.45 1.08 0.89 200 16 16.7 1.04 0.36 0.38 0.43 1.06 0.88 400 12.5 11.7 0.94 0.48 0.44 0.48 0.92 0.92 600 6.6 6.0 0.91 0.52 0.46 0.49 0.88 0.93
G2-I3
800 2.2 2.0 0.91 0.52 0.46 0.49 0.88 0.93 20 17.9 19.2 1.07 0.25 0.28 0.33 1.12 0.86 200 16.9 18.5 1.09 0.25 0.28 0.33 1.12 0.86 400 12.6 11.3 0.90 0.32 0.27 0.32 0.84 0.86 600 7.1 6.2 0.87 0.39 0.32 0.36 0.82 0.88
G2-I4
800 2.4 2.4 1.00 0.39 0.39 0.42 1.00 0.94 20 18.9 20.7 1.10 0.19 0.24 0.28 1.26 0.86 200 18.3 19.6 1.07 0.19 0.23 0.27 1.21 0.85 400 12.1 12.7 1.05 0.21 0.25 0.28 1.19 0.89 600 7.2 6.5 0.90 0.28 0.26 0.29 0.93 0.90
G2-I5
800 2.5 2.2 0.88 0.30 0.26 0.29 0.87 0.90 Mean 0.97 0.99 0.91 COV 0.079 0.128 0.038
20 56.1 57.2 1.02 0.66 0.67 0.67 1.02 1.00 200 53.5 54.1 1.01 0.64 0.65 0.65 1.02 1.00 400 43.2 40.1 0.93 0.63 0.57 0.59 0.90 0.97 600 23.5 21.6 0.92 0.55 0.50 0.52 0.91 0.95
G5-I1
800 6.3 5.6 0.89 0.56 0.48 0.51 0.86 0.94 20 60.1 61.4 1.02 0.42 0.43 0.46 1.02 0.93 200 57.2 58.1 1.02 0.40 0.41 0.45 1.03 0.92 400 42.1 43.2 1.03 0.35 0.36 0.39 1.03 0.92 600 24.1 23.1 0.96 0.32 0.30 0.34 0.94 0.89
G5-I2
800 6.2 6.1 0.98 0.31 0.30 0.33 0.97 0.91 20 64.1 66.7 1.04 0.30 0.32 0.35 1.07 0.91 200 60.1 63.2 1.05 0.29 0.31 0.35 1.07 0.88 400 44.1 45.2 1.02 0.24 0.25 0.28 1.04 0.89 600 23.9 24.1 1.01 0.20 0.20 0.24 1.00 0.85
G5-I3
800 6.3 6.4 1.02 0.20 0.20 0.24 1.00 0.85 20 13.7 15.1 1.10 0.24 0.28 0.32 1.17 0.88 200 12.9 14.1 1.09 0.22 0.26 0.30 1.18 0.87 400 9.3 9.8 1.05 0.17 0.19 0.22 1.12 0.85 600 4.9 5.0 1.02 0.13 0.14 0.17 1.08 0.83
G5-I4
800 1.3 1.3 1.00 0.14 0.15 0.17 1.07 0.86 20 13.7 15.2 1.11 0.16 0.20 0.23 1.25 0.85 200 12.5 14.0 1.12 0.14 0.18 0.21 1.29 0.84 400 9.1 9.8 1.08 0.10 0.12 0.14 1.14 0.86 600 5.1 5.2 1.02 0.08 0.09 0.11 1.09 0.85
G5-I5
800 1.3 1.4 1.05 0.08 0.10 0.12 1.13 0.82 Mean 1.03 1.05 0.89 COV 0.057 0.098 0.057
Table 7.4 Comparison of Ultimate Strength and Effective Width from Tests and Half Length FEA Model at Elevated Temperatures
Note: FEA1 and FEA2 were based on “fully effective lips” and “same reduction as the web”, respectively
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-10
In Table 7.4, the effective widths of the web and flanges based on the two
assumptions were compared. The differences between the FEA1 (assuming fully
effective lips) and FEA2 (assuming the same rate of effective width reduction for all
the elements of lips, web and flanges) effective widths were 9% and 11% for low
and high strength steel sections, respectively. The actual effective width of lips may
have to be based on an assumption between these two. The effective widths of the
web and flanges were overestimated when the second assumption was used (FEA2),
ie., unconservative. Therefore the effective width based on the first assumption
(FEA1) was considered to be adequate for the purpose of design rule verification.
Further the contribution of lips to the effective width of the web and flanges was also
small. The experimental effective widths given in Table 7.4 were obtained based on
the assumption of fully effective lips. The mean values of the ratio of FEA and
experimental effective width were found to be 0.99 and 1.05 for low and high
strength steel sections, respectively.
(a) Test
(c) FEA (half length)
Figure 7.6 Comparison of Local Buckling Failure Modes at 600°C
(b) FEA (quarter-wave buckling length)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-11
The typical local buckling failure modes obtained from test, half length model and
quarter-wave buckling length model were compared in Figures 7.6(a) to (c). The
failure modes from FEA agrees well with the experimental failure mode. Figures
7.7(a) and (b) present a comparison of typical axial compression load versus axial
shortening curves obtained from tests and the half length experimental model.
Further typical axial compression versus out-of-plane displacement curves obtained
from tests and FEA were compared in Figures 7.8(a) and (b). Due to the limitation in
the displacement measuring device, the experimental out-of-displacement was
recorded only up to about 1.3 mm. However, the comparison between the test and
FEA results shows a reasonably good agreement. All these comparisons confirmed
that the half length model can be used to simulate the interactive local buckling
behaviour of test sections used in this study.
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3
Test
FEA
(a) Test Series G2-I5 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial shortening
Figure 7.7 Axial Load versus Axial Shortening Curves from Test and Half
Length FEA Model
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-12
0
10
20
30
40
50
60
70
80
0 0.5 1 1.5 2 2.5 3
Test
FEA
Figure 7.7 Axial Load versus Axial Shortening Curves from Test and Half
Length FEA Model
(b) Test Series G5-I3 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial shortening
0
5
10
15
20
0 1 2 3 4 5 6
Test
FEA
(a) 200°C Out-of-plane displacement
mm
kN
Axi
al c
ompr
essi
on lo
ad
Figure 7.8 Axial Compression Load versus Out-of-plane Displacement Curves
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-13
7.3.4 Quarter-wave buckling length model
The half length model simulates the test conditions more closely and hence can be
validated using the test results in the last section. The validation of half length model
provided the confidence in using FEA model for developing and validating design
rules. However, the results from the quarter-wave buckling length model were also
compared with experimental results to confirm its accuracy.
Figure 7.9 Quarter-wave Buckling Length Model
0
1
2
3
4
0 1 2 3 4 5 6
Test
FEA
(b) 800°C
Out-of-plane displacement
mm
kN
Axi
al c
ompr
essi
on lo
ad
Figure 7.8 Axial Compression Load versus Out-of-plane Displacement Curves
P (SPC, 12456)
S4R5 elements
Plane of symmetry (SPC345)
R3D4 elements (rigid surface)
Elastic strip
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-14
Figure 7.9 describes the quarter-wave buckling length model including elements and
boundary conditions. A rigid surface was used to distribute an applied axial load and
create the ideal pin-end boundary condition using the R3D4 element. Single point
constraints (SPC) with proper boundary conditions were used to create the
symmetric half buckling length. As used in Chapters 5 and 6, the quarter-wave
buckling length model with appropriate boundary conditions including the reduced
mechanical properties, appropriate residual stresses and geometric imperfections was
used to assess the design rules developed in Chapter 5. This is because the quarter-
wave buckling length model simulates the ideal theoretical conditions more closely.
Tables 7.5 and 7.6 present the local buckling stress (fcr,T) and ultimate strength (Pu,T)
at elevated temperatures obtained from the quarter-wave buckling length model and
tests. The comparison of the numerical local buckling stress with experimental local
bucking stress gives a mean value of 0.94 for both low and high strength steel
members. The mean values of the ultimate strength ratio were 0.94 and 0.98 for low
and high strength steel sections, respectively. The results obtained from the quarter-
wave buckling length model were overall slightly lower than the half length model
by about 4.5% as expected. As discussed in Section 6.4.6, the difference is due to the
end conditions. The experimental and numerical effective widths given in Table 7.6
were obtained by assuming that lips were fully effective as discussed in Section
7.3.3. The comparison of the numerical and experimental effective widths gave a
mean ratio of 0.93 and 0.99 for low and high strength steel sections, respectively.
The local buckling failure modes obtained from test, half length model and quarter-
wave buckling length model agreed reasonably well with each other as seen in
Figures 7.6(a) to (c). These figures also show that there was no change in the
buckling mode at high temperatures, e.g. local buckling to distortional buckling.
Figures 7.10(a) and (b) compare typical axial compression load versus axial
shortening curves predicted from the quarter-wave buckling length model with those
from tests at elevated temperatures. The ultimate strength and axial stiffness obtained
from FEA are overall in good agreement with test results at both low and high
temperatures.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-15
fcr,T(MPa) kT λT Test
series Temp. (°C) Test FEA FEA/Test Test FEA Test FEA 20 298 294 1.03 3.61 3.57 1.031 1.038 200 280 261 0.97 3.83 3.56 1.028 1.065 400 190 188 1.03 3.69 3.55 1.147 1.153 600 97 89 0.95 3.92 3.54 1.180 1.232
G2-I1
800 40 31 0.80 4.85 3.50 1.095 1.244 20 105 107 1.06 3.84 3.93 1.738 1.721 200 91 94 1.08 3.69 3.89 1.804 1.775 400 71 67 0.99 4.61 3.95 1.876 1.932 600 37 32 0.89 4.92 3.87 1.910 2.054
G2-I2
800 10 11 1.10 5.23 3.87 2.191 2.089 20 76 71 0.97 4.13 3.88 2.084 2.156 200 72 65 0.94 4.35 3.83 2.098 2.208 400 49 42 0.90 4.44 3.83 2.312 2.498 600 26 21 0.85 4.71 3.83 2.279 2.535
G2-I3
800 8.5 7 0.82 5.13 4.07 2.828 3.117 20 44 42 0.98 4.24 4.04 2.739 2.803 200 40 37 0.95 4.28 3.91 2.815 2.927 400 29 24 0.86 4.57 3.86 3.006 3.304 600 14 12 0.86 4.49 3.71 3.105 3.354
G2-I4
800 5 4 0.80 5.35 4.12 3.688 4.123 20 29 27 0.97 4.36 4.11 3.373 3.496 200 26 24 0.96 4.35 4.02 3.492 3.634 400 18 16 0.94 4.51 4.10 3.815 4.047 600 9 8 0.89 4.51 3.86 3.873 4.108
G2-I5
800 3.2 3 0.94 5.35 4.02 4.610 4.761 Mean 0.94 COV 0.087
20 457 429 0.95 3.83 3.60 1.180 1.218 200 435 387 0.91 4.04 3.60 1.190 1.262 400 281 252 0.92 4.18 3.75 1.343 1.418 600 148 135 0.91 4.51 4.12 1.433 1.501
G5-I1
800 37 32 0.89 4.43 3.46 1.480 1.591 20 202 191 0.96 3.99 3.77 1.774 1.825 200 185 171 0.94 4.05 3.74 1.825 1.898 400 115 108 0.95 4.02 3.78 2.100 2.167 600 61 51 0.87 4.38 3.66 2.232 2.441
G5-I2
800 17 13 0.76 4.79 3.53 2.183 2.496 20 119 116 1.02 4.18 4.10 2.312 2.342 200 96 104 1.13 3.74 4.12 2.533 2.434 400 65 68 1.09 4.05 4.08 2.793 2.731 600 37 32 0.89 4.73 4.06 2.866 3.082
G5-I3
800 10 8 0.80 5.02 3.86 2.846 3.182 20 95 87 0.95 4.08 3.82 2.757 2.881 200 86 79 0.95 4.13 3.79 2.863 2.987 400 60 51 0.88 4.42 3.75 3.144 3.410 600 26 22 0.88 4.69 3.82 3.695 4.017
G5-I4
800 8 7 0.88 4.77 3.44 3.391 3.625 20 57 58 1.05 3.96 4.03 3.559 3.528 200 50 50 1.04 3.89 4.04 3.755 3.755 400 35 34 1.00 4.17 4.02 4.116 4.176 600 14 15 1.14 4.09 3.93 5.036 4.865
G5-I5
800 5 4 0.80 4.83 3.72 4.290 4.796 Mean 0.94 COV 0.098
Table 7.5 Comparison of Local Buckling Stress and Buckling Coefficient from Tests and Quarter-wave Buckling Length FEA Model at Elevated Temperatures
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-16
Pu,T be,T/b Test
series Temp. (°C) Test
(kN) FEA (kN)
FEA Test Test FEA FEA
Test 20 44.1 40.1 0.91 0.81 0.72 0.89 200 40.2 37.0 0.92 0.79 0.71 0.90 400 32 29.1 0.91 0.73 0.65 0.89 600 18.2 15.8 0.87 0.78 0.66 0.84
G2-I1
800 6.9 5.9 0.86 0.86 0.70 0.82 20 46.5 47.4 1.02 0.47 0.48 1.03 200 45.1 41.9 0.93 0.49 0.45 0.91 400 31.7 33.6 1.06 0.39 0.43 1.10 600 19.5 18.3 0.90 0.46 0.42 0.92
G2-I2
800 7.3 6.4 0.84 0.50 0.42 0.84 20 17.1 17.6 1.03 0.37 0.38 1.04 200 16 16.3 1.02 0.36 0.37 1.03 400 12.5 11.1 0.89 0.48 0.42 0.86 600 6.6 5.8 0.88 0.52 0.44 0.84
G2-I3
800 2.2 1.9 0.86 0.52 0.43 0.82 20 17.9 18.6 1.04 0.25 0.27 1.07 200 16.9 17.9 1.06 0.25 0.27 1.10 400 12.6 11.2 0.87 0.32 0.27 0.83 600 7.1 6.0 0.84 0.39 0.30 0.77
G2-I4
800 2.4 2.2 0.92 0.39 0.35 0.88 20 18.9 20.0 1.06 0.19 0.22 1.16 200 18.3 19.4 1.06 0.19 0.21 1.13 400 12.1 12.2 1.01 0.21 0.21 1.02 600 7.2 6.3 0.86 0.28 0.23 0.80
G2-I5
800 2.5 2.2 0.82 0.30 0.26 0.87 Mean 0.94 0.93 COV 0.089 0.124
20 56.1 54.4 0.97 0.66 0.63 0.96 200 53.5 51.4 0.96 0.64 0.61 0.95 400 43.2 37.6 0.87 0.63 0.53 0.84 600 23.5 20.2 0.86 0.55 0.46 0.82
G5-I1
800 6.3 5.4 0.85 0.56 0.45 0.80 20 60.1 57.7 0.96 0.42 0.39 0.94 200 57.2 54.3 0.95 0.40 0.38 0.93 400 42.1 40.8 0.97 0.35 0.33 0.95 600 24.1 21.7 0.90 0.32 0.28 0.85
G5-I2
800 6.2 6.0 0.96 0.31 0.29 0.94 20 64.1 62.8 0.98 0.30 0.29 0.97 200 60.1 59.5 0.99 0.29 0.28 0.98 400 44.1 43.2 0.98 0.24 0.23 0.96 600 23.9 23.7 0.99 0.20 0.20 0.98
G5-I3
800 6.3 6.4 1.02 0.20 0.20 1.03 20 13.7 14.1 1.03 0.24 0.25 1.06 200 12.9 13.2 1.02 0.22 0.23 1.04 400 9.3 9.5 1.02 0.17 0.18 1.05 600 4.9 5.0 1.02 0.13 0.13 1.03
G5-I4
800 1.3 1.3 1.02 0.14 0.14 1.03 20 13.7 14.7 1.07 0.16 0.18 1.14 200 12.5 13.9 1.11 0.14 0.17 1.23 400 9.1 9.8 1.08 0.10 0.12 1.21 600 5.1 5.2 1.02 0.08 0.09 1.04
G5-I5
800 1.3 1.3 1.02 0.08 0.08 1.05 Mean 0.98 0.99 COV 0.066 0.107
Table 7.6 Comparison of Ultimate Strength and Effective Width from Tests and Quarter-wave Buckling Length FEA Model at Elevated Temperatures
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-17
Figure 7.10 Axial Load versus Axial Shortening Curves from Test and Quarter-wave Buckling Length Model
0
10
20
30
40
50
60
70
0 0.5 1 1.5 2 2.5 3
Test
FEA
(b) Test Series G5-I3 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial shortening
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3
Test
FEA
(a) Test Series G2-I5 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-18
These comparisons of the local buckling stress, ultimate strength, failure mode and
axial compression load versus axial shortening curve indicate that the quarter-wave
buckling length model provides suitable and reliable results of the elastic bucking
stress and ultimate strength at elevated temperatures and thus can be used to verify
the adequacy of the current design rules, and the new design rules developed in
Chapter 5.
7.4 Results and Discussions
7.4.1 Local buckling stress
The variation of the local buckling stress was investigated for low and high strength
cold-formed steel lipped channel sections subject to the interactive local buckling
effects at elevated temperatures. Using the experimental and numerical bucking
coefficients, the effects of temperature on the rotational restraint were investigated.
Based on Equation 5.2, the experimental and numerical buckling coefficients at
elevated temperatures were calculated and plotted in Figures 7.11(a) and (b).
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0 200 400 600 800 1000
G2-I1(Test)
G2-I2(Test)
G2-I3(Test)
G2-I4(Test)
G2-I5(Test)
G2-I1(FEA)
G2-I2(FEA)
G2-I3(FEA)
G2-I4(FEA)
G2-I5(FEA)
kT
°C
Buc
klin
gco
effic
ient
atte
mpe
ratu
reT
Temperature
Figure 7.11 Variation of Buckling Coefficient at Elevated Temperatures (a) G250 steels
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-19
Figure 7.11 Variation of Buckling Coefficient at Elevated Temperatures
For some cases at the temperature of 800ºC, the buckling coefficient slightly
increased even though the reduction of the elasticity modulus was considered. This
might have been caused by the lower reduction in the local buckling stress when
compared with the reduction of elasticity modulus at 800ºC. Further, even small
experimental errors with low buckling loads obtained at high temperature caused
larger differences in the buckling coefficient obtained from these loads.
As for the variation of the local buckling coefficient of stiffened web and unstiffened
flange elements, the local bucking coefficients of the web and flange elements in the
lipped channel sections subject to the local buckling interaction were reasonably
constant when the reduced elasticity modulus was considered. These results showed
that the local buckling stress was predominantly governed by the reduced elasticity
modulus at elevated temperatures for both low and high strength steel members.
Therefore it can be stated that the use of the simply modified elastic buckling stress
equation (Equation 5.1) considering the appropriate reduction factors of the elasticity
modulus provides reasonable accuracy in the prediction of local buckling stress of
lipped channel sections subject to interactive local buckling effects.
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0 200 400 600 800 1000
G5-I1(Test)
G5-I2(Test)
G5-I3(Test)
G5-I4(Test)
G5-I5(Test)
G5-I1(FEA)
G5-I2(FEA)
G5-I3(FEA)
G5-I4(FEA)
G5-I5(FEA)
kT
°C
Buc
klin
gco
effic
ient
atte
mpe
ratu
reT
Temperature (b) G550 steels
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-20
Figure 7.12 shows the experimental failure modes at different temperatures. As
expected, all the sections failed by local buckling without any transition to other
buckling modes at elevated temperatures. Therefore it can be stated that there is no
buckling mode change at elevated temperatures for cold-formed steel compression
members subject to interactive local buckling. This observation is similar to that of
Feng et al. (2003b).
Figure 7.12 Local Buckling Failure Modes at Different Temperatures
7.4.2 Effective width at elevated temperatures
The experimental and numerical effective width ratios of cold-formed lipped channel
sections subject to the local buckling interaction effects were directly derived from
the ultimate strength and reduced yield strength values at elevated temperatures.
Figure 7.13 presents the effective width variations obtained from tests and FEA. A
similar trend to that of stiffened web and unstiffened flange elements is seen in
Figure 7.13. Therefore the simply modified design rules as given in Equation 5.5 are
adequate for low strength steel sections, but are inadequate for high strength cold-
formed steel sections subject to the interactive local buckling effects.
(a) 200°C (b) 400°C (c) 800°C
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-21
In order to examine the reliability and adequacy of the new effective width design
rules developed in Chapter 5 (Equations 5.10, 5.27 and 5.31), additional numerical
analyses were conducted at elevated temperatures to cover a larger range of
slenderness ratio (b/t) of web and flange elements and obtain more results for high
strength steel sections subject to the local buckling interaction effects. For this
purpose, the quarter-wave buckling length model was used. Table 7.7 presents the
additional FEA results of the ultimate strength and effective width at elevated
temperatures.
Figure 7.13 Variation of Effective Width of Stiffened Web and Flange Elements
0.0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800 1000
G2-I1(Test)
G2-I2(Test)
G2-I3(Test)
G2-I4(Test)
G2-I5(Test)
G2-I1(FEA)
G2-I2(FEA)
G2-I3(FEA)
G2-I4(FEA)
G2-I5(FEA)
be,T/b
°C
(a) G250 steels
Effe
ctiv
e w
idth
ratio
at
Temperature
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 200 400 600 800 1000
G5-I1(Test)
G5-I2(Test)
G5-I3(Test)
G5-I4(Test)
G5-I5(Test)
G5-I1(FEA)
G5-I2(FEA)
G5-I3(FEA)
G5-I4(FEA)
G5-I5(FEA)
(b) G550 steels
be,T/b
°C
Effe
ctiv
e w
idth
ratio
at
Temperature
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-22
Table 7.7 Additional FEA results of Effective Width for High Strength
Stiffened Web and Flange Elements
Web/flange
b/t ratio Temp. (°C)
fy.T (MPa)
fcr,T (MPa) λ Pu,T
(kN) be,T
(mm) be,T/b
20 636 290 1.48 57.8 23.7 0.49 200 616 274 1.50 55.4 23.3 0.49 400 507 168 1.74 38.8 18.6 0.39 600 304 84 1.90 20.4 15.2 0.32
51 (t=0.936)
800 81 21 1.96 5.6 16.0 0.33 20 636 143 2.11 58.4 22.7 0.33 200 616 130 2.18 55.2 21.9 0.32 400 507 82 2.49 39.2 17.5 0.26 600 304 40 2.77 20.6 14.1 0.21
73 (t=0.936)
800 81 10 2.81 5.6 14.7 0.22 20 636 93 2.62 59.5 20.0 0.24 200 616 84 2.71 55.6 18.8 0.22 400 507 53 3.09 40.1 14.8 0.17 600 304 26 3.42 21.2 11.5 0.14
91 (t=0.936)
800 81 7 3.50 5.7 11.8 0.14 20 650 64 3.19 44.5 15.7 0.19 200 630 58 3.30 42.1 15.0 0.18 400 510 37 3.71 26.8 9.0 0.11 600 307 19 4.02 14.8 7.2 0.08
108 (t=0.785)
800 92 5 4.37 4.4 7.1 0.08 20 650 42 3.93 46.2 13.5 0.14 200 630 38 4.07 44.0 13.0 0.13 400 510 24 4.61 30.7 8.9 0.09 600 307 13 4.72 16.9 6.7 0.07
127 (t=0.785)
800 92 3 4.83 5.2 7.4 0.07 Note: t is the measured base metal thickness.
Figures 7.14(a) to (e) compare the FEA results with the simply modified design rules
(Equations 5.5 and 5.6) and the theoretical design method developed in Chapter 5. In
a few cases, the numerical effective widths of low strength steel sections are slightly
below the current design rules (Equation 5.5). This is due to the conservative
assumption of the fully effective lips. The current design rules could therefore be
considered to be adequate for low strength cold-formed steel sections subject to the
interactive local buckling effects, but they overestimate the effective widths for high
strength steel sections at elevated temperatures as seen in Figures 7.14(a) to (e).
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-23
Figure 7.14 Comparisons of Theoretical Design Curves with FEA results of
Stiffened Web and Flange Elements at Elevated Temperatures
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq. (5.6)
Eq.(5.10)
G2-I20
G5-I20
(a) 20 °C
be,T/b
n=3
n=7
n=5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
n=15
Tcr
Ty
ff
,
,
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq.(5.6)
Eq.(5.10)
G2-I200
G5-I200
(b) 200 °C
be,T/b
n=3
n=7
n=5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
n=15
Tcr
Ty
ff
,
,
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-24
Figure 7.14 Comparisons of Theoretical Design Curves with FEA results of
Stiffened Web and Flange Elements at Elevated Temperatures
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq.(5.6)
Eq.(5.10)
G2-I400
G5-I400
(c) 400 °C
be,T/b
Tcr
Ty
ff
,
,
n=3
n=7
n=5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
n=15
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq.(5.6)
Eq.(5.10)
G2-I600
G5-I600
(d) 600 °C
be,T/b
Tcr
Ty
ff
,
,
n=3
n=7
n=5
Slenderness ratio at temperature T
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
n=15
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-25
Figure 7.14 Comparisons of Theoretical Design Curves with FEA results of
Stiffened Web and Flange Elements at Elevated Temperatures
Due to the inadequacy of the current effective width design rules for high strength
steel sections at elevated temperatures, a theoretical design method was developed
considering the inelastic mechanical characteristics in Section 5.4.3.1. The
theoretical design method was found to be adequate for unstiffened flange and
stiffened web elements in Chapters 5 and 6. The design method was therefore used
for cold-formed steel lipped channel sections subject to the local buckling
interaction. The comparison in Figures 7.14(a) to (e) shows the adequacy of the
theoretical design method. The use of a parameter n of 3 should be recommended for
high temperatures more than 400°C even though slightly conservative results are
expected for non-slender sections. The parameter n of 5 or 7 can be used in the low
temperature range. The use of n equal to 15 may produce slightly unconservative
results.
The local buckling stress was mainly dependant on the reduced elasticity modulus
for both low and high strength steel sections whereas the ultimate strength decreased
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.5)
Eq.(5.6)
Eq.(5.10)
G2-I800
G5-I800
(e) 800 °C
be,T/b
Tcr
Ty
ff
,
,
n=3
n=7
n=5
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
n=15
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-26
severely for high strength steel sections at elevated temperatures. These observations
were similar to those of stiffened web and unstiffened flange elements in the earlier
chapters. Elevated temperatures led to the presence of a greater post-buckling
strength of high strength steel sections. Further the inelastic mechanical behaviour
and the larger reduction ratio of the elasticity modulus to yield strength at elevated
temperatures had an important role in the ultimate strength of cold-formed steel
sections subject to interactive local buckling effects.
The semi-empirical design methods (Equations 5.27 and 5.29) were also used to
predict the effective width of stiffened web and flange elements at elevated
temperatures. Figure 7.15 compares the effective width obtained from the semi-
empirical design method with FEA results. The comparison showed that the semi-
empirical design methods can be used to predict the effective widths of stiffened
elements in the lipped channel members subject to interactive local buckling effects.
Figure 7.15 Comparisons of Effective Widths from Equations 5.27 and 5.29 with FEA Results of Stiffened Web and Flange Elements at Elevated Temperatures
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
G5-I20
G5-I200
G5-I400
G5-I600
G5-I800
be,T/b
Tcr
Ty
ff
,
,
Equation 5.5
Equation 5.27
Slenderness ratio at temperature T
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Equation 5.29
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-27
Figure 7.16 Comparisons of Effective Widths from Equation 5.31 with
FEA Results of Stiffened Web and Flange Elements at Elevated
Temperatures
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-I20
be,T/b
Tol
Ty
ff
,
,
(a) 20 C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-I200
be,T/b
Tol
Ty
ff
,
,
(b) 200 C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-28
Figure 7.16 Comparisons of Effective Widths from Equation 5.31 with
FEA Results of Stiffened Web and Flange Elements
at Elevated Temperatures
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-I400
Tol
Ty
ff
,
,
be,T/b
(c) 400 C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-I600
be,T/b
Tol
Ty
ff
,
,
(d) 600 C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-29
Figures 7.16(a) to (e) compare the numerical effective widths with the empirical
design curve based on Equation 5.31. This design method also provides a reasonable
good agreement with the numerical results. However, slightly unconservative
prediction is seen in a few cases, particularly for slender sections. This may be due to
larger contribution of lips to the effective width of the web and flanges in slender
sections when compared with non-slender sections. Overall, the design rules
developed in Chapter 5 appeared to be applicable to cold-formed steel lipped channel
members subject to interactive local buckling effects at elevated temperatures.
From the experimental and numerical studies of cold-formed steel compression
members subject to local buckling of stiffened web element, unstiffened flange
element and stiffened web and flange elements, it appears that at 400°C, these cold-
formed compression members retained 74% of their capacity at ambient temperature.
Therefore the Eurocode 3 Part 1.2 (ECS, 1995) which recommends a critical
Figure 7.16 Comparisons of Effective Widths from Equation 5.31 with
FEA Results of Stiffened Web and Flange Elements
at Elevated Temperatures
be,T/b
Tol
Ty
ff
,
,
(e) 800 C°
Effe
ctiv
e w
idth
ratio
at t
empe
ratu
re T
Slenderness ratio at temperature T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Eq.(5.31)
G5-I800
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 7-30
temperature of 350°C for cold-formed steel members seems to be conservative.
However, the critical temperature is highly dependant on the load ratio, i.e., the ratio
of the member capacity at ambient temperature to the applied load at elevated
temperatures. Therefore further study is required to recommend suitable critical
temperature for cold-formed steel members.
7.5 Summary
This chapter has presented the details of experiments and finite element analyses
undertaken to investigate the behaviour of lipped channel members subject to
interactive local buckling effects under uniform axial compression. This study
showed that the elastic local buckling behaviour of cold-formed steel sections was
predominantly governed by the reduced elasticity modulus for both low and high
strength steel members and thus the current elastic buckling formula is applicable at
elevated temperatures provided the reduced elasticity modulus values are used.
The simply modified current design rules and the new effective width design rules
developed in Chapter 5 were used in this chapter to assess their adequacy in
predicting the ultimate strengths and effective widths of the lipped channel members
considered here. As observed with the unstiffened flange and stiffened web
elements, the simply modified effective width rules (Equation 5.5) were found to be
reasonably adequate for low strength steel sections, but provided unconservative
predictions for high strength steel sections. Therefore the theoretical, semi-empirical
and empirical effective width design rules developed in Chapter 5 were used to
predict the effective width of high strength cold-formed steel sections subject to the
interactive local buckling behaviour. The comparison of numerical results with
predictions based on the new methods demonstrated their adequacy for high strength
steel sections at elevated temperature up to 800°C.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 8-1
Chapter 8. Conclusions and Recommendations
This thesis has presented a detailed investigation into the local buckling behaviour of
light gauge cold-formed steel members subject to axial compression at elevated
temperatures. Aiming at improving the knowledge and understanding of the
behaviour and hence the design of cold-formed steel compression members subject
to fire effects, both experiments and finite element analyses were conducted under
simulated fire conditions. Prior to undertaking the investigations at elevated
temperatures, the structural behaviour of cold-formed steel compression members
subject to local buckling effects at ambient temperature and the mechanical
properties of cold-formed steels at elevated temperatures were investigated in the
first phase. In the second phase, more than 300 experiments including three repeats
were conducted in a specially designed furnace to study their structural behaviour
involving local buckling effects (unstiffened flange element, stiffened web element,
stiffened web and flange interactive local buckling) at elevated temperatures up to
800°C. Finite element models were also developed considering the temperature
effects on cold-formed steels and then validated by comparing the ultimate strength,
load-deflection curves and failure mode with corresponding results obtained from
experiments. Finally, suitable theoretical and empirical design rules were developed
for predicting the effective width of cold-formed steel compression members subject
to local buckling effects at elevated temperatures, and verified using experimental
and numerical results.
Chapter 3 of this thesis has presented the details of the experimental investigation
into the mechanical characteristics of both low and high strength light gauge steels at
elevated temperatures. Chapter 4 has presented the details of the local buckling
investigation of unstiffened flange elements at ambient temperature, whereas
Chapter 5 has presented the simulated fire tests and finite element analyses of cold-
formed steel unstiffened flange elements at elevated temperatures. Chapters 6 and 7
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 8-2
have discussed the local buckling behaviour of stiffened web elements and the
interactive local buckling behaviour of stiffened web and flange elements.
The most valuable outcomes achieved in this research were to provide the
fundamental data of mechanical properties of light gauge cold-formed steels and to
develop the effective width design rules (theoretical, semi-empirical and simplified
empirical methods) for cold-formed steel compression members at elevated
temperatures up to 800°C. The design rules were developed for unstiffened flange
and stiffened web elements subject to local buckling and stiffened web and flange
elements subject to interactive local buckling. The following conclusions have been
drawn from this research project.
1. An experimental investigation using tensile coupon tests at elevated temperatures
showed the presence of severe reduction to the yield strength and elasticity
modulus of light gauge cold-formed steels. The reduction factors given in the
current steel design standards for hot-rolled steels were found to be
unconservative for light gauge cold-formed steels.
2. There was no significant variation in the elasticity modulus between low and high
strength steels, whereas the yield strength was influenced by steel grade in the
temperature range of 400°C to 800°C. Based on these investigations, suitable
mathematical equations were developed to predict the mechanical properties of
light-gauge steels at elevated temperatures up to 800°C. These observations and
developments are very important and useful to the researchers, manufacturers
and designers involved in the fire safety design of cold-formed steel structures
because the deterioration of mechanical properties is the primary element
affecting the structural behaviour under fire.
3. Based on the Ramberg-Osgood equation, a stress-strain model showing the non-
linear elastic-plastic relationship at elevated temperature was developed and
verified using experimental results of stress-strain curves. It was further found
from recent research that the yield strength is very sensitive in the temperature
range of 500°C to 650°C due to material recrystallisation. Hence the
investigation of the mechanical properties at high temperatures is currently
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 8-3
reinvestigated by another researcher at Queensland University of Technology.
Even though there would be the sensitivity of mechanical properties at high
temperatures, outcomes achieved in this thesis are still reliable because all the
experimental investigations of tensile and compressive tests were conducted
under the same conditions using the same furnace. For instance, the compressive
strengths at elevated temperatures were predicted using the mechanical
properties obtained at the corresponding temperatures and compared with
experimental compressive strength results.
4. This research showed that the current effective width rules were adequate in
predicting the ultimate strength of low strength cold-formed steel members, but
were inadequate for high strength steel compression members subject to the local
buckling effects at ambient temperature. This might be due to the different
mechanical behaviour of high strength steels when compared with low strength
steels e.g. lack of ductility and the Bauschinger effect. Therefore, a modified
effective width rule was proposed for high strength unstiffened elements based
on the experimental results.
5. The effect of the strain hardening was investigated using the finite element
analysis. It was found that the ultimate strength of cold-formed steel compression
members subject to the local buckling effects was hardly increased even though
the strain hardening effect was included due to the failure of members prior to
reaching the ultimate strength of materials. Therefore, it is reasonable to ignore
the strain hardening effect to simplify finite element modelling.
6. A sensitivity study of initial geometric imperfections at ambient temperature
showed that the local buckling behaviour of unlipped channel members subject
to the flange local buckling was not sensitive to generally used imperfection
magnitudes proposed by Walker (1975) and, Schafer and Pekoz (1996, 1998).
7. The direct strength method that was developed to avoid the complexity of current
design procedures for cold-formed steel structures by Schafer and Pekoz (1998)
and Schafer (2002) was able to safely predict the ultimate strength of unlipped
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 8-4
channel members subject to local buckling effects. However, the method was
found to provide slightly conservative predictions for slender sections.
8. Accurate finite element models were successfully developed to investigate the
local buckling behaviour of cold-formed steel compression members at elevated
temperatures using field option variables based on true stress-strain curves,
which simulate the plastically behaved mechanical characteristics of cold-formed
steels. The initial geometric imperfections and residual stresses were also
successfully included at elevated temperatures.
9. Three types of finite element models were developed for unlipped and lipped
channel members subject to the local buckling of unstiffened flange elements,
stiffened web elements and interactive local buckling of stiffened web and flange
elements at elevated temperatures. These models were full length model
(experimental model), half length model (experimental model) and quarter-wave
buckling length model (ideal model). It was found that there is very little
difference in the ultimate strength and buckling stress from the three modelling
methods for unlipped channel members subject to the flange local buckling
effects. However there was a small difference (4 to 8 percent) in the ultimate
strength between the experimental and ideal models for lipped channel members
subject to the local buckling of stiffened web and the interactive local buckling
of stiffened web and flange elements. This might be due to the end boundary
conditions created by the rigid body for the ideal model and the multi point
constraints for the experimental model. However this effect was minimal for the
unlipped channel members subject to the flange local buckling due to lower
rotational restraint. The half length model was used in the comparisons with
experimental results while the quarter-wave buckling length model was used in
the assessment and development of design rules as the latter simulates ideal
conditions.
10. The experimental and numerical results demonstrated that the local buckling
stress at elevated temperatures was dependant on the deterioration of the
elasticity modulus, and thus the theoretical equation for the local buckling stress
could be simply modified by considering the reduced elasticity modulus.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 8-5
11. The simply modified effective width design rule using the reduced yield strength
and elasticity modulus was found to be adequate for low strength steel members,
but was inadequate for high strength cold-formed steel members subject to the
local buckling effects. The reasons for the inadequacy were that the severe
reduction of the ultimate strength at elevated temperatures occurred in the post
buckling strength range, and the reduced ratio of the elasticity modulus to the
yield strength had a significant role in the ultimate strength of cold-formed steel
compression members.
12. By employing the Ramberg-Osgood law, a theoretical effective width design rule
was developed for the local buckling of unstiffened flange elements by
considering the inelastic behaviour of cold-formed steels at elevated
temperatures. The comparison of the theoretical curves with the experimental
and numerical results showed that the theoretical method provided good
predictions using a parameter n that controls the non-linear stress-strain curve in
the inelastic range. Based on the Winter’s effective width approach, a semi-
empirical design method was further developed. This design method was also
adequate to predict the effective width. Due to the complexity of the theoretical
and semi-empirical design methods, a simplified empirical method was proposed
on the basis of the experimental and numerical results. The empirical effective
width rule also provided reasonably accurate predictions at elevated
temperatures.
13. The behaviour of lipped channel members subject to the local buckling of
stiffened web elements and the interactive local buckling of stiffened web and
flange elements was similar to that of unlipped channel members subject to the
local buckling of unstiffened flange elements at elevated temperatures. The
elastic buckling stress equation considering the reduced elasticity modulus was
therefore adequate for both stiffened and unstiffened elements made of low and
high strength steels. The current effective width rule was adequate for low
strength steel sections and yet inadequate for high strength steel sections, even
though the reduced mechanical properties were considered. The theoretical,
semi-empirical and empirical design proposals developed for the unstiffened
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 8-6
flange elements were found to be adequate to predict the effective width of
stiffened web and flange elements subject to local buckling at elevated
temperatures.
Further research is recommended to investigate the effect of temperature on the
distortional buckling of cold-formed steel compression members. The distortional
buckling of cold-formed steel members is also one of the important failure modes.
The interaction of local and distortional buckling also requires further research
because most cold-formed steel sections are subject to these buckling modes.
Therefore, research on the distortional buckling and the interaction of local and
distortional buckling is required to enhance the understanding of the structural
behaviour of cold-formed steel members under fire conditions.
Additional research is still required to clearly define the Bauschinger effect on high
strength cold-formed steel members at ambient and elevated temperatures. The
predictions calculated at elevated temperatures in this study were based on the
mechanical properties obtained from tensile coupon tests. Thus, it would be useful to
conduct additional experimental study on the mechanical properties under
compression to improve the knowledge of the structural behaviour of cold-formed
steel members.
By understanding the effect of elevated temperatures on bare steel structures, fire
protection materials and their attachment systems can be used to achieve the required
level of fire resistance. The significant factors of fire protection materials are thermal
conductivity, specific heat, density and moisture content. These characteristics and
their interaction between steel structures and fire protection materials under fire
conditions can be evaluated using small scale tests and the standard fire test. Further,
the integrity of attachment systems should be carefully assessed to ensure the fire
resistant design is appropriate.
In general, the finite element models do not include the mechanical properties
including the Bauschinger effect. Advanced finite element models are required that
will include the variation of yield strength in compression and tension at the same
time. Further, it is also useful to investigate thermal effects of heat transferring
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures 8-7
factors such as conductivity, convection and radiation on structural members using
advanced finite element models under the transient state analysis.
This research investigated the degree of rotational restraints provided by the plate
elements to the adjoining plate elements at elevated temperatures. However, further
research is required to thoroughly understand the variations in the end support
restraint conditions of cold-formed steel structures under fire conditions.
Cold-formed steel compression members are subject to not only local buckling
effects, but also to distortional buckling and flexural/flexural torsional buckling
effects. Therefore, further research into the flexural and flexural torsional buckling
behaviour at elevated temperatures is required for better understanding of the
structural behaviour cold-formed steel compression members under fire conditions.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-1
APPENDIX A Stress-Strain Curves of Tensile Coupon Tests at Elevated
Temperatures
0
100
200
300
400
500
600
700
800
0 0.5 1 1.5 2 2.5 3 3.5
G550 – 0.6 mm (20, 100, 200, 300, 400, 500, 600, 700, 800 °C)
Stre
ss (M
Pa)
Strain (%)
0
100
200
300
400
500
600
700
0 0.5 1 1.5 2 2.5 3 3.5
G500 – 1.2 mm (20, 100, 200, 300, 400, 500, 600, 700, 800 °C)
Stre
ss (M
Pa)
Strain (%)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-2
0
100
200
300
400
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
G250 – 0.4 mm (20, 100, 200, 300, 400, 500, 600, 700, 800 °C)
Stre
ss (M
Pa)
Strain (%)
G250 – 0.6 mm (20, 100, 200, 300, 400, 500, 600, 700, 800 °C)
0
100
200
300
400
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Stre
ss (M
Pa)
Strain (%)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-3
0
100
200
300
400
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
G250 – 1.0 mm (20, 100, 200, 300, 400, 500, 600, 700, 800 °C)
Stre
ss (M
Pa)
Strain (%)
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-4
APPENDIX B
ABAQUS Residual Stress Subroutines
B1. Abaqus subroutine used for unstiffened elements
*INITIAL CONDITIONS, TYPE=STRESS, USER
*USER SUBROUTINES
SUBROUTINE
SIGINI(SIGMA,COORDS,NTENS,NCRDS,NOEL,NPT,LAYER,KSPT)
INCLUDE 'ABA_PARAM.INC'
REAL X,Y,Z,nipt,ipt,fy,fn
DIMENSION SIGMA(NTENS), COORDS(NCRDS)
C
nipt=5.
fy=636.
X=COORDS(1)
Y=COORDS(2)
Z=COORDS(3)
C
IF (KSPT.EQ.1) THEN
ipt=1.
ELSEIF (KSPT.EQ.2) THEN
ipt=2.
ELSEIF (KSPT.EQ.3) THEN
ipt=3.
ELSEIF (KSPT.EQ.4) THEN
ipt=4.
ELSEIF (KSPT.EQ.5) THEN
ipt=5.
ENDIF
C
IF ((NOEL.GE.1).AND.(NOEL.LE.2199)) THEN
fn=0.1*fy
ELSEIF ((NOEL.GE.2200).AND.(NOEL.LE.2449)) THEN
fn=0.08*fy
ELSEIF ((NOEL.GE.2450).AND.(NOEL.LE.2949)) THEN
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-5
fn=0.17*fy
ELSEIF ((NOEL.GE.2950).AND.(NOEL.LE.3249)) THEN
fn=0.33*fy
ELSE
fn=0.
ENDIF
C
IF (fn.NE.0.) THEN
SIGMA(2)=fn*(1.-2.*(nipt-ipt)/(nipt-1.))
ELSE
SIGMA(2)=0.
ENDIF
C
SIGMA(1)=0.
SIGMA(3)=0.
RETURN
END
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-6
B2. Abaqus subroutine used for stiffened elements
*INITIAL CONDITIONS, TYPE=STRESS, USER
*USER SUBROUTINES
SUBROUTINE
SIGINI(SIGMA,COORDS,NTENS,NCRDS,NOEL,NPT,LAYER,KSPT)
INCLUDE 'ABA_PARAM.INC'
REAL X,Y,Z,nipt,ipt,fy,fn
DIMENSION SIGMA(NTENS), COORDS(NCRDS)
C
nipt=5.
fy=317.
X=COORDS(1)
Y=COORDS(2)
Z=COORDS(3)
C
IF (KSPT.EQ.1) THEN
ipt=1.
ELSEIF (KSPT.EQ.2) THEN
ipt=2.
ELSEIF (KSPT.EQ.3) THEN
ipt=3.
ELSEIF (KSPT.EQ.4) THEN
ipt=4.
ELSEIF (KSPT.EQ.5) THEN
ipt=5.
ENDIF
C
IF ((NOEL.GE.1).AND.(NOEL.LE.1556)) THEN
fn=0.08*fy
ELSEIF ((NOEL.GE.3010).AND.(NOEL.LE.4255)) THEN
fn=0.08*fy
ELSEIF ((NOEL.GE.1557).AND.(NOEL.LE.2899)) THEN
fn=0.17*fy
ELSEIF ((NOEL.GE.4256).AND.(NOEL.LE.5688)) THEN
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-7
fn=0.33*fy
ELSEIF ((NOEL.GE.2900).AND.(NOEL.LE.3009)) THEN
fn=0.17*fy
ELSE
fn=0.
ENDIF
C
IF (fn.NE.0.) THEN
SIGMA(2)=fn*(1.-2.*(nipt-ipt)/(nipt-1.))
ELSE
SIGMA(2)=0.
ENDIF
C
SIGMA(1)=0.
SIGMA(3)=0.
RETURN
END
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-8
B3. Abaqus subroutine used for members subject to local buckling
interaction
*INITIAL CONDITIONS, TYPE=STRESS, USER
*USER SUBROUTINES
SUBROUTINE
SIGINI(SIGMA,COORDS,NTENS,NCRDS,NOEL,NPT,LAYER,KSPT)
INCLUDE 'ABA_PARAM.INC'
REAL X,Y,Z,nipt,ipt,fy,fn
DIMENSION SIGMA(NTENS), COORDS(NCRDS)
C
nipt=5.
fy=722.
X=COORDS(1)
Y=COORDS(2)
Z=COORDS(3)
C
IF (KSPT.EQ.1) THEN
ipt=1.
ELSEIF (KSPT.EQ.2) THEN
ipt=2.
ELSEIF (KSPT.EQ.3) THEN
ipt=3.
ELSEIF (KSPT.EQ.4) THEN
ipt=4.
ELSEIF (KSPT.EQ.5) THEN
ipt=5.
ENDIF
C
IF ((NOEL.GE.1).AND.(NOEL.LE.120)) THEN
fn=0.08*fy
ELSEIF ((NOEL.GE.241).AND.(NOEL.LE.360)) THEN
fn=0.08*fy
ELSEIF ((NOEL.GE.121).AND.(NOEL.LE.240)) THEN
fn=0.17*fy
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-9
ELSEIF ((NOEL.GE.360).AND.(NOEL.LE.408)) THEN
fn=0.33*fy
ELSE
fn=0.
ENDIF
C
IF (fn.NE.0.) THEN
SIGMA(2)=fn*(1.-2.*(nipt-ipt)/(nipt-1.))
ELSE
SIGMA(2)=0.
ENDIF
C
SIGMA(1)=0.
SIGMA(3)=0.
RETURN
END
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-10
APPENDIX C
Example Calculations
C1. Direct Strength Method
Unstiffened flange element series G2-C9 (Chapter 4)
b (flange) = 74.0 mm, d (web) = 46.0 mm, t (thickness) = 1.15 mm fy = 312 MPa
corner length = 3.76 mm based on a radius of 1.2 mm
The ultimate strength Pn of cold-formed steel columns subject to local buckling is
−
=
4.04.0
15.01y
cr
y
cryn P
PPPPP for λ > 0.776
The local buckling load Pcr obtained from the finite element analysis is 11.6 kN.
The squash load Py is
(74 × 2 + 46 + 3.76) × 1.15 ×312 / 1000 = 70.96 kN
The ultimate strength of G2-C9 specimen is as follows.
kN9.319.706.1115.01
9.706.1196.70
4.04.0=
−
Therefore Pn/Py = 0.450.
The experimental ultimate strength of G2-C9 is 35.4 kN.
Pn,test/Py = 0.499.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-11
Unstiffened flange element series G5-C3 (Chapter 4)
b (flange) = 34.0 mm, d (web) = 32.0 mm, t (thickness) = 0.936 mm fy = 636 MPa
corner length = 3.76 mm based on a radius of 1.2 mm
The local buckling load Pcr obtained from the finite element analysis is 11.7 kN.
The squash load Py is
(34 × 2 + 32 + 3.76) × 0.936 × 636 / 1000 = 61.77 kN
The ultimate strength of G5-C3 specimen is as follows.
kN3.297.617.1115.01
7.617.1177.61
4.04.0=
−
Therefore, Pn/Py = 0.475.
The experimental ultimate strength of G5-C3 is 30.7 kN.
Pn,test/Py = 0.498.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-12
C2. Effective Width
Unstiffened flange element series G2-C2 at 400 °C (Chapter 5)
b (flange) = 47.5 mm, d (web) = 45.1 mm, t (thickness) = 1.56 mm
Pu (test) = 36.3 kN, fy,400 = 298 × 0.779 (reduction factor) = 232.14 MPa
corner length = 3.76 mm based on a radius of 1.2 mm
Ae × fy =Pu
(2be + d + 3.76) t fy = Pu
(2be + 45.1 + 3.76) × 1.56 × 232.14 = 36300 N
be = 25.69 mm
Therefore be/b of flange element = 0.54
Stiffened web element series G5-L2 at 400 °C (Chapter 6)
b (flange) = 14.1 mm, d (web) = 57.5mm, L (lip) = 9.5 mm, t (thickness) = 0.936
Pu (test) = 36.29 kN, fy,400 = 636 × 0.797 (reduction factor) = 506.89 MPa
corner length = 7.54 mm based on a radius of 1.2 mm
Ae × fy =Pu
(2b + de + 2L + 7.54) t fy = Pu
(2×14.1 + de + 2×9.5 + 7.54) × 0.936 × 506.89 = 36290 N
de = 21.74 mm
Therefore de/d of stiffened web element = 0.37
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-13
Stiffened web and flange element series G2-I1 at 200 °C assuming fully
effective lips (Chapter 7)
b (flange) = 47.7 mm, d (web) = 47.3 mm, L (lip) = 11.0 mm, t (thickness) = 0.964
Pu (FEA) = 38.7 kN, fy,200 = 317 × 0.935 (reduction factor) = 296.4 MPa
corner length = 7.54 mm based on a radius of 1.2 mm
Ae × fy =Pu
(2be + de + 2L + 7.54) t fy = Pu
(3be + 2×11 + 7.54) × 0.964 × 296.4 = 38700 N
be = 35.3 mm
Therefore de/d of stiffened web and flange element = 0.75
Stiffened web and flange element series G2-I1 at 200 °C assuming the
same reduction as the web (Chapter 7)
b (flange) = 47.7 mm, d (web) = 47.3 mm, L (lip) = 11.0 mm, t (thickness) = 0.964
Pu (FEA) = 38.7 kN, fy,200 = 317 × 0.935 (reduction factor) = 296.4 MPa
corner length = 7.54 mm based on a radius of 1.2 mm
Ae × fy =Pu
(2be + de + 2Le + 7.54) t fy = Pu
Since the effective width ratios for all the element (web, flanges and lips)
are same,
(2ρb + ρd + 2ρL + 7.54) × 0.964 × 296.4 = 38700 N
where ρ is the effective width ratio.
(2×47.7×ρ + 47.3×ρ + 2×11×ρ + 7.54) × 0.964 × 296.4 = 38700 N
Therefore ρ (=be/b) of stiffened web and flange element = 0.78
Note: Same method is used for both FEA and test loads in all the cases.
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-14
APPENDIX D
Axial Compression Load versus Axial Shortening Curves
at Ambient Temperatures (Chapter 4)
0
10
20
30
40
50
60
70
80
0 1 2 3
Test
FEA
Test Series G2-C1, G2-C2 and G2-C6
mm
Axi
al C
ompr
essi
on L
oad
kN
G2-C1
G2-C2
G2-C6
Axial Shortening
0
5
10
15
20
25
30
35
40
45
0 1 2 3
Test
FEA
Test Series G2-C7, G2-C8 and G2-C11
mm
Axi
al C
ompr
essi
on L
oad
kN
G2-C7
G2-C8
G2-C11
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-15
0
5
10
15
20
25
30
35
40
45
0 1 2 3
Test
FEA
Test Series G5-C1, G5-C5 and G5-C7
mm
Axi
al C
ompr
essi
on L
oad
kN
G5-C1
G5-C7
G5-C5
Axial Shortening
0
5
10
15
20
25
30
0 1 2 3
Test
FEA
Test Series G5-C9, G5-C10 and G5-C14
mm
Axi
al C
ompr
essi
on L
oad
kN
G5-C9
G5-C10
G5-C14
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-16
APPENDIX E
Axial Compression Load versus Axial Shortening Curves
at Elevated Temperatures
E1. Chapter 5
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5 3
Test
FEA
kN
Test Series G2-1 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
Axial Shortening
0
10
20
30
40
50
60
70
80
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G2-3 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-17
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G5-1 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-18
E2. Chapter 6
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G2-L1 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G2-L2 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-19
0
5
10
15
20
25
30
35
40
45
50
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G2-L3 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
0
5
10
15
20
25
30
35
40
45
50
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G5-L1 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-20
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G5-L2 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
0
10
20
30
40
50
60
70
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G5-L3 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-21
E3. Chapter 7
0
5
10
15
20
25
30
35
40
45
50
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G2-I1 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G2-I2 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-22
0
2
4
6
8
10
12
14
16
18
20
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G2-I3 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
0
2
4
6
8
10
12
14
16
18
20
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G2-I4 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-23
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G2-I5 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G5-I1 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-24
0
10
20
30
40
50
60
70
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G5-I2 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
0
2
4
6
8
10
12
14
16
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G5-I4 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-25
0
2
4
6
8
10
12
14
16
0 0.5 1 1.5 2 2.5 3
Test
FEA
Test Series G5-I5 (20, 200, 400, 600, 800 °C)
mm
Axi
al C
ompr
essi
on L
oad
kN
Axial Shortening
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-26
APPENDIX F
Ultimate Strength of Stub Columns including Three Repeats
Table F.1 Unstiffened Flange Elements at Ambient Temperature (Chapter 4)
Pu (kN) Pu (kN) Test series 1 2 3 Ave.
Test series 1 2 3 Ave.
G2-C1 33.5 33.7 34.2 33.8 G5-C3 29.7 31.2 31.1 30.7 G2-C2 45.8 44.9 45.0 45.2 G5-C4 34.1 33.1 33.1 33.4 G2-C3 52.1 53.2 53.1 52.8 G5-C5 38.9 38.1 38.4 38.5 G2-C4 58.3 57.1 57.9 57.8 G5-C6 39.4 40.6 40.8 40.3 G2-C5 65.5 66.4 65.4 65.8 G5-C7 16.9 16.7 16.6 16.7 G2-C6 72.4 71.3 72.2 72.0 G5-C8 19.8 18.8 18.8 19.1 G2-C7 30.9 31.8 32.3 31.7 G5-C9 23.7 23.2 23.5 23.5 G2-C8 24.3 24.7 23.2 24.1 G5-C10 4.6 4.9 4.9 4.8 G2-C9 35.4 35.2 35.7 35.4 G5-C11 7.0 7.1 7.2 7.1 G2-C10 37.2 37.7 38.1 37.7 G5-C12 9.2 8.6 8.7 8.8 G2-C11 39.9 39.1 40.2 39.7 G5-C13 8.8 9.1 8.9 8.9 G5-C1 25.1 25.8 25.3 25.4 G5-C14 13.6 12.9 13 13.2 G5-C2 26.1 26.9 26.6 26.5 G5-C15 14.4 14.1 13.9 14.1
Table F.2 Unstiffened Flange Elements at Elevated Temperatures (Chapter 5)
Pu,T (kN) Pu,T (kN) Test series
Temp (°C) 1 2 3 Ave.
Test series
Temp(°C) 1 2 3 Ave.
20 34.9 35.4 35.1 35.1 20 25.9 25.1 25.3 25.4 200 33.9 33.7 33.1 33.6 200 23.3 24.1 24.5 24.0 400 27.4 27.5 29.8 27.2 400 19.8 20.3 20.3 20.1 600 17.2 17.7 17.5 17.5 600 11.5 11.6 11.5 11.5
G2-1
800 4.1 4.2 4.2 4.2
G5-1
800 2.5 2.5 2.7 2.6 20 50.7 49.9 50 50.2 20 35.5 35.3 34.7 35.2 200 48.2 48.2 47.7 48.0 200 33.1 33.4 33.2 33.2 400 36.1 36.5 36.4 36.3 400 24.2 23.8 23.9 24.0 600 21.2 21.3 21.0 21.1 600 14.7 14.5 14.4 14.5
G2-2
800 6.2 6.3 6.3 6.3
G5-2
800 4.1 3.8 3.9 3.9 20 66.7 67.8 67.2 67.2 20 39.2 38.2 38.2 38.5 200 64.1 64.8 64.7 64.5 200 37.4 38.1 36.9 37.5 400 45.2 45.6 44.7 45.2 400 31.5 32.2 32 31.9 600 24.6 25.2 25.5 25.1 600 17.0 17.3 17.1 17.1
G2-3
800 7.2 7.7 7.7 7.5
G5-3
800 4.8 4.9 4.8 4.8
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-27
Table F.3 Stiffened Web Elements at Elevated Temperatures (Chapter 6)
Pu,T (kN) Pu,T (kN) Test series
Temp(°C) 1 2 3 Ave.
Test series
Temp(°C) 1 2 3 Ave.
20 28.8 27.9 28.1 28.3 20 42.1 43.2 43.6 43.0 200 27.4 27.8 27.2 27.5 200 40.2 40.9 40.6 40.6 400 22.5 22.9 23.8 23.1 400 34.2 34.8 33.4 34.1 600 11.1 11.0 11.4 11.2 600 19.5 19.2 19.8 19.5
G2- L1
800 3.7 3.8 3.8 3.8
G5-L1
800 4.7 4.6 5.0 4.8 20 36.3 36.9 37.7 37.0 20 47.5 48.3 48 47.9 200 35.9 35.8 34.9 35.5 200 46.3 46 46.1 46.1 400 28.1 28.3 28.7 28.4 400 38.1 38.1 38.5 38.2 600 15.1 15.3 15 15.1 600 21.2 21.7 21.7 21.5
G2-L2
800 5.2 5.3 5.3 5.3
G5-L2
800 5.5 5.8 5.3 5.5 20 44.6 46.1 45.7 45.5 20 59.9 59.5 58.3 59.2 200 42.8 42.9 41.9 42.5 200 56.4 57.2 57.7 57.1 400 34.3 34.6 33.8 34.2 400 45.9 45.1 45.8 45.6 600 18.5 18.5 18.6 18.5 600 25.4 26.2 26.3 26.0
G2-L3
800 6.4 6.6 6.5 6.5
G5-L3
800 6.8 6.7 6.8 6.8 20 21.2 21.3 20.9 21.0 20 52.2 52.9 52.4 52.5 200 20.5 20.4 20.4 20.4 200 52.0 52.1 52.4 52.2 400 12.6 12.1 11.7 12.1 400 40.2 40.5 39.8 40.2 600 6.6 6.1 6.9 6.5 600 22.1 23.1 23.4 22.9
G2-L4
800 2.2 2.2 2.2 2.2
G5-L4
800 7.2 7.2 6.8 7.1
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures A-28
Table F.3 Interactive Local Buckling of Stiffened Web and Flange Elements
at Elevated Temperatures (Chapter 7)
Pu,T (kN) Pu,T (kN) Test series
Temp (°C) 1 2 3 Ave.
Test series
Temp(°C) 1 2 3 Ave.
20 43.6 44.6 44.2 44.1 20 55.2 56.6 56.4 56.1 200 40.6 40.5 39.5 40.2 200 53.9 53.1 53.4 53.5 400 32.1 32.4 31.6 32.0 400 43.0 43.2 43.5 43.2 600 18.0 18.4 18.3 18.2 600 23.4 23.6 23.6 23.5
G2- I1
800 6.9 6.8 6.9 6.9
G5-I1
800 6.2 6.5 6.2 6.3 20 46.5 46.7 46.5 46.5 20 59.0 60.7 60.5 60.1 200 45.2 45.5 44.7 45.1 200 57.8 57.5 56.4 57.2 400 31.5 31.4 32.1 31.7 400 41.6 41.9 42.7 42.1 600 19.3 19.6 19.6 19.5 600 24.4 24.5 23.5 24.1
G2-I2
800 7.3 7.3 7.2 7.3
G5-I2
800 6.2 6.2 6.1 6.2 20 17.0 17.3 17.0 17.1 20 65.1 63.8 63.5 64.1 200 16.2 15.7 16.3 16.0 200 60.1 60.8 59.4 60.1 400 12.4 12.6 12.6 12.5 400 44.6 43.6 44.2 44.1 600 6.5 6.5 6.9 6.6 600 24.6 23.7 23.4 23.9
G2-I3
800 2.1 2.2 2.2 2.2
G5-I3
800 6.3 6.1 6.4 6.3 20 17.5 18.2 18.0 17.9 20 13.3 13.8 14.1 13.7 200 17.3 17.1 16.4 16.9 200 12.7 13.1 13.0 12.9 400 12.5 12.8 12.4 12.6 400 9.3 9.3 9.4 9.3 600 7.1 7.0 7.3 7.1 600 5.0 4.8 4.8 4.9
G2-I4
800 2.3 2.4 2.4 2.4
G5-I4
800 1.3 1.3 1.3 1.3 20 18.9 18.9 19 18.9 20 13.9 13.6 13.7 13.7 200 18.6 18.1 18.2 18.3 200 12.2 12.8 12.6 12.5 400 11.4 12.5 12.5 12.1 400 9.0 9.1 9.1 9.1 600 7.3 7.4 7.0 7.2 600 5.2 5.2 5.0 5.1
G2-I5
800 2.5 2.3 2.6 2.5
G5-I5
800 1.3 1.3 1.3 1.3
Local buckling behaviour and design of cold-formed steel compression members at elevated temperatures R-1
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