a deformation based approach to structural steel design€¦ · statically determinate and...

A deformation based approach to structural

steel design

A thesis submitted to Imperial College London

for the degree of Doctor of Philosophy

By

Facheng Wang

Department of Civil and Environmental Engineering

Imperial College London

London SW7 2AZ

United Kingdom

February 2011

2

ABSTRACT Current structural steel design codes, such as EN 1993-1-1, were developed on the

basis of a bi-linear (elastic, perfectly-plastic) material model, which lends itself to the

idea of cross-section classification. This step-wise design concept is a useful, but

somewhat artificial simplification of the true behaviour of structural steel since the

relationship between the resistance of a structural cross-section and its slenderness is,

in reality, continuous. The aim of this study is therefore to develop a more efficient

structural steel design method recognising this relationship and rationally exploiting

strain-hardening, whilst maintaining, where possible, consistency with current design

approaches.

As part of the present study, laboratory tests were carried out on cold-formed and hot-

rolled steel hollow sections. A total of 6 simple beams and 12 continuous beams (with

two configurations) and corresponding material coupon tests were conducted. These

experimental results were added to existing collected test data to develop and calibrate

a new structural steel design method. The test results indicated that capacities beyond

the yield load in compression and the plastic moment capacity in bending could be

achieved due to strain-hardening. The new design approach, termed the continuous

strength method (CSM), enables this extra capacity to be harnessed.

The developed deformation based steel design method employs a continuous ‘base

curve’ to provide a relationship between cross-section slenderness and deformation

capacity in conjunction with a strain-hardening material model. The material model is

elastic, linear-hardening and has been calibrated on the basis of collected stress-strain

data from a range of structural sections. The CSM has been developed for both

statically determinate and indeterminate structures utilising both experimental data

and that generated through sophisticated numerical modelling. Comparisons between

test results and predictions according to EN 1993-1-1 and the proposed method were

made. The results revealed that the CSM provides a more accurate prediction of test

response and enhanced structural capacity over current design methods.

3

ACKNOWLEDGEMENTS This thesis is dedicated to my parents with all my love and respect.

The work reported in this thesis was carried out under the supervision of Dr Leroy

Gardner, Reader in the Department of Civil and Environmental Engineering. I would

express my sincere gratitude for his guidance, patience and continuous

encouragement throughout the course of this study.

I would like to acknowledge the funding and supply of test specimens from Corus

Tubes UK. The experimental works were carried out in the Structures Laboratory in

the Department and would not have been possible without the efforts made by many

technicians, especially by Gordon Herbert. Special thanks should be extended to Dr

Marios Theofanous and fellow PhD student Nadiah Saari for their assistance in the

laboratory.

The inspiring and quiet working environment provided by both staff and students of

the Department, in particular Fionnuala Ni Dhonnabhain, Antonia Szigeti, Dr Ahmer

Wadee, Dr Andrew Phillips, Dr Rafee Mohamed Ali, Dr Ka Ho Nip, Dr Daisuke

Saito, Ada Law, Mohammad Haidarali, Christian Malaga, Mukesh Kumar, Panagiotis

Stylianidis, Yanzhi Liu, Adelaja Osofero, Aneeka Ahmed, Xiao Ban and other fellow

researchers in Room 424, has significantly contributed to the successful completion of

the thesis. Special thanks are also due to Professor David Nethercot for his advices on

my transfer report, Dr Jeanette Abela for her valuable comments and tremendous help

on the thesis and Dr Stylianos Yiatros for his kind academic and social support

including organising weekly football games.

Finally, I will always be grateful to my parents for their unconditional love and

support throughout the course of my studies.

Contents

4

CONTENTS ABSTRACT

ACKNOWLEDGEMENTS

CONTENTS

NOTATION

LIST OF FIGURES

LIST OF TABLES

CHAPTER 1 INTRODUCTION

1.1 Background........................................................................................................24

1.2 Limitations of the current codes of practice ........................................................25

1.2.1 Strain-hardening ..........................................................................................25

1.2.2 Cross-section classification..........................................................................25

1.3 Scope of the study and research innovation ........................................................26

1.4 Outline of thesis .................................................................................................26

CHAPTER 2 LITERATURE REVIEW

2.1 Introduction........................................................................................................29

2.2 Existing design guidance ....................................................................................29

2.2.1 Determinate structures.................................................................................30

Contents

5

2.2.2 Indeterminate structures ..............................................................................31

2.3 Laboratory testing ..............................................................................................32

2.3.1 Stub column tests ........................................................................................32

2.3.2 Simple beam tests........................................................................................33

2.3.3 Continuous beam tests.................................................................................34

2.3.4 Frame tests ..................................................................................................34

2.4 Material modelling .............................................................................................35

2.4.1 Factors influencing material properties ........................................................35

2.4.2 Existing material models .............................................................................36

2.5 Numerical modelling ..........................................................................................37

2.5.1 Element type ...............................................................................................38

2.5.2 Geometric imperfections .............................................................................38

2.6 Concluding remarks ...........................................................................................39

CHAPTER 3 LABORATORY TESTING

3.1 Introduction........................................................................................................40

3.2 Material testing ..................................................................................................41

3.2.1 Details of material supply............................................................................41

3.2.2 Tensile coupon tests ....................................................................................41

3.2.2.1 Preparation of coupons ......................................................................41

3.2.2.2 Instrumentation and testing ................................................................42

3.2.2.3 Results...............................................................................................42

3.3 Simple beam tests...............................................................................................44

3.3.1 Test specimens and measured dimensions ...................................................44

3.3.2 Test configuration and loading rates ............................................................45

3.3.3 Instrumentation ...........................................................................................46

3.3.4 Test results ..................................................................................................47

3.4 Continuous beam tests........................................................................................51

3.4.1 Test specimens and measured dimensions ...................................................51

3.4.2 Test configuration and loading rates ............................................................51

3.4.3 Instrumentation ...........................................................................................53

3.4.4 Test results ..................................................................................................54

3.5 Analysis of experimental results .........................................................................57

Contents

6

3.5.1 Material properties ......................................................................................57

3.5.1.1 Flat material properties ......................................................................57

3.5.1.2 Corner material properties .................................................................58

3.5.1.3 Revised corner material predictive model ..........................................60

3.5.2 Simple beam tests........................................................................................64

3.5.2.1 Evaluation of slenderness limits.........................................................64

3.5.2.2 Relationship between moment capacity and slenderness ....................67

3.5.3 Continuous beam tests.................................................................................68

3.5.3.1 Study on slenderness b/tε limit by continuous beam tests...................68

3.5.4 Discussion...................................................................................................69

3.6 Concluding remarks ...........................................................................................70

CHAPTER 4 MATERIAL MODELLING

4.1 Introduction........................................................................................................71

4.2 Review of key factors influencing material properties ........................................72

4.2.1 Stress-strain curve .......................................................................................72

4.2.2 Tensile coupon versus compressive section tests..........................................72

4.2.3 Strain rate....................................................................................................74

4.2.4 Forming route..............................................................................................75

4.2.5 Material thickness .......................................................................................78

4.2.6 Variation of material properties around a cross-section................................79

4.2.7 Residual stress.............................................................................................80

4.2.8 Steel grades .................................................................................................81

4.2.9 Discussion...................................................................................................81

4.3 Appraisal of existing material models.................................................................81

4.3.1 Rigid-plastic model .....................................................................................81

4.3.2 Elastic, perfectly-plastic model....................................................................82

4.3.3 Elastic, linearly-hardening model ................................................................82

4.3.4 Tri-linear model ..........................................................................................84

4.3.5 Piecewise nonlinear models.........................................................................87

4.3.5.1 Models with simple power functions..................................................87

4.3.5.2 Ramberg-Osgood models...................................................................89

4.3.5.3 Models with exponential functions ....................................................90

Contents

7

4.3.6 Discussion...................................................................................................91

4.4 Collection of existing experimental data .............................................................92

4.5 Analysis of existing experimental data................................................................95

4.5.1 Variation in stress-strain characteristics .......................................................95

4.5.1.1 Hot-rolled I-sections ..........................................................................95

4.5.1.2 Hot-rolled hollow sections .................................................................96

4.5.1.3 Comparison between hot-rolled I and hollow sections........................97

4.5.2 Variation in strain-hardening properties of coupon tests...............................98

4.5.2.1 Comparison of hot-rolled I-sections and hollow sections....................99

4.5.2.2 Comparison of hot-rolled and cold-formed hollow sections ...............99

4.5.2.3 Other sections (cruciform and plate sections) ...................................100

4.5.3 Variation in the strain-hardening properties from stub column tests ...........100

4.5.3.1 Comparison of hot-rolled and cold-formed hollow sections .............101

4.5.3.2 Cruciform sections...........................................................................101

4.5.4 Distinction between tensile coupon and stub column tests .........................102

4.5.4.1 Hot-rolled hollow sections ...............................................................102

4.5.4.2 Cold-formed hollow sections ...........................................................103

4.5.4.3 Cruciform sections...........................................................................104

4.5.4.4 Summary .........................................................................................105

4.6 Proposals of material models ............................................................................105

4.6.1 Hot-rolled I-sections..................................................................................106

4.6.2 Hot-rolled hollow sections.........................................................................107

4.6.3 Cold-formed hollow sections.....................................................................108

4.6.4 Other sections............................................................................................109

4.6.5 Summary...................................................................................................110

4.7 Concluding remarks .........................................................................................112

CHAPTER 5 DETERMINATE STRUCTURES

5.1 Introduction......................................................................................................113

5.2 Collection of existing test data..........................................................................114

5.2.1 Stub column tests ......................................................................................114

5.2.1.1 Hot-rolled sections...........................................................................114

5.2.1.2 Cold-formed sections.......................................................................116

Contents

8

5.2.1.3 Welded sections...............................................................................118

5.2.1.4 Press-formed and seam-welded sections ..........................................119

5.2.2 Simple beam tests......................................................................................121

5.2.2.1 Hot-rolled I-sections ........................................................................121

5.2.2.2 Hot-rolled SHS and RHS .................................................................123

5.2.2.3 Cold-formed SHS and RHS .............................................................123

5.2.3 Discussion.................................................................................................126

5.3 Design approach...............................................................................................126

5.3.1 Cross-section classification........................................................................126

5.3.1.1 Methodology ...................................................................................126

5.3.1.2 Shortcomings of cross-section classification ....................................129

5.3.2 Other existing design methods...................................................................133

5.3.2.1 Stress-based methods.......................................................................133

5.3.2.2 Strain-based methods.......................................................................138

5.3.2.3 Other models ...................................................................................139

5.3.3 The continuous strength method (CSM) ....................................................141

5.3.3.1 Background .....................................................................................141

5.3.3.2 General methodology and application range.....................................141

5.3.3.3 Cross-section compression resistance...............................................142

5.3.3.4 Cross-section bending resistance......................................................145

5.3.3.5 Application flow chart .....................................................................148

5.4 Assessment of the CSM....................................................................................149

5.4.1 Compression .............................................................................................149

5.4.1.1 Hot-rolled sections...........................................................................149

5.4.1.2 Cold-formed sections.......................................................................150

5.4.1.3 Welded sections...............................................................................151

5.4.1.4 Pressed-formed and seam-welded sections.......................................152

5.4.1.5 Summary .........................................................................................153

5.4.2 Bending.....................................................................................................154

5.4.2.1 Hot-rolled I-sections ........................................................................154

5.4.2.2 Hot-rolled SHS and RHS .................................................................156

5.4.2.3 Cold-formed SHS and RHS .............................................................156

5.4.2.4 Summary .........................................................................................158

5.5 Reliability study ...............................................................................................160

Contents

9

5.6 Discussion and concluding remarks ..................................................................167

CHAPTER 6 INDETERMINATE STRUCTURES

6.1 Introduction......................................................................................................169

6.2 Collection of existing test data..........................................................................170

6.2.1 Continuous beam tests...............................................................................170

6.2.2 Full-scale frame tests.................................................................................173

6.3 Numerical modelling ........................................................................................176

6.3.1 Modelling..................................................................................................176

6.3.1.1 Shell elements model.......................................................................176

6.3.1.2 Beam element models......................................................................178

6.3.2 Validation .................................................................................................180

6.3.2.1 Shell element models .......................................................................180

6.3.2.2 Beam element model .......................................................................183

6.4 Design approach...............................................................................................187

6.4.1 Traditional plastic analysis method............................................................187

6.4.1.1 Deleterious influence of second order effects...................................189

6.4.1.2 Beneficial influence of strain-hardening effects ...............................190

6.4.1.3 Sensitive balance between second order and strain-hardening ..........191

6.4.2 Development of the CSM for indeterminate structures...............................191

6.4.3 Parametric studies .....................................................................................193

6.4.3.1 Validation of kinematic assumption for determining deformation

demand .......................................................................................................193

6.4.3.2 Partial moment redistribution...........................................................195

6.4.3.3 Second order effects ........................................................................197

6.5 Assessment of the CSM....................................................................................205

6.5.1 Comparison of continuous beam test results with design models ...............205

6.5.2 Comparison of full-scale frame test results with design models .................208

6.6 Discussion and concluding remarks ..................................................................209

CHAPTER 7 CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK

7.1 Conclusions......................................................................................................210

Contents

10

7.2 Suggestions for further work ............................................................................213

7.2.1 Member buckling ......................................................................................213

7.2.2 Fire design.................................................................................................214

7.2.3 Other metallic materials.............................................................................214

7.2.4 Composite construction .............................................................................214

7.2.5 Experiments on frames ..............................................................................214

REFERENCES.....................................................................................................216

Notation

11

NOTATION

A Cross-sectional area

ag Geometric shape factor = Wpl/Wel

aCSM Load factor calculated by the CSM

af Ultimate load factor at failure

ap Load factor calculated by simple plastic analysis

B Outer cross-section width

b Internal flat element width

bcf Correction factor

Bc Parameter for corner material predictive model

Bcp Width of cover plate

c Offset strain at a proof stress

C1 Constant for power material model

C2 Constant for power material model

CF Cold-formed

CHS Circular hollow section

Ci Parameter determined from tests

COV Coefficient of variation

CSM Continuous strength method

D Outer cross-section depth

Notation

12

DSM Direct strength method

e Exponential constant where ln(e) = 1

E Young’s modulus

EC Eurocode

Eexp Measured tensile Young’s modulus

Esh Strain-hardening modulus

Fcr Elastic critical load

FEd Design load on structure

f0.2 0.2% proof stress

fcr Elastic buckling stress

FE Finite element

fnom Nominal stress

fLB Local buckling stress

fLB,web Local buckling stress at flange-to-web junction

fy Material yield strength

fy,c,AISI Corner material yield strength predicted by AISI model (AISI, 1996)

fy,c,exp Measured tensile yield stress of corner material

fy,exp Measured tensile yield stress

fy,mill Mill certificate yield stress

fp Proof stress

ftrue True stress

fu Material ultimate tensile strength

fu,mill Mill certificate ultimate tensile stress

fu,exp Measured tensile ultimate tensile stress

h Section height

hw Internal web height between flanges

HR Hot-rolled

I Second moment of area

k Dimensionless strain-hardening parameter

kc Curvature

kd,n Design fractile factor

ku Ultimate curvature value

kel Elastic limit curvature

K Constant for Ramberg-Osgood function

Notation

13

L Length

LC Load combinations

LRFD Load and resistance factor design

Ls1 Span between loading points

Ls2 Span between end support and loading point

LVDT Linear Variable Differential Transformer

m Parameter for corner material predictive model

Mc,Rd Cross-section bending resistance

MCSM Bending resistance predicted by the CSM

MEC3 Bending resistance predicted by EC3

MEd Design bending moment

Mel Elastic moment capacity

Mhingei Bending moment at plastic hinge i

MR Merchant-Rankine

Mpl Plastic moment capacity

Mu Ultimate test moment capacity

n Constant for power material model

nt Number of tests

N Applied load

Ncoll Plastic collapse load

Nc,Rd Cross-section compression resistance

NCSM Compression resistance/collapse load predicted by the CSM

NEC3 Compression resistance/collapse load predicted by EC3

NEd Design axial force

Nh1 Total theoretical load at which the first hinge forms

NHL Notional horizontal load

Nu Ultimate test load

Ny Yield load

Qi Parameter determined from tests

R Rotation capacity

R2 Coefficient of determination

RHS Rectangular hollow section

ri Internal corner radius

Notation

14

rt Theoretical resistance as a function of all the relevant independent

variables

SHS Square hollow section

t Thickness

tf Flange thickness

tref Reference thickness used for a method proposed by Lechner et al.

(2008)

tw Web thickness

Vδ Coefficient of variation of error

Vr Overall coefficient of variation

Wel Elastic section modulus

Wpl Plastic section modulus

y1 Distance to neutral axis

αcr Factor by which the design loading would have to be increased to

cause elastic instability of a structure in a global mode according to EN

1993-1-1 (2005)

αcr,CSM Factor by which the design loading would have to be increased to

cause elastic instability of a structure in a global mode according to the

CSM

α10/α5 Section generalised shape factor related to ultimate curvature values

αe Slope of elastic part of curve (either moment-end rotation or load-

deflection)

αp Slope of plastic part of curve (either moment-end rotation or load-

deflection)

δ End shortening; virtual displacement

δM Increase in bending moment above Mpl due to strain-hardening

δL Displacement at loading points

δM Displacement at midspan

δu End shortening at ultimate load

εe Elastic strain

εf Plastic strain at fracture

εf,exp Measured tensile plastic strain at fracture

εhingei Strain at hinge i

Notation

15

εLBi Local buckling strain at hinge i

εLB Local buckling strain

εmax Strain at ultimate stress

εnom Nominal strain pllnε Log plastic strain

εp Plastic strain

εsh Strain at the onset of strain-hardening

εtotal Total strain

εy Yield strain

γM Partial safety factor

pλ Plate slenderness

θ Rotation

θi Rotation at hinge i

θmax Maximum recorded test rotation where θrot was not attained

θpl Elastic rotation at the plastic moment

θrot Total rotation upon reaching the plastic moment on the unloading path

θu Total rotation at ultimate moment

List of figures

16

LIST OF FIGURES Fig. 1.1: Typical stress-strain relationship for hot-finished structural steel................25

Fig. 2.1: Simple material models ..............................................................................37

Fig. 3.1: Section labelling convention and locations of flat and corner tensile coupons

................................................................................................................................42

Fig. 3.2: Typical stress-strain curves from hot-rolled and cold-formed tensile coupons

(SHS 60×60×3) ........................................................................................................44

Fig. 3.3: Simple beam test set-up..............................................................................46

Fig. 3.4: Roller support and LVDT set-up in bending test.........................................46

Fig. 3.5: A typical deformed simple beam test specimen (SHS 40×40×3-CF) ...........47

Fig. 3.6: Definition of rotation capacity from moment-rotation graphs .....................48

Fig. 3.7: Normalised moment-rotation curves for simple SHS 60×40×4 beams.........48



Fig. 3.10: Continuous beam setup 1 – loading applied centrally between supports

(dimensions in mm) .................................................................................................52


(dimensions in mm) .................................................................................................53

Fig. 3.12: General view of continuous beam test set-up ............................................54

Fig. 3.13: Location of plastic hinges in deformed continuous beam (RHS 60×40×4-

HR1) ........................................................................................................................54

List of figures

17

Fig. 3.14: Normalised load-end rotation curves for RHS 60×40×4 continuous beams

................................................................................................................................55

Fig. 3.15: Normalised load-end rotation curves for SHS 40×40×4 continuous beams56

Fig. 3.16: Normalised load-end rotation curves for SHS 40×40×4 continuous beams56

Fig. 3.17: Corner material strength data and AISI predictive model..........................60

Fig. 3.18: Corner material strength data and revised predictive model ......................63

Fig. 3.19: Mu/Mel versus b/tε for assessment of Class 3 slenderness limit .................65

Fig. 3.20: Mu/Mpl versus b/tε for assessment of Class 2 slenderness limit .................66

Fig. 3.21: Rotation capacity R versus b/tε for assessment of Class 1 slenderness limit

................................................................................................................................67

Fig. 3.22: Mu/Mel versus pλ for available tests on simple SHS and RHS beams........68

Fig. 3.23: Nu/Ncoll versus b/tε for assessment of Class 1 slenderness limit.................69

Fig. 4.1: Stub column and coupon tests - RHS 60×40×4-HR ....................................73

Fig. 4.2: Stub column and coupon tests - RHS 60×40×4-CF.....................................74

Fig. 4.3: Tensile f-ε curves for hot-rolled and cold-formed material .........................76

Fig. 4.4: Stress-strain diagram illustrating the effects of cold-working......................77

Fig. 4.5: Schematic non-dimensional stress-strain curves from various locations

around on I-section...................................................................................................79

Fig. 4.6: Rigid-plastic model ....................................................................................82

Fig. 4.7: Elastic-perfectly plastic model....................................................................82

Fig. 4.8: Elastic, linear-hardening model ..................................................................83

Fig. 4.9: Elastic, linear-hardening model based on equal energy dissipation .............83

Fig. 4.10: Tri-linear model .......................................................................................84

Fig. 4.11: Methods employed by various researchers for obtaining Esh in tri-linear

material models........................................................................................................85

Fig. 4.12: Summary of tri-linear material models adopted by various researchers .....87

Fig. 4.13: Simplest model with power function ........................................................88

Fig. 4.14: Elastic-linear power hardening model.......................................................88

Fig. 4.15: Gehring and Saal’s model of the strain-hardening behaviour of structural

steel (Ramanto, 2009) ..............................................................................................90

Fig. 4.16: Stress-strain characteristics of hot-rolled I-sections ..................................96

Fig. 4.17: Stress-strain characteristics of hot-rolled hollow sections .........................97

List of figures

18

Fig. 4.18: Mean normalised stress-strain curves for hot-rolled I-sections and hollow

sections ....................................................................................................................98

Fig. 4.19: Strain-hardening properties of hot-rolled I-sections, hot-rolled hollow

sections and cold-formed hollow sections.................................................................99

Fig. 4.20: Strain-hardening property on cruciform and plate sections......................100

Fig. 4.21: Stub column tests – hot-rolled versus cold-formed hollow sections.........101

Fig. 4.22: Stub column tests – Cruciform sections ..................................................102

Fig. 4.23: Coupon versus stub column tests – hot-formed hollow sections..............103

Fig. 4.24: Coupon versus stub column tests – cold-formed hollow sections ............104

Fig. 4.25: Tensile coupon versus stub column tests – cruciform sections ................105

Fig. 4.26: Proposed material model for hot-rolled I-sections...................................107

Fig. 4.27: Proposed material model for hot-rolled hollow sections..........................108

Fig. 4.28: Proposed material model for cold-formed hollow sections......................109

Fig. 4.29 Strain-hardening properties of cruciform sections ....................................110

Fig. 4.30 Strain-hardening properties of plated sections..........................................110

Fig. 4.31 Summary of the proposed material models ..............................................112

Fig. 5.1: Section notation for hot-rolled steel SHS and RHS...................................115

Fig. 5.2: Section notation for cold-formed steel SHS and RHS ...............................117

Fig. 5.3: Section labelling convention and location of welds (between four plates) .118

Fig. 5.4: Section labelling convention and location of welds (between two channels)

..............................................................................................................................120

Fig. 5.5: Section designation of test specimens .......................................................121

Fig. 5.6: Test arrangement for four-point bending tests...........................................121

Fig. 5.7: Section designation of test specimens with cover plates............................122

Fig. 5.8: Moment-rotation response of four behavioural classes of cross-section ....127

Fig. 5.9: Idealised bending stress distributions (symmetric section) ........................127

Fig. 5.10: Cross-section compression and bending resistances according to EN 1993-

1-1 .........................................................................................................................128

Fig. 5.11: Cross-section resistances for aluminium sections according to Annex F of

EN 1999-1-1 (2007) ...............................................................................................129

Fig. 5.12: Stub column test results..........................................................................130

Fig. 5.13: Normalised load-end shortening graphs from stocky stub column tests...131

Fig. 5.14: Simple beam test results .........................................................................132

Fig. 5.15: Normalised moment–end rotation graphs from stocky simple beam tests133

List of figures

19

Fig. 5.16: Stress distributions in a rectangular aluminium section in bending..........134

Fig. 5.17: Contributions of element groups to the total moment capacity ................136

Fig. 5.18: Design proposal for Class 3 sections.......................................................137

Fig. 5.19: Comparison between Kemp et al. (2002) model and existing bending test

data ........................................................................................................................139

Fig. 5.20: ‘Base curve’ – relationship between cross-section deformation capacity and

slenderness.............................................................................................................143

Fig. 5.21: Bending response of I-section with elastic, linear strain-hardening material

model.....................................................................................................................145

Fig. 5.22: Design model for elastic-plastic stage ( 3ε/ε1 yLB ≤< ).........................146

Fig. 5.23: Design model for strain-hardening stage ( 15ε/ε3 yLB ≤< ) ....................147

Fig. 5.24: CSM bending moment resistance model.................................................148

Fig. 5.25: Stub column test data and comparison with design models .....................154

Fig. 5.26: Simple beam test data and comparison with design models ....................159

Fig. 5.27: Comparison between experimental and theoretical results from Eurocode 3

for compression resistance .....................................................................................165

Fig. 5.28: Comparison between experimental and theoretical results from CSM (Eq.

(5.12)) for compression resistance ..........................................................................165


for bending resistance.............................................................................................166


(5.13)) for bending resistance .................................................................................166

Fig. 6.1: Schematic continuous beam test arrangements (dimensions in mm) .........173

Fig. 6.2: Geometry and loading arrangements for frame tests (dimensions in mm) .176

Fig. 6.3: Method of obtaining moment–curvature relationship................................179

Fig. 6.4: Comparison between the experimental and numerical results for the simple

beam RHS 60×40×4-CF.........................................................................................181

Fig. 6.5: Comparison between experimental and numerical results for the continuous

beam RHS 60×40×4-CF1.......................................................................................181

Fig. 6.6: Comparison between the experimental and numerical failure modes for the

simple beam SHS 40×40×3-CF ..............................................................................182

Fig. 6.7: Continuous beam SHS 40×40×3 CF1 .......................................................184

Fig. 6.8: Continuous beam SHS 40×40×3-CF2.......................................................185

List of figures

20

Fig. 6.9: Continuous beam SHS 40×40×3-HR1 ......................................................185

Fig. 6.10: Continuous beam SHS 40×40×3-HR2 ....................................................186

Fig. 6.11: Numerical modelling of full-scale portal frame test (Charlton, 1960)......187

Fig. 6.12: Plastic collapse mechanism for two-span continuous beam.....................188

Fig. 6.13: Collapse bending moment diagram from traditional plastic analysis .......189

Fig. 6.14: Illustration of second-order effects .........................................................190

Fig. 6.15: Collapse bending moment diagram (CSM) .............................................193

Fig. 6.16: Deformation demands for continuous beam configuration 1 ...................194

Fig. 6.17: Deformation demands for continuous beam configuration 2 ...................194

Fig. 6.18: Numerical study of partial moment redistribution...................................197

Fig. 6.19: Frame to be analysed in numerical study ................................................198

Fig. 6.20: Comparisons of results against simple plastic analysis for LC1...............202

Fig. 6.21: Comparisons of results against simple plastic analysis for LC2...............202

Fig. 6.22: Comparisons of results against CSM for LC1 .........................................204

Fig. 6.23: Comparisons of results against CSM for LC2 .........................................205

Fig. 6.24: Continuous beam test and FE data compared with Eurocode 3 design model

..............................................................................................................................207

Fig. 6.25: Continuous beam test and FE data compared with the CSM design model

..............................................................................................................................207

Fig. 6.26: Frame test data compared with the CSM design model...........................209

List of tables

21

LIST OF TABLES

Table 2.1: Stub column tests.....................................................................................33

Table 2.2: Simple beam tests....................................................................................33

Table 2.3: Continuous beam tests .............................................................................34

Table 2.4: Frame tests ..............................................................................................35

Table 3.1: Chemical composition (% by mass) of two steel grades employed ...........41

Table 3.2: Mill certificate (virgin) and measured tensile material properties of test

specimens ................................................................................................................43

Table 3.3: Measured dimensions of simply supported beam specimens (three-point

bending)...................................................................................................................45

Table 3.4: Ultimate flexural capacities of simple beam tests (three-point bending) ...50

Table 3.5: Ultimate rotation capacities of simple beam tests (three-point bending) ...50

Table 3.6: Measured dimensions of continuous beams (five-point bending) .............51

Table 3.7: Summary of results from continuous beam tests ......................................57

Table 3.8: Summary of corner material properties ....................................................59

Table 3.9: Comparison of corner material properties with predictive models ............62

Table 4.1: Summary of tri-linear material models.....................................................86

Table 4.2: Parameters for the equation proposed by Gehring and Saal (2008)...........91

Table 4.3: Stub columns – hot-rolled hollow sections ...............................................93

Table 4.4: Stub columns – cold-formed hollow sections ...........................................93

List of tables

22

Table 4.5: Tensile coupons – hot-rolled I-sections ....................................................93

Table 4.6: Tensile coupons - hot-rolled hollow sections ...........................................94

Table 4.7: Tensile coupons - cold-formed hollow sections........................................94

Table 4.8: Stub columns - other sections ..................................................................94

Table 4.9: Tensile coupons - other sections ..............................................................94

Table 4.10: Summary of proposed material models ................................................111

Table 5.1: Geometric properties and ultimate capacities of the hot-rolled stub

columns .................................................................................................................115

Table 5.2: Geometric properties and ultimate capacities of cold-formed stub columns

..............................................................................................................................117

Table 5.3: Geometric properties and ultimate capacities of welded box stub columns

..............................................................................................................................119

Table 5.4: Geometric properties and key results of press-formed and seam-welded

stub column tests....................................................................................................120

Table 5.5: Geometric properties and key results for I-beam test specimens.............122

Table 5.6: Summary of simple beam tests on hot-rolled SHS and RHS...................123

Table 5.7: Summary of simple beam tests on cold-formed SHS and RHS...............124

Table 5.8: Comparison of hot-rolled stub column test results with design models...150

Table 5.9: Comparison of cold-formed stub column test results with design models

..............................................................................................................................151

Table 5.10: Comparison of welded stub column test results with design models .....152

Table 5.11: Comparison of press-formed and seam welded stub column test results

with design models.................................................................................................153

Table 5.12: Comparison of the CSM and Eurocode methods with stub column test

results ....................................................................................................................154

Table 5.13: Comparison of hot-rolled I-section simple beam test results with design

models ...................................................................................................................155

Table 5.14: Comparison of hot-rolled SHS and RHS simple beam test results with

design models ........................................................................................................156

Table 5.15: Comparison of SHS and RHS simple beam test results with design

models ...................................................................................................................156

Table 5.16: Comparison of the CSM and Eurocode methods with bending test results

..............................................................................................................................159

List of tables

23

Table 5.17: Summary of statistical evaluation results for predictive model equations

..............................................................................................................................167

Table 6.1: Geometric and material properties and ultimate capacities of I-section

continuous beams...................................................................................................171

Table 6.2: Cross-section dimensions of beam specimens from frame tests..............173

Table 6.3: Ultimate capacities of frame tests ..........................................................173

Table 6.4: Validation of FE models against simple beam test results considering 4

imperfection amplitudes .........................................................................................182

Table 6.5: Validation of FE models against continuous beam test results considering 4

imperfection amplitudes .........................................................................................183

Table 6.6: Frames analysed under LC1...................................................................199

Table 6.7: Frames analysed under LC2...................................................................199

Table 6.8: Ultimate results of frames under LC1 ....................................................201

Table 6.9: Ultimate results of frames under LC2 ....................................................201

Table 6.10: Comparison of continuous beam test results with design methods........206

Table 6.11: Comparison of frame test result with design methods...........................208

Chapter 1 Introduction

24

CHAPTER 1

INTRODUCTION 1.1 BACKGROUND

Structural steel is widely employed by engineers in buildings, bridges and other civil

infrastructure applications owing to its many advantages, including excellent strength-

to-weight ratio, efficient cross-section shapes and rapid construction. However, its

extensive usage however has a negative impact on the environment. The current

global transition towards a reduction in the environmental impact of construction and

sustainable development requires a structural design method capable of efficiently

utilising materials by exploiting the maximum capacity of structural elements. The

aim of this thesis is therefore to propose a more efficient design method for structural

steel which maximises the full potential of the material, by taking account of strain-

hardening, a property that is essentially unutilised in current design practice.


25

1.2 LIMITATIONS OF THE CURRENT CODES OF PRACTICE

1.2.1 Strain-hardening

Strain-hardening refers to the increase in strength of metallic materials beyond yield

as a result of plastic deformation as shown in Fig. 1.1. This increase in strength is not

systematically utilised in current international steel design codes, though allowance is

made for the spread of plasticity through cross-sections (i.e. use of the plastic moment

capacity) and redistribution of moments within an indeterminate structural frame until

a collapse mechanism forms (i.e. plastic design). Both of these design techniques are

synonymous with simplified elastic-plastic and rigid-plastic material modelling. This

simplification limits the resistance of cross-sections in compression to the yield load

Ny (defined as the cross-sectional area A multiplied by the material yield strength fy)

and the resistance of cross-sections in bending to the plastic moment capacity Mpl

(defined as the plastic section modulus Wpl multiplied by the material yield strength

fy). A material model that accounts for strain-hardening is required in the proposed

design method.

Fig. 1.1: Typical stress-strain relationship for hot-finished structural steel

1.2.2 Cross-section classification

The ability of a cross-section to sustain increased loading, and indeed to develop

strain-hardening, is limited by the effects of local buckling. Susceptibility to local

Stress

fu

fy

Upper yield stress

εy εsh

Elastic Range

Strain-hardening Range Plastic Range

Esh

E

Strain


26

buckling is currently assessed by means of cross-section classification in most

international steel design codes, including AS 4100 (1998), EN 1993-1-1 (2005) and

AISC 360 (2005). In such a classification method, structural cross-sections are

assigned to discrete behavioural classes depending on the slenderness of the

constituent elements. However, the resistance of structural cross-sections is in reality,

a continuous function of the slenderness of the constituent plate elements and stresses

beyond yield can be sustained. Resistance based on the assignment of cross-sections

into this discrete classification system is useful but artificial, and sometimes a

conservative simplification.

1.3 SCOPE OF THE STUDY AND RESEARCH INNOVATION

The aim of this research is to develop a new structural steel design method to exploit

strain-hardening in steel as a construction material, while maintaining design

simplicity and safety. The proposed approach is referred to as the continuous strength

method (CSM) (Gardner, 2008; Gardner and Wang, 2010) – the development and

benefits of the method over current design practice are described in the thesis.

Most of the structural steel members used in construction, such as universal columns

and beams (UCs and UBs) specified in BS 4-1 (2005), are stocky sections made of

carbon steel. In order to cover this practical application range, the focus of the present

study is on stocky (i.e. non-slender) carbon steel sections in both statically

determinate and indeterminate structures.

1.4 OUTLINE OF THESIS

This chapter briefly introduces the drawbacks of the current steel design method in

determining the load-carrying capacity of structural steel members. A general

overview of the thesis follows.

Chapter 2 reviews the general literature that is relevant to the present research project.

The review contains a brief introduction of important topics including laboratory

testing, material modelling, existing design guidance and numerical modelling, with

more specific and detailed discussion on these topics being given in Chapters 3-6.


27

After selecting the most important parameters required in the development of the

design method through the literature review, an extensive laboratory testing

programme was undertaken. This comprised tensile coupon tests on flat and corner

material and a series of simple beam tests and continuous beam tests (with two

loading configurations), performed on the hot-rolled and cold-formed steel square and

rectangular hollow sections (SHS and RHS, respectively). These are presented in

detail in Chapter 3.

Chapter 4 describes a study of material modelling, conducted to develop a simple

material model for structural steel that allows for strain-hardening. A significant pool

of material data was collected from the literature and analysed. Based on the findings,

a bi-linear material model has been proposed with a set of tables defining the ratio of

strain-hardening modulus to Young’s modulus (Esh/E) as a function of the ratio of

ultimate stress to yield stress fu/fy for various cross-sectional shapes.

The development of a more efficient structural steel design method for determinate

structures is described in Chapter 5. The proposed approach employs a continuous

relationship between cross-section slenderness and cross-section deformation capacity,

rather than the step-wise approach of cross-section classification, and utilises the

material model proposed in Chapter 4. A comparison of the proposed design method

against test results and the European design rules given in EN 1993-1-1 (2005) is also

made and followed by the reliability study on the method.

Chapter 6 extends the steel design method proposed in Chapter 5 to indeterminate

structures. A numerical modelling programme involving both shell and beam finite

elements was carried out on the basis of data generated in the laboratory testing

programme reported in Chapter 3 and the collection of test data from the literature.

Following the successful replication of experimental results, parametric studies were

conducted to investigate various structural issues including second-order effects.

Assessment of the method was carried out by comparing the experimental test data

with the predictions provided by the proposed method and EN 1993-1-1 (2005).


28

Chapter 7 summarises the key findings from the research project and the respective

conclusions and discusses the scope for further work in this area.

Chapter 2 Literature review

29

CHAPTER 2

LITERATURE REVIEW 2.1 INTRODUCTION

This chapter presents a brief review of previous literature and research that are

pertinent to the present study. From the compilation and re-evaluation of existing

laboratory tests, encompassing the re-appraisal of available material and numerical

modelling methods, to the reviewing of current steel design guidance, a thorough

investigation has been carried out.

2.2 EXISTING DESIGN GUIDANCE

The first UK structural steel design code, BS 449 (BS 449, 1932), which was based on

the allowable stress concept, was introduced in 1932. The first limit states design code

for steel structures was published in 1985 when the first edition of BS 5950 (BS 5950,

1985) was released. This evolved to the most recent edition that was published in

2000 (BS 5950, 2000).


30

The current Eurocode EN 1993-1-1 (2005) for the design of steel structures is also

based on limit states principles and is reviewed in the first instance as being

representative of modern design practice. A review of other design methods follows.

2.2.1 Determinate structures

EN 1993-1-1 (2005) defines four classes of cross-section in order to identify the

extent to which structural resistance and rotation capacity is affected by the influence

of local buckling. Cross-section classification is discussed further in Section 5.3.1 of

this thesis. Although the determination of structural resistance in EN 1993-1-1 (2005)

does not explicitly include strain-hardening, it is a necessary component of the cross-

section classification system, and is required, for instance, to enable the development

of the plastic moment resistance Mpl at finite strains.

Similar to EN 1993-1-1 (2005), EN 1999-1-1 (2007) for aluminium design also

employs cross-section classification to categorise the influence of local buckling on

load-carrying resistance. It also however allows design resistances of cross-sections

beyond the yield load, in the case of compression and beyond the plastic moment

capacity in the case of bending.

Mazzolani (1995) coined the term ‘generalized shape factor’ which is multiplied by

the elastic moment capacity in order to obtain the moment resistance of an aluminium

cross-section, allowing the moment capacity to reach beyond the plastic moment. In

this method, the ‘generalized shape factor’ is determined in relation to the ductility of

the material. Similarly, Kim and Peköz (2008) proposed a method that rearranges the

nonlinear plastic stress distribution of aluminium through the depth of a cross-section

via the introduction of a ‘yield/ultimate shape factor’. In this approach, this shape

factor is multiplied by the elastic moment capacity to obtain the moment resistance.

This factor differs from the one proposed by Mazzolani (1995) since it is dependent

on the material strength rather than the ductility.

Kemp (2002) proposed a design approach considering strain-hardening and allowing

moment capacity to be determined up to 8% beyond plastic moment resistance on the


31

basis of a bi-linear moment-curvature relationship. Critical curvature can be obtained

as a function of local and lateral buckling parameters and steel properties.

Lechner et al. (2008) investigated the resistance of class 3 cross-sections in bending

and recommended a linear transition between the plastic and elastic moment

capacities for class 3 sections.

In addition to the methods reviewed above, which are primarily developed for

relatively stocky sections, there are other design methods focusing on slender sections

such as the ‘Winter’ effective width approach (Kalyanaraman et al., 1977) and the

direct strength method proposed by Schafer (2008). Both allow for the occurrence of

local buckling prior to yielding. The methods are discussed further in Chapter 5.

2.2.2 Indeterminate structures

Traditional plastic analysis (employed in EN 1993-1-1), which is based on the

formation and subsequent rotation of plastic hinges at their full plastic moment

capacity, is generally used to design indeterminate steel structures constructed of

Class 1 sections. A progressive reduction in stiffness of the structure results from the

formation of each plastic hinge; collapse occurs when sufficient hinges form to create

a mechanism. Each hinge is assumed to operate at the plastic moment capacity.

Enhanced capacity beyond the plastic collapse load can be attained in steel frames due

to strain-hardening, as demonstrated in the frame tests carried out by Baker and

Eickhoff (1955). However, Wood (1958) warned that the benefits arising from

plasticity in structures may be curtailed in multi-storey frames because of

simultaneous deterioration of elastic stability such as side-sway frame instability.

The importance of strain-hardening in indeterminate structures has been described by

Davies (1966, 2002 and 2006). Davies (1966) proposed an approach to calculate the

increase in bending moment above the plastic moment Mpl at a hinge due to strain-

hardening, and its link with plastic hinge rotation. He showed that, provided that local

and lateral-torsional buckling are eliminated, the additional capacity in steel frames

arising from strain-hardening could be accurately predicted. Davies (2002) reviewed

recent developments in the elastic-plastic analysis and design of steel frames. The


32

sensitive balance between strain-hardening and second-order effects was highlighted;

this has also been investigated in Section 6.4.3.3 of this thesis. Davies (2006)

investigated the deleterious influence of local buckling, lateral-torsional buckling and

their interaction on the contributions of strain-hardening to structures’ load-carrying

capacity. He concluded that the current knowledge was insufficient to incorporate

strain-hardening reliably into design calculations due to the limited information

regarding its influence on the structural behaviour.

The aforementioned methods are described in more detail in Chapter 6.

2.3 LABORATORY TESTING

As mentioned in the previous section, neither EN 1993-1-1 (2005) nor any other steel

design codes offer systematic exploitation of strain-hardening for stocky sections

(Classes 1-3). Given the possible extra load-carrying capacity available due to strain-

hardening of steel structures, a new design method considering strain-hardening is

required. Central to the development of this design method is high quality

experimental data. Prior to developing the method, test data available in previous

investigations that are relevant to the present study on stub columns, simple beams,

continuous beams and frames have been gathered.

Most of the available tests on indeterminate structures were carried out in the 1950s

and 1960s, when plastic analysis and tall building stability issues were investigated

extensively. However, the current research interests on steel structures are mostly

focused on stainless steel, aluminium and slender cold-formed sections. This is

evident when assessing the availability of test data on indeterminate steel structures.

A summary of the available experimental data relevant to this thesis is given in the

following sub-sections.

2.3.1 Stub column tests

A total of 63 stub column test results have been collected from existing published

resources. All these data have been regrouped into 4 categories: hot-rolled, cold-

formed, welded and press-formed and seam-welded sections and are summarised in


33

Table 2.1. Reference resources, section type and number of tests conducted are also

tabulated. These data are employed in Chapter 5 to develop and validate the new

approach.

Table 2.1: Stub column tests

Section category Reference resource Section type No. of tests

Hot-rolled sections Gardner et al. (2010) SHS RHS 10

Gardner et al. (2010) SHS RHS 10

Akiyama et al. (1992) SHS 5

Zhao and Hancock (1991) RHS SHS 7

Cold-formed sections

Wilkinson and Hancock (1997) RHS 1 Rasmussen and Hancock (1992) SHS 4

Welded sections Akiyama et al. (1992) SHS 10 Akiyama et al. (1992) SHS 15 Press-formed and seam-

welded sections Gao et al. (2009) RHS 1

2.3.2 Simple beam tests

In addition to stub column tests, flexural member tests have also been collected. A

total of 90 bending tests were gathered and may be divided into three categories: hot-

rolled I-sections, hot-rolled SHS/RHS and cold-formed SHS/RHS and are

summarised in Table 2.2. Resource reference, loading method and number of tests are

also shown here. The results from these tests are discussed in Chapter 6 and were used

to validate the development of the proposed design method for beams.

Table 2.2: Simple beam tests

Section category Reference resource Loading method

No. of tests

Byfield and Nethercot (1998) Four-point 32 Hot-rolled I sections

Popov and Willis (1957) Three-point 2 Hot-rolled SHS/RHS Gardner et al. (2010) Three-point 3

Gardner et al. (2010) Three-point 3 Wilkinson and Hancock (1998) Four-point 41 Cold-formed SHS/RHS Zhao and Hancock (1991) Four-point 9


34

2.3.3 Continuous beam tests

In addition to gathering experimental results on determinate structures, test data on

indeterminate structures were also collected for development and validation of the

proposed design method. Recent experiments performed on indeterminate steel

structures consisting of stocky sections are relatively scarce. From the ASCE ‘Plastic

design in steel – a guide and commentrary’ (ASCE, 1972), a total of 7 continuous

beam test results have been gathered from published papers, and a further 12 were

carried out by the author and are reported in detail in Chapter 3. The section types and

number of tests are shown in Table 2.3. These test data were used in Chapter 6 to

develop the proposed method for indeterminate structures and assess its suitability in

representing the structural behaviour of continuous beams.

Table 2.3: Continuous beam tests

Reference resource Section type No. of tests

Gardner et al. (2010) RHS SHS 12

Popov and Willis (1957) I-section I-section with cover plate 5

Yang et al. (1952) I-section 1 Driscoll et al. (1957) I-section 1

2.3.4 Frame tests

As discussed above, because of the limited availability of full-scale frames

constructed of stocky sections, only 5 full-scale and 34 model frame tests carried out

between 1950 and 1960 are gathered and summarised in Table 2.4. Reference

resource, section type and number of tests are also tabulated. These valuable data

were used in Chapter 6 to validate the suitability of the application of the proposed

method to frames, and address second-order effects.


35

Table 2.4: Frame tests

Reference resource Section type No. of tests Charlton (1960) I-section 1 Baker and Eickhoff (1955) I-section 2 Driscoll et al. (1957) I-section 1 Ruzek et al. (1954) I-section 1

Low (1959) Square solid section Rectangular solid section 34

2.4 MATERIAL MODELLING

The present generation of structural steel design codes treats material nonlinearity

through simplified elastic, perfectly-plastic or rigid-plastic material models. However,

the actual stress-strain response of structural carbon steel is more complex than these

simplified models, in particular with respect to strain-hardening. EN 1993-1-5 (2006)

suggests a bi-linear model with a strain-hardening modulus of E/100 for use in finite

element models. Accurate material modelling is a key aspect of the proposed design

method. Relevant literature on this topic has therefore been reviewed.

2.4.1 Factors influencing material properties

A number of key factors including section forming route, strain rate, material

thickness and influence of residual stresses and steel grade on stress-strain response

have been studied. The results of the review suggested that the ratio of ultimate stress

to yield stress fu/fy was a fundamental variable in determining the level of strain-

hardening and should therefore be included in any proposed model. An overview of

the key literature is given below, while more detailed consideration is presented in

Chapter 4.

The distinction of material performance between tensile coupon and compressive

section tests have been studied by McDermott (1969), Doane (1969) and ASCE

(1971). The material properties in compression and tension were found to be

practically identical in the initial elastic region, while in the post-yielding stage,

strain-hardening generally occurs earlier in compression than in tension.


36

The mechanical properties of steel are sensitive to the strain rate at which they were

obtained. As a general rule, as the strain rate increases, the material strength increases

but the ductility is reduced (Bruneau et al., 1998; Brockenbrough and Merritt, 1999;

Kemp et al., 2002; Trahair et al., 2008). This study does however focus on the static

design of steel structures where strain rate effects are generally insignificant.

Different methods of manufacture of steel cross-sections also have an effect on the

finished material, the two principal forming methods being hot-rolling and cold-

forming, each producing considerably distinct material properties. Hot-rolled sections

have good ductility, consistent hardness, a well-defined yielding point with a

relatively long plastic plateau, homogeneous material properties and low residual

stresses (Lay and Ward, 1969; Madugula, 1997; Chan and Gardner, 2008a). Cold-

formed sections have lower ductility, rounded yielding and enhanced material

strength (Schafer and Peköz, 1998; Chou et al., 2000; Dubina and Ungureanu, 2002;

Guo et al., 2007). Direct comparisons between hot-rolled and cold-formed structural

steel sections are made in Chapter 4 to study the effect of processing routes.

The effect of variation in cross-section thickness on strain-hardening properties has

also been investigated extensively and there exist different opinions on its influence:

Alpsten (1972) concluded that the larger the plate thickness, the higher strain-

hardening capacity; Byfield and Dhanalakshmi (2002) however concluded that

variable material thickness has no direct influence on rate of strain-hardening. On

reviewing other researchers’ studies on strain-hardening properties (Hasan and

Hancock, 1989; Kemp et al., 2002), the author supports the opinion that the variation

in cross-section thickness does have a significant influence on the strain-hardening

properties. This topic will be discussed in more detail in Chapter 4.

2.4.2 Existing material models

A wide range of material models for structural steel have been adopted in design

codes and employed by researchers. An overview of these models is presented below,

while a more detailed assessment of their relative merits is given in Chapter 4.


37

The rigid-plastic and elastic, perfectly-plastic material models in EN 1993-1-1 (2005)

and other steel design codes are the simplest representation of the f-ε response of

structural steel. These models shown in Fig. 2.1, do not allow for strain-hardening.

The simplest allowance for strain-hardening comes from an elastic, linear hardening

material model. Such a model is recommended in EN 1993-1-5 for use in finite

element analysis with a strain-hardening modulus of E/100, where E is Young’s

modulus. Other values ranging from E/200 to E/20 were suggested by Bruneau et al.

(1998). This model offers a good representation of material behaviour with little

increase in complexity when compared to the simplest models. Elastic, piecewise-

linear models have been also investigated by other researchers (Roderick, 1954;

Haaijer, 1957; Lay and Smith, 1965; Alpsten, 1972; Rogers, 1976; Kato, 1990;

Byfield and Dhanalakshmi, 2002). These models generally provide more accurate

representation of hot-rolled material behaviour since features like the plastic plateau

can be included. Nonlinear material models, including Ramberg-Osgood functions

(Ramberg and Osgood, 1943) can accurately capture the shape of rounded stress-

strain curves, such as those of aluminium alloys and stainless steel, but at the expense

of increased complexity. These moduli are reviewed in detail in Chapter 4.

Fig. 2.1: Simple material models

2.5 NUMERICAL MODELLING

Owing to the increasing computational power and the development of sophisticated

finite element (FE) software, FE analysis is now widely employed to generate

supplementary information to that available experimentally, by both academic

researchers and practising engineers, in investigating the structural behaviour of steel

components. Furthermore, parametric studies based on validated FE models can be

f

E = 0

ε

fy

f

E E = 0

ε

fy

εy (a) Rigid-plastic (b) Elastic, perfectly-plastic

E ∞


38

utilised to provide a basis for increasing the efficiency of proposed design provisions.

A combination of laboratory tests and numerical modelling is nowadays common in

research on steel structures. To this end, selected experiments are first conducted and

FE models are then generated to replicate the experimental results. Once the FE

models are validated, parametric studies are carried out to generate further results and

investigate key issues. These studies can be conducted by changing key parameters of

the model such as geometrical and mechanical properties. In so doing, they provide a

relatively quick and low-cost approach in comparison to laboratory testing. Such a

combination of laboratory testing and finite element modelling is employed in this

thesis.

2.5.1 Element type

Shell elements are generally employed to model thin-walled structural steel

components, where local buckling and softening are expected. The element library in

ABAQUS (2007) includes various shell elements. The element adopted in the present

study is S4R, a 4-noded doubly curved shell element with finite membrane strains and

reduced integration; this element has been successfully employed in similar previous

studies (Chan and Gardner, 2008a; Chan and Gardner, 2008b).

Beam elements are not customarily used in structural engineering research to simulate

and analyse structures where local buckling is expected. This is because local

buckling effects cannot generally be represented by beam elements because the cross-

section geometry of the beam element cannot change (Mirambell and Real, 2000).

However, ABAQUS (2007) does provide the possibility of mimicking the effects of

local buckling in a beam elements based model, using the ‘M1’ command to the input

moment-curvature response, which can include both strain-hardening and softening.

Therefore, the 2-noded linear beam element B21 has been adopted to perform

numerical simulations in conjunction with shell elements based models.

2.5.2 Geometric imperfections

Initial geometric imperfections of structural sections induced during fabrication and

production can considerably alter the structural behaviour of any component. The

lowest local elastic buckling mode shape, into which perfect structures would buckle,


39

is usually assumed as the initial form of the geometric imperfections. In order to

obtain the required buckling mode pattern, eigenvalue analyses can be performed.

In addition to the buckling mode shape which can only provide a perturbation pattern,

an imperfection amplitude is required to be incorporated into FE models. The value of

the amplitude can (1) be determined from experimental measurements, (2) be derived

from an analytical predictive model such as that proposed by Dawson and Walker

(1972), or (3) be assumed as a fraction of the component thickness as successfully

adopted by Schafer and Peköz (1998), Chan and Gardner (2008b) and Chacón et al.

(2009).

The results generated by FE models can be very sensitive to imperfection distribution

and amplitude (Wadee, 2000; Schafer et al., 2010). Comparisons of the results

produced by FE models against those from carefully conducted tests are required in

order to validate the models.

2.6 CONCLUDING REMARKS

The objective of this chapter has been to offer a general overview of the recent

developments in design guidance, laboratory testing, and material and numerical

modelling. More detailed consideration of relevant literature is made within each

individual chapter of this thesis.

In general, the cross-section classification system employed by current design codes,

in conjunction with an elastic, perfectly-plastic material model, cannot predict

accurately the ultimate capacity of both determinate and indeterminate structures

comprising stocky cross-sections, where strain-hardening increases resistance. There

is no current steel design method that can simply and rationally exploit strain-

hardening. Development of such a method is the focus of this thesis.

Chapter 3 Laboratory testing

40

CHAPTER 3

LABORATORY TESTING

3.1 INTRODUCTION

An experimental programme comprising tensile coupon tests on flat and corner

material and a series of simple beam tests and continuous beam tests (with two

loading configurations) were performed on the hot-rolled and cold-formed steel

square and rectangular hollow sections (SHS and RHS, respectively) to assess

moment capacity, rotation capacity and collapse loads. A total of ten SHS and RHS

specimens were considered – 5 hot-rolled and 5 cold-formed of the same section sizes

– 100×100×4, 60×60×3, 60×40×4, 40×40×4 and 40×40×3. All experiments were

carried out in the Structures Laboratory of the Department of the Civil and

Environmental Engineering at Imperial College London.

The experimental results are discussed in the present chapter, but are further utilised

for validation of numerical models in Chapter 6 and for development and verification

of the continuous strength method in Chapters 5 and 6.


41

3.2 MATERIAL TESTING

3.2.1 Details of material supply

The steels grades of the hot-rolled and cold-formed specimens employed in the tests

were grades S355 J2H (EN 10210-1, 2006) and S235 JRH (EN 10219-1, 2006). The

nominal chemical compositions of these grades according to EN 10210-1 (2006) and

EN 10219-1 (2006) are given in Table 3.1.

Table 3.1: Chemical composition (% by mass) of two steel grades employed

% by mass (maximum permitted) Steel grade C Si Mn P S N

S355 J2H 0.22 0.55 1.60 0.30 0.30 0.009 S235 JRH 0.17 - 1.40 0.040 0.040 0.009

3.2.2 Tensile coupon tests

The basic stress-strain properties of the investigated hot-rolled and cold-formed

sections were obtained through tensile coupon tests. These tests were conducted in

accordance with EN 10002-1 (1990).

3.2.2.1 Preparation of coupons

For each of the ten SHS and RHS specimens, one flat parallel coupon was machined

from the face opposite the weld. Corner coupons were also extracted and tested for

each of the five cold-formed sections in order to examine the influence of the high

localised cold-work, and for one of the hot-rolled sections to confirm uniformity of

properties. Figure 3.1 shows the locations of the flat and corner tensile coupons

extracted from the hot-rolled and cold-formed box sections for this study, together

with the adopted dimensioning and labelling system. The nominal dimensions of the

flat coupons were 350×15 mm for the smaller cross-section sizes (40×40×4 mm and

40×40×3 mm) and 320×20 mm for the larger cross-sections.


42

Fig. 3.1: Section labelling convention and locations of flat and corner tensile coupons

3.2.2.2 Instrumentation and testing

Linear electrical strain gauges were affixed at the midpoint of each side of the tensile

coupons and a series of overlapping proportional gauge lengths was marked onto the

surface of the coupons to determine the elongation at fracture. Load, strain,

displacement and input voltage were all recorded using the data acquisition equipment

DATASCAN and logged using the DALITE and DSLOG computer packages.

All tensile tests were performed using an Amsler 350 kN hydraulic testing machine.

The strain rate employed was according to EN 10002-1 (2001).

3.2.2.3 Results

Typical measured stress-strain curves from hot-rolled and cold-formed material (SHS

60×60×3-HR and SHS 60×60×3-CF) are shown in Fig. 3.2, with the hot-rolled

material displaying the anticipated sharply defined yield point, yield plateau and

subsequent strain-hardening whilst the cold-formed material exhibited a more

rounded response. The key results from all tensile coupon tests, together with the

corresponding mill certificate (virgin) material properties, are given in Table 3.2. The

specimens were labelled according to their different section geometries and

production routes (HR = hot-rolled and CF = cold-formed), while a ‘C’ was appended

ri

Weld

Corner coupon

t

B

D y y

z

z

Flat coupon


43

to the specimen designation to indicate corner coupon. In Table 3.2, fy and fu refer to

the yield and ultimate strengths of the material, respectively, E denotes Young’s

modulus and εf is the plastic strain at fracture. For the hot-rolled sections a distinct

yield stress was observed (with the lower yield stress being reported in Table 3.2), but

for the cold-formed sections the yield stress was taken as the 0.2% proof stress (as

marked in Fig. 3.2).

Table 3.2: Mill certificate (virgin) and measured tensile material properties of test

specimens

Mill certificate (virgin) material

properties Measured tensile material properties

Tensile test specimen fy,mill

(N/mm2) fu,mill

(N/mm2) fy,exp

(N/mm2) fu,exp

(N/mm2) Eexp

(N/mm2) εf,exp (-)

SHS 100×100×4-HR 491 569 488 570 212600 0.33 SHS 100×100×4-CF 378 423 482 500 208300 0.29 SHS 100×100×4-CF-C - - 522 567 199900 0.15 SHS 60×60×3-HR 478 574 449 555 215200 0.31 SHS 60×60×3-CF 395 423 361 402 207400 0.49 SHS 60×60×3-CF-C - - 442 471 208000 0.21 RHS 60×40×4-HR 482 561 468 554 213800 0.37 RHS 60×40×4-CF 445 471 400 452 212000 0.21 RHS 60×40×4-CF-C - - 480 570 202400 0.15 SHS 40×40×4-HR 523 576 496 572 212300 0.34 SHS 40×40×4-HR-C 523 576 499 578 215500 0.37 SHS 40×40×4-CF 383 413 410 430 201600 0.38 SHS 40×40×4-CF-C - - 479 507 210900 0.17 SHS 40×40×3-HR 520 565 504 581 219600 0.36 SHS 40×40×3-CF 430 456 451 502 212900 0.24 SHS 40×40×3-CF-C - - 534 589 196700 0.16


44

Fig. 3.2: Typical stress-strain curves from hot-rolled and cold-formed tensile coupons

(SHS 60×60×3)

3.3 SIMPLE BEAM TESTS

A total of six simply-supported beam tests were carried out. The purpose of the tests

was to obtain moment capacities and rotation capacities for both hot-rolled and cold-

formed structural sections.

3.3.1 Test specimens and measured dimensions

The nominal section sizes tested as beams were RHS 60×40×4, SHS 40×40×4 and

SHS 40×40×3; one hot-rolled and one cold-formed specimen of each size were

examined. The measured geometric properties of each specimen have been recorded

in Table 3.3; the symbols employed are as defined previously in this chapter. The

calculated elastic and plastic moduli, Wel and Wpl, respectively (determined taking

suitable account of the corner geometry) are also presented in Table 3.3.

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8 Strain (%)

Stre

ss (N

/mm

2 )

SHS 60×60×3.0-HR SHS 60×60×3.0-CF


45

Table 3.3: Measured dimensions of simply supported beam specimens (three-point

bending)

Simple beam specimen

D (mm)

B (mm)

t (mm)

ri (mm)

Wel (mm3)

Wpl (mm3)

RHS 60×40×4-HR 60.09 40.24 3.90 1.91 10600 13400 RHS 60×40×4-CF 60.04 40.09 3.93 2.07 10600 13400 SHS 40×40×4-HR 39.75 40.00 3.91 2.16 5650 7080 SHS 40×40×4-CF 40.31 40.42 3.70 3.10 5520 6900 SHS 40×40×3-HR 39.87 40.20 3.05 2.07 4820 5900 SHS 40×40×3-CF 40.16 40.11 2.80 2.63 4520 5500

3.3.2 Test configuration and loading rates

The symmetrical three-point simply-supported bending test arrangement is shown in

Fig. 3.3. The span of the beams was fixed at 1100 mm and testing was displacement-

controlled at a rate of 3.0 mm/min. Simple support conditions were achieved by

means of steel rollers. The specimens extended approximately 50 mm beyond each

end support. Wooden blocks were inserted into the tubular specimens at the loading

point and supports to prevent local bearing failure. Steel plates (50 mm wide and 10

mm thick) were also employed at the points of support and load introduction, as

shown in Fig. 3.4. For each specimen, two strain gauges were adhered to the tensile

and compressive flanges of the beams, at a distance of 60 mm from the loading point.

(a) General view of bending test configuration


46

(b) Schematic three-point bending test arrangement (dimensions in mm)

Fig. 3.3: Simple beam test set-up

3.3.3 Instrumentation

End rotations and mid-span deflections were recorded digitally throughout the tests by

affixing five linear displacement transducers on the specimens, locations of which are

shown in Figs 3.3 and 3.4. LVDTs 1 and 2 and LVDTs 4 and 5 measured the rotation

of the specimens at the supports, while LVDT 5 was employed to obtain the mid-span

deflection.

Fig. 3.4: Roller support and LVDT set-up in bending test

Loading jack LVDT1 LVDT2

50 550 550 50

Strain gauge

60

LVDT5

Beam specimen

60 LVDT3 LVDT4

60


47

3.3.4 Test results

The failures of all beams were due to in-plane bending with inelastic local buckling

observed in the compression flanges and upper region of the web. A typical deformed

test specimen (SHS 40×40×3-CF), exhibiting a local buckling failure mode, is shown

in Fig. 3.5.

Fig. 3.5: A typical deformed simple beam test specimen (SHS 40×40×3-CF)

The normalised bending moment-rotation curves for the six simple beam tests are

presented in Figs 3.7-3.9. The graphs are arranged such that a direct comparison

between the hot-rolled and cold-formed sections of similar nominal dimensions can

be made. In one out of the six simple bending tests, the bending moment fell below

Mpl ,which is the product of plastic section modulus and yield stress, on the unloading

path prior to the termination of the experiment, whilst for the remaining specimens

rotation capacity was calculated on the basis of θmax (the maximum attained rotation

prior to the test being terminated), though this does not necessarily reflect the full

rotation capacity of the specimens (Gardner and Theofanous, 2008). Despite the full

rotation capacity not being attained in some tests, all specimens were deemed to have

sufficient rotation capacity (R > 3) for plastic design (Wilkinson and Hancock, 1998),

as shown in Table 3.5.

The rotation capacity R of the test specimens was determined from Eq. (3.1) on the

basis of the recorded moment-rotation curves.

Inelastic local buckling of compression flange and upper portion of web

Overall deformed beam geometry


48

1θθ

Rpl

rot −= (3.1)

Symbols and determination of rotation capacity are illustrated in Fig. 3.6, where M is

the bending moment at mid-span and θ is the rotation of the plastic hinge (taken as the

sum of the two end rotations).

Fig. 3.6: Definition of rotation capacity from moment-rotation graphs

Fig. 3.7: Normalised moment-rotation curves for simple SHS 60×40×4 beams

θpl θrot

Applied moment M

Mpl

1Rpl

rot −θθ

=

Rotation θ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10 12 14 16 θ/θpl

M/M

pl

RHS 60×40×4-HR RHS 60×40×4-CF


49



The ultimate flexural capacities achieved in the six three-point bending tests are

summarised in Table 3.4. The following values are presented: the measured ultimate

test bending moment Mu (at mid-span), the calculated elastic and plastic moment

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7 8

SHS 40×40×4-HR SHS 40×40×4-CF

θ/θpl

M/M

pl

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7 8

SHS 40×40×3-HR SHS 40×40×3-CF

θ/θpl

M/M

pl


50

capacities, Mel and Mpl respectively and the normalised ultimate moment (Mu/Mel and

Mu/Mpl). Note that Mel and Mpl have been calculated on the basis of the measured

geometry and the measured tensile yield strength (or 0.2% proof strength) of the flat

material for each section.

Table 3.4: Ultimate flexural capacities of simple beam tests (three-point bending)


Mu (kNm)

Mpl (kNm)

Mel (kNm) Mu/Mpl Mu/Mel

RHS 60×40×4-HR 7.14 6.27 4.97 1.14 1.44 RHS 60×40×4-CF 7.59 5.37 4.25 1.42 1.79 SHS 40×40×4-HR 3.84 3.51 2.80 1.09 1.37 SHS 40×40×4-CF 3.61 2.83 2.26 1.28 1.59 SHS 40×40×3-HR 3.44 2.97 2.43 1.16 1.42 SHS 40×40×3-CF 3.09 2.48 2.04 1.25 1.52

The ultimate rotation capacities attained in the simple beam tests are presented in

Table 3.5, where the following values are included: the elastic rotation at the plastic

moment θpl, the total rotation at ultimate moment θu, the total rotation upon reaching

the plastic moment on the unloading path θrot, and where θrot was not attained, the

maximum recorded test rotation θmax. Note that θpl has been determined on the basis

of the measured flexural rigidity (EI) from the bending tests, which varied from the

theoretical (EI) – tensile Young’s modulus multiplied by second moment of area as

calculated from measured geometry – by a maximum of 5%.

Table 3.5: Ultimate rotation capacities of simple beam tests (three-point bending)


θpl (rad)

θu (rad)

θrot or θmax (rad) R

RHS 60×40×4-HR 0.044 0.46 0.74 >15.9 RHS 60×40×4-CF 0.041 0.24 0.55 >12.5 SHS 40×40×4-HR 0.083 0.51 0.51 >5.1 SHS 40×40×4-CF 0.061 0.17 0.50 >7.2 SHS 40×40×3-HR 0.094 0.19 0.60 >5.3 SHS 40×40×3-CF 0.085 0.25 0.39a 3.5 Note: a θrot attained; θmax reported for remaining sections


51

3.4 CONTINUOUS BEAM TESTS

A total of 12 continuous beam tests were conducted. The aim of the tests was to

generate experimental results for both hot-rolled and cold-formed materials for further

development of the CSM for indeterminate structures, which is described in Chapter 6.

3.4.1 Test specimens and measured dimensions

As in the simply-supported arrangement, three nominal section sizes – RHS 60×40×4,

SHS 40×40×4 and SHS 40×40×3 – were considered; for each section size, two hot-

rolled and two cold-formed specimens were tested. The measured geometric

properties of the test specimens have been presented in Table 3.6; symbols have been

previously defined.

Table 3.6: Measured dimensions of continuous beams (five-point bending)

Continuous beam specimen Configuration D

(mm) B

(mm) t

(mm) ri

(mm) Wel

(mm3) Wpl

(mm3)

RHS 60×40×4-HR1 1/2 Span 60.09 40.27 3.85 1.91 10500 13300 RHS 60×40×4-CF1 1/2 Span 60.14 40.20 3.89 2.07 10600 13400 RHS 60×40×4-CF2 1/2 Span 60.15 40.08 3.87 2.07 10500 13300 SHS 40×40×4-HR1 1/2 Span 39.79 39.98 3.85 2.16 5600 7010 SHS 40×40×4-CF1 1/2 Span 40.37 40.36 3.72 3.10 5550 6930 SHS 40×40×3-HR1 1/2 Span 39.90 40.22 3.01 2.07 4780 5850 SHS 40×40×3-CF1 1/2 Span 40.08 40.20 2.72 2.63 4420 5370 RHS 60×40×4-HR2 1/3 Span 60.06 40.33 3.82 1.91 10500 13200 SHS 40×40×4-HR2 1/3 Span 39.93 39.78 3.90 2.16 5650 7090 SHS 40×40×4-CF2 1/3 Span 40.43 40.36 3.71 3.10 5550 6930 SHS 40×40×3-HR2 1/3 Span 40.21 39.91 3.02 2.07 4820 5890 SHS 40×40×3-CF2 1/3 Span 40.12 40.14 2.76 2.63 4470 5430

3.4.2 Test configuration and loading rates

Two symmetrical five-point bending test configurations were employed as shown in

Figs 3.10 and 3.11. Similar to the three-point bending tests, symmetrical span lengths

of 1100 mm and displacement-controlled testing at a rate of 3 mm/min were chosen.

In the first configuration, designated ‘1/2 span’ in Table 3.6, the loads were applied at


52

the centre of the two spans, as shown in Fig. 3.10. In the second configuration,

designated ‘1/3 span’ in Table 3.6, the loads were applied at 366.7 mm from the

central support, as shown in Fig. 3.11. The principal purpose of adopting these two

test configurations was to vary the ratio of load levels between the formation of the

first hinge and the final collapse mechanism, thus placing differing rotation demands

on the first-forming plastic hinge.


(dimensions in mm)

Loading jack

LVDT6 LVDT5 LVDT3

LVDT4

LVDT1

Spreader beam

Load cell LVDT2

550

200

550 550 550

Strain gauge

LVDT7 LVDT8

100

Beam specimen

100


53


(dimensions in mm)

3.4.3 Instrumentation

As for the simply supported beams, steel rollers were employed to achieve

rotationally-free conditions at the beam ends and central support, and steel plates and

wooden blocks were introduced at the loading points and central support to prevent

local bearing failure, as shown in Fig. 3.12. Four displacement transducers (LVDTs 1,

2, 3 and 4) were used to measure the end rotations of the beams, while two additional

transducers (LVDTs 5 and 6) measured rotation at the central support. Two further

LVDTs (LVDTs 7 and 8) were employed to obtain the vertical deflections at the

loading points. The locations of the displacement transducers are shown in Figs 3.10

and 3.11.

Loading jack

LVDT6 LVDT5 LVDT3 LVDT4

LVDT1

Spreader beam

Load cell

LVDT2

733.3

200

366.7 366.7 733.3

LVDT7 LVDT8

Strain gauge Beam specimen

100 100


54

Fig. 3.12: General view of continuous beam test set-up

3.4.4 Test results

All test specimens failed by the formation of a three-hinge plastic collapse mechanism.

A deformed continuous beam specimen (RHS 60×40×4-HR1), exhibiting the three

distinct plastic hinges (the central hinge forming first and the two hinges at the

loading points forming simultaneously and precipitating collapse), is shown in Fig.

3.13.

Fig. 3.13: Location of plastic hinges in deformed continuous beam (RHS 60×40×4-

HR1)

Hinge 1 forms at central support

Hinges 2 and 3 form simultaneously at loading points

θ1 θ2


55

The normalised load-end rotation curves for the 12 continuous beam tests are shown

in Figs 3.14-3.16, where Ncoll is the plastic collapse load calculated by Eq. (6.5). As

for the simple beam tests, the arrangement of the graphs allows direct comparison

between nominally similar sections from the two different production routes. The end

rotation θ is taken as the average of the rotations at the two ends of the beam, as

shown in Fig. 3.13.

Fig. 3.14: Normalised load-end rotation curves for RHS 60×40×4 continuous beams

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 End rotation θ (rad.)

N/N

coll

RHS 60×40×4-HR1 RHS 60×40×4-HR2 RHS 60×40×4-CF1 RHS 60×40×4-CF2


56

Fig. 3.15: Normalised load-end rotation curves for SHS 40×40×4 continuous beams

Fig. 3.16: Normalised load-end rotation curves for SHS 40×40×4 continuous beams

The key results from the continuous beam tests are presented in Table 3.7; the total

theoretical load at which the first hinge forms Nh1, the total theoretical plastic collapse

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


N/N

coll

SHS 40×40×4-HR1 SHS 40×40×4-HR2 SHS 40×40×4-CF1 SHS 40×40×4-CF2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


N/N

coll

SHS 40×40×3-HR1 SHS 40×40×3-HR2 SHS 40×40×3-CF1 SHS 40×40×3-CF2


57

load Ncoll, the total ultimate load attained in the tests Nu and the ratio of the ultimate

test load to the plastic collapse load are included.

Table 3.7: Summary of results from continuous beam tests

Continuous beam specimen

Ultimate test load Nu (kN)

Theoretical first hinge load Nh1(kN)

Theoretical plastic collapse load Ncoll (kN)

Nu / Ncoll

RHS 60×40×4-HR1 78.1 60.2 67.8 1.15 RHS 60×40×4-CF1 83.4 51.8 58.3 1.43 RHS 60×40×4-CF2 83.3 51.5 57.9 1.44 SHS 40×40×4-HR1 44.6 33.7 37.9 1.18 SHS 40×40×4-CF1 40.6 27.5 31.0 1.31 SHS 40×40×3-HR1 38.1 28.6 32.1 1.18 SHS 40×40×3-CF1 34.2 23.5 26.4 1.30 RHS 60×40×4-HR2 98.4 60.6 84.2 1.17 SHS 40×40×4-HR2 55.2 34.5 47.9 1.15 SHS 40×40×4-CF2 51.5 27.9 38.7 1.33 SHS 40×40×3-HR2 49.0 29.1 40.5 1.21 SHS 40×40×3-CF2 42.3 24.0 33.4 1.27 3.5 ANALYSIS OF EXPERIMENTAL RESULTS

3.5.1 Material properties

3.5.1.1 Flat material properties

From Table 3.2, it may be seen that the virgin material properties (from the mill

certificate) and the measured material properties of the flat coupons extracted from

the complete sections are similar. This would be expected for the hot-rolled sections,

since essentially the same material is being tested. However, for the cold-formed

sections, the mill test is carried out on sheet material prior to section forming, whereas

the tensile coupon tests reported herein are performed on material extracted from the

complete section. The limited results presented in this study indicate modest levels of

strength enhancement in the flat faces of square and rectangular steel hollow sections

during the cold-forming process – an average increase in strength of around 4% over

the mill certificate value was observed. It would be anticipated that stockier sections

exhibit greater strength enhancements owing to the higher strain input required during

forming – this has been observed in studies on cold-formed circular hollow sections


58

(Kurobane et al., 1989). In the case of cold-formed stainless steel hollow sections,

more considerable strength enhancements have been observed following section

forming owing to the pronounced strain-hardening nature of the material (Ashraf et al.,

2005; Cruise and Gardner, 2008).

3.5.1.2 Corner material properties

The results from the corner coupon tests (Table 3.2) have been combined with those

from Guo et al. (2007); Key et al. (1988); Wilkinson and Hancock (1997); Zhao and

Hancock (1992) and presented in Table 3.8 and Fig. 3.17. In Table 3.8, the internal

radius ri, coupon thickness t and mill yield and ultimate stress are reported. In Fig.

3.17, the measured yield strength of the corner material fy,c,exp has been normalised by

the yield strength of the virgin material fy,mill to indicate the level of strength

enhancement due to corner forming, and plotted against the corner ri/t ratio. The

thickness of the test specimens ranged between about 3 mm and 12 mm, while the

corner radii ranged between about 2 mm and 20 mm. For some of the test data (Key et

al., 1988; Wilkinson and Hancock, 1997; Zhao and Hancock, 1992), the mill

certificate information was not available, and the corner yield strengths were instead

normalised by the measured yield strength of the flat material taken from the

corresponding sections. For the test data from (Zhao and Hancock, 1992), since no

internal corner radius was given, it was assumed to be equal to the material thickness.

In Fig. 3.17, the test results have been categorised by their ratio of ultimate tensile

strength to yield strength of the virgin material (fu,mill/fy,mill), which is indicative of the

potential for cold-work, whilst curves from the predictive model given in the AISI

Specification for the Design of Cold-formed Steel Structural Members (Karren, 1967;

AISI, 1996) are also plotted on the basis of the following equations:

mi

c

mill,y

AISI,c,y

)t/r(B

ff

= (3.2)

in which fy,c,AISI is the predicted corner yield strength according to the AISI

specification,


59

79.1ff

819.0ff

69.3B2

mill,y

mill,u

mill,y

mill,uc −

−= (3.3)

and

068.0ff

92.0mmill,y

mill,u −= (3.4)

Table 3.8: Summary of corner material properties

Resource Specimens ri (mm)

t (mm) ri/t

fy,mill (N/mm2)

fu,mill (N/mm2)

SHS 100×100×4.0 3.65 6.50 1.78 378 423 SHS 60×60×3.0 2.76 4.00 1.45 395 423 RHS 60×40×4.0 3.88 1.75 0.45 445 471 SHS 40×40×4.0 3.75 3.50 0.93 383 413

Gar

dner

et a

l. (2

010)

SHS 40×40×3.0 2.79 3.00 1.07 430 456 RHS 320×200×8.0 8.21 9.90 1.21 259 377 RHS 320×200×10.0 10.01 12.13 1.21 261 379 RHS 320×200×12.0 12.35 13.70 1.11 258 430 RHS 200×180×8.0 7.87 11.87 1.51 261 381 SHS 300×300×10.0 9.69 15.28 1.58 260 433

Guo

et a

l. (2

007)

SHS 300×300×12.0 11.96 19.30 1.61 256 427 SHS 76×76×2.0-S1 2.00 3.00 1.50 425 499 SHS 76×76×2.0-S2 2.00 3.00 1.50 370 449 SHS 152×152×4.9 4.90 7.35 1.50 416 475 SHS 203×203×6.3 6.30 9.45 1.50 395 494 SHS 254×254×6.3 6.30 9.45 1.50 405 479 RHS 102×51×2.0 2.00 3.00 1.50 422 494 RHS 127×51×3.6 3.60 5.40 1.50 388 456 RHS 127×64×4.0 4.00 6.00 1.50 418 479 RHS 152×76×4.9 4.90 7.35 1.50 372 437 RHS 203×102×4.9 4.90 7.35 1.50 371 429

Key

et a

l. (1

988)

RHS 254×152×6.3 6.30 9.45 1.50 397 458 RHS 102×51×4.9 4.73 4.73 1.00 437 470 RHS 102×51×3.2 3.16 3.16 1.00 425 482 RHS 102×51×2.0 2.04 2.04 1.00 407 474 SHS 102×102×9.5 9.64 9.64 1.00 482 559 Zh

ao a

nd

Han

cock

(1

988)

SHS 102×102×6.3 6.10 6.10 1.00 428 488


60

Table 3.8: Summary of corner material properties (continued)

Resource Specimens ri (mm)

t (mm) ri/t

fy,mill (N/mm2)

fu,mill (N/mm2)

RHS 150×50×5.0 C450 4.89 6.91 1.41 441 495 RHS 150×50×4.0 C450 3.88 3.22 0.83 457 527 RHS 150×50×3.0 C450 2.94 2.86 0.97 444 513 RHS 150×50×2.5 C450 2.55 1.45 0.57 446 523 RHS 150×50×2.3 C450 2.23 2.17 0.97 444 518 RHS 100×50×2.0 C450 2.07 2.33 1.13 449 499 RHS 75×50×2.0 C450 1.93 2.47 1.28 411 457 RHS 75×25×2.0 C450 1.98 2.12 1.07 457 515 RHS 75×25×1.6 C450 1.55 1.65 1.06 439 511 RHS 75×25×1.6 C350 1.55 1.85 1.19 422 456 RHS 150×50×3.0 C350 2.98 2.92 0.98 370 429 RHS 100×50×2.0 C350 2.05 2.15 1.05 400 450

Wik

inso

n an

d H

anco

ck (1

997)

RHS 125×75×3.0 C350 2.94 3.36 1.14 397 449

Fig. 3.17: Corner material strength data and AISI predictive model

3.5.1.3 Revised corner material predictive model

The experimental data shown in Fig. 3.17 broadly exhibits the anticipated trends, with

greater corner yield strength enhancements being observed for sections with higher

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

ri/t

f y,c

,exp

/ fy,

mill

fu/fy 1.0 - 1.1 fu/fy 1.1 - 1.2 fu/fy 1.2 - 1.3 fu/fy 1.4 - 1.5 fu/fy 1.6 - 1.7

Test results

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0


61

ratios of fu,mill/fy,mill and tighter corner radii. On applying Eqs (3.2) to (3.4), the mean

predicted divided by measured corner strength for the 40 corner test results (5 from

the present study and 35 from the literature (Guo et al., 2007; Key et al., 1988;

Wilkinson and Hancock, 1997; Zhao and Hancock, 1992)) was found to be 1.13, with

a coefficient of variation of 0.11, as shown in Table 3.9. Note however that the

corners in cold-formed tubular sections experience a particular, two-stage strain

history, arising from the fact that the sections are initially formed into a circular

profile, seam welded and then subsequently pressed into their final square or

rectangular shapes; this is unlike the more common single continuous operation

typically applied to cold-formed open sections, such as channels and lipped channels.

Based on the collated test data, revised values of the coefficients of the predictive

model for the corner regions of square and rectangular hollow sections, formed in the

above-described fashion, are proposed – see Eqs (3.5) and (3.6):

09.1ff

752.0ff

90.2B2

mill,y

mill,u

mill,y

mill,uc −

−= ， (3.5)

and

041.0ff

23.0mmill,y

mill,u −= . (3.6)

The tests results are compared to the revised predictive model in Fig. 3.18.

Application of the revised predictive model to the 40 corner test results provides a

mean predicted divided by measured corner strength equal to unity and a coefficient

of variation of 0.09, as shown in Table 3.9, in which fy,c,prop is the predicted corner

yield strength according to the revised model.


62

Table 3.9: Comparison of corner material properties with predictive models

Resource Specimens fy,c,exp (N/mm2)

fy,c,AISI (N/mm2)

fy,c,prop. (N/mm2)

fy,c,AISI fy,c,exp

fy,c,prop fy,c,exp

SHS 100×100×4.0 522 456 405 0.87 0.78 SHS 60×60×3.0 442 459 422 1.04 0.95 RHS 60×40×4.0 480 594 592 1.24 1.23 SHS 40×40×4.0 479 478 451 1.00 0.94

Gar

dner

et a

l. (2

010)

SHS 40×40×3.0 534 512 482 0.96 0.90 RHS 320×200×8.0 372 459 377 1.24 1.01 RHS 320×200×10.0 390 462 379 1.19 0.97 RHS 320×200×12.0 375 524 412 1.40 1.10 RHS 200×180×8.0 343 443 357 1.29 1.04 SHS 300×300×10.0 386 483 368 1.25 0.95

Guo

et a

l. (2

007)

SHS 300×300×12.0 370 474 361 1.28 0.98 SHS 76×76×2.0-S1 531 564 495 1.06 0.93 SHS 76×76×2.0-S2 476 513 444 1.08 0.93 SHS 152×152×4.9 498 530 472 1.07 0.95 SHS 203×203×6.3 520 569 487 1.09 0.94 SHS 254×254×6.3 487 542 475 1.11 0.98 RHS 102×51×2.0 551 557 490 1.01 0.89 RHS 127×51×3.6 451 515 452 1.14 1.00 RHS 127×64×4.0 485 536 476 1.10 0.98 RHS 152×76×4.9 459 494 434 1.08 0.94 RHS 203×102×4.9 481 482 426 1.00 0.89

Key

et a

l. (1

988)

RHS 254×152×6.3 476 514 455 1.08 0.96 RHS 102×51×4.9 498 538 506 1.08 1.02 RHS 102×51×3.2 535 570 523 1.07 0.98 RHS 102×51×2.0 439 568 515 1.29 1.17 SHS 102×102×9.5 536 669 608 1.25 1.13 Zh

ao a

nd

Han

cock

(1

988)

SHS 102×102×6.3 538 579 530 1.08 0.98


63

Table 3.9: Comparison of corner material properties with predictive models

(continued)

Resource Specimens fy,c,exp (N/mm2)

fy,c,AISI (N/mm2)

fy,c,prop. (N/mm2)

fy,c,AISI fy,c,exp

fy,c,prop fy,c,exp

RHS 150×50×5.0 C450 499 553 498 1.11 1.00 RHS 150×50×4.0 C450 557 647 597 1.16 1.07 RHS 150×50×3.0 C450 542 615 561 1.14 1.03 RHS 150×50×2.5 C450 539 688 646 1.28 1.20 RHS 150×50×2.3 C450 494 624 567 1.26 1.15 RHS 100×50×2.0 C450 498 573 527 1.15 1.06 RHS 75×50×2.0 C450 459 516 469 1.12 1.02 RHS 75×25×2.0 C450 521 601 550 1.15 1.06 RHS 75×25×1.6 C450 559 607 548 1.09 0.98 RHS 75×25×1.6 C350 506 511 474 1.01 0.94 RHS 150×50×3.0 C350 479 515 468 1.08 0.98 RHS 100×50×2.0 C350 493 526 483 1.07 0.98

Wik

inso

n an

d H

anco

ck (1

997)

RHS 125×75×3.0 C350 473 520 473 1.10 1.00 Mean 1.13 1.00 COV 0.11 0.09

Fig. 3.18: Corner material strength data and revised predictive model

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ri/t

f y,c

,exp

/ f y

,mill

fu/fy 1.0 - 1.1 fu/fy 1.1 - 1.2 fu/fy 1.2 - 1.3 fu/fy 1.4 - 1.5 fu/fy 1.6 - 1.7

Test results


64


3.5.2.1 Evaluation of slenderness limits

The results from the simple bending tests carried out in the present study have been

combined with those from existing studies (Zhao and Hancock, 1991; Zhao and

Hancock, 1992; Hancock and Zhao, 1992; Wilkinson and Hancock, 1998) on square

and rectangular hollow sections and plotted in Figs 3.19-3.21. In each of these figures,

the slenderness of the compression flange of the beam b/tε is plotted on the horizontal

axis. For all presented data points, the compression flange is the most slender element

in the section, having taken due account of the different stress distributions that exist

in the flange and web through the buckling coefficient kσ (EN 1993-1-5, 2006). The

dimension b is that flat element width (i.e. b = B - 2t - 2ri) and ε = yf/235 .

In Fig. 3.19, the maximum moment achieved in the tests Mu normalised by the elastic

moment capacity Mel is plotted on the vertical axis, allowing assessment of the Class

3 slenderness limit. The results indicate that the current slenderness limit of 42ε given

in EN 1993-1-1 (2005) is suitable for both hot-rolled and cold-formed sections,

though a wider range of data is required for hot-rolled sections. This is in contrast to

the findings from stub column tests (Gardner et al., 2010), though more favourable

performance would be anticipated from bending tests due to the less onerous stress

distribution in the web and therefore additional support offered to the compression

flange and possible partial plastification of the tension flange. Similar findings have

been observed for structural stainless steel sections.


65

Fig. 3.19: Mu/Mel versus b/tε for assessment of Class 3 slenderness limit

In Fig. 3.20, the ultimate test moment Mu has been normalised the plastic moment

resistance Mpl in order to assess the Class 2 slenderness limit of 38ε, given in EN

1993-1-1 (2005). As for the Class 3 limit, the general trend of the test data indicates

that the current Class 2 slenderness limit is appropriate. Ultimate moment capacities

beyond the plastic moment capacity may be observed in the figure and this is due to

strain-hardening (Byfield and Dhanalakshmi, 2002; Kemp et al., 2002; Gardner and

Wang, 2010) and a method for capturing this resistance is developed in subsequent

chapters of this thesis.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 10 20 30 40 50 60

b/tε

Hot-rolled Cold-formed

Class 3 slenderness limit = 42ε

Mu/M

el


66

Fig. 3.20: Mu/Mpl versus b/tε for assessment of Class 2 slenderness limit

Rotation capacity is examined in Fig. 3.21 in order to assess the Class 1 slenderness

limit; the rotation capacity requirement for a Class 1 section in Eurocode 3 is that R >

3, though some design codes require R > 4 (Wilkinson and Hancock, 1998). The

collected test data is somewhat scattered, as is often the case when considering

rotation capacity, and there are a number of tests, which were performed in a 4-point

bending arrangement and reported by Wilkinson and Hancock (1998), that meet the

current requirement for a Class 1 section (i.e. b/t < 33ε), but show a rotation capacity

of less than 3. This is attributed to the fact that rotation capacities for beams under

uniform bending tend to exhibit a moment plateau at approximately Mpl and can be

sensitive to small variations in the calculation of Mpl. To overcome this problem,

rotation capacity can be determined at a reduced plastic moment, such as 0.95Mpl,

providing a more stable measure of ductility, as adopted by Lay and Galambos (1965),

Sedlacek et al. (1998) and Chan and Gardner (2008b). The general trend of the data

does however support the current class 1 slenderness limit.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 10 20 30 40 50 60 b/tε



Mu/M

pl


67

Fig. 3.21: Rotation capacity R versus b/tε for assessment of Class 1 slenderness limit

3.5.2.2 Relationship between moment capacity and slenderness

By considering test series on RHS beams that include sections where the web is the

most slender element in the section, further data can be studied. A total of 65 test

results are plotted in Fig. 3.22, 6 from the present study and the remaining results

from the literature (Zhao and Hancock, 1991; Wilkinson and Hancock, 1997;

Wilkinson and Hancock, 1998). On the vertical axis, the maximum attained test

moment has been normalised by the elastic moment capacity Mel, whilst the non-

dimensional plate slenderness pλ of the most slender element in the cross-section,

taking due account of the different stress distributions that exist in the flange and web

through the buckling coefficient kσ (EN 1993-1-5, 2006), is plotted on the horizontal

axis. The data display a continuous trend of increasing normalised moment capacity

with reducing plate slenderness; this is examined further in subsequent chapters of

this thesis. The slenderness pλ is defined by Eq. (3.7).

επ

ν−==λ

σ tb

kE235)1(12

ff 2

cr

yp (3.7)

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50 60 b/tε

Rot

atio

n ca

paci

ty R


R = 3



68

where fcr is the elastic critical buckling stress of the plate element, b and t are the plate

width and thickness respectively, E is Young’s modulus, ν is Poisson’s ratio, ε =

(235/fy)1/2 and kσ is the familiar buckling coefficient allowing for differing loading

and boundary conditions (Allen and Bulson, 1980).

Fig. 3.22: Mu/Mel versus pλ for available tests on simple SHS and RHS beams


3.5.3.1 Study on slenderness b/tε limit by continuous beam tests

The results from the continuous beam tests conducted on SHS and RHS in the present

study are presented in Fig. 3.23, where the slenderness of the compression flange of

the specimen b/tε is plotted on the horizontal axis and the test ultimate failure load Nu

normalised by plastic collapse load Ncoll is plotted on the vertical axis. For all the

presented results, the compression flange is the most slender element in the section.

Figure 3.23 shows a similar trend for continuous beams as observed previously for

simple beams, with increasing normalised capacity with reducing cross-section

slenderness. The ultimate load-carrying capacities achieved in the tests may also be

seen to be higher than the collapse load Ncoll predicted by traditional plastic analysis.

The additional capacity beyond Ncoll may be attributed primarily to strain-hardening,

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Mu/M

el

Slenderness pλ


69

with enhanced corner properties also contributing in the case of cold-formed sections.

This issue is explored further is subsequent chapters of this thesis.

Fig. 3.23: Nu/Ncoll versus b/tε for assessment of Class 1 slenderness limit

3.5.4 Discussion

On the basis of the results generated in the present study, together with those collected

from existing studies, a number of differences between the behaviour of hot-rolled

and cold-formed hollow sections have been found. It is generally observed that the

current slenderness limits and plastic design approaches are equally applicable to hot-

rolled and cold-formed sections. Of the various features investigated, it is concluded

that the strain-hardening characteristics represent the most influential factor in

shaping the individual structural responses of hot-rolled and cold-formed sections,

particularly for non-slender sections. The inability of existing codes to exploit the

strain-hardening of the material rationally is evident, with stocky sections achieving

load-carrying capacities significantly beyond those predicted by current design

approaches. A new design approach, the continuous strength method (CSM) (Gardner,

2008; Gardner and Wang, 2010), has been developed to overcome these shortcomings,

offering a systematic means of utilising strain-hardening, based on cross-section

deformation capacity. The details of this method will be discussed from the

0.0

0.5

1.0

1.5

0 5 10 15 20 25 30 35

b/tε

Nu/N

coll




70

perspectives of material modelling and application to determinate and indeterminate

structures in Chapters 4, 5 and 6 of this thesis.


A series of tests on hot-rolled and cold-formed steel hollow sections has been

performed in order to assess the influence of the two different production routes on

material and structural responses. Material tests revealed modest increases in strength

in the flat regions of cold-formed sections during forming, but marked strength

enhancements in the corner regions; these were compared with the AISI predictive

model and revised coefficients have been proposed. Current codified slenderness

limits were evaluated on the basis of bending tests on hot-rolled and cold-formed

sections. The results of simple beam tests revealed that current limits are generally

acceptable for both production routes, though further test data, particularly for hot-

rolled sections, are required to confirm this point. The results of continuous beam

tests showed that plastic design was equally applicable to stocky hot-rolled and cold-

formed sections. A feature of many of the tests was the pronounced over-strength in

comparison to current design guidance as a result of strain-hardening; a new design

approach, the continuous strength method (CSM), is further developed in this thesis to

address this issue.

Chapter 4 Material modelling

71

CHAPTER 4

MATERIAL MODELLING 4.1 INTRODUCTION

Material modelling is a key aspect of structural analysis and design. Accurate

representation of the material behaviour is required for the development of any

different design method. The aim of this chapter is to develop an accurate, yet simple,

model to represent the material stress-strain characteristics of structural steel,

including allowance for strain-hardening.

There exists a wide variety of parameters, including grade of steel, cross-section

shape, forming process, type of loading and plate thicknesses, which can cause

significant variations in material properties. The following investigation is conducted

to determine factors which alter the stress-strain response of structural steel, and

corresponding parameters which may quantitatively account for these factors.

This chapter presents 1) studies on material properties, especially strain-hardening

properties including a review of key factors affecting them and an appraisal of


72

existing material models; 2) collection and analysis of existing material test data from

published resources and; 3) proposals of material models based on the findings of

these above studies.

4.2 REVIEW OF KEY FACTORS INFLUENCING MATERIAL PROPERTIES

Material properties, especially strain-hardening, will be investigated in this section by

reviewing the key factors influencing these properties. An appraisal of existing

material models will also be undertaken. A suitable format for the proposed material

model based on these studies will be suggested.

4.2.1 Stress-strain curve

The mechanical properties of hot-finished structural steels under static uniaxial load

may be schematically illustrated by the idealised tensile stress-strain diagram shown

in Fig. 4.1. The initial part of a stress-strain curve is linear, where the slope is the

Young’s modulus of elasticity E. Values of the Young’s modulus of carbon steel from

available test data collected from literature are in the range of 180000-220000 N/mm2,

as documented by Byfield and Nethercot (1998). Typically E = 205000 N/mm2 or E =

210000 N/mm2 is adopted in structural design codes. In this thesis, a value 210000

N/mm2, employed in EN 1993-1-1 (2005), is adopted in cases where the Young’s

modulus is not available.

The linear elastic range is limited by the yield stress fy and the corresponding yield

strain εy = fy/E. The elastic range is then followed by a plastic plateau. In this range,

there is no increase in stress until the strain-hardening strain εsh is reached. At this

point, the stress increases again at a lower rate than the initial Young’s modulus. The

slope of the curve at this point is called the strain-hardening modulus Esh, which is

generally taken as the tangent value at the onset of strain-hardening εsh (Fig. 1.1).

Further discussion on the measure of Esh is presented later in this chapter.

4.2.2 Tensile coupon versus compressive section tests

The material properties obtained from tensile coupon tests are generally used to

evaluate the structural performance of the corresponding complete sections. However,


73

as the experimental comparisons carried out in this sub-section show complete cross-

sections generally demonstrate a shorter plastic plateau and an earlier onset of strain-

hardening.

Stub column test results reported by Gardner et al. (2010) have been compared with

those from corresponding material tensile coupon tests; two such comparisons on hot-

rolled and cold-formed RHS 60×40×4 are shown in Figs 4.1 and 4.2 respectively. The

normalised stress from the stub column tests refers to the ratio of the applied load to

the corresponding yield stress multiplied by the cross-sectional area. For the coupon

tests it refers to the ratio of the measured stress to the corresponding yield stress. The

designations of the specimens have been introduced in Chapter 3. As shown in Fig.

4.1, it is clear for hot-rolled specimens that the stub column curves exhibit a shorter

plastic plateau with earlier onset of strain-hardening. In the case of the cold-formed

specimens, as shown in Fig. 4.2, the stress-strain behaviour of the stub columns are

between the responses of the flat and corner materials.

Fig. 4.1: Stub column and coupon tests - RHS 60×40×4-HR

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Strain

Nor

mal

ised

stre

ss f/

f y

Stub RHS 60×40×4-HR1 Stub RHS 60×40×4-HR2 Coupon 60×40×4-HR

Onset of strain-hardening (Section tests)

Onset of strain-hardening (Coupon tests)


74

Fig. 4.2: Stub column and coupon tests - RHS 60×40×4-CF

McDermott (1969) performed compression tests on cruciform specimens and

corresponding longitudinal tensile tests. Similar to the observations above, strain-

hardening occurred earlier in the compression tests than in the corresponding material

coupon tests (McDermott, 1969), and higher strain-hardening moduli were observed.

Similar conclusions have been reached in other previous studies (ASCE, 1971; Doane,

1969).

At present, the tensile coupon test is widely accepted as the measure to investigate

material properties and for hot-finished material, the f-ε response may be closely

replicated with a tri-linear model. However, the full section properties of hot-finished

sections exhibit a shorter yield plateau than the corresponding tensile coupon, while

for cold-formed sections, both tensile coupon tests and full section tests in

compression exhibit no distinct yield point and more rounded behaviour. Therefore, a

similar bi-linear material model with adjustable strain-hardening modulus is

considered more suitable and will be implemented in this study.

4.2.3 Strain rate

To obtain approximate information for designing structures subject to static loads,

tensile properties of structural steels are generally measured at relatively low strain

rates (Brockenbrough, 1999). However, higher strain rates normally cause a loss of

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Strain

Nor

mal

ised

stre

ss f/

f y

Stub RHS 60×40×4-CF1 Stub RHS 60×40×4-CF2 Coupon 60×40×4-CF-Flat Coupon 60×40×4-CF-Corner


75

material ductility and also increase the yield stress and rate of strain-hardening, except

at elevated temperatures where the reverse is true (Trahair et al., 2008; Bruneau et al.,

1998; Brockenbrough, 1999; Kemp et al., 2002). Tensile coupon tests are generally

conducted within described strain ranges, within which the variation in material

properties is relatively small, and will not considered further in this study.

In fact, for the purpose of normal structural steel design, the rate of loading will not be

considered the same as the strain rates commonly used in tests. The yield stress of a

particular steel grade under laboratory conditions can be far higher than the nearly

static loading rates often encountered in actual structures (Brockenbrough, 1999).

4.2.4 Forming route

Structural steel sections are produced in a variety of ways, the principal two being

hot-rolling and cold-forming. The other two less common alternative techniques for

forming hollow sections involve welding two channel sections tip-to-tip or welding

four flat plates at the corners. Cold-formed sections may be subsequently stress-

relieved. Owing to the different strain histories and thermal actions that may be

experienced during production, cross-sections of nominally similar geometries, but

from the two different production routes, may vary significantly in terms of their

general material properties, geometric imperfections, residual stresses, corner

geometry, material response, general structural behaviour and load-carrying capacity.

Stress-strain curves from standard tensile coupon tests on material extracted from a

flat portion of a hot-rolled (HR) square hollow section (SHS) 60×40×4 and a cold-

formed (CF) SHS 60×40×4 at ambient temperature are presented in Fig. 4.3. The

graph demonstrates the differences between hot-rolled and cold-formed steels, mainly

the rounded yielding and lack of a plastic yield plateau in cold-formed steel.


76

Fig. 4.3: Tensile f-ε curves for hot-rolled and cold-formed material

According to EN 10210-1 (2006), hot-rolled sections are generally produced above

re-crystallisation temperature of the material (normally around 850 ˚C). At ambient

temperature, the rolled sections have consistent hardness, good ductility,

homogeneous material properties and relatively low residual stresses, as may be seen

in Fig. 4.3. Residual stresses and their effect on material properties will be discussed

in Section 4.2.7.

Conversely, the cold-forming of structural steel sections is normally carried out at

ambient temperatures according to EN 10219-2 (2006). This involves plastic

deformation through a set of roll stations, during forming into the required profile.

The plastic deformation induced causes cold-working, resulting in an enhanced

material strength and a loss of ductility. To illustrate the general effect of the plastic

deformation on material strength and ductility, the basic behaviour of steel tension

specimens subject to such deformation and subsequent unloading is studied and

shown in Fig. 4.4.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.02 0.04 0.06 0.08 0.1 Strain ε

Nor

mal

ised

stre

ss f/

f y

Coupon - SHS 60×40×4-HR Coupon - SHS 60×40×4-CF


77

Fig. 4.4: Stress-strain diagram illustrating the effects of cold-working

As the figure illustrates, the unloading of a steel member after being strained into the

post-yielding range (whether in the plastic or strain-hardening ranges), follows a path

parallel to the initial elastic part of the stress-strain curve. Thus a residual strain is

induced and remains even after the load is removed. If the amount of plastic

deformation is beyond the onset of strain-hardening, the yield stress will be enhanced.

Otherwise, the yield stress stays close to that of the virgin material, although the

ductility in both these cases is less. The level of cold-working therefore influences the

resulting material properties. Cruise and Gardner (2008) proposed a measure of the

level of cold-working by utilising the ratio of (fu,mill/fy,mill) to (fu/fy) as a parameter in

the material model, where the former stresses are from the mill certificate and the

latter ones are obtained experimentally from complete sections.

Enhanced material strength in the corner regions of a cross-section induced by cold-

forming is another factor influencing the structural behaviour of the corresponding

complete section. This is generally due to the fact that the material in the corner area

experiences more plastic deformation during the cold-forming process than the flat

region. Ashraf et al. (2006a) proposed a weighted average method to calculate the

compressive resistance of stainless steel cross-sections. The corner enhancements

Stress

fu

fy

Elastic Range

Strain-hardening Range Plastic Range

Strain Ductility after strain-hardening

Ductility after deformation within plastic range Residual strain

Ductility of virgin material

Increase in yield strength


78

induced by cold-working have been found to extend into the flat regions of cross-

sections and a study to assess the extent of these enhancements has been carried out

by Ashraf et al. (2006b). For the studied stainless steel sections, enhancements were

found to extend between one and two times the material thickness beyond the corners

into the flats. However, for carbon steel material, Karren (1967) concluded that the

corner enhancements were localised in the corners only, with only very minor

extension into the flat regions.

In general, the material strength enhancement caused by cold-working increases both

the yield and ultimate stresses, although this increase is not proportional and the ratio

of the ultimate stress to yield stress fu/fy typically reduced with cold-work level

(Macdonald et al., 1997). To some extent, the more cold-working, the higher the yield

stress and the lower the rate of strain-hardening (Adams and Galambos, 1969).

Therefore, the ratio fu/fy is able to reflect both the level of cold work and the rate of

strain-hardening.

4.2.5 Material thickness

The effect of thickness variation on material properties has been studied by many

researchers. There hence exist different opinions on its effect on strain-hardening.

Sawyer (1961) and Alpsten (1972) individually concluded that the strain-hardening

capacity of a particular steel increases, on average, with increasing plate thickness. On

the other hand, Byfield and Dhanalakshmi (2002) believed that material thickness has

no direct influence on the rate of strain-hardening.

Byfield and Nethercot (1997) carried out a sensitivity study on the effect of material

thickness on yield stress, based on 7000 mill tests, with particular reference to

sections with thickness less than 20 mm. It was found that both yield stress and

ultimate stress increase with reducing thickness, but not at the same rate, with yield

stress showing greater variation. This means that the ratio of ultimate stress to yield

stress fu/fy decreases with decreasing thickness. In addition, there is a general trend

that the rate of strain-hardening increases with an increasing ratio of ultimate stress to

yield stress fu/fy, as stated previously by Alpsten (1972) and Kemp (2002). Section


79

thickness is therefore indirectly correlated with the rate of strain-hardening via the

ratio of ultimate stress to yield stress fu/fy.

4.2.6 Variation of material properties around a cross-section

The variation of material properties around the cross-section of structural members

has been extensively investigated (Alpsten, 1972; Byfield and Nethercot, 1997;

Roderick, 1954). Since most sections are initially made from the same virgin material

having the same material properties, the variations in material strength suggest that

there are operations, such as the variable rate of cooling around the cross-section,

which may alter.

Roderick (1954) investigated variation in strain-hardening characteristics in I-sections

by conducting a series of coupon tests on material cut from different locations. After

replotting the stress-strain curves non-dimensionally (shown schematically in Fig.

4.5), stress over yield stress f/fy against strain over yield strain ε/εy, the effect of the

variation on strain-hardening property is evident. The curves prior to the onset of

strain-hardening are similar, but vary considerably in the strain-hardening region. The

variation was attributed to non-uniform cooling following the hot-forming process.

Fig. 4.5: Schematic non-dimensional stress-strain curves from various locations

around on I-section

1.0

f/fy

ε/εy 5 10 15 20 0 25


80

4.2.7 Residual stress

Residual stresses are induced into structural steel member as a result of the non-

uniform cooling following hot-rolling or welding operations or due to plastic

deformation in processes such as cold-rolling (Trahair et al., 2008). Residual

compressive stresses in the early-cooling regions are established by the shrinking of

the late-cooling regions, and these are balanced by equilibrating tensile stresses in the

late-cooling regions. In hot-rolled I-sections, the flange-web junctions have a

relatively low ratio of surface area to volume and therefore cool more slowly resulting

in tensile residual stresses. Conversely, those areas that cool more rapidly, such as

flange tips are left in residual compression (Trahair et al., 2008).

Over the past decades, considerable work has been carried out on assessing the

influence of residual stresses on structural steel members. Studies have been

performed on welded and hot-rolled carbon steel elements (Alpsten, 1968; Nishino

and Tall, 1969; Bjorhovde et al., 1972; Nethercot, 1974; Machacek, 1988; Gardner et

al., 2010) and on cold-formed carbon steel profiles (Key and Hancock, 1993; Gardner

et al., 2010). Residual stresses will generally cause earlier yielding and strain-

hardening (Bredenkamp et al., 1992), as shown in Fig. 4.4. For instance, if an

operation causes strains beyond the onset of strain-hardening of virgin material, the

onset of strain-hardening of the worked material would be shifted earlier just after

attainment of the yield stress.

Key and Hancock (1993) investigated the influence of residual stresses on cold-

formed carbon steel members. An experimental study was performed to measure both

membrane and bending residual stress. The analysis concluded that membrane

residual stresses were negligible in terms of the influence on section behaviour, while

the bending residual stresses were more significant and could cause reductions in

ultimate load and pre-ultimate stiffness.

In the light of these results, the ratio of fu/fy and the section-forming methods can be

used to consider the effect of residual stress on material property.


81

4.2.8 Steel grades

The effect of variable steel grades on material property, especially in the strain-

hardening range, has been studied by Byfield and Dhanalakshmi (2002) who

concluded, from a survey of 50 different mill tests, that the rate of strain-hardening

does not significantly differ between S275 and S355 steel grades. However, higher

grades have lower ratios of ultimate stress to yield stress fu/fy and have been observed

to require greater deformation to reach the onset of strain-hardening (Kuhlmann,

1989). Therefore, the ratio of fu/fy could once again be taken as a parameter to

consider the effect of steel grade on strain-hardening properties.

4.2.9 Discussion

The ratio of ultimate stress to yield stress fu/fy has been found, from the sections

above considering the forming method, thickness, variation of strain-hardening

properties around the cross-section, residual stress and steel grades, to be a key

parameter in determining the general shape of the stress-strain curve, especially in the

strain-hardening region. Furthermore, this ratio has already been widely accepted as a

measure of steel’s strain-hardening ability by Hasan and Hancock (1989), Byfield and

Nethercot (1997) and Kemp et al. (2002). This ratio is therefore employed in the

material model developed later in this chapter.

4.3 APPRAISAL OF EXISTING MATERIAL MODELS

This section describes existing approaches to representing the stress-strain behaviour

of structural steel, and discusses their relative merits.

4.3.1 Rigid-plastic model

The rigid-plastic model is illustrated in Fig. 4.6, where there is no strain ε until the

yield stress is reached (producing an infinite value of Young’s modulus E) and

subsequently no further increase in stress f as the strain increases to infinity and E

changes to 0. This model is commonly used in plastic analysis of determinate and

indeterminate structures. For plastic analysis, the rigid-plastic model enables


82

determination of the collapse load of a structure but provides no information on

deflections prior to collapse. (Trahair et al., 2008)

Fig. 4.6: Rigid-plastic model

4.3.2 Elastic, perfectly-plastic model

For most applications, in particular when information on deformations prior to

structural failure is required or when an appropriate estimation of deformations such

as deflections in the inelastic regime is desired, an elastic, perfectly-plastic model is

generally employed; this model forms the basis of EN 1993-1-1 (2005) and is shown

in Fig. 4.7. There are two stiffness stages in this model: elastic and plastic. Choosing

zero stiffness for the plastic state is appropriate for many applications where strain-

hardening is not anticipated, although the perfectly-plastic stage may lead to overly

safe predictions of plastic collapse loads and deformations when strain-hardening is

expected to develop (Bruneau et al., 1998).

Fig. 4.7: Elastic-perfectly plastic model

4.3.3 Elastic, linearly-hardening model

The elastic, linearly-hardening model offers the simplest consideration of strain-

hardening, as shown in Fig. 4.8, where Esh represents the strain-hardening modulus.

f

E E = 0

ε

fy

εy

f

E ∞ E = 0

ε

fy


83

EN 1993-1-5 (2006) recommends this model for FE analysis of steel structures, with a

strain-hardening modulus Esh of E/100.

Fig. 4.8: Elastic, linear-hardening model

There are various methods for determining suitable values for the slope of the linear

hardening region. One of them is known as the equal energy dissipation method as

shown in Fig. 4.9. In this method, a straight line bisects the actual stress-strain curve

and a suitable slope is defined to achieve an equal amount of plastic energy dissipated

at a given limiting strain εmax. Depending on the εmax, widely variable values of Esh

ranging from E/200 to E/20 have been found (Bruneau, 1998).

Fig. 4.9: Elastic, linear-hardening model based on equal energy dissipation

Such a bi-linear model has been proposed and studied by McDermott (1969) on A514

grade steel according to the ASTM specification (2000). Similar to the elastic, linear-

hardening model given in Fig. 4.10, McDermott (1969) proposed an idealised bi-

linear stress-strain curve which consists of a linear elastic part followed by a linear

f

E Esh

ε

fy

εy

f

ε

εmax Area 1 = Area 2

Ultimate stress Actual stress-strain curve


84

strain-hardening part. The yield stress of the linear elastic portion is as the stress at

0.5% strain or the 0.2% offset yield stress, both of which were observed to be very

similar, while the slope for the linear strain-hardening part was measured between

approximately 2% and 3% strain. The strain-hardening moduli measured were close

to E/100 as suggested by EN 1993-1-5 (2006).

4.3.4 Tri-linear model

More complex, tri-linear models can be used to more accurately represent the stress-

strain response of steel – see Fig. 4.10. A number of previous studies have been

conducted, in which it has been observed, as noted in Section 4.2.2, that higher values

of strain-hardening modulus generally result from compression tests than from tension

tests on the same material. Furthermore, the strain-hardening property can be

influenced by using different definitions of the strain-hardening modulus. Fig. 4.11

gives three commonly employed methods (ASCE, 1972).

Fig. 4.10: Tri-linear model

f

E Esh

ε

fy

εy εsh


85

Fig. 4.11: Methods employed by various researchers for obtaining Esh in tri-linear

material models

Values of strain-hardening modulus Esh obtained by the following researchers are

presented in Table 4.1: Roderick (1954), Haaijer (1957), Lay and Smith (1965),

Alpsten (1972), Rogers (1976), Kato (1990) and Byfield and Dhanalakshmi (2002).

These collected test data have also been presented in non-dimensional format in Fig.

4.12. All results are based on tensile tests which, as discussed earlier, tend to exhibit a

0.003 0.007 Reported value of εsh

Apparent onset of strain-hardening

0.005 Modified onset of strain-hardening

Esh1 Tangent drawn by eye

Esh2

Esh3

(b) Definition of Esh1

(c) Definition of Esh2

(d) Definition of Esh3

ε

σ

(a) Stretch of material stress-strain curve


86

far longer yield plateau than full section compression tests. Since full structural

sections one of primary interest in practice, the simpler bi-linear material model (Fig.

4.8) is considered adequate and more appropriate for design.

Table 4.1: Summary of tri-linear material models

Source Steel grade fu/fy εsh/εy E/Esh Alpsten (1972) ASTM A 7 1.26 11 52 ASTM A 36 1.61 16 66 ASTM A 441 1.40 11 45 ASTM A 572(50) 1.20 6 41 ASTM A 572(65) 1.23 9 54 Horne (1981) unclassified unknown 8 20 Lay and Smith (1965) ASTM A 36 1.26 11 80 Byfield and Dhanalakshmi (2002) S275 S355 1.50 6 74 Haaijer (1957) ASTM A 36 1.26 6 39 Kato (1990) SM41 1.53 11 52 SM50 1.38 9 63 SM58L 1.38 2 38 SM58H 1.14 6 116 Kuhlmann (1989) S235 S355 1.40 10 50 Rogers (1976) S275 1.79 9 54-136 Esh by Alpsten (1972) is Esh2 (see Fig. 4.11) Esh by Lay and Smith (1965) is Esh2 (see Fig. 4.11) Esh by Byfield and Dhanalakshmi (2002) measured from 1.5% - 4% Esh by Haaijer (1957) is Esh3 (see Fig. 4.11) Esh by Kato (1990) measured from the onset of the strain-hardening and the point (f/fy = 0.95) Esh by Kuhlmann (1989) is measured at the onset of strain-hardening The methods for measuring Esh by Rogers (1976) and Horne (1981) are unknown SM41, SM 50, SM58L and SM58H are Japanese steel grades


87

Fig. 4.12: Summary of tri-linear material models adopted by various researchers

4.3.5 Piecewise nonlinear models

As discussed in Section 4.2.4, the non-uniform plastic deformation during cold-

forming process causes cold-working of the material which results in enhanced

strength and reduced ductility. Non-homogeneity of material properties and variation

in hardness around the section typically arise due to the uneven levels of plastic

deformation experienced during forming; the corner regions of cold-formed sections,

in particular, undergo high levels of cold-work. The length of the plastic plateau may

also be greatly reduced, often to zero. This leads to rounded yielding characteristics,

where a specific yield point is not evident and cannot be defined. These features make

models for cold-formed steel distinct from the models for hot-rolled steel, especially

in the plastic stage. Hence, the linear models described above are generally not

suitable for precise modelling of cold-formed steel. Other commonly used methods

are summarised below.

4.3.5.1 Models with simple power functions

The simplest model with a power function is given as follows:

f = C1εn (4.1)

ASTM A7 (Alpsten, 1972) ASTM A36 (Alpsten, 1972) ASTM A441 (Alpsten, 1972) ASTM A572(50) (Alpsten, 1972) ASTM A572(65) (Alpsten, 1972) Unclassified steel (Horne, 1981) ASTM A36 (Lay and Smith, 1965)

S275 S355 (Byfield and Dhanalakshmi, 2002)

ASTM A36 (Haaijer, 1957) SM41 (Kato, 1990) SM50 (Kato, 1990) SM58L (Kato, 1990) SM58H (Kato, 1990)

S275 (Rogers, 1976) S235 S355 (Kuhlmann, 1989)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25 30 35 40ε/εy

f/fy1.0

f/fy


88

where C1 and n are constants and n is generally less than unity and larger than zero

(see Fig. 4.13).

Fig. 4.13: Simplest model with power function

This model is commonly used to represent material stress-strain response and is

explicitly solvable for stress. However, this model is inaccurate at low strains because

the function is tangential to the stress axis for |n| ≤ 1 (Gardner, 2002).

Improvements to overcome the inaccuracies at low strains can be achieved by

adopting an elastic, power-hardening model (see Eqs (4.2a) and (4.2b) and Fig. 4.14),

but this induces a discontinuity at the yield stress.

f = Eε (f ≤ fy) (4.2a)

f = C2εn (f > fy) (4.2b)

where E is the Young’s Modulus, C2 and n are constants and n is generally less than

unity and larger than zero.

Fig. 4.14: Elastic-linear power hardening model

n = 1

n = 0.5

n = 0

C1

f

ε

Eq. (4.1)

f

ε

fy

Eq. (4.2a)

Eq. (4.2b)


89

4.3.5.2 Ramberg-Osgood models

The Ramberg-Osgood model may be applied to describe the stress-strain relationship

for cold-formed carbon steel material in the form ε = ε(f). This model is also given in

EN 1999-1-1 (2007) for aluminium alloys.

In addition to the simple-power hardening model described above, models with power

functions can be said to be the summation of elastic and plastic parts. Ramberg and

Osgood (1943) proposed Eq. (4.3) for representing nonlinear material stress-strain

behaviour:

n

petotal EfK

Efεεε

+=+= (4.3)

where εtotal is the total strain, εe is the elastic strain, εp is the plastic strain and K and n

are constants.

The basic expression was revised by Hill (1944) to obtain Eq. (4.4)

n

pffc

Efε

+= (4.4)

where fp is a proof stress and c is the corresponding offset strain at this stress.

The adopted proof stress, f0.2, is generally evaluated by means of the 0.2% offset

approach, leading to:

n

2.0ff002.0

Efε

+= (4.5)

The value of the constant n shown in the equation should be calibrated on the basis of

the actual mechanical material properties. Methods for accomplishing this are given in

Annex E in EN 1999-1-1 (2007).


90

4.3.5.3 Models with exponential functions

In addition to the models with power functions, Gehring and Saal (2008) proposed a

model using exponential functions, based on more than 300 tensile coupon tests on

cold-formed S320GD grade steel. Eq. (4.6) below was used to derive the stress-strain

relationship.

[ ]∑=

−−+=n

1i

)εC(iy

pie1(Qff (4.6)

where εp is plastic strain, i is usually between 1 and 3 and Qi and Ci are determined

from tests. By fitting Eq. (4.6) to 300 stress-strain curves, Gehring and Saal (2008)

created lines representing maximum, minimum and mean values by adjusting the

parameters Qi and Ci in the equation, as shown in Fig. 4.15 and Table. 4.2.

Fig. 4.15: Gehring and Saal’s model of the strain-hardening behaviour of structural

steel (Ramanto, 2009)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 Strain

Nor

mal

ised

stre

ss f/

f y

Minimum Average Maximum


91

Table 4.2: Parameters for the equation proposed by Gehring and Saal (2008)

Level Q1 (MPa) C1 (-) Q2 (MPa) C2 (-) Q3 (MPa) C3 (-) Minimum 359 10.0 -223 17.5 3.1 2094 Average 201 10.2 -14 436.1 0.7 2171 Maximum 204 10.0 64 52.2 -3.3 2345

Eq. (4.6) has been studied by Ramanto (2009) and compared to tensile coupon test

results taken from hot-rolled RHS and SHS, which are reported in Chapter 3, to

investigate if the model could be used for hot-rolled materials. The comparison

showed that the model parameters (Qi and Ci) had to be adjusted to suit the nature of

the different structural steel grades.

4.3.6 Discussion

The models described above have been assessed and compared for their suitability to

represent material of hot-rolled and cold-formed steel through consideration the five

key points below, which were modified from Gardner (2002):

� Number of parameters

� Overall accuracy of stress-strain description on hot-rolled and cold-formed steel

� Consideration of strain-hardening

� Explicit solvability for stress

� Consistency with the current design code

The rigid-plastic and elastic, perfectly-plastic models have the minimum number of

parameters and are explicitly solvable for stress and consistent with the current design

code, but they do not consider strain-hardening, which is an essential requirement for

the proposed design method. These are therefore excluded.

The tri-linear model does allow for strain-hardening and represents well the long

plastic plateau and strain-hardening features associated with tensile coupon test results

on hot-rolled material. However, based on full section behaviour in compression, hot-

rolled material exhibits a far shorter yield plateau (if any) while cold-formed material

exhibits generally no yield plateau. The tri-linear model is less appropriate in this case.


92

Nonlinear material models have been found to represent the rounded f-ε features of

cold-formed material accurately, but generally require more parameters than other

models and are more complex. In addition, some of the models with Ramberg-Osgood

functions or exponential functions are not explicitly solvable for stress, which

complicates their use in resistance functions.

The elastic, linear-hardening model is considered to most effectively meet the five

requirements. Firstly, the model can provide sufficiently accurate representation of the

behaviour of both hot-rolled and cold-formed materials. Secondly, the model is

consistent with current practice (in the sense that the bi-linear model is already given

in EN 1993-1-5 (2005)). The model is explicitly solvable for stress and considers

strain-hardening. In addition, there is only one extra parameter beyond E and fy,

which is strain-hardening modulus Esh, and the same basic model can be applied to

both hot-rolled and cold-formed material.

On the basis of the above discussion, the elastic, linear-hardening model with a strain-

hardening modulus defined as a function of the ratio fu/fy, is selected for further

development and use in the proposed design method.

4.4 COLLECTION OF EXISTING EXPERIMENTAL DATA

Existing tensile coupon test and stub column test data have been gathered and

analysed to aid the development of the chosen material model. Data collected from

various published papers have been grouped in appropriate categories and reported in

Tables 4.3-4.9, where parameters from various section types including I-sections,

rectangular hollow sections (RHS), square hollow sections (SHS), elliptical hollow

sections (EHS) and circular hollow sections (CHS) are included.

Stub column test results available in the literature which are useful for the present are

somewhat limited; this is because the main interest of the current study is on stocky

sections, which progress into the strain-hardening regime before buckling, rather than

slender sections which fail earlier due to local buckling. Hence, although there are

many stub column tests available (McDermott, J. F., 1969; Gardner et al., 2010;


93

Wilkinson and Hancock, 1997; Guo et al., 2007; Chan and Gardner, 2008a), only a

limited number (where ultimate resistance is more than yield strength) can be used in

the present study. These are summarised in Tables 4.3 and 4.4 below.

Table 4.3: Stub columns – hot-rolled hollow sections

Source Section Type No. of tests


Chan and Gardner (2008a) EHS 7

Table 4.4: Stub columns – cold-formed hollow sections



Zhao and Hancock (1991) SHS 4 Tensile coupon test data are more readily available and are summarised in Tables 4.5-

4.7. Stress-strain graphs provided in the literature (Driscoll and Beedle, 1957; Sawyer,

1961; Kato, 1970; Kuhlmann, 1989; Wilkinson and Hancock, 1997; Byfield et al.,

2005; Guo et al., 2007; Liu et al., 2010; Packer et al., 2010) were manually extracted

and digitized. Further test data available electronically from (Byfield and Nethorcot,

1998; Chan and Gardner, 2008a; Gardner el al., 2010; Law, 2010) were also added in

the test database.

Table 4.5: Tensile coupons – hot-rolled I-sections

Source Steel Grade No. of Tests Byfield et al. (2005) S275 22 Byfield et al. (2005) S355 22 Byfield and Nethercot (1998) S275 12 Driscoll and Beedle (1957) Unknown 8 Kuhlmann (1989) Unknown 10 Sawyer (1961) Unknown 12


94

Table 4.6: Tensile coupons - hot-rolled hollow sections

Source Section Type No. of tests Chan and Gardner (2008a) EHS 19


Law (2010) EHS 12 Packer et al. (2010) CHS 3 Liu et al. (2010) CHS 3

Table 4.7: Tensile coupons - cold-formed hollow sections


Wilkinson and Hancock (1997) RHS SHS 36

Gardner et al. (2010) RHS 5

Guo et al. (2007) RHS SHS 3

Packer et al. (2010) CHS 1

In addition to the commonly used steel sections, welded cruciform section stub

columns and tensile coupons, and plate tensile coupon test data were also collected

and summarized in Tables 4.8 and 4.9.

Table 4.8: Stub columns - other sections

Source Section Type No. of tests McDermott (1969) Cruciform welded section 7

Table 4.9: Tensile coupons - other sections

Source Section Type No. of tests Kato (1970) Plate 4 McDermott (1969) Cruciform welded section 8

In order to keep the data from various resources consistent for the following study,

some assumptions were drawn. In all cases where Young’s modulus is not available,

the value of 210000 N/mm2 was assumed in accordance with EN 1993-1-1 (2005).

The ultimate stress was not reported by Byfield et al. (2005) and hence the same

results as those presented by Byfield and Nethercot (1998) were assumed, given that

the same grade of material was being studied.


95

The stress-strain behaviour of materials tested was not always fully documented, and

in some cases only the strain-hardening modulus Esh was given. These Esh were

measured in a variety of ways, as reported by Alpsten (1972); this was the case with

data obtained from Sawyer (1961), McDermott (1969) and Kuhlmann (1989). In order

to achieve a consistent test database, all the data in this case was converted using the

method suggested by Alpsten (1972), where the stress values at 0.5% and 3% strain

form the linear slope. The details of the definition of the strain-hardening modulus

will be presented later in this chapter.

4.5 ANALYSIS OF EXISTING EXPERIMENTAL DATA

Complementary to these discussed in Section 4.2, the key factors influencing material

properties are reviewed experimentally in this section, based on the collected test data

presented in Section 4.4. Hot-rolled I-sections, hot-rolled hollow sections and cold-

formed sections are the focus of the following analysis.

The strain-hardening modulus Esh for the proposed material model was measured as

the linear slopes of the stress-strain curves between 0.5% and 3% strain. This was due

to the fact that the stress at 0.5% strain has been widely used as the yield stress by

other researchers (Kemp et al., 2002) and was shown to be similar to the 0.2% proof

stress for most steel grades (McDermott, 1969). 3% strain corresponds approximately

roughly to 15 times the yield strain, which is the minimum ductility requirement given

in EN 1993-1-1 (2005). For curves that peaked before the attainment of 3% strain, the

strains at peak loads were used in the calculation of Esh. However, if the strain at peak

load was too close to 0.5% strain, the data was excluded.

4.5.1 Variation in stress-strain characteristics

4.5.1.1 Hot-rolled I-sections

A total of 60 normalised tensile stress-strain curves reported by Driscoll and Beedle

(1957), Byfield and Nethercot (1998) and Byfield et al. (2005) have been plotted in

Fig. 4.16, where f denotes applied stress, fy denotes yield stress, ε denotes strain and

εy denotes yield strain (= fy/E). The data exhibits a relatively wide envelope and show

variation both between and within grades.


96

Fig. 4.16: Stress-strain characteristics of hot-rolled I-sections

4.5.1.2 Hot-rolled hollow sections

In order to assess the variation of material properties in hot-rolled hollow sections, a

total of 46 normalised tensile stress-strain curves from Gardner et al. (2010), Chan

and Gardner (2008a), Law (2010), Liu (2010) and Packer et al. (2010) have been

plotted in Fig. 4.17. CHS were seen to display a shorter plastic plateau and an earlier

onset of strain-hardening in comparison to data plotted from RHS and SHS. The onset

of strain-hardening in EHS was observed to lie between these two sets of data.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25 30 ε/εy

f/fy

S275 Byfield and Nethercot (1998) S275 Byfield et al. (2005) S355 Byfield et al. (2005)

Unknown grade Driscoll and Beedle (1957)


97

Fig. 4.17: Stress-strain characteristics of hot-rolled hollow sections

4.5.1.3 Comparison between hot-rolled I and hollow sections

To distinguish between the variation in mechanical properties of hot-rolled I-sections

and hollow sections, the mean normalised stress-strain curves for the two collected

data sets was determined, as illustrated in Fig. 4.18. The mean curves indicate an

earlier onset of strain-hardening in I-sections than in hollow sections. The reduced

plastic plateau in I-sections is believed to be due to the presence of residual stress and

non-uniform strength induced uneven during cooling of the sections. The geometry of

hollow sections gives rise to more uniform cooling and hence lower residual stresses

and consistent strength around the cross-section (Chan and Gardner, 2008a; Gardner

et al., 2010).

RHS and SHS (Gardner et al., 2010) EHS (Chan and Gardner, 2008a) EHS (Law, 2010) CHS (Liu, 2010) CHS (Packer et al., 2010)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25 30 35 40 45 50 ε/εy

f/fy


98

Fig. 4.18: Mean normalised stress-strain curves for hot-rolled I-sections and hollow

sections

4.5.2 Variation in strain-hardening properties of coupon tests

The strain-hardening moduli Esh (based on 0.5% and 3% strain) have been determined

for all coupon tests results summarised in Tables 4.5 to 4.7 and have been plotted in

Fig. 4.19. Linear regression curves based on the least square method (Douglas et al.,

2001) for each set of data have been added. The strain-hardening modulus suggested

by Kemp et al. (2002) (equivalent to Esh/E = 0.013) and Esh/E = 0.01, as given in the

Annex C in EN 1993-1-5 (2006), are also illustrated.

ε/εy

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25

f/fy

Hot-rolled I-sections Hot-rolled hollow sections

Onset of strain-hardening


99

Fig. 4.19: Strain-hardening properties of hot-rolled I-sections, hot-rolled hollow

sections and cold-formed hollow sections

There is a general trend that the rate of strain-hardening increases with an increasing

ratio of fu/fy, though the scatter in the data is relatively high. This scatter is attributed

to the sensitivity of the strain-hardening modulus to variations experienced, within a

given section type, in the forming process, cooling conditions, straightening

operations, section geometry and so on. For the purpose of the study, which is to

propose a practical material model which balances present accuracy with minimum

increase in application complexity, the observed trends shown in Fig. 4.19 have

offered sufficient information for further development.

4.5.2.1 Comparison of hot-rolled I-sections and hollow sections

Comparing the strain-hardening property between hot-rolled I-sections and hollow

sections shown in Fig. 4.19, it is clear that I-sections exhibit a higher degree of strain-

hardening than hollow sections with similar ratios of fu/fy.

4.5.2.2 Comparison of hot-rolled and cold-formed hollow sections

In Fig. 4.19, the comparison between hot-rolled and cold-formed hollow sections

reveals that most hot-rolled sections have higher fu/fy ratios than the cold-formed ones,

0

0.005

0.010

0.015

0.020

0.025

0.030 E s

h/E

1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 fu/fy

Kemp et al. (2002) Esh/E = 0.013

EN 1993-1-5 (2006) Esh/E = 0.01

Hot-rolled I-sections Hot-rolled hollow sections Cold-formed hollow sections

Linear (Cold-formed hollow sections) Linear (Hot-rolled hollow sections) Linear (Hot-rolled I-sections)


100

but cold-formed sections usually exhibit higher strain-hardening properties than the

hot-rolled ones. This is due to the significantly higher plastic deformation undergone

during the cold-forming process; this reduces the fu/fy ratio and induces an earlier

onset of strain-hardening. The values of the strain-hardening modulus are

subsequently increased.

4.5.2.3 Other sections (cruciform and plate sections)

In addition to the aforementioned cross-section shapes, tensile test results on

cruciform and plate sections are plotted in Fig. 4.20. The database is clearly rather

limited, through a general trend of Esh/E increasing with fu/fy.

Fig. 4.20: Strain-hardening property on cruciform and plate sections

4.5.3 Variation in the strain-hardening properties from stub column tests

Owing to the absence of stub column tests on hot-rolled I-sections, only the hot-rolled

and cold-formed hollow sections and cruciform section tests are considered here in

the investigation of the variation in strain-hardening properties.

0

0.005

0.01

0.015

0.02

0.025

0.03

1.00 1.10 1.20 1.30 1.40 1.50 1.60 fu/fy

E/E s

h

EN 1993-1-5 (2006) Esh/E = 0.01 Kemp et al. (2002) Esh/E = 0.013

Cruciform sections Plate sections


101

4.5.3.1 Comparison of hot-rolled and cold-formed hollow sections

Strain-hardening moduli Esh, calculated from a total of 19 stub column tests,

summarised in Tables 4.3 and 4.4 have been plotted in Fig. 4.21. These sections were

all sufficiently stocky that local buckling occurred after 3%, thus enabling Esh to be

determined, as for the tensile coupons, on the basis of strength at 0.5% and 3% strain.

Linear regression occurs for both sets of data and the strain-hardening modulus

recommendations given in Annex C in EN 1993-1-5 (2006) and Kemp et al. (2002)

are also depicted. Similar to the trend observed in coupon tests, cold-formed sections

have shown a higher rate of strain-hardening than hot-rolled ones, though in the full

section tests, the influence will also be present from the enhanced strength corner

regions.

Fig. 4.21: Stub column tests – hot-rolled versus cold-formed hollow sections

4.5.3.2 Cruciform sections

A total of 7 stub column tests on cruciform sections carried out by McDermott (1969)

are plotted in Fig. 4.22. Similar to the observation above, the trend that the higher the

ratios of fu/fy the higher the rate of strain-hardening has also been found.

0

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

1.00 1.10 1.20 1.30 1.40 1.50

Hot-rolled Cold-formed Linear (Hot-rolled) Linear (Cold-formed)

EN 1993-1-5 (2006) Esh/E = 0.01

Kemp et al. (2002) Esh/E = 0.013

fu/fy

E sh/E


102

Fig. 4.22: Stub column tests – Cruciform sections

4.5.4 Distinction between tensile coupon and stub column tests

Direct comparison of the strain-hardening properties between tensile coupon and stub

column tests has been made, and the results presented in the following sub-sections.

Owing to the absence of stub column tests on hot-rolled I-sections, the comparisons

were carried out on hot-rolled and cold-formed hollow sections only.

4.5.4.1 Hot-rolled hollow sections

Tensile coupon and stub column tests on hot-rolled hollow sections are plotted in Fig.

4.23, together with existing recommendations (EN 1993-1-5, 2006; Kemp et al., 2002)

for strain-hardening slopes. The stub column tests may be seen to exhibit higher

strain-hardening moduli than the tensile coupon tests. As discussed previously in

Section 4.2.7, this is attributed to an earlier onset of strain-hardening due to the

variable material properties and residual stresses that occur around the complete

cross-section.

0

0.005

0.010

0.015

0.020

1.00 1.05 1.10 1.15 1.20

EN 1993-1-5 (2006) Esh/E = 0.01

Kemp et al. (2002) Esh/E = 0.013

fu/fy

E/E s

h


103

Fig. 4.23: Coupon versus stub column tests – hot-formed hollow sections

Again, the stub column tests may be seen to exhibit higher strain-hardening moduli

than the tensile coupon tests.

4.5.4.2 Cold-formed hollow sections

Tensile coupon and stub column tests from cold-formed hollow sections are plotted in

Fig. 4.24 for comparison of their strain-hardening properties, where existing

recommendations (EN 1993-1-5, 2006; Kemp et al., 2002) are also shown.

E sh/E

0

0.002

0.004

0.006

0.008

0.010

0.012

0.014

1.00 1.10 1.20 1.30 1.40 1.50 1.60 fu/fy

Stub column tests Coupon tests Linear (Stub) Linear (Coupon)

EN 1993-1-5 (2006) Esh/E = 0.01

Kemp et al. (2002) Esh/E = 0.013


104

Fig. 4.24: Coupon versus stub column tests – cold-formed hollow sections

4.5.4.3 Cruciform sections

Comparisons between the strain-hardening properties from tensile coupon and stub

column tests on cruciform sections have been made and presented in Fig. 4.25, where

existing recommendations (EN 1993-1-5, 2006; Kemp et al., 2002) are also shown.

As before, it is clear that stub column tests exhibit higher rates of strain-hardening

than tensile coupon tests.

0.000

0.005

0.010

0.015

0.020

1.00 1.05 1.10 1.15 1.20 1.25 1.30 fu/fy

E sh/E


EN 1993-1-5 (2006) Esh/E = 0.01

Kemp et al. (2002) Esh/E = 0.013


105

Fig. 4.25: Tensile coupon versus stub column tests – cruciform sections

4.5.4.4 Summary

The results show that using tensile coupon test data to evaluate the strain-hardening

characteristics of structural steel, which is a common approach by researchers

(Byfield and Nethercot, 1998; Byfield and Nethercot, 1997; Kemp et al., 2002;

Gardner et al., 2010) and codes (ASTM, 2000; EN 10025-1, 2004), is safe.

4.6 PROPOSALS OF MATERIAL MODELS

The ratio of ultimate stress to yield stress fu/fy has been found to be a key factor in

influencing material strain-hardening characteristics as described in Section 4.2.

Section 4.3 suggests that the elastic, linear-hardening material model adopted in

Annex C in EN 1993-1-5 (2006) offers a good representation of structural steel

behaviour at the material level, as well as for complete structures and with little

increase in complexity over the traditional elastic, perfectly-plastic model (Gardner

and Wang, 2010). However, the model given in EN 1993-1-5 has a fixed strain-

hardening modulus of E/100, which does not take into account the influences of

different steel grades, processing routes and section types. Also, results show that this

0

0.005

0.010

0.015

0.020

1.00 1.05 1.10 1.15 1.20


EN 1993-1-5 (2006) Esh/E = 0.01

Kemp et al. (2002) Esh/E = 0.013

fu/fy

E sh/E


106

value generally underestimates stress corresponding to a given strain. Therefore, the

strain-hardening modulus needs to be re-evaluated.

As discussed in Sections 4.2.2 and 4.5.4, tensile coupon tests give more conservative

estimates of Esh than stub column tests, and have been widely used to represent

material characteristics in the past. Further, there is significantly more tensile test data

and hence these will be used as the foundation of the proposed models rather than

stub column tests.

On the basis of the above discussion, a new bi-linear material model with strain-

hardening modulus as a function of the ratio fu/fy has been proposed for hot-rolled I-

sections, hot-rolled hollow sections, cold-formed hollow sections and other sections,

and will be described in detail in the following sub-sections.

4.6.1 Hot-rolled I-sections

A total of 88 tensile coupon results from hot-rolled I-sections (Driscoll and Beedle,

1957; Sawyer, 1961; Kuhlmann, 1989; Byfield and Nethercot, 1998; Byfield et al.,

2005) have been gathered and are reported in Table 4.5. They have also been plotted

in Fig. 4.26, where E/Esh is replaced by Esh/E on the ordinate axis to facilitate

comparison of the extensive range of the values (including zero values). A simple

model has been proposed on the basis of the linear regression fitting of these 88 test

results defined in Eqs (4.7a-b) and is shown in Fig. 4.26.

Esh/E 0.17.1

)0.1f/f(015.0 yu

−

−= for fu/fy ≤ 1.70 (4.7a)

Esh/E = 0.015 for fu/fy > 1.70 (4.7b)


107

Fig. 4.26: Proposed material model for hot-rolled I-sections

4.6.2 Hot-rolled hollow sections

Similar to hot-rolled I-sections, a simple model based on a total of 42 tensile coupon

results (Chan and Gardner, 2008a; Gardner et al., 2010; Law, 2010; Liu et al., 2010;

Packer et al., 2010) for hot-rolled hollow sections has been proposed. The expression

of the model is given in Eqs (4.8a-c) and plotted in Fig. 4.27, where test data and

existing recommendations for strain-hardening modulus are also shown. The figure

shows that most of the test data give lower values for the strain-hardening moduli than

recommended by EN 1993-1-5 (2006) and Kemp et al. (2002). For low values of fu/fy,

both stub column and tensile coupon data are considered for hot-rolled hollow

sections, since the tensile coupon data give unduly long yield plateaus that do not

represent the true response of complete structural cross-sections.

Esh/E 0.13.1

)0.1f/f(003.0 yu

−

−= for fu/fy ≤ 1.30 (4.8a)

Esh/E 3.16.1

)3.1f/f(007.0003.0 yu

−

−+= for 1.30 < fu/fy ≤ 1.60 (4.8b)

Esh/E = 0.01 for fu/fy > 1.60 (4.8c)

E sh/E

0.000

0.005

0.010

0.015

0.020

0.025

1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 fu/fy

Test data Linear regression Proposed model


108

Fig. 4.27: Proposed material model for hot-rolled hollow sections

4.6.3 Cold-formed hollow sections

A total of 42 tensile coupon tests on cold-formed hollow sections reported in Table

4.7, excluding 3 tests carried by Guo et al. (2007), which were ignored due to the

discrepancies in stress-strain results, have been plotted in Fig. 4.28. The two values of

strain-hardening moduli given by EN 1993-1-5 (2006) and Kemp et al. (2002) are also

included. A simple model based on a linear regression fit to this data has been

proposed in Eqs (4.9a-b).

Esh/E 00.125.1

)00.1f/f(015.0 yu

−

−= for fu/fy ≤ 1.25 (4.9a)

Esh/E = 0.015 for fu/fy > 1.25 (4.9b)

0

0.002

0.004

0.006

0.008

0.010

0.012

0.014

1.00 1.10 1.20 1.30 1.40 1.50 1.60 fu/fy

E sh/E

EN 1993-1-5 (2006) Esh/E = 0.01

Kemp et al. (2002) Esh/E = 0.013

Stub column test

Linear regression (Stub column test)

Linear regression (Coupon test)

Coupon test

Proposed model


109

Fig. 4.28: Proposed material model for cold-formed hollow sections

4.6.4 Other sections

Limited test results were collected on welded and plated sections, as already

summarised in Tables 4.8 and 4.9, and have been plotted in Fig. 4.20. The range and

number of the available test data are too limited to propose a material model

applicable to both cruciform and plate sections. The current recommendation Esh/E =

0.01 given by EN 1993-1-5 (2006) approximately represents the material behaviour as

shown in Figs 4.29 and 4.30 for cruciform and plated sections respectively. For these

two sections, the recommended material model therefore remains unchanged from EN

1993-1-5 (2006), as given by Eq. (4.10).

Esh/E = 0.01 for any fu/fy (4.10)

0.000

0.005

0.010

0.015

0.020

1.00 1.05 1.10 1.15 1.20 1.25 1.30 fu/fy

E sh/E

EN 1993-1-5 (2006) Esh/E = 0.01

Kemp et al. (2002) Esh/E = 0.013

Stub column test

Linear regression (Stub column test)

Linear regression (Coupon test)

Coupon test

Proposed model


110

Fig. 4.29 Strain-hardening properties of cruciform sections

Fig. 4.30 Strain-hardening properties of plated sections

4.6.5 Summary

The proposed models for various section types are summarised in Table 4.10 and

illustrated in Fig. 4.31 They are more accurate in representing each category of

section type in comparison to the elastic, perfectly-plastic or bi-linear material model

with a fixed strain-hardening modulus of Esh/E = 0.01 employed by EN 1993-1-5

1.00 1.05 1.10 1.15 1.20

EN 1993-1-5 (2006) Esh/E = 0.01

Kemp et al. (2002) Esh/E = 0.013

fu/fy

E sh/E

Test data

0.000

0.005

0.010

0.015

0

0.005

0.01

0.015

0.02

0.025

0.03

1.00 1.10 1.20 1.30 1.40 1.50 1.60

fu/fy

E sh/E

EN 1993-1-5 (2006) Esh/E = 0.01

Kemp et al. (2002) Esh/E = 0.013


111

(2006). For section types out of the categories below, Esh/E = 0.01 as adopted in

Annex C EN 1993-1-5 (2006) is employed.

Depending on (1) the ratio fu/fy, which is readily available to designers, (2) the cross-

section type and, (3) the forming routes, designers can easily obtain an elastic, linear-

hardening model according to Table 4.10.

Table 4.10: Summary of proposed material models

Hot-rolled I-sections

Esh/E 0.17.1

)0.1f/f(015.0 yu

−

−= for fu/fy ≤ 1.70 (4.7a)

Esh/E = 0.015 for fu/fy > 1.70 (4.7b)

Hot-rolled hollow sections

Esh/E 0.13.1

)0.1f/f(003.0 yu

−

−= for fu/fy ≤ 1.30 (4.8a)

Esh/E 3.16.1

)3.1f/f(007.0003.0 yu

−

−+= for 1.30 < fu/fy ≤ 1.60 (4.8b)

Esh/E = 0.01 for fu/fy > 1.60 (4.8c)

Cold-formed hollow sections

Esh/E 00.125.1

)00.1f/f(015.0 yu

−

−= for fu/fy ≤ 1.25 (4.9a)

Esh/E = 0.015 for fu/fy > 1.25 (4.9b)

Cruciform sections


Plated sections



112

Fig. 4.31 Summary of the proposed material models


The key factors influencing strain-hardening have been investigated in this chapter,

and existing material models have been appraised. The review suggested that a bi-

linear model, which a strain-hardening slope as a function of the ratio of fu/fy provides

a simple, yet reasonably accurate representation of observed stress-strain

characteristics.

Based on a total of 204 tensile coupon test results, proposed material model has been

calibrated, and a summary presented in Table 4.10. This model will be employed in

Chapters 5 and 6 for the development of a new deformation based design method for

steel structures.

1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 fu/fy

E sh/E

Cold-formed hollow sections

Hot-rolled hollow sections

Cruciform and plated sections (the same as EN 1993-1-5)

Hot-rolled I-sections

0.020

0.010

0.015

0.005

0

Chapter 5 Determinate structures

113

CHAPTER 5

DETERMINATE STRUCTURES

5.1 INTRODUCTION

Current steel design codes, including EN 1993-1-1 (2005), use cross-section

classification to identify the extent to which the compression and bending resistances

of cross-sections are limited by local buckling. The current approach generally leads

to a conservative prediction of load-carrying capacities for stocky cross-sections. This

conservatism has been demonstrated in Chapters 3 and 4 and attributed to the

occurrence of strain-hardening which refers to the increase in strength of metallic

materials beyond yield as a result of plastic deformation. Strain-hardening is not

systematically utilised in current international steel design codes. This limits the

resistance of cross-sections in compression to the yield load Ny (defined as the cross-

sectional area A multiplied by the material yield strength fy) and the resistance of

cross-sections in bending to the plastic moment capacity Mpl (defined as the plastic

section modulus Wpl multiplied by the material yield strength fy).


114

A new design approach, the continuous strength method (CSM), has been developed

to enable more efficient design of steel structures offering a systematic means of

utilising strain-hardening, based on cross-section deformation capacity (Gardner,

2008). The method allows the attainment of compression resistances greater than the

yield load as well as bending resistances beyond the plastic moment capacity,

resulting in better prediction of observed structural behaviour.

In this chapter, existing test data on determinate structures including stub columns and

simple beams are first collected and added to that generated in the present study.

Existing design methods are then evaluated and the development of the continuous

strength method (CSM) is described. Finally, both approaches are subjected to

reliability analysis.

5.2 COLLECTION OF EXISTING TEST DATA

In this section, existing experimental data have been gathered and analysed to validate

the developments of the CSM. The scope of the existing data is fairly comprehensive

and covers compression and bending of I-sections, as well as RHS and SHS, both hot-

rolled and cold-formed. The assembled database provides valuable information for the

validation of the CSM and the corresponding reliability study. The focus of the

investigation is on non-slender cross-sections, since these are the ones that benefit

most from consideration of strain-hardening. The present study is also currently

restricted to cross-section capacity.

5.2.1 Stub column tests

5.2.1.1 Hot-rolled sections

Available test results on hot-rolled stub columns are those generated in Chapter 3 and

described in Gardner et al. (2010). A total of 10 stub column tests were carried out –

two repeated tests on each section size, denoted ‘1’ and ‘2’ in Table 5.1. The

specimen designation begins with the section size, e.g. SHS 100×100×4, followed by

the production route – HR for hot-rolled, and finally the test number – 1 or 2. The

geometric properties, as defined in Fig. 5.1, tensile material yield stress fy and

ultimate load carrying capacities Nu are reported in Table 5.1, in which B and D are


115

the outer cross-section dimensions, t is the section thickness, ri is the internal corner

radius, b/t is the slenderness of the most slender constituent element in the cross-

section and Nu is the ultimate test load.

Fig. 5.1: Section notation for hot-rolled steel SHS and RHS

Table 5.1: Geometric properties and ultimate capacities of the hot-rolled stub

columns

Stub column specimen D (mm)

B (mm)

t (mm)

ri (mm)

fy (N/mm2) b/t Nu

(kN) Nu Afy

SHS 100×100×4-HR1 100.01 100.89 4.09 2.75 488 21.3 706 1.09 SHS 100×100×4-HR2 99.83 100.84 4.11 2.75 488 21.2 707 1.06 SHS 60×60×3-HR1 60.21 60.18 3.35 2.38 449 14.6 353 1.26 SHS 60×60×3-HR2 60.22 60.23 3.38 2.44 449 14.4 363 1.25 RHS 60×40×4-HR1 59.84 40.09 3.83 1.88 468 12.6 344 1.19 RHS 60×40×4-HR2 59.72 40.17 3.83 1.94 468 12.5 346 1.20 SHS 40×40×4-HR1 40.00 39.94 3.91 2.06 496 7.2 333 0.94 SHS 40×40×4-HR2 40.05 39.94 3.91 2.25 496 7.1 335 0.94 SHS 40×40×3-HR1 40.25 40.23 3.05 2.13 504 9.8 263 1.08 SHS 40×40×3-HR2 40.16 40.10 3.05 2.00 504 10.0 257 1.08

ri

t

B

D y y

z

b

d


116

5.2.1.2 Cold-formed sections

A total of 23 cold-formed stub column tests have been collected from the literature

(Gardner et al., 2010; Zhao and Hancock, 1991; Wilkinson and Hancock, 1997;

Akiyama et al., 1996). In the series of stub column tests conducted by Gardner et al.

(2010) to investigate the distinction in structural behaviour between hot-rolled and

cold-formed materials, 10 cold-formed stub column tests have been collected, with

specimens cut from the same nominal section dimensions and tested in the same

configurations as the 10 hot-rolled ones reported in the previous sub-section. 7 out of

the 10 stub tests reported by Zhao and Hancock (1991) have been added to the

database, with the remaining 3 being slender sections and exhibiting local buckling

failure prior to yielding. For the same reason, only 1 of the 13 cold-formed specimens

tested by Wilkinson and Hancock (1997) was collected. A further 5 SHS tests

conducted by Akiyama et al. (1996) have been gathered, although detailed geometric

data were not reported; only information on the B/t ratios were given, where B is the

outer width of plate and t is the thickness as defined in Fig. 5.2. The internal corner

radii were assumed to be equal to the thickness and hence the B/t ratios could be

converted into b/t ratios, where b is the width of the flat part of the flange, as defined

in Fig. 5.2. The geometric and material properties and ultimate capacities of the cold-

formed stub columns are summarised in Table 5.2, where dimensions have been

defined previously and f0.2 denotes 0.2% proof strength, taken as an equivalent yield

strength.


117

Fig. 5.2: Section notation for cold-formed steel SHS and RHS



B (mm)

t (mm)

ri (mm) b/t f0.2

(N/mm2) Nu

(kN) Nu Afy

SHS 100×100×4.0-CF1a 100.55 100.56 3.59 6.38 22.5 482 660 1.02 SHS 100×100×4.0-CF2a 100.75 100.68 3.61 6.13 22.5 482 663 1.02 SHS 60×60×3.0-CF1a 60.30 60.14 2.78 3.75 17.0 361 249 1.12 SHS 60×60×3.0-CF2a 60.17 60.17 2.79 3.88 16.8 361 250 1.13 RHS 60×40×4.0-CF1a 60.09 40.07 3.95 2.19 12.1 400 370 1.32 RHS 60×40×4.0-CF2a 60.06 40.00 3.97 1.94 12.2 400 370 1.31 SHS 40×40×4.0-CF1a 40.36 40.32 3.76 3.13 7.1 410 256 1.21 SHS 40×40×4.0-CF2a 40.32 40.31 3.79 3.06 7.0 410 256 1.20 SHS 40×40×3.0-CF a 40.12 40.13 2.76 2.56 10.7 451 224 1.26 SHS 40×40×3.0-CF2a 40.04 40.07 2.75 2.69 10.6 451 230 1.30 SHS 100×100×3.8-CFb 100.38 100.12 3.80 5.70 21.4 459 766 1.18 SHS 100×100×3.3-CFb 100.07 100.15 3.46 4.54 24.3 435 662 1.17 SHS 75×75×3.3-CFb 75.25 75.13 3.38 4.12 17.8 462 494 1.14 SHS 75×75×2.8-CFb 75.15 75.13 2.79 2.71 23.0 490 415 1.08 SHS 75×75×2.3-CFb 75.23 75.05 2.29 2.21 28.9 469 285 0.93 SHS 65×65×2.3-CFb 64.92 65.07 2.28 2.22 24.6 479 281 1.05 RHS 125×75×3.8-CFb 125.02 75.10 3.79 5.71 28.0 448 665 1.05 RHS 150×50×5.0-CFc 150.99 50.11 4.89 6.91 26.1 441 878 1.11 SHS-CF1d - - - - 21.0 435 - 1.13 SHS-CF2d - - - - 11.8 411 - 1.25

ri

t

B

D y y

z

b

d

z


118


(continued)


B (mm)

t (mm)

ri (mm) b/t f0.2

(N/mm2) Nu

(kN) Nu Afy

SHS-CF3d - - - - 17.1 440 - 1.16 SHS-CF4d - - - - 21.0 382 - 1.14 SHS-CF5d - - - - 29.3 367 - 1.04 a Gardner et al. (2010); b Zhao and Hancock (1991); c Wilkinson and Hancock (1997); d Akiyama et al. (1996)

5.2.1.3 Welded sections

A total of 15 stub column tests on welded sections reported by Akiyama et al. (1996),

Rasmussen and Hancock (1992) and Gao et al. (2009) have been collected. 10 tests

carried out by Akiyama et al. (1996) and 4 tests carried by Rasmussen and Hancock

(1992) were performed on welded box sections which were fabricated by welding 4

equal length plates at the corners, as shown in Fig. 5.3, where the section notation and

locations of the welds are given. The geometric and material properties and the

ultimate capacities of the collected welded stub columns are summarised in Table 5.3,

where dimensions and notation have been previously defined.

Fig. 5.3: Section labelling convention and location of welds (between four plates)

(Akiyama et al., 1996; Rasmussen and Hancock, 1992)

t

B

D

b Weld

d


119

Table 5.3: Geometric properties and ultimate capacities of welded box stub columns

Stub column specimens

D (mm)

B (mm)

t (mm) b/t f0.2

(N/mm2) Nu

(kN) Nu Afy

Welded SHS-1a - - - 14.0 353 - 1.28 Welded SHS-2a - - - 16.2 377 - 1.03 Welded SHS-3a - - - 18.5 372 - 1.00 Welded SHS-4a - - - 21.7 328 - 1.02 Welded SHS-5a - - - 26.1 345 - 1.00 Welded SHS-6a - - - 14.0 267 - 1.39 Welded SHS-7a - - - 16.0 262 - 1.33 Welded SHS-8a - - - 18.5 270 - 1.17 Welded SHS-9a - - - 21.7 285 - 1.03 Welded SHS-10a - - - 26.1 285 - 0.96 Welded SHS-11b 89.36 89.36 4.98 15.9 750 1174 0.96 Welded SHS-12b 89.52 89.52 4.96 16.0 750 1146 0.94 Welded SHS-13b 118.52 118.52 4.96 21.9 750 1499 0.91 Welded SHS-14b 118.62 118.62 4.96 21.9 750 1508 0.91 a Akiyama et al. (1996); b Rasmussen and Hancock (1992)

5.2.1.4 Press-formed and seam-welded sections

A total of 15 press-formed and seam-welded stub column test results have been

gathered from Akiyama et al. (1996). The specimen dimensions and weld locations

are as shown in Fig. 5.4. In the absence of the detailed reported data on dimensions,

the internal corner radii have been assumed to be equal to the section thickness in the

calculation of the b/t ratios. The available geometric properties and key test results of

these 15 specimens have been summarised in Table 5.4. A further test reported by

Gao et al. (2009) has also been collected. This specimen was produced by welding

two channel sections, which were cold-formed from Grade 18Mn2CrMoBA steel (a

high strength Chinese steel grade), tip-to-tip to form a box section, as shown in Fig.

5.4.


120

Fig. 5.4: Section labelling convention and location of welds (between two channels)

(Gao et al., 2009)

Table 5.4: Geometric properties and key results of press-formed and seam-welded

stub column tests

Stub column specimens

D (mm)

B (mm)

t (mm) b/t f0.2

(N/mm2) Nu

(kN) Nu Afy

Press-seam SHS-1a - - - 14.2 422 - 1.16 Press-seam SHS-2a - - - 19.7 368 - 1.13 Press-seam SHS-3a - - - 14.0 336 - 1.29 Press-seam SHS-4a - - - 19.7 327 - 1.16 Press-seam SHS-5a - - - 12.0 385 - 1.27 Press-seam SHS-6a - - - 14.2 402 - 1.22 Press-seam SHS-7a - - - 16.5 392 - 1.13 Press-seam SHS-8a - - - 19.7 343 - 1.14 Press-seam SHS-9a - - - 24.1 345 - 1.02 Press-seam SHS-10a - - - 12.0 313 - 1.38 Press-seam SHS-11a - - - 14.0 295 - 1.36 Press-seam SHS-12a - - - 16.5 309 - 1.24 Press-seam SHS-13a - - - 19.7 296 - 1.22 Press-seam SHS-14a - - - 24.1 296 - 0.99 Press-seam SHS-15a - - - 33.5 296 - 0.91 Press-seam SHS-16b 61.08 58.62 3.28 16.6 793 647.8 1.10 a Akiyama et al. (1996); b Gao et al. (2009)

Weld

t

B

D y y

z

z b

Weld

d

ri


121



Byfield and Nethercot (1998) carried out a total of 32 four-point bending tests on

laterally restrained stocky hot-rolled I-beams with sections: 203×102×23 Universal

Beam and 152×152×30 Universal Column. The section notation of the test specimens

has been illustrated in Fig. 5.5 and the test configuration has been sketched in Fig. 5.6.

A further two I-beam tests were reported by Popov and Willis (1957) and have also

been collected and added to the database. Cover plates were connected to the two

flanges of section ‘I beam-2’ with intermittent welding, as shown in Fig. 5.7. The

material properties and ultimate moment capacities Mu achieved, together with the

ratio of Mu to the plastic moment capacity Mpl, are reported in Table 5.5.

Fig. 5.5: Section designation of test specimens

Fig. 5.6: Test arrangement for four-point bending tests

tf

D

tf

tw

B

N/2

Ls2

N/2

Ls2 Ls1


122

Fig. 5.7: Section designation of test specimens with cover plates

Table 5.5: Geometric properties and key results for I-beam test specimens

Specimen D (mm)

B (mm)

tf (mm)

tw (mm)

fy (N/mm2)

Mu (kNm)

Mu Mpl

203×102×23-HR1a1 202.50 102.25 8.84 5.74 310 79.9 1.13 203×102×23-HR2a1 202.50 102.75 8.85 5.71 324 80.9 1.09 203×102×23-HR3a1 202.50 102.50 8.89 5.71 323 87.5 1.18 203×102×23-HR4a1 203.00 102.00 8.83 5.72 329 79.4 1.06 203×102×23-HR5a1 202.50 102.00 8.85 5.73 315 79.3 1.10 203×102×23-HR6a1 202.00 102.25 8.74 5.73 322 79.2 1.09 203×102×23-HR7a1 202.25 101.75 8.86 5.64 315 89.1 1.25 203×102×23-HR8a1 202.50 102.00 8.87 5.65 315 82.3 1.15 203×102×23-HR9a1 202.00 102.25 8.74 5.70 317 80.2 1.12 203×102×23-HR10a1 202.25 102.50 8.54 5.69 317 82.9 1.17 152×152×30-HR1a1 156.75 152.00 9.06 5.81 286 78.0 1.16 152×152×30-HR2a1 157.00 151.50 9.04 6.09 288 80.0 1.18 152×152×30-HR3a1 157.25 151.25 9.08 6.05 289 81.9 1.20 152×152×30-HR4a1 156.75 151.50 9.02 6.03 293 81.3 1.18 152×152×30-HR5a1 156.25 151.00 9.05 6.03 290 81.5 1.20 152×152×30-HR6a1 157.50 151.25 9.05 6.04 287 81.2 1.21 152×152×30-HR7a1 157.25 151.50 9.09 6.05 294 81.2 1.17 152×152×30-HR8a1 156.75 151.50 9.03 6.16 299 82.3 1.17 152×152×30-HR9a1 156.75 151.50 9.05 6.03 291 80.7 1.18 152×152×30-HR10a1 156.50 151.50 9.10 6.18 299 80.0 1.13 203×102×23-HR11a2 202.00 102.00 8.74 5.67 303 86.1 1.26 203×102×23-HR12a2 203.00 102.75 8.77 5.77 330 94.2 1.25 203×102×23-HR13a2 202.50 102.25 8.88 5.57 330 87.7 1.17 203×102×23-HR14a2 202.50 103.00 8.74 5.73 317 91.9 1.27 203×102×23-HR15a2 202.75 103.00 8.79 5.66 314 88.4 1.23 203×102×23-HR16a2 203.00 102.00 8.89 5.57 316 85.7 1.19 152×152×30-HR11a2 157.50 151.75 9.05 6.05 284 83.0 1.24

tf

D

tf

tw

B

Bcp

tc

cover plate

cover plate


123

Table 5.5: Geometric properties and key results for I-beam test specimens (continued)

Specimen D (mm)

B (mm)

tf (mm)

tw (mm)

fy (N/mm2)

Mu (kNm)

Mu Mpl

152×152×30-HR12a2 157.25 151.75 9.07 5.80 290 83.3 1.22 152×152×30-HR13a2 156.50 151.00 9.05 6.04 284 82.0 1.23 152×152×30-HR14a2 156.75 150.75 9.09 6.11 285 82.4 1.23 152×152×30-HR15a2 156.50 151.25 9.03 5.93 292 81.7 1.20 152×152×30-HR16a2 156.50 151.25 9.05 6.06 283 84.7 1.28 I Beam-1b 127.00 76.30 5.44 8.28 279 33.9 1.33 I Beam-2b* 127.00 76.30 5.44 8.28 279 53.7 1.28 a1 Byfield and Nethercot (1998), the configuration with Ls1 = 800 mm and Ls2 = 650 mm in Fig. 5.6; a2 Byfield and Nethercot (1998), the configuration with Ls1 = 1100 mm and Ls2 = 1100 mm in Fig. 5.6; b Popov and Willis (1957); * Cover plates employed, as shown in Fig. 5.7 with Bcp = 101.6 mm, tc = 4.8 mm and fy = 256 N/mm2

5.2.2.2 Hot-rolled SHS and RHS

A total of 3 simply supported beam tests were conducted, as reported in Chapter 3, (in

three-point bending) to obtain the basic flexural response characteristics and ultimate

moment capacities of simple beams of hot-rolled SHS and RHS. Full details of the

tests are reported by Gardner et al. (2010) and in Chapter 3, while a summary of the

results is presented in Table 5.6, in which geometric dimensions have been defined in

Fig. 5.1. Note that all the test specimens are Class 1 or 2 according to EN 1993-1-1

(2005).

Table 5.6: Summary of simple beam tests on hot-rolled SHS and RHS


D (mm)

B (mm)

t (mm)

ri (mm)

fy (N/mm2)

Mu (kNm)

Mu Mpl

SHS 40×40×4-HR 39.75 40.00 3.91 2.16 496 3.84 1.09 SHS 40×40×3-HR 39.87 40.20 3.05 2.07 504 3.44 1.16 RHS 60×40×4-HR 60.09 40.24 3.90 1.91 468 7.14 1.14 5.2.2.3 Cold-formed SHS and RHS

A total of 53 simple beam tests on cold-formed SHS and RHS have been collected

from the literature (in three-point and four-point bending). Similar to the 3 hot-rolled


124

beam tests reported in the previous section, there were 3 cold-formed SHS and RHS

simple beam tests in three-point bending reported by Gardner et al. (2010) and in

Chapter 3, while a summary of the tests is given in Table 5.7. There were 41 simply

supported beam tests in four-point bending on cold-formed SHS and RHS conducted

by Wilkinson and Hancock (1998). The test arrangements, designed to avoid lateral-

torsional buckling was similar to the one shown in Fig. 5.6, where Ls1 = 800 mm and

Ls2 = 450 mm for RHS with nominal depth D ≥ 100 mm and Ls1 = 500 mm and Ls2 =

400 mm for RHS with nominal depth D ≤ 75 mm. A further 9 simple beams tests,

with a four-point bending test arrangement similar to that in Fig. 5.6 with Ls1 = 500

mm and Ls2 = 250 mm, were carried out by Zhao and Hancock (1991). These results

have been collected and added to the database. The geometric properties, as defined in

Fig. 5.2, and ultimate moment capacities are given in Table 5.7.

Table 5.7: Summary of simple beam tests on cold-formed SHS and RHS

Simple beam specimen D (mm)

B (mm)

t (mm)

ri (mm)

fy (N/mm2)

Mu (kNm)

Mu Mpl

SHS 40×40×4-CFa 40.31 40.42 3.70 3.10 410 3.61 1.28 SHS 40×40×3-CFa 40.16 40.11 2.80 2.63 451 3.09 1.25 RHS 60×40×4-CFa 60.04 40.09 3.93 2.07 400 7.59 1.41 RHS 100×50×2.0-CF1b 100.83 50.52 2.05 1.75 400 7.75 1.00 RHS 100×50×2.0-CF2b 100.91 50.43 2.06 1.54 400 7.70 0.99 RHS 100×50×2.0-CF3b 100.46 50.24 2.04 2.66 449 9.30 1.10 RHS 100×50×2.0-CF4b 100.49 50.55 2.07 1.83 449 8.80 1.01 RHS 100×50×2.0-CF5b 100.45 50.22 2.04 1.36 423 8.75 1.11 RHS 100×50×2.0-CF6b 100.45 50.70 2.06 1.74 449 9.30 1.07 RHS 125×75×2.5-CF1b 125.40 75.10 2.53 1.37 374 16.3 1.06 RHS 125×75×3.0-CF1b 125.40 75.56 2.91 4.19 397 18.7 1.03 RHS 125×75×3.0-CF2b 125.40 75.74 2.93 3.97 397 19.1 1.04 RHS 125×75×3.0-CF3b 125.56 75.84 2.92 3.68 397 18.9 1.03 RHS 150×50×2.3-CF1b 150.37 50.70 2.26 2.54 444 17.3 0.98 RHS 150×50×2.3-CF2b 150.65 50.64 2.25 2.35 444 17.4 0.98 RHS 150×50×2.3-CF2b 150.51 50.57 2.28 1.92 444 18.2 1.01 RHS 150×50×2.5-CF1b 150.31 50.40 2.64 2.66 440 22.6 1.11 RHS 150×50×2.5-CF2b 150.35 50.23 2.59 2.21 446 21.8 1.08 RHS 150×50×2.5-CF3b 150.43 50.15 2.60 2.00 446 20.8 1.02 RHS 150×50×2.5-CF4b 150.39 50.41 2.57 2.03 446 20.2 1.00


125

Table 5.7: Summary of simple beam tests on cold-formed SHS and RHS (continued)

Simple beam specimen D (mm)

B (mm)

t (mm)

ri (mm)

fy (N/mm2)

Mu (kNm)

Mu Mpl

RHS 150×50×3.0-CF1b 150.45 50.51 3.00 3.80 382 23.2 1.18 RHS 150×50×3.0-CF2b 150.50 50.19 2.96 3.54 370 21.7 1.15 RHS 150×50×3.0-CF3b 150.46 50.13 3.00 3.20 370 23.2 1.21 RHS 150×50×3.0-CF4b 150.38 50.51 3.00 3.30 382 23.9 1.21 RHS 150×50×3.0-CF5b 150.47 50.22 2.97 2.93 444 26.2 1.15 RHS 150×50×3.0-CF6b 150.79 50.01 2.95 2.85 444 26.3 1.16 RHS 150×50×3.0-CF7b 150.80 50.34 2.96 2.74 444 25.8 1.13 RHS 150×50×4.0-CF1b 150.32 50.21 3.90 4.00 349 29.7 1.30 RHS 150×50×4.0-CF2b 150.39 50.57 3.85 3.65 410 31.8 1.19 RHS 150×50×4.0-CF3b 150.42 50.11 3.89 3.41 457 37.3 1.25 RHS 150×50×4.0-CF4b 150.44 50.40 3.87 3.43 457 35.5 1.19 RHS 150×50×4.0-CF5b 150.43 50.27 3.92 2.88 457 38.6 1.27 RHS 150×50×4.0-CF6b 150.21 50.16 3.89 1.51 423 33.0 1.18 RHS 150×50×5.0-CF1b 150.92 50.41 4.90 5.80 441 41.1 1.17 RHS 150×50×5.0-CF2b 151.04 50.25 4.92 4.98 441 43.8 1.23 RHS 75×25×1.6-CF1b 74.90 25.20 1.54 1.86 439 3.25 1.15 RHS 75×25×1.6-CF2b 75.27 25.12 1.55 1.85 422 2.90 1.03 RHS 75×25×1.6-CF3b 75.19 25.25 1.56 1.84 422 2.82 1.00 RHS 75×25×1.6-CF4b 74.98 25.08 1.56 2.34 439 3.10 1.10 RHS 75×25×1.6-CF5b 75.24 25.12 1.54 1.56 439 3.16 1.11 RHS 75×50×2.0-CF1b 75.33 25.23 1.95 2.05 457 4.25 1.13 RHS 75×50×2.0-CF2b 75.31 25.28 1.98 1.72 457 4.24 1.11 RHS 75×50×2.0-CF3b 75.63 50.31 1.95 2.45 411 4.96 1.02 RHS 75×50×2.0-CF4b 75.48 50.10 1.94 2.46 411 5.00 1.04 SHS 100×100×3.8-CFc 100.27 100.17 3.80 5.70 459 29.0 1.25 SHS 100×100×3.3-CFc 100.18 100.20 3.47 4.53 435 26.7 1.31 SHS 75×75×3.3-CFc 75.17 75.13 3.37 4.13 462 14.0 1.21 SHS 75×75×2.8-CFc 75.13 75.17 2.78 2.72 490 12.2 1.18 SHS 75×75×2.3-CFc 75.17 75.08 2.30 2.20 469 8.70 1.04 SHS 65×65×2.3-CFc 64.93 65.10 2.28 2.22 479 6.98 1.12 SHS 125×75×3.8-CFc 125.02 75.12 3.76 5.74 448 33.1 1.29 RHS 125×75×3.3-CFc 125.03 75.10 3.27 4.23 452 28.6 1.24 RHS 100×50×2.8-CFc 100.02 50.18 2.82 2.68 451 15.0 1.31 a Gardner et al. (2010); b Wilkinson and Hancock (1998); c Zhao and Hancock (1991)


126

5.2.3 Discussion

A total of 62 stub column test results and 90 simple beam test results have been

collected from the present study and the literature. These assembled data are

employed in the following sections for the development and assessment of a new

proposed design approach for steel structures.

5.3 DESIGN APPROACH

This section begins with a brief description of the cross-section classification system,

which is widely employed by current structural steel design codes. A review of other

proposed design methods including stress-based, strain-based and other approaches

follow. Finally, the continuous strength method (CSM) is presented.

5.3.1 Cross-section classification

5.3.1.1 Methodology

Most structural steel design codes, including EN 1993-1-1 (2005), define four classes

of cross-section – Class1 (plastic), Class 2 (compact), Class 3 (semi-compact) and

Class 4 (slender). The moment-rotation characteristics and idealised bending stress

distributions associated with the four classes of cross-section are illustrated in Figs 5.8

and 5.9, respectively. Class 1 cross-sections are fully effective under pure

compression and are capable of reaching and maintaining their full plastic moment

Mpl in bending (and may therefore be used in plastic design). Class 2 cross-sections

have a somewhat lower deformation capacity, but are also fully effective in pure

compression and are capable of reaching their full plastic moment in bending. Class 3

cross-sections are fully effective in pure compression, but local buckling prevents

attainment of the full plastic moment in bending; bending moment resistance is

therefore limited to the elastic (yield) moment Mel. For Class 4 cross-sections, local

buckling occurs below the yield stress. The loss of effectiveness due to local buckling

(in the elastic material range) is generally accounted for by the determination of

effective cross-section properties based on the width-to-thickness ratios, boundary

conditions and loading conditions of the individual plate elements. The resulting

effective area Aeff (for compression) and effective modulus Weff (for bending) is then


127

used to determine cross-section resistance. The compressive and flexural resistance

according to EN 1993-1-1 (2005) are illustrated in Fig. 5.10, where fy is the material

yield stress, Wel is elastic section modulus and Wpl is plastic section modulus. Note

that for cold-formed sections, the 0.2% proof stress f0.2 is used as the equivalent yield

stress.

Fig. 5.8: Moment-rotation response of four behavioural classes of cross-section

Fig. 5.9: Idealised bending stress distributions (symmetric section)

fy

fy

(a) Class 1 and 2 (b) Class 3 (c) Class 4

Loss of effectiveness due to local buckling

fy

fy

fy

Neutral axis

Mpl

Mel Class 1

Class 2

Class 4

Class 3

Rotation θ

App

lied

mom

ent M


128

Fig. 5.10: Cross-section compression and bending resistances according to EN 1993-

1-1

Although not explicitly included in the determination of resistance, strain-hardening is

an essential component of the described section classification system, for example, to

enable the attainment of the plastic moment Mpl at finite strains. This feature has been

recognised by the European design code for aluminium – EN 1991-1-1 (2007). Annex

F of EN 1999-1-1 (2007) provides a method, considering strain-hardening, for

determining the post-elastic resistance of cross-sections according to the mechanical

properties of the material and the geometrical features of the section. Compared with

steel, aluminium typically has lower ductility and a lower ratio of fu/fy, so the use of

strain-hardening would be expected to have a greater impact on steel design. A

summary of the EN 1999-1-1 (2007) method is illustrated in Fig. 5.11, where α10 and

α5 are two section generalized shape factors related to ultimate curvature values (EN

1999-1-1, 2007).

(a) Cross-section compression resistance

NEd

Afy

Class 1, 2 and 3 Class 4

Slenderness

Welfy

Class 1 and 2 Class 4

MEd

Wplfy

Class 3

Slenderness (b) Cross-section bending resistance


129

Fig. 5.11: Cross-section resistances for aluminium sections according to Annex F of

EN 1999-1-1 (2007)

5.3.1.2 Shortcomings of cross-section classification

To illustrate the shortcomings of cross-section classification, test data for structural

steel sections in compression and bending are plotted in Figs 5.12 and 5.14

respectively. Fig. 5.12 shows the results of stub column tests (Gardner et al., 2010;

Zhao and Hancock, 1992; Wilkinson and Hancock, 1997; Akiyama et al., 1996;

Rasmussen and Hancock, 1992; Gao et al., 2009; Feng et al., 2003; Ge and Usami,

1992; Han et al., 2004; Tao et al., 2004; Uy, 1998) on structural steel square and

rectangular hollow sections and lipped channels and the slenderness limit 0.673 for

internal plate elements according to EN 1993-1-5 (2006) has been presented.

(b) Cross-section bending resistance

Welf0.2

Class 1 Class 4

MEd

Slenderness

Class 2

Wplf0.2

Class 3

α10(α5)Welf0.2

Af0.2

NEd

Slenderness

Class 1 Class 4 Class 2 and 3

Afu

(a) Cross-section compression resistance


130

Fig. 5.12: Stub column test results

The maximum load-carrying capacity of the stub columns Nu has been normalised by

the yield load (determined as the gross cross-sectional area A multiplied by the

material yield strength fy) and plotted against the maximum slenderness of the

constituent plate elements pλ , defined by Eq. (3.7).

The slenderness limit beyond which cross-sections are deemed not to be fully

effective, together with the curve representing reduction factors for loss of

effectiveness from Eurocode 3 are indicated in Fig. 5.12, and may be seen to accord

well with the test data. However, the test data also reveal significant conservatism

when the resistance of stocky cross-sections is limited to the yield load; this is due to

the occurrence of strain-hardening.

Typical normalised load-end shortening responses for both hot-rolled and cold-

formed sections – SHS 40×40×3-HR1 and SHS 40×40×3-CF1 – reported by Gardner

et al. (2010) are plotted in Fig. 5.13 to demonstrate the origin of the conservatism. On

the vertical axis, the test load N has been normalised by the yield load (Afy) and on

the horizontal axis, the end shortening δ of the test specimens has been normalised by

the stub column length L. The influence of strain-hardening, characterised by load-

carrying capacities in excess of the yield load (i.e. the Eurocode resistance for stocky

sections), may be seen for both the hot-rolled and cold-formed sections. The hot-

0.00

0.25

0.50

0.75

1.00

1.25

1.50

0.0 0.5 1.0 1.5 2.0 2.5

Nu/A

f y

pλPlate slenderness

673 . 0 = λ p


131

rolled sections exhibit a yield plateau before the commencement of strain-hardening,

while the cold-formed sections display a more rounded response. For the cold-formed

sections, part of the additional capacity beyond the yield load may be attributed to the

enhanced strength in the corner regions of the sections arising from high localised

plastic deformations during production.

Fig. 5.13: Normalised load-end shortening graphs from stocky stub column tests

Fig. 5.14 shows the results of bending tests on structural steel square and rectangular

hollow sections. These results were collated from a series of three and four point

bending tests (Gardner et al., 2010; Wilkinson and Hancock, 1998; Zhao and Hancock,

1991). In Fig. 5.14, the maximum bending moment from the beam tests Mu has been

normalised by the elastic moment Mel (determined as the elastic modulus Wel

multiplied by the material yield strength fy) and plotted against the cross-section

slenderness pλ , defined in Eq. (3.7). The Eurocode 3 cross-section classes are also

indicated in Fig. 5.14, where the plastic moment Mpl (determined as the plastic

modulus Wpl multiplied by the material yield strength fy) applies to Class 1 and 2

cross-sections, the elastic moment Mel applies to Class 3 cross-sections and an

effective moment should be determined for Class 4 cross-sections.

Fig. 5.14 generally indicates the cross-section classification system is conservative

and that its step wise nature does not reflect the observed physical response and

slenderness limits for classification on internal flange element according to EN 1993-

1-1 (2005) are also included..

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0 0.02 0.04 0.06 0.08 0.1 δ/L

N/A

f y

SHS 40×40×3.0-HR1 SHS 40×40×3.0-CF1

Cold-formed Hot-rolled


132

Fig. 5.14: Simple beam test results

Typical normalised moment-rotation curves for hot-rolled and cold-formed sections –

RHS 60×40×4-HR and RHS 60×40×4-CF – reported in Chapter 3 are plotted in Fig.

5.15 to demonstrate how the conservatism develops. On the vertical axis, the test

moment M has been normalised by the plastic moment capacity Mpl, and on the

horizontal axis, the central rotation θ (calculated as the sum of the end rotations) has

been normalised by the elastic rotation at the plastic moment θpl. Similar strain-

hardening characteristics to those seen in the stub column tests may be observed, with

ultimate test moments in excess of the plastic moment capacity.

0.0

0.5

1.0

1.5

2.0

0 0.2 0.4 0.6 0.8 1

Wplfy

Welfy

Wefffy

Class 3

Class 4 Class 1

Class 2

Mu/W

elf y

pλPlate slenderness

58 . 0 = λ p

67 . 0 = λ p 74 . 0 = λ p


133

Fig. 5.15: Normalised moment–end rotation graphs from stocky simple beam tests

5.3.2 Other existing design methods

5.3.2.1 Stress-based methods

Two stress-based design proposals that are relevant to the present study are reviewed

in this section. The first one proposed by Kim and Peköz (2008) for aluminium

structures is based on rearrangement of the plastic stress distribution through the

depth of a cross-section achieved through the introduction of ‘yield’ and ‘ultimate’

shape factors, αy and αu, respectively. The second, proposed by Lechner et al. (2008),

is a modification to the existing cross-section classification approach, with a linear

transition between the plastic and elastic moment capacities for Class 3 sections. A

modified effective width design approach has also been proposed by Bambach and

Rasmussen (2004a; 2004b) for the design of unstiffened elements with stress

gradients.

The approach of Kim and Peköz (2008)

The design proposal of Kim and Peköz (2008) for aluminium sections introduced a

so-called ‘ultimate shape factor’ to transfer the actual nonlinear stress distribution

experienced in bending in practice, with stresses at both outer-fibres at the ultimate

tensile stress fu, as shown in Fig. 5.16(e), to an equivalent linear stress distribution.

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0 2 4 6 8 10 12 14 16

Rotation θ/θpl

M/M

pl

RHS 60×40×4-HR RHS 60×40×4-CF

Cold-formed

Hot-rolled


134

Note that the ultimate tensile stress is used for both extreme fibres due to the absence

of an ultimate stress in compression and the general symmetry of the f-ε response.

A key part of this approach is the introduction of the ultimate shape factor αu, which is

similar to the ‘generalized shape factor’ employed EN 1999-1-1 (2007), but based on

the situation where the stresses at both extreme fibres are at the ultimate tensile stress

(and strain), while the generalized shape factor is used to predict the maximum

bending moment resistance at any given outer fibre strain limit. The distinct

difference between the ultimate shape factor and the traditional geometric shape

factor is that the latter is dependent only on the geometric shape of the section and

assumes elastic, perfectly-plastic stress-strain (f-ε) characteristics. The geometric

shape factor is not entirely suitable for aluminium material (see Figs 5.16 (c) and (e))

which exhibits strain-hardening and no clearly defined yield point. The moment

capacity corresponding to Fig. 5.16 (e) may be calculated by integration of the stress-

distribution through the depth of the section. The ultimate shape factor αu is defined as

the ratio of the ultimate moment capacity (based on fu at the extreme fibres) to the

yield moment capacity Mel (based on the 0.2 % proof stress at the extreme fibres) (Fig.

5.16 (b)). In a similar way, the moment capacity corresponding to the 0.2% proof

stress f0.2 at the outer fibres and the ‘yield shape factor’ can be obtained. Both the

ultimate shape factor and yield shape factor result in the linear stress distributions

shown as dashed lines in Figs 5.16 (c) and (e).

Fig. 5.16: Stress distributions in a rectangular aluminium section in bending

(a) (b) (c) (d) (e)

f0.2 f0.2

fp,0.2 = αyf0.2

M=Mel M=αyMel M=Mpl=1.5

M

M=Mu=αuMel

f0.2 fu

fp,u=αuf0.2

Linear material Elastic, perfectly-plastic material


135

An expression for the ultimate shape factor (Eq. (5.1)) for solid rectangular sections

was derived by analytical integration (Kim, 2000) based on the stress-strain

relationship approximated by the Ramberg-Osgood equation (Ramberg and Osgood,

1943):

++

+

==

Ef

ff

1n21n

5001

Ef

31

ff

ε3

MMα u

n

2.0

u2

u

2.0

u2uel

uu (5.1)

where εu is the material ultimate strain, fu is the material ultimate tensile stress, f0.2 is

the 0.2% proof yield stress, E is Young’s modulus and n is a strain-hardening

exponent employed in the Ramberg-Osgood equation (Ramberg and Osgood, 1943).

A simplified expression derived by Kim (2000) is given by Eq. (5.2)

2.0f/f25.1 cyuu +=α (5.2)

where fu is ultimate tensile stress and fcy is compression yield stress.

Since the above ultimate shape factor αu expression is for a solid rectangular section,

in order to calculate bending resistance of practical cross-sections such as I-sections,

channels and rectangular hollow sections, an improved Weighted Average Stress

Approach (WASA2) and the Total Moment Capacity Approach (TMCA) have been

developed (Kim, 2000). The concept of these two approaches is illustrated in Fig.

5.17; further detailed information is available from Kim and Peköz (2008).


136

Fig. 5.17: Contributions of element groups to the total moment capacity

The approach proposed by Lechner et al. (2008)

Lechner et al. (2008) investigated the resistance of semi-compact (Class 3) cross-

sections in bending and presented proposals covering design under the load

combinations of axial compression and biaxial bending. In place of the current system

(EN 1993-1-1, 2005) where moment resistance drops from Mpl to Mel at the Class 2-3

boundary, a gradual reduction in resistance (see Fig. 5.18 (a)) was proposed.

fflange

fweb

fweb Entire cross-section

(Nonlinear) Flange Web

(Two linear approximations)

fflange


137

`

Fig. 5.18: Design proposal for Class 3 sections

A linear transition (Lechner et al., (2008)) from the plastic bending moment Mpl to the

elastic bending moment Mel was proposed to determine the resistances in the range of

Class 3, as shown in Figs 5.16 (a) and (b). In order to calculate the relative bending

resistance, non-dimensional relative Class 3 slenderness c/tref, where c is the flat

element width and tref is a reference measurement of the corresponding thickness, is

determined for each of the separate stress distributions corresponding to design

bending moment MEd. This results in the slenderness c/tref, whereby c/tref = 0 at the

c/tref

fy fy fy fy fy

0.0 1.0

0.0

0.5

1.0

1.5

2.0

Class 1 and 2 Class 3 Class 4 Wefffy

Mel=Welfy Mpl=Wplfy

Lechner et al. (2008)

c/tref

Mu/Welfy

0.0 1.0 (a) Lechner et al. (2008) method for Class 3 section

(b) Comparison between the Lechner et al. (2008) proposal and Eurocode 3

Neutral axis

(i) EN 1993-1-1 (2007)

(ii) Lechner et al. (2008) method

Neutral axis


138

Class 2/3 border and c/tref = 1 at the Class 3/4 border. Further information regarding

the detailed calculation approach, the M-N interaction and biaxial bending interaction

has been reported by Lechner et al. (2008).

5.3.2.2 Strain-based methods

Strain-based structural steel design proposals have been made by Mazzolani (1995),

Kemp et al (2002) and Knobloch and Fontana (2006). The proposals of Mazzolani

(1995) and Kemp et al. (2002) are most relevant to the present research and are

reviewed in this sub-section.

The approach proposed by Mazzolani (1995)

Prior to the work of Kim and Peköz (2008) described in the previous sub-section,

Mazzolani (1995) had previously also proposed a similar shape factor called the

‘generalized shape factor’ α to predict ultimate moment in aluminium alloys as Mu=

αM0.2, where M0.2 is elastic moment based on the 0.2% proof yield stress, and

described the relationships between α and the cross-section geometry, the f-ε law and

the assumed limiting curvature limk . For a Class 1 section, Annex F in EN 1999-1-1

(2007) employs α5 and α10, which are the section generalized shape factors

corresponding respectively to ultimate curvature values elu k5k = and elk10 ( elk is the

elastic limit curvature) to take the effect of strain-hardening into account. The

curvature limit depends on the ductility properties of the alloy under consideration.

The approach proposed by Kemp et al. (2002)

Kemp et al. (2002) proposed the use of a bi-linear moment-curvature relationship to

predict the ductility and moment capacity of steel beams, allowing for strain-

hardening, as shown in Fig. 5.19. The transition point between elastic and inelastic

behaviour is at 90% of the full plastic moment Mpl and a flexural rigidity EshI in the

strain-hardening region where E/Esh = 75 (based on coupon test results). The

importance of both local buckling and lateral torsional buckling on the maximum

curvature that a beam could sustain was recognised.


139

The model (Kemp et al., 2002) has been plotted with gathered existing test data

(Byfield and Nethercot, 1998; Wilkinson and Hancock, 1998; Zhao and Hancock,

1991) in Fig. 5.19, where the deformation capacity of the test specimens has been

determined using the expressions provided by Kemp et al. (2002) to calculate critical

strain limited by lateral-torsional buckling or lateral-torsional and local buckling. The

figure highlights the importance of strain-hardening and the suitability of a

deformation based approach.

Fig. 5.19: Comparison between Kemp et al. (2002) model and existing bending test

data

5.3.2.3 Other models

The approach proposed by Davies (2006)

Davies (2006) also recognised that moment capacities could exceed Mpl as a result of

strain-hardening, and proposed a method for capturing this increment in the moment

δM. The formula for calculating the increase in bending moment above Mpl due to

strain-hardening, δM, associated with a plastic hinge rotation θpl which is defined as

the elastic rotation at the plastic moment capacity, is given by Eq. (5.3)

plθkhEIMδ = (5.3)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

5 10 15 20

Test data Model

0εLB/εy

Mu/M

pl

0.9Mpl

Mu=1.08Mpl

εLB/εy =1

εLB/εy = 16


140

where E is the Young’s modulus, I is the second moment of area of the section, h is

the “equivalent cantilever” which depends on the shape of the bending moment

diagram in the vicinity of the plastic hinge and k is a dimensionless strain-hardening

parameter which is a property of the steel.

Eq. (5.3) was derived from a proposal made originally by Horne (1960) and later

reported in Davies (1966). The most important part of this method is the introduction

of the strain-hardening parameter k to take into account strain-hardening under load.

The parameter k in Eq. (5.3) can be determined either from the material stress-strain

curve or from the load-deflection or moment-end rotation characteristics from

bending tests. From the stress-strain curve, k is equal to the ratio of the strain at the

onset of strain-hardening to the strain at yield (Horne, 1960), whilst from the load-

deflection or moment-end rotation relationships it can be determined directly from the

relative slopes of the elastic and plastic regions (Byfield and Dhanalakshmi, 2002).

Byfield and Dhanalakshmi (2002) show that k = αe/(3αp) for a load-deflection curve

and k = (αe/2αp) for a moment-end rotation curve, where αe is the slope of the elastic

part of the curve and αp is the slope of the plastic part of the curve.

Byfield and Dhanalakshmi (2002) explain how the moment-deflection or moment-end

rotation characteristics of a beam can be generated using the mean stress-strain curves

obtained from mill tests. The generation of the relationships is a two-stage process of

numerical integration. Firstly, it is assumed that the strain distribution is linear

through the depth of the cross-section. The integration of the product of stress and

distance from the neutral axis of the cross-section is then used to obtain the

corresponding bending moment. The curvature is increased step by step and a

moment-curvature relationship generated for a given cross-section. Secondly, the

curvature can be integrated once to find the distribution of slope and twice to get the

distribution of deflection. Then, the moment-deflection and moment-end rotation

relationships have been generated.


141

5.3.3 The continuous strength method (CSM)

5.3.3.1 Background

The influence of strain-hardening on the capacity of structural steel members has been

illustrated through the stub column and simple beam tests examined earlier in this

chapter. The results have shown that the limiting resistances adopted in present design

practice of the yield load in compression for stub columns and the plastic moment

capacity in bending for determinate beams are conservative in the case of stocky

sections.

The ability of a cross-section to sustain increased loading and indeed deform into the

strain-hardening regime is limited by the effects of local buckling. Susceptibility to

local buckling is currently assessed by means of cross-section classification, where

structural cross-sections are assigned to discrete behavioural classes depending on the

slenderness of the constituent elements. The continuous strength method (CSM)

(Gardner, 2008) is an alternative approach to calculating cross-section resistance,

which is based on a continuous relationship between cross-section slenderness and

deformation capacity and a rational exploitation of strain-hardening. Previous studies

have confirmed the suitability of a deformation base approach to the design of

metallic structures. Development of the method for determinate structures is described

below, while development of the method for indeterminate structures is described in

Chapter 6.

5.3.3.2 General methodology and application range

The CSM recognises that the resistance of structural cross-sections is a continuous

function of their deformation capacity, as controlled by the slenderness (and hence

propensity to local buckling) of the constituent plate elements. The method employs a

continuous ‘base curve’ (Fig. 5.20), defining the relationship between cross-section

slenderness and cross-section deformation capacity, together with a material model

that allows for the influence of strain-hardening. The CSM currently applies only to

fully effective (i.e. non-slender) sections, though extension of the method to allow for

partial plasticity within slender sections is under consideration. Determination of


142

cross-section capacities in compression and bending, incorporating recent

developments to the method, are summarised in the following sections.

5.3.3.3 Cross-section compression resistance

Within the continuous strength method, cross-section slenderness is defined through

Eq. (3.7) by the plate slenderness of the most slender constituent element in the

section, as set out in EN 1993-1-5 (2006). Alternatively, as in the direct strength

method (DSM) (Schafer, 2008), the slenderness of the full cross-section rather than

that of the most slender constituent element may be employed. This approach has

been found to offer modest improvements in the accuracy of the method for the

sections considered herein (I-sections, SHS and RHS); the benefits are more

significant in the case of slender sections with more complex geometries where

element interaction and the influence on the elastic buckling behaviour is more

pronounced.

Having established the cross-section slenderness pλ , the corresponding normalised

deformation capacity of the cross-section εLB/εy is then obtained through the base

curve, given by Eqs (5.4a and 5.4b), and shown in Fig. 5.20.

15εε

y

LB = for pλ ≤ 0.328 (5.4a)

15λ

4.0εε

25.3py

LB ≤= for 0.328 < pλ ≤ 0.748 (5.4b)

in which εy=fy/E is the yield strain of the material, where E is Young’s modulus, εLB is

the local buckling strain of the section and an upper bound limit of 15 times the yield

strain has been set according to the ductility requirement in EN 1993-1-1 (2005).


143

Fig. 5.20: ‘Base curve’ – relationship between cross-section deformation capacity and

slenderness

The base curve (Eq. (5.4)) was generated, as described in (Gardner, 2008), by means

of stub column test data, including those described herein (Gardner et al., 2010)

together with further data collected from previous studies (Rasmussen and Hancock,

1992; Akiyama et al., 1996; Wilkinson and Hancock, 1997; Gao et al., 2009). In

interpreting the test data, for stocky sections, where the ultimate load Nu is greater

than the yield load Ny, the local buckling strain is defined as the end shortening at

ultimate load δu normalised by the stub column length L, as given by Eq. (5.5), while

for slender sections (Nu<Ny), where the response is influenced by elastic post-

buckling behaviour, the normalised local buckling strain εLB/εy is defined as the ratio

of the ultimate load Nu to the yield load Ny, as given by Eq. (5.6). Since slender

sections fail below their yield load, where stress is directly proportional to strain,

adoption of Eq. (5.6) yields a normalised relationship between deformation capacity

and slenderness that is similar to that between strength and slenderness given by the

familiar Winter curve (Kalyanaraman et al., 1977).

E/fL/δ

εε

y

u

y

LB = for yu NN ≥ (5.5)

y

u

y

LB

NN

εε

= for yu NN < (5.6)

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1.0 1.2

CSM Effective width Test data

CSM DSM or effective width method

Def

orm

atio

n ca

paci

ty ε

LB/ε

y

Plate slenderness

Class 3-4 limit

15λ

4.0εε

25.3py

LB ≤=

pλ


144

Following recent developments, the base curve defined by Eq. (5.4) now differs from

that presented previously (Gardner, 2008) due to:

(1) Availability of further test data (Rasmussen and Hancock, 1992; Wilkinson,

Hancock, 1997; Gao et al., 2009; Gardner et al., 2010) upon which to establish the

curve;

(2) Element slenderness is defined using flat plate widths, in line with EN 1993-1-1

(2005) rather than centreline dimensions;

(3) Applicability of the method has been limited to sections where pλ < 0.748, with

more slender sections being covered by the existing effective width (EN 1993-1-5,

2006) or direct strength methods (DSM) (Schafer, 2008);

(4) A limitation has been placed on the normalised local buckling strain εLB/εy of 15,

which corresponds to the material ductility requirement expressed in EN 1993-1-1

(2005).

Having established the local buckling strain of the section, the local buckling stress

fLB is determined directly from the strain-hardening material model proposed in

Chapter 4, for which a bi-linear elastic, linear-hardening representation, with a strain-

hardening slope that depends on the section type and the ratio of fu/fy. Where the ratio

of fu/fy is not available Esh=E/100, as recommended in EN 1993-1-5 (2006), has been

adopted. Finally, the cross-section compression resistance NCSM is given by Eq. (5.7)

as the product of the gross cross-section area A and the local buckling stress fLB

where the local buckling stress fLB is determined from the strain-hardening material

model given by Eq. (5.8):

LBCSM AfN = (5.7)

shy

LByyLB E1

εε

εff

−+= (5.8)


145

5.3.3.4 Cross-section bending resistance

In-plane bending resistance may be calculated on a similar basis to compression

resistance, whereby the deformation capacity εLB of the cross-section is limited either

by local buckling of the web in bending or the compression flange in pure

compression. Assuming a linearly varying strain distribution, the bending moment

resistance may be determined by direct integration of the stresses through the depth of

the section, which, for I-sections, results in the explicit expression given by Eq. (5.9),

which is a corrected version of that presented by Gardner (2008):

where symbols are defined by reference to Fig. 5.21.

Fig. 5.21: Bending response of I-section with elastic, linear strain-hardening material

model

However, this expression is specific to I-sections and may not be appropriate for

practical design; in order to facilitate the bending resistance calculation, a general

simplified direct relationship between normalised bending resistance MCSM/Mpl and

normalised local buckling strain εLB/εy has been developed for plated sections. Three

different stages of behaviour have been identified:

)yh2yhh(h6

)ff(t

)y4h(4ft

3ytf2

)th(btfM

21w

2w

3w

yweb,LBw

21

2w

yw21wy

fwfLBCSM

−−−

+

−+++= (5.9)

(a) Cross-section (b) Strain (c) Stress

εLB

εLB

fy

fy

εy

εy

fLB

fLB

tf

y1 y

tf

hw/2

tw

b

fLB,web hw/2


146

(1) Elastic stage;

(2) Elastic-plastic stage;

(3) Strain-hardening stage.

The continuous strength method addresses stages (2) and (3); in the elastic range,

where εLB/εy < 1, the moment capacity may be calculated by existing methods, such as

the effective width approach (EN 1993-1-5, 2006) or direct strength method (Schafer

2008).

In the elastic-plastic range, where 1 < εLB/εy ≤ 3, a nonlinear reduction in moment

capacity from the full plastic moment Mpl at εLB/εy = 3 to the elastic moment Mel at

εLB/εy = 1, as given by Eq. (5.10) and illustrated in Fig. 5.22, is proposed.

For a solid rectangular section, integration of stress from an elastic, perfectly-plastic f-

ε model through the depth of the section shows that the moment resistance is 8/9th of

way towards Mpl from Mel at εLB/εy = 3. For practical structural steel sections, where

the shape factor (ag = Wpl/Wel) will range between approximately 1.1 and 1.3, this will

mean that the moment will be between about 96% and 99% of Mpl at εLB/εy = 3. Thus,

with any modest strain-hardening, the moment will be Mpl at εLB/εy = 3.

Fig. 5.22: Design model for elastic-plastic stage ( 3ε/ε1 yLB ≤< )

Elastic-plastic stage:

−−−−=

181)ε/ε(

)εε

()MM(MM yLB2

LB

yelplplCSM

3εε1

y

LB ≤<(5.10)

fy

Strain Stress

εLB εy

fy


147

This point at which Mpl is reached, namely εLB/εy = 3, has also been recommended by

Bruneau et al. (1998) and further supported by the available experimental data.

Alternative transitions between the elastic and fully plastic responses have also been

proposed: Juhas (2007) presented a strain based approach similar to that described

herein, while Lechner et al. (2008) proposed a linear transition with slenderness and

also considered combined loading.

In the strain-hardening range, where εLB/εy > 3, capacities beyond the full plastic

moment can be achieved. The associated strain and stress distributions are shown in

Fig. 5.23, together with the proposed design model, which comprises the full plastic

moment capacity Mpl plus the additional moment capacity due to strain-hardening.

The strain-hardening component is derived from a linearly varying stress distribution

with an outer fibre stress equal to fLB-fy. The design model is given by Eq. (5.11),

Fig. 5.23: Design model for strain-hardening stage ( 15ε/ε3 yLB ≤< )

Fig. 5.24 shows the normalised moment capacity (Mc,Rd/Mpl) versus normalised local

buckling strain (εLB/εy) for the elastic-plastic (Eq. (5.10)) and strain-hardening (Eq.

(5.11)) stages of the continuous strength method (CSM) model for a typical I-section

beam. The corresponding analytical response of the beam, as determined by direct

integration, is also shown in Fig. 5.24, together with a previous model proposed by

Kemp et al. (2002) that also allows capacities beyond Mpl due to strain-hardening.

The CSM design model (Eqs (5.10) and (5.11)) may be seen to closely follow the

analytical response, which has also been verified numerically.

Strain-hardening stage: )3εε

(E

EMMM

y

LBshelplCSM −+= 15

εε3

y

LB ≤< (5.11)

fLB-fy

fLB fy

Strain Stress

εy εLB

Design model

fy fy fLB


148

Fig. 5.24: CSM bending moment resistance model

5.3.3.5 Application flow chart

The basic design steps of the CSM for compression may be illustrated as follow:

1.

Determine cross-section slenderness from the most slender element

cr

yp f

f=λ

2.

Obtain corresponding deformation capacity

15λ

4.0εε

25.3py

LB ≤=

3.

Determine resulting local buckling stress fLB from material model (see Chapter 4)

shy

LByyLB E1

εεεff

−+=

4.

Cross-section compression resistance is the product of local buckling stress fLB and the gross cross-section area

LBCSM AfN =

The corresponding steps for bending resistance may be summarised as follows:

Mc,

Rd/M

pl

Analytical model CSM model Kemp et al. (2002) model

εLB/εy 1 3 5 7 9 11 13 15

0.6

0.7

0.8

0.9

1

1.1

1.2


149

1.

Determine cross-section slenderness from the most slender element cr

yp σ

fλ =

2.

Obtain corresponding deformation capacity

15λ

4.0εε

25.3py

LB ≤=

3.

Determine moment capacity: If εLB/εy ≤ 3, Eq. (5.10) applies, otherwise Eq. (5.11) applies

−−

−−=

181)ε/ε(

εε

)MM(MM yLB2

LB

yelplplCSM

−+= 3

εε

EE

MMMy

LBshelplCSM

5.4 ASSESSMENT OF THE CSM

Comparison of the predictions of the CSM with the results of stub column and simple

beam tests are presented in the following sub-sections, in which the Eurocode 3

design model is also compared.

5.4.1 Compression

5.4.1.1 Hot-rolled sections

Numerical comparisons, including the mean and coefficient of variation (COV) of the

predictions, of the CSM and Eurocode 3 with the ultimate capacities from hot-rolled

steel stub column tests (Gardner et al., 2010) are presented in Table 5.8. The

comparisons show that the CSM provides the same results as Eurocode 3. The

comparisons show that the CSM provides slightly improved prediction accuracy with

a similar scatter.


150

Table 5.8: Comparison of hot-rolled stub column test results with design models

5.4.1.2 Cold-formed sections

A total of 23 cold-formed stub column test results were compared with the predictions

of the CSM and Eurocode 3. The numerical comparisons have been reported in Table

5.9. The 5 specimens from Akiyama et al. (1996) provided no information on the ratio

of fu/fy, so the material model proposed for cold-formed SHS and RHS in Chapter 4

could not be applied; hence, the material model given in EN 1993-1-5 (2006) with a

strain-hardening slope Esh of E/100 was employed. Table 5.9 shows that a 4%

improvement in prediction accuracy and a reduced scatter of prediction is obtained

from the CSM relative to Eurocode 3.

Stub column specimen pλ Nu NEC3

Nu NCSM

NCSM NEC3

SHS 100×100×4-HR1 0.54 0.94 0.93 1.00 SHS 100×100×4-HR2 0.53 0.94 0.93 1.00 SHS 60×60×3-HR1 0.35 1.06 1.04 1.03 SHS 60×60×3-HR2 0.35 1.09 1.06 1.03 RHS 60×40×4-HR1 0.31 1.08 1.05 1.03 RHS 60×40×4-HR2 0.31 1.08 1.05 1.03 SHS 40×40×4-HR1 0.18 1.25 1.24 1.01 SHS 40×40×4-HR2 0.18 1.26 1.25 1.01 SHS 40×40×3-HR1 0.25 1.20 1.18 1.01 SHS 40×40×3-HR2 0.25 1.19 1.17 1.01 Mean 1.11 1.09 1.02 COV 0.10 0.11 -


151

Table 5.9: Comparison of cold-formed stub column test results with design models

Stub column specimen pλ Nu NEC3

Nu NCSM

NCSM NEC3

SHS 100×100×4.0-CF1a 0.57 1.02 1.02 1.00 SHS 100×100×4.0-CF2a 0.57 1.02 1.01 1.00 SHS 60×60×3.0-CF1a 0.37 1.12 1.06 1.06 SHS 60×60×3.0-CF2a 0.36 1.13 1.06 1.07 RHS 60×40×4.0-CF1a 0.28 1.32 1.19 1.11 RHS 60×40×4.0-CF2a 0.28 1.31 1.18 1.11 SHS 40×40×4.0-CF1a 0.17 1.21 1.16 1.04 SHS 40×40×4.0-CF2a 0.17 1.20 1.15 1.04 SHS 40×40×3.0-CF a 0.26 1.26 1.15 1.09 SHS 40×40×3.0-CF2a 0.26 1.30 1.19 1.09 SHS 100×100×3.8-CFb 0.54 1.18 1.16 1.01 SHS 100×100×3.3-CFb 0.59 1.17 1.16 1.01 SHS 75×75×3.3-CFb 0.43 1.14 1.10 1.04 SHS 75×75×2.8-CFb 0.59 1.08 1.07 1.01 SHS 75×75×2.3-CFb 0.71 0.93 0.93 1.00 SHS 65×65×2.3-CFb 0.62 1.05 1.04 1.01 RHS 125×75×3.8-CFb 0.72 1.05 1.05 1.00 RHS 150×50×5.0-CFc 0.65 1.11 1.11 1.00 SHS-CF1d* 0.50 1.13 1.10 1.03 SHS-CF2d* 0.27 1.25 1.09 1.14 SHS-CF3d* 0.41 1.16 1.10 1.06 SHS-CF4d* 0.47 1.14 1.10 1.04 SHS-CF5d* 0.64 1.04 1.03 1.01 Mean 1.14 1.10 1.04 COV 0.09 0.06 - a Gardner et al. (2010); b Zhao and Hancock (1991); c Wilkinson and Hancock (1997); d Akiyama et al. (1996)

5.4.1.3 Welded sections

A total of 14 welded stub column test results were compared with the predictions of

the CSM and Eurocode 3 and presented in Table 5.10. The CSM provides a 6%

increase in average capacity predictions and a reduced scatter.


152

Table 5.10: Comparison of welded stub column test results with design models

Stub column specimen pλ Nu

NEC3 Nu

NCSM NCSM NEC3

Welded SHS-1a 0.30 1.28 1.13 1.14 Welded SHS-2a 0.36 1.03 0.93 1.10 Welded SHS-3a 0.41 1.00 0.94 1.06 Welded SHS-4a 0.45 1.02 0.98 1.04 Welded SHS-5a 0.56 1.00 0.98 1.02 Welded SHS-6a 0.26 1.39 1.22 1.14 Welded SHS-7a 0.30 1.33 1.17 1.14 Welded SHS-8a 0.35 1.17 1.05 1.11 Welded SHS-9a 0.42 1.03 0.98 1.06 Welded SHS-10a 0.51 0.96 0.93 1.03 Welded SHS-11b 0.50 0.96 0.93 1.03 Welded SHS-12b 0.50 0.94 0.91 1.03 Welded SHS-13b 0.69 0.91 0.90 1.00 Welded SHS-14b 0.69 0.91 0.91 1.00 Mean 1.07 1.00 1.06 COV 0.15 0.10 - a Akiyama et al. (1996); b Rasmussen and Hancock (1992)

5.4.1.4 Pressed-formed and seam-welded sections

The experimental results of a total of 16 press-formed and seam-welded stub column

tests (Akiyama et al., 1996; Gao et al., 2009) were numerically compared with the

predictions provided by the CSM and Eurocode 3, as shown in Table 5.11. Again, the

CSM provides better prediction accuracy and less scatter.


153

Table 5.11: Comparison of press-formed and seam welded stub column test results

with design models

Stub column specimen pλ Nu

NEC3 Nu

NCSM NCSM NEC3

Press-seam SHS-1a 0.33 1.16 1.03 1.13 Press-seam SHS-2a 0.43 1.13 1.07 1.05 Press-seam SHS-3a 0.29 1.29 1.13 1.14 Press-seam SHS-4a 0.41 1.16 1.09 1.06 Press-seam SHS-5a 0.27 1.27 1.12 1.14 Press-seam SHS-6a 0.33 1.22 1.07 1.14 Press-seam SHS-7a 0.37 1.13 1.04 1.09 Press-seam SHS-8a 0.42 1.14 1.08 1.06 Press-seam SHS-9a 0.51 1.02 1.00 1.02 Press-seam SHS-10a 0.24 1.38 1.21 1.14 Press-seam SHS-11a 0.28 1.36 1.20 1.14 Press-seam SHS-12a 0.33 1.24 1.09 1.13 Press-seam SHS-13a 0.39 1.22 1.13 1.08 Press-seam SHS-14a 0.48 0.99 0.96 1.03 Press-seam SHS-15a 0.66 0.91 0.90 1.01 Press-seam SHS-16b 0.57 1.10 1.08 1.02

Mean 1.17 1.07 1.09 COV 0.11 0.07 -

a Akiyama et al. (1996); b Gao et al. (1992)

5.4.1.5 Summary

The comparison of the predictions of the CSM with the results of a total of 63 stub

column tests are shown in Fig. 5.25, in which the Eurocode 3 design model is also

depicted. Numerical comparisons, including the mean and coefficient of variation

(COV) of the predictions, of the CSM and Eurocode with a total of 63 stub column

tests are presented in Table 5.12. The results show that the CSM offers more accurate

prediction of the test data with an average of 5% increase in capacity and a significant

reduction in scatter.


154

Fig. 5.25: Stub column test data and comparison with design models

Table 5.12: Comparison of the CSM and Eurocode methods with stub column test

results

Section type No. of tests

Nu NEC3

Nu NCSM

NCSM NEC3

Hot-rolled 10 1.11 1.09 1.02 Cold-formed 23 1.14 1.10 1.04 Welded 14 1.07 1.00 1.06 Press-formed and seam-welded 16 1.17 1.07 1.09

Mean 63 1.13 1.07 1.05 COV 63 0.11 0.09 -

5.4.2 Bending


The results of a total of 34 simple beam tests on hot-rolled steel I-sections were

compared with the predictions of the CSM and Eurocode 3 and are presented in Table

5.13. A 6% average increase in capacity is achieved by the CSM over Eurocode 3

with similar scatter.

0.5 0.6 0.7 0.8 0.9

1 1.1 1.2 1.3 1.4 1.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

748 . 0 = λ p

15 y

LB = ε ε

Slenderness

Nu/A

f y

EC3 yield limit CSM Test data

pλ

y

LB

εε

= 1.0


155

Table 5.13: Comparison of hot-rolled I-section simple beam test results with design

models

Simple beam specimen pλ Mu

MEC3 Mu

MCSM MCSM MEC3

203×102×23-HR1a 0.34 1.13 1.03 1.10 203×102×23-HR2a 0.36 1.09 1.01 1.08 203×102×23-HR3a 0.35 1.18 1.08 1.09 203×102×23-HR4a 0.36 1.06 0.98 1.08 203×102×23-HR5a 0.34 1.10 1.01 1.09 203×102×23-HR6a 0.34 1.09 0.99 1.10 203×102×23-HR7a 0.32 1.25 1.12 1.11 203×102×23-HR8a 0.33 1.15 1.04 1.11 203×102×23-HR9a 0.34 1.12 1.03 1.09 203×102×23-HR10a 0.38 1.17 1.10 1.06 152×152×30-HR1a 0.49 1.16 1.15 1.01 152×152×30-HR2a 0.50 1.18 1.17 1.01 152×152×30-HR3a 0.48 1.20 1.18 1.02 152×152×30-HR4a 0.49 1.18 1.17 1.01 152×152×30-HR5a 0.48 1.20 1.18 1.02 152×152×30-HR6a 0.48 1.21 1.19 1.02 152×152×30-HR7a 0.49 1.17 1.15 1.01 152×152×30-HR8a 0.49 1.17 1.16 1.01 152×152×30-HR9a 0.48 1.18 1.16 1.01 152×152×30-HR10a 0.49 1.13 1.12 1.01 203×102×23-HR11a 0.35 1.26 1.15 1.09 203×102×23-HR12a 0.35 1.25 1.15 1.09 203×102×23-HR13a 0.37 1.17 1.10 1.06 203×102×23-HR14a 0.35 1.27 1.16 1.10 203×102×23-HR15a 0.35 1.23 1.13 1.09 203×102×23-HR16a 0.34 1.19 1.09 1.09 152×152×30-HR10a 0.48 1.24 1.21 1.02 152×152×30-HR11a 0.51 1.22 1.21 1.01 152×152×30-HR12a 0.47 1.23 1.21 1.02 152×152×30-HR13a 0.47 1.23 1.21 1.02 152×152×30-HR14a 0.48 1.20 1.18 1.02 152×152×30-HR15a 0.47 1.28 1.25 1.02 I Beam-1b 0.25 1.33 1.18 1.13 I Beam-2b 0.25 1.28 1.14 1.13 Mean 1.19 1.13 1.06 COV 0.05 0.06 - a Byfield and Nethercot (1998); b Popov and Willis (1957)


156

5.4.2.2 Hot-rolled SHS and RHS

Owing to the limited available test data on simple beams with hot-rolled SHS and

RHS, only 3 tests results (Gardner et al., 2010) have been compared with the CSM

and Eurocode 3 as shown in Table 5.14. The limited comparisons still reflect the

improved accuracies in predicting test capacities provided by the CSM in comparison

to Eurocode 3.

Table 5.14: Comparison of hot-rolled SHS and RHS simple beam test results with

design models

Simple beam specimen pλ Mu MEC3

Mu MCSM

MCSM MEC3

SHS 40×40×4-HR 0.18 1.09 1.08 1.01 SHS 40×40×3-HR 0.25 1.16 1.14 1.01 RHS 60×40×4-HR 0.18 1.14 1.12 1.02 Mean 1.13 1.11 1.02 COV 0.03 0.03

5.4.2.3 Cold-formed SHS and RHS

A total of 53 test results on simple beams with cold-formed SHS and RHS were

compared with the predictions of the CSM and Eurocode 3, and are presented in

Table 5.15. The comparisons clearly show the significantly improved accuracy of the

CSM and the reduced scatter.


models


Mu MCSM

MCSM MEC3

SHS 40×40×4-CFa 0.17 1.28 1.17 1.09 SHS 40×40×3-CFa 0.26 1.25 1.10 1.14 RHS 60×40×4-CFa 0.17 1.41 1.17 1.21 RHS 100×50×2.0-CF1b 0.48 1.00 0.95 1.05 RHS 100×50×2.0-CF2b 0.48 0.99 0.94 1.06 RHS 100×50×2.0-CF3b 0.49 1.10 1.06 1.04


157


models (continued)


Mu MCSM

MCSM MEC3

RHS 100×50×2.0-CF4b 0.50 1.01 0.97 1.05 RHS 100×50×2.0-CF5b 0.50 1.11 1.05 1.06 RHS 100×50×2.0-CF6b 0.51 1.07 1.02 1.05 RHS 125×75×2.5-CF1b 0.59 1.06 1.03 1.03 RHS 125×75×3.0-CF1b 0.48 1.03 0.99 1.04 RHS 125×75×3.0-CF2b 0.48 1.04 1.00 1.04 RHS 125×75×3.0-CF3b 0.49 1.03 0.99 1.04 RHS 150×50×2.3-CF1b 0.62 1.25 0.97 1.28 RHS 150×50×2.3-CF2b 0.62 1.26 0.98 1.28 RHS 150×50×2.3-CF3b 0.62 1.30 1.00 1.29 RHS 150×50×2.5-CF1b 0.52 1.11 1.05 1.06 RHS 150×50×2.5-CF2b 0.54 1.08 1.01 1.07 RHS 150×50×2.5-CF3b 0.54 1.02 0.96 1.07 RHS 150×50×2.5-CF4b 0.54 1.00 0.94 1.07 RHS 150×50×3.0-CF1b 0.42 1.18 1.10 1.07 RHS 150×50×3.0-CF2b 0.42 1.15 1.05 1.09 RHS 150×50×3.0-CF3b 0.42 1.21 1.10 1.10 RHS 150×50×3.0-CF4b 0.42 1.21 1.12 1.08 RHS 150×50×3.0-CF5b 0.46 1.15 1.06 1.08 RHS 150×50×3.0-CF6b 0.47 1.16 1.07 1.08 RHS 150×50×3.0-CF7b 0.47 1.13 1.04 1.08 RHS 150×50×4.0-CF1b 0.30 1.30 1.02 1.28 RHS 150×50×4.0-CF2b 0.33 1.19 1.04 1.15 RHS 150×50×4.0-CF3b 0.35 1.25 1.07 1.16 RHS 150×50×4.0-CF4b 0.35 1.19 1.02 1.16 RHS 150×50×4.0-CF5b 0.35 1.27 1.09 1.17 RHS 150×50×4.0-CF6b 0.35 1.18 1.00 1.18 RHS 150×50×5.0-CF1b 0.26 1.17 1.01 1.15 RHS 150×50×5.0-CF2b 0.26 1.23 1.06 1.17 RHS 75×25×1.6-CF1b 0.44 1.15 1.06 1.09 RHS 75×25×1.6-CF2b 0.43 1.03 0.98 1.05 RHS 75×25×1.6-CF3b 0.42 1.00 0.95 1.05 RHS 75×25×1.6-CF4b 0.42 1.10 1.01 1.09 RHS 75×25×1.6-CF5b 0.44 1.11 1.02 1.10 RHS 75×50×2.0-CF1b 0.35 1.13 1.00 1.13 RHS 75×50×2.0-CF2b 0.34 1.11 0.97 1.14


158


models (continued)


Mu MCSM

MCSM MEC3

RHS 75×50×2.0-CF3b 0.50 1.02 0.97 1.05 RHS 75×50×2.0-CF4b 0.50 1.04 0.99 1.05 SHS 100×100×3.8-CFc 0.54 1.25 1.20 1.04 SHS 100×100×3.3-CFc 0.59 1.31 1.25 1.05 SHS 75×75×3.3-CFc 0.43 1.21 1.12 1.08 SHS 75×75×2.8-CFc 0.59 1.18 1.13 1.04 SHS 75×75×2.3-CFc 0.70 1.26 1.00 1.25 SHS 65×65×2.3-CFc 0.63 1.12 1.07 1.04 SHS 125×75×3.8-CFc 0.38 1.29 1.21 1.07 RHS 125×75×3.3-CFc 0.45 1.24 1.17 1.06 RHS 100×50×2.8-CFc 0.34 1.31 1.13 1.17 Mean 1.14 1.03 1.11 COV 0.09 0.05 - a Gardner et al. (2010); b Wilkinson and Hancock (1998); c Zhao and Hancock (1991)

5.4.2.4 Summary

Comparisons of the predictions of the CSM with the results of a total of 90 simple

beam tests are shown in Fig. 5.26, in which the Eurocode 3 design model is also

depicted.


159

Fig. 5.26: Simple beam test data and comparison with design models

In Fig. 5.26, the CSM design model is displayed for two geometric shape factors ag,

which is defined as the ratio of Mpl to Mel, – 1.14 and 1.25 – which correspond to the

average shape factors of the examined I-section and SHS/RHS test data, respectively.

The responses may be seen to be similar over the majority of the slenderness range in

the presented normalised form.

Numerical comparisons, including the mean and coefficient of variation (COV) of the

predictions, of the CSM and Eurocode with the simple beam tests are presented in

Table 5.16. The results show that the CSM offers more accurate prediction of the test

data with an 8% average increase in ultimate moment capacity and slightly reduced

scatter.

Table 5.16: Comparison of the CSM and Eurocode methods with bending test results

Section type No. of tests

Mu MEC3

Mu MCSM

MCSM MEC3

Hot-rolled I-sections 34 1.18 1.13 1.05 Hot-rolled SHS/RHS 3 1.13 1.11 1.02 Cold-formed SHS/RHS 53 1.14 1.03 1.11 Mean 90 1.17 1.08 1.08 COV 90 0.08 0.08 -

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.1 0.2 0.3 0.4 0.5 0.6 0.7

EC3 design model Cold-formed Hot-rolled CSM (ag = 1.25) CSM (ag = 1.14)

Class 2-3 limit

Class 3-4 limit

15 y

LB = ε ε

Mu/M

pl

Slenderness pλ


160

5.5 RELIABILITY STUDY

In order to verify the proposed method, statistical evaluations according to the

procedures provided by Annex D in EN 1990 (2002) have been performed on the

basis of test results. The evaluations were carried out following the steps as given in

EN 1990 (2002), which are summarised below:

Step 1: Develop a design model rt

A design model for determining theoretical resistance rt that is a function of all the

relevant independent variables X is firstly developed. The prediction models have

been summarised in Section 5.3.3 and can be briefly written as follow:

Compression predictive model: rt=grt[B, D, t, ri, fy, fu, kσ, Esh] (5.12)

Bending predictive model: rt=grt[B, D, t, ri, fy, fu, kσ, ag, Esh] (5.13)

where terms are all defined earlier in this chapter.

Step 2: Compare experimental re and theoretical values rt

The theoretical values rti are obtained by substituting the actual measured properties

into the resistance function. These values are plotted against the corresponding

experimental test results re to check the deviation from the line re = rt.

Step 3: Estimate the mean value correction factor bcf

The probabilistic model of the resistance R is represented in the following format:

R=bcfrtδ (5.14)

where bcf is the ‘least squares’ regression fitting to the slope, as given in Eq. (5.15)

and δ is an error term giving information on the scatter of the plotted points from the

mean value of the strength function.


161

∑∑= 2

t

tecf r

rrb (5.15)

The values of bcf obtained in the present study are presented in Table 5.17. A value of

bcf being greater than unity indicates that the proposed design model underestimates,

on average, the test results.

Step 4: Estimation the coefficient of variation Vδ of the errors of the design model

An estimated value of the coefficient of variation of the errors is determined in a log-

normal distribution as follow:

1)sexp(V 2Δδ −= (5.16)

( )∑=

∆ ∆−∆−

=tn

1i

2i

t

2

1n1s (5.17)

)δln(Δ ii = (5.18)

ticf

eii rb

rδ = (5.19)

∑=

∆=∆tn

1ii

tn1 (5.20)


162

where 2Δs is the variance of the error terms, iΔ is the ith error in the log-normal

distribution, δi is the error term for the ith experimental value, Δ is the average value

of the error terms and nt is the number of tests.

Step 5: Analyse compatibility

The degree of scatter of the test data (re, rt) is assessed by using the coefficient of

determination R2. This coefficient indicates the quality of the approximation of the

regression line on the test data – the approximation improves as R2 approaches 1.0. R2

is defined by:

0.1ssss

ssR

yyxx

2xy2 ≤= (5.21)

where

( )∑ −≡2

ttixx rrss (5.22)

( )∑ −≡2

eeiyy rrss (5.23)

( )( )∑ −−≡ eeittixy rrrrss (5.24)

where tr and er are theoretical and experimental resistances respectively.

The values of R2 are given in Table 5.17. These values show that the proposed design

model accurately predicts the compressive and flexural resistances.


163

Step 6: Determine the coefficients of variation Vxi of the basic variables

The coefficient of variation Vxi is generally determined on the basis of prior

knowledge. The values determined by an analysis of more than 7000 samples (Byfield

and Nethercot, 1998) have been employed in the current study.

Yield strength material Vfy = 0.05

Major axis plastic section modulus VWpl = 0.02

Step 7: Determination of the design value of the resistance

Depending on the number of tests nt, the design value of the resistance rd should be

obtained from:

)Q5.0QαkQαkexp()X(gbr 2δδn,drtrt,dmrtcfd −−−= ∞ for nt < 100 (5.25)

)Q5.0Qkexp()X(gbr 2,dmrtcfd −−= ∞ for nt ≥ 100 (5.26)

where b is the mean value correction factor from tests, ∞,dk is the value of kd,n for nt =

∞, kd,n is the design fractile factor, Xm is the mean value of basic variables measured

in tests, αrt is the weighting factor for Qrt as defined by Eq. (5.27), αδ is the weighting

factor for Qδ as defined by Eq. (5.28) and Qrt, Qδ and Q are defined by Eq. (5.29) –

(5.31) respectively.

QQα rt

rt = (5.27)

QQα δ

δ = (5.28)

)1Vln(σQ 2rt)rtln(rt +== (5.29)


164

)1Vln(σQ 2δ)δln(δ +== (5.30)

)1Vln(σQ 2r)rln( +== (5.31)

For small values of 2δV and 2

xiV , the overall error 2rV can be obtained using the

following approximation:

2rt

2δ

2r VVV += (5.32)

∑=

=j

1i

2xi

2rt VV (5.33)

where j is the total number of basic variables.

Step 8: Determining the partial safety factor γM

The partial safety factor γM for the proposed design model is determined as the

nominal resistance rn normalised by the design resistance rd:

d

nM r

rγ = (5.34)

The nominal resistance rn is obtained by substituting the nominal values of all the

independent basic variables Xn into the resistance function:

rn = grt(Xn) (5.35)

Following the process described above, comparisons between the load-carrying

capacities achieved in the tests and those predicted by Eurocode 3 and Eqs (5.12) and

(5.13) have been made, and are shown in Figs 5.27 - 5.30. The statistical evaluations

are summarised in Table 5.17, where design fractile factor kd,n, correction factor bcf,


165

coefficient of determination R2, coefficient of variation of error Vδ, overall coefficient

of variation Vr and partial safety factor γM are reported.


for compression resistance


(5.12)) for compression resistance

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700 800 900 rt (kN)

r e (k

N)

re = rt

re = 1.10 rt

R2 = 0.818

0 100 200 300 400 500 600 700 800 900

0 100 200 300 400 500 600 700 800 900

re = rt

re = 1.05 rt

R2 = 0.859

rt (kN)

r e (k

N)


166


for bending resistance


(5.13)) for bending resistance

0

20

40

60

80

100

0 20 40 60 80 100

r e (k

Nm

)

rt (kNm)

re = rt

re = 1.18 rt R2 = 0.993

0

20

40

60

80

100

0 20 40 60 80 100

re = rt

re = 1.11 rt R2 = 0.989

r e (k

Nm

)

rt (kNm)


167

Table 5.17: Summary of statistical evaluation results for predictive model equations

Method Number of tests kd,n bcf R2 Vδ Vr γM

Eurocode 3 for compression 63 3.252 1.095 0.818 0.115 0.129 1.20

Eq. (5.12) for compression 63 3.252 1.054 0.859 0.089 0.106 1.15

Eurocode 3 for bending 90 3.202 1.184 0.993 0.078 0.096 0.99

Eq. (5.13) for bending 90 3.202 1.113 0.989 0.076 0.095 1.05

The reliability study shows that, for compression, the CSM provides more reliable

results than Eurocode 3, as indicated by a lower value of γM, which results primarily

from the reduced scatter in the prediction. Given that γM0 = 1.0 has been used to date

for cross-section resistance in compression, it is proposed that this is retained for the

CSM.

In bending, the reliability analysis suggests a value of γM = 1.05 for the CSM; this is

considered sufficiently close to unity for γM = 1.0 to be used, in line with current

practice. Numerous similar examples of such an approach exist within Eurocode 3.

5.6 DISCUSSION AND CONCLUDING REMARKS

In this chapter, the shortcomings of the cross-section classification system and the

importance of strain-hardening in the response of determinate steel structures have

been highlighted. It has been shown, both through experimentation and the analysis of

existing test data, that the limiting resistances adopted in present design practice of the

yield load in compression for stub columns and the plastic moment capacity in

bending are conservative in the case of stocky sections, due to the influence of strain-

hardening. As an alternative treatment, the continuous strength method (CSM), which

offers a rational means of exploiting strain-hardening in steel design, has been

developed to overcome this conservatism. Comparisons have been made against test

results on stub columns and simple beams. These comparisons, together with the

corresponding reliability analysis, show that the CSM provides a more accurate

prediction of test response and enhanced structural capacity over current design

methods and suggest partial safety factors γM = 1.00 for compression and bending.


168

Furthermore, since cross-section deformation capacity is explicitly determined in the

calculations, this enables a more sophisticated and informed assessment of ductility

supply and demand as an additional benefit of the proposed approach.

Chapter 6 Indeterminate structures

169

CHAPTER 6

INDETERMINATE STRUCTURES 6.1 INTRODUCTION

Subject to prescribed limitations on cross-section slenderness and member restraint

conditions being met, indeterminate steel structures are generally designed using

traditional plastic analysis methods, which are based on the formation and subsequent

rotation of plastic hinges at their full plastic moment capacity. The formation of each

plastic hinge causes a progressive reduction in stiffness of the structure until the final

hinge forms resulting in a collapse mechanism. In reality though, plastic hinges do not

rotate at a constant moment equal to Mpl of the section due to the occurrence of strain-

hardening, with stockier sections often achieving resistances significantly beyond

those predicted by current design approaches. The importance of strain-hardening in

indeterminate structures has been described by Davies (2006) who observed that

enhanced capacity could be attained in steel frames by considering strain-hardening

provided local and lateral-torsional buckling were eliminated.


170

In this chapter, existing experimental data on indeterminate steel structures are first

collected and added to that produced as part of the present study (Chapter 3). Further

data are then generated by means of validated numerical models, after which existing

and new proposed design methods are assessed.

6.2 COLLECTION OF EXISTING TEST DATA

Experimental results on indeterminate steel structures consisting of members with

stocky cross-sections, especially on full-scale frames, are relatively scarce, with much

of the relevant research being conducted during the 1950s and 1960s when plastic

analysis and tall building stability issues were studied extensively. Existing test data is

exploited where possible, but it is not always reported in sufficient detail to allow

meaningful comparisons to be made. Seven continuous beam tests reported by Yang

et al. (1952), Driscoll et al. (1957) and Popov and Willis (1957) and 5 full-scale frame

tests conducted by Ruzek et al. (1954), Baker and Eickhoff (1955), Driscoll et al.

(1957) and Charlton (1960) have been collected. The details of these tests have been

gathered, along with the 12 continuous beam tests conducted by the author, to form

the experimental database for the development and application of the CSM to

indeterminate structures.


The geometric and material properties and test results of the 7 I-section continuous

beam specimens from the literature, including section depth D, section width B,

flange thickness tf, web thickness tw, yield stress fy and ultimate load achieved Nu,

have been reported in Table 6.1. The adopted dimension labelling system and the

locations of cover plates, are illustrated in Fig. 5.7; the width and thickness of the

cover plate – Bcp and tc – are reported in the same column as flange width and

thickness (B and tf, respectively) of Table 6.1. The test configurations of all 7 tests are

illustrated in Fig. 6.1, where the longitudinal locations of the cover plates along the

specimens are also presented.


171

Table 6.1: Geometric and material properties and ultimate capacities of I-section

continuous beams

Specimen Description D (mm)

B or Bcp (mm)

tf or tc (mm)

tw (mm)

fy (N/mm2)

fu (N/mm2)

Nu (kN)

Beam 1a Main beam 152.40 84.63 5.89 9.12 271 - 117.0 Main beam 127.00 76.30 5.44 8.28 279 -

Beam 2a Cover plate - 101.60 4.76 - 256 -

109.9

Main beam 101.60 67.64 4.90 7.44 271 - Outer cover

plate - 88.90 6.35 - 284 - Beam 3a Inner cover

plate - 76.20 6.35 - 284 -

110.3

Main beam 127.00 76.30 5.44 8.28 279 - Beam 4a

Cover plate - 88.90 11.11 - 261 - 109.0

Main beam 127.00 76.30 5.44 8.28 279 - Beam 5a

Cover plate - 88.90 11.11 - 261 - 161.9

Beam 6b Main beam 211.33 204.72 14.02 9.40 259 447 500.8 Beam 7c Main beam 312.42 168.28 13.06 8.56 234 412 1051.7 a Popov and Willis (1957); b Yang et al. (1952); c Driscoll et al. (1957)


172

N N

N N

N N

N N

N

Inner cover plate Outer cover plate

(a) Beam 1

(L = 1067)

(b) Beam 2

(L = 1270)

(c) Beam 3

(L = 2438)

(d) Beam 4

(L = 2438)

(e) Beam 5

(L = 2438)

L/2 L/2 L/2 L/2

N/2

(f) Beam 6

(L = 4267)

N/2

L/3 L/3 L/3 L/2 L/2

L L/2 L/2

Cover plate


173

Fig. 6.1: Schematic continuous beam test arrangements (dimensions in mm)

6.2.2 Full-scale frame tests

A total of 5 tests on full-scale steel frames have been collected from the literature. The

geometric and material properties of the sections employed in the frame tests are

presented in Table 6.2. The labelling system is as given in Fig. 5.5. Table 6.3 provides

the source of the frame test data and the ultimate collapse loads achieved. These

values should be considered with reference to the overall frame geometry and loading

presented in Figs 6.2 (a)–(f).

Table 6.2: Cross-section dimensions of beam specimens from frame tests

Section Ref. D (mm)

B (mm)

tf (mm)

tw (mm)

fy (N/mm2)

fu (N/mm2)

Mpl (kNm)

I-section 1a 127.00 76.20 7.78 10.45 272 - 28.0 I-section 2b 127.00 76.20 10.38 5.36 257 - 27.6 I-section 3c 312.42 168.28 13.06 8.56 234 412 211.3 I-section 4d 211.33 204.72 14.02 9.40 259 447 168.5 a Charlton (1960); b Baker and Eickhoff (1955); c Driscoll et al. (1957); d Ruzek et al. (1954)

Table 6.3: Ultimate capacities of frame tests

Frame specimen Section Resource Total collapse

load Nu (kN) Frame 1 I-section 1 Charlton (1960) 107.1 Frame 2 I-section 2 Baker and Eickhoff (1955) 62.8 Frame 3 I-section 2 Baker and Eickhoff (1955) 105.9 Frame 4 I-section 3 Driscoll et al. (1957) 402.1 Frame 5 I-section 4 Ruzek et al. (1954) 462.1

N/4

(g) Beam 7

(L = 3048)

N/4

L/3 L/3 L/3 L/3 L/3 L/3

N/4 N/4


174

(a) Frame 1

(b) Frame 2

A

B

4162 520 1041 1041 1041 1235

520

22.5˚

714

N/4 N/4

N/4 N/4

N/2

1524

1143 2286 1143 1143 2286 1143

473

947

473

N/4

N/4

N/4

N/4


175

(c) Frame 3

(d) Frame 4

2L

L/4 L/2 L/4 L/4 L/2 L/4

L/6 L/6 L/6

L/2

L/3

N/4 N/4

N/4 N/4 N/4

N/4

L = 6096

4162 520 1041 1041 1041 1235

22.5˚

714

N/4 N/4

N/4 N/4

520


176

(e) Frame 5

Fig. 6.2: Geometry and loading arrangements for frame tests (dimensions in mm)

6.3 NUMERICAL MODELLING

A numerical study was performed in parallel with the test programme reported in

Chapter 3, using the finite element (FE) analysis package ABAQUS, Version 6.7-1.

The primary aims of the investigation were to validate the numerical models against

the generated experimental data, and, once validated, to perform parametric studies to

provide results to support the development of the CSM for indeterminate structures,

which will be described later in this chapter. The key issues to be investigated include

a kinematic deformation capacity assumption, moment redistribution and second

order effects. With the focus being indeterminate structures, validation was performed

against existing tests on determinate and indeterminate beams and frames.

6.3.1 Modelling

Finite element models were developed using both shell and beam elements, with local

buckling being artificially simulated in the case of the latter elements by means of

specifying moment–curvature characteristics with an unloading branch.

6.3.1.1 Shell elements model

For the shell element based models, the reduced integration 4-noded shell elements

designated S4R, which are suitable for thin or thick shell applications (ABAQUS,

2007) have been employed for the initial modelling of bending tests. Mesh

convergence suggested a uniform mesh density throughout the models could achieve

1600

2134

1067 1600

N/2 N/2


177

accurate results, while maintaining reasonable computational times. A suitable mesh

size was found to be 5 mm with the deviation factor for the curvature control being set

at 0.05. This provided approximately 6 elements across the width of the flat parts of

each cross-section and 4 elements to approximate the curved corner geometry.

For each specimen considered, only half the cross-section was modelled, with suitable

symmetry boundary conditions applied to reflect the symmetry in material properties,

boundary conditions, loading and failure modes observed in the tested beams. The end

supports and loads were applied at the junctions of the webs with the corner radii in

the lower parts of the beams to avoid web crippling. The full length of each beam was

modelled to allow anti-symmetric as well as symmetric local buckling modes.

Measured geometry was incorporated into the models.

For material properties employed in the FE models, the continuous engineering stress-

strain curve measured in the coupon tests was converted into the true stress ftrue–log

plastic strain pllnε format required by ABAQUS, in two steps: (1) Firstly, the

engineering stress-strain curve (fnom–εnom) was discretised and represented by a total

of 20 points; (2) and secondly, these points were converted into true stress–log plastic

strain by means of Eqs (6.1) and (6.2).

)ε1(ff nomnomtrue += (6.1)

Ef)ε1ln(ε true

nomplln −+= (6.2)

For cold-rolled sections, plastic deformation during forming leads to significant

strength enhancements in the corner regions of the cross-section; these strength

enhancements extend beyond the curved corners into the flat regions. For accurate

results, account of these enhancements is necessary. Gardner and Nethercot (2004)

and Ashraf et al. (2006) concluded that numerical models of press-braked stainless

steel cross-sections produce the closest predictions of test response when the

enhanced corner properties are extended up to a distance equal to the plate thickness

beyond the corner region. For cold-rolled section, this extension was found to be

twice the plate thickness. In order to assess how much extension of the enhanced

corner properties beyond curved regions was appropriate for cold-formed steel


178

specimens, extensions of corner material properties into the flat parts up to three

distances equal to 0, 1 and 2 times plate thickness were considered.

Initial geometric imperfections exist in all structural members and influence their

structural responses. In this study, for each model, initially a linear eigenvalue

buckling analysis was carried out to obtain the lowest local buckling mode. This

shape was then taken as the initial geometric imperfection and incorporated into a

subsequent geometrically nonlinear analysis. There were 4 different imperfection

amplitudes considered: the mean measured imperfection, t/10, t/100 (where t is the

section thickness) and the Dawson and Walker model (Dawson and Walker, 1972;

Gardner et al., 2010).

6.3.1.2 Beam element models

When local buckling of thin-walled structures and softening behaviour are expected in

an analysis, shell elements have conventionally been chosen. However, although shell

elements enable local failure modes to be accurately captured, they are

computationally expensive, and can become impractical for the modelling of full

structural frames. In this study, linear beam elements (designated B21 in ABAQUS,

2007) have been employed for the modelling of frames, with local buckling

incorporated by assigning suitable moment–curvature characteristics to the elements

(in place of material stress-strain characteristics).

Moment–curvature response

ABAQUS (2007) offers ‘Axial’ and ‘M1’ commands under the Nonlinear General

Section input option, which can be input by editing the keywords of the input file. The

moment–curvature response of a cross-section can first be obtained from a shell

element based model simulating a symmetrical 4-point bending test, as shown in Fig.

6.3 where a region of constant moment and curvature between the two loading points

exists. In this region, the average curvature of the beam kc can be calculated from the

output displacement values at midspan and at the loading points as follows:


179

21s

2LM

LMc L)δδ(4

)δδ(8r1k

+−−

== (6.3)

where r is the radius of curvature, δM is the displacement at midspan, δL is the

displacement at the loading points and Ls1 is the distance between the two loading

points. Eq. (6.3) was employed by Chan and Gardner (2008b) for a similar purpose

and can be derived on the assumption of a circular deflected shape between the

loading points. The values of curvature obtained from this equation were found to be

identical to those outputs directly from ABAQUS at the element level (Brown, 2010).

(a) Symmetrical 4-point bending test simulated in ABAQUS

(b) Curvature obtained from displacements via Eq. (6.3)

Fig. 6.3: Method of obtaining moment–curvature relationship

0

1

2

3

4

5

0.000 0.005 0.010 0.015 Curvature

Mom

ent (

kNm

)

Ls2 Ls2 Ls1

Ls2 Ls2 Ls1

r δL δM

N N

N N

N N


180

Hence, the output from the shell element models was used as input to the beam

element models.

6.3.2 Validation

The accuracy of the numerical simulations was assessed by comparing the ultimate

moments and initial stiffness achieved in the FE models with the test values. The

general shapes of the moment–rotation curves and the modes of failure have also been

compared.

6.3.2.1 Shell element models

For the shell element based models, the results show that a 2t extension of the corner

material properties into the flat regions of cold-formed sections, which were used for

all the following modelling studies, provides best agreement with test values, and

every one thickness extension leads to about a 4% increase in the ultimate load. This

percentage will vary though with section thickness. The results of the numerical

simulations of the simple and continuous beams are tabulated in Tables 6.4 and 6.5

respectively, with 4 imperfection amplitudes being considered, as previously

discussed. The t/100 amplitude, which was generally slightly higher than measured

values, was chosen for the parametric studies because this provided the most accurate

representation of the test results and may be considered to compensate for the non-

inclusion of residual stresses. Figures 6.4 and 6.5, where Ncoll denotes the plastic

collapse load, show good agreement between the experimental and numerical

performances of a typical simple and continuous beam, respectively. Typical

experimental and numerical failure modes are displayed in Fig. 6.6.


181

Fig. 6.4: Comparison between the experimental and numerical results for the simple

beam RHS 60×40×4-CF

Fig. 6.5: Comparison between experimental and numerical results for the continuous

beam RHS 60×40×4-CF1

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0 2 4 6 8 10 12 14

Test

FE

θ/θpl

M/M

pl

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

End rotation θ (rad.)

N/N

coll

Test FE


182

(a) Failure mode in test

(b) Failure mode in FE model

Fig. 6.6: Comparison between the experimental and numerical failure modes for the

simple beam SHS 40×40×3-CF

Table 6.4: Validation of FE models against simple beam test results considering 4

imperfection amplitudes

Measured amplitude t/10 t/100 Dawson and

Walker Beam specimen designation

Test Mu / FE Mu SHS 40×40×4-HR 0.91 0.97 0.92 0.91 SHS 40×40×4-CF 1.05 1.05 1.05 1.05 SHS 40×40×3-HR 1.05 1.12 1.05 1.05 SHS 40×40×3-CF 1.00 1.02 1.00 1.00 RHS 60×40×4-HR 0.92 0.98 0.93 0.92 RHS 60×40×4-CF 1.03 1.04 1.03 1.03 Mean 0.99 1.03 0.99 0.99 COV 0.06 0.05 0.06 0.06


183

Table 6.5: Validation of FE models against continuous beam test results considering 4

imperfection amplitudes

Measured amplitude t/10 t/100 Dawson and

Walker Beam specimen designation

Test Nu / FE Nu RHS 60×40×4-CF1 1.04 1.05 1.04 1.04 RHS 60×40×4-CF2 1.04 1.05 1.04 1.04 SHS 40×40×3-CF1 1.03 1.06 1.03 1.03 SHS 40×40×4-HR1 0.98 1.03 0.98 0.98 SHS 40×40×3-HR1 1.06 1.13 1.06 1.05 SHS 40×40×4-CF1 1.06 1.07 1.06 1.06 RHS 60×40×4-HR1 0.97 1.00 0.97 0.97 SHS 40×40×4-HR2 1.00 1.02 1.00 1.00 SHS 40×40×3-HR2 1.09 1.13 1.09 1.09 SHS 40×40×3-CF2 1.02 1.03 1.02 1.03 SHS 40×40×4-CF2 1.07 1.08 1.07 1.07 RHS 60×40×4-HR2 1.05 1.08 1.05 1.05 Mean 1.03 1.06 1.03 1.03 COV 0.03 0.04 0.03 0.04

6.3.2.2 Beam element model

Validation for the method of replacing the shell element models by beam element

models to consider local buckling effects consisted of three steps. Note that, as

expected, the beam element based models (with moment–curvature characteristics

extracted from prior shell element based models) yielded identical results to the

original shell element based models. The accuracy of the beam element based

numerical models when representing the physical behaviour of arrangements different

to those from which the moment–curvature characteristics were originally generated

was assessed by considering continuous beams. Firstly, the moment–curvature

relationships were obtained from the validated shell element based models using the

approach described in Section 6.3.1.2. Secondly, these relationships were used as

input data for the beam element based models to simulate the behaviour of continuous

beams of the same section as the original simple beams, and load–rotation responses

was obtained. Finally, the load–rotation relationships from the beam element based

models were compared with the corresponding shell element based models and the

test results. Four comparisons have been made, as shown in Figs 6.7–6.10.


184

Figures 6.7 to 6.10 enable comparisons of the ultimate moment, initial stiffnesses and

the general shape of the moment–rotation curves from the beam element based FE

models with those from the corresponding shell element based models and the tests.

Overall, good general agreement was found in all comparisons. However, there was a

consistent tendency for the beam element based models to unload more rapidly than

the corresponding shell element based models. This is believed to be due to the fact

that the beam element based models were developed on the basis of the most severe

condition of pure bending (i.e. as experienced in the 4-point bending arrangement),

whilst in the continuous beam arrangement, moment gradients exist. For the shell

element based models, the additional support (i.e. restriction of the local buckling

wavelength) from the regions adjacent to those at peak moment will be explicitly

modelled, but this is not the case for the beam element based models. The beam

element based models would therefore be expected to provide a conservative

representation of the unloading response, and may hence be employed safely in

subsequent parametric studies. Differences between the test and FE results in the early

stages of loading seen in Figs 6.9 and 6.10 (i.e. the hot-rolled continuous beams) is

believed to relate to neglection of the upper yield point in the FE models.

Fig. 6.7: Continuous beam SHS 40×40×3 CF1

0

5

10

15

20

25

30

35

40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 End rotation (rad.)

Load

(kN

)

Test FE (Shell) FE (Beam)


185

Fig. 6.8: Continuous beam SHS 40×40×3-CF2

Fig. 6.9: Continuous beam SHS 40×40×3-HR1

0

5

10

15

20

25

30

35

40

45

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 End rotation (rad.)

Load

(kN

)

Test FE (shell) FE (beam)

0

5

10

15

20

25

30

35

40

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 End rotation (rad.)

Load

(kN

)



186

Fig. 6.10: Continuous beam SHS 40×40×3-HR2

Having demonstrated the suitability of the beam element based models for safely

representing the behaviour of continuous beams, the approach was extended to the

simulation of full-scale frames to provide more data for the development of the CSM

for practical indeterminate structures. A portal frame test conducted by Charlton

(1960) was simulated using the previously described method of inputting the

moment–curvature relationship from a validated shell element based FE models of a

simple beam into a beam element based model of the full frame.

Hence, the moment–curvature response from the validated shell element based models

simulating a control beam with I-section 1, as shown in Table 6.2 (Charlton, 1960)

was obtained and input into a beam element based model of the full-scale portal frame

test (see Fig. 6.2 (a)) carried out by Charlton (1960). The results from the numerical

simulation have been plotted together with the simple plastic analysis results and test

results in Fig. 6.11. The unloading response of the test frame was not recorded but

comparison with the available data shows good agreement between FE and test results,

and the conservatism of plastic design.

0

10

20

30

40

50

60

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 End rotation (rad.)

Load

(kN

)



187

Fig. 6.11: Numerical modelling of full-scale portal frame test (Charlton, 1960)

6.4 DESIGN APPROACH

6.4.1 Traditional plastic analysis method

As described at the beginning of this chapter, traditional plastic analysis methods,

which are based on the formation and subsequent rotation of plastic hinges at their full

plastic moment capacity, are generally employed to design indeterminate steel

structures.

The aim of the plastic analysis method is to calculate the ultimate capacity at which a

plastic collapse mechanism first forms. Hence, the critical collapse mechanism needs

to be found. In the basic approach of plastic analysis, the locations of a series of

plastic hinges are assumed and three conditions: equilbrium, mechanism and plasticity;

are required to be met. The static equilbrium condition means that the externally

applied loads must be in equilbrium with the internal forces and moments that resist

the loads. The compatibility condition states that when the ultimate plastic load is

reached, a collapse mechanism is formed and the number of the plastic hinges are just

sufficient to form the mechanism. The plasticity condition states that the calculated

moment at any cross-section should not exceed the full plastic moment capacity of the

0

30

60

90

120

0 80 160 240 320 400 Apex vertical deflection (mm)

Load

(kN

)

Simple plastic design Test

FE simulation

Maximum Load Carried = 107.1 kN

Plastic collapse load Ncoll = 94.7 kN


188

section. If these three conditions are satisified, the defined mechanism is the correct

one, as shown in Eq. (6.4).

Once the correct critical collapse mechanism has been obtained, the plastic collapse

load can be calculated using the kinematic method (also known as the virtual-work

method) (Bruneau et al., 1998). This method determines the plastic collapse load by

considering that plastic deformation is restricted to discrete hinges and that rigid links

exist between these hinges. In the collapse mechanism, the internal work in the plastic

hinges must be equal to the external work done by the applied loads. Therefore, the

design approach of the kinematic method can be summarised as follows, which is

illustrated (Fig. 6.12) on a two-span continuous beam with a Class 1 cross-section.

Fig. 6.12: Plastic collapse mechanism for two-span continuous beam

(1) Determine the collapse mechanism by satisfying the three conditions

(Equilibrium, Mechanism, and Plasticity) and identify the locations of

plastic hinges – see Fig. 6.12.

(2) Calculate the cross-section plastic bending moment capacity Mpl at the

plastic hinges.

(3) Determine kinematically the rotations θ1 and θ2 at each plastic hinge location.

(4) Using virtual work, determine the final collapse load by equating the

external work done by the loads to the internal work resulting from rotation

(True collapse mechanism) satisfies Equilibrium CompatibilityPlasticity

(6.4) Conditions

δ

θ2

N θ1 N

L1 L2 L2 L1


189

of the plastic hinges, given for the continuous beam shown in Fig. 6.13 by

Eq. (6.5).

Fig. 6.13: Collapse bending moment diagram from traditional plastic analysis

2Nδ = Mplθ1 + 2Mplθ2 (6.5)

The distinction between continuous beams and frames, in terms of structural

behaviour, is that the former are much less sensitive to second order effects. Therefore,

in order to extend the application range of the CSM to all indeterminate structures,

relevant existing frame test data should be analysed and the beneficial influence of

strain-hardening considered alongside the deleterious influence of second order

effects, which is described below.

6.4.1.1 Deleterious influence of second order effects

A first order (linear) analysis is based on the initial geometry of the structure and

ignores deformation of the structure under load. In reality, however, during the

loading process, there will be favourable or unfavourable changes in geometry which

will lead to the collapse load exceeding or falling short of that predicted by first order

analysis (Wood, 1958).

The second order effects caused by the changes in geometry are generally considered

in two categories: ‘P–δ’ effects arise from deflections within the length of members

and ‘P–∆’ effects arise from displacements of the overall frame (Trahair et al., 2008),

where δ and ∆ have been illustrated in Fig. 6.14. EN 1993-1-1 (2005) requires

consideration of the latter (sway effects) when the αcr parameter, which is the factor

Mpl

Mpl Mpl


190

by which the design loading would have to be increased to cause elastic instability of

the frame in a global mode, exceeds a limiting value.

Fig. 6.14: Illustration of second-order effects

The αcr parameter is defined in EN 1993-1-1 (2005) as:

Ed

crcr F

Fα = (6.6)

where FEd is the design load on the structure (calculated, assuming the frame is loaded

to its full design capacity, as equal to the plastic collapse load in this study) and Fcr is

the elastic critical buckling load for global sway instability, based on initial elastic

stiffnesses. According to Eurocode 3, if αcr is larger than 10 for elastic analysis or 15

for plastic analysis, second order effects do not need to be considered. As described

later, these limiting values are modified in the UK National Annex to EN 1993-1-1

(NA to BS EN 1993-1-1, 2005).

6.4.1.2 Beneficial influence of strain-hardening effects

There are two distinct beneficial effects of strain-hardening in the formation of a

plastic hinge. Firstly, in a region of approximately constant flexural moment, the

moment of resistance may rise above the value of the full plastic moment and remain

approximately constant as the rotation at the hinge increases (Byfield and Nethercot,

1998; Kemp et al., 2002; Lim et al., 2005). Secondly, when a hinge forms in a region

of significant bending moment gradient, it first forms at the calculated value of the

δ

∆


191

full plastic moment and then steadily rises in capacity as the hinge rotates in the

plastic stage of loading to collapse (Lim et al., 2005).

There are existing approaches for utilising strain-hardening, such as those suggested

by Kemp et al. (2002) and Davies (1966; 2002; 2006), both of which have been

summarised in Chapter 5. However, the method in Kemp et al. (2002) does not cover

indeterminate structures or allow for second order effects, while the method in Davies

(2006) relies on a sophisticated computer programme.

6.4.1.3 Sensitive balance between second order and strain-hardening

On the basis of the above introductions, it can be seen that there is a sensitive balance

between the (generally) opposing influences of second order effects and strain-

hardening effects. In many frames of practical proportions, the benefits of strain-

hardening in the hinges may outweigh the deleterious influence of second order

effects. The current treatment in EN 1993-1-1 (2005) effectively implies that the

balance point is at αcr = 15 for plastic analysis.

Consideration of a range of αcr values is therefore required in order to study the

sensitive balance between these two effects and their impact on the load capacities of

indeterminate structural steel frames. This matter is raised further in Section 6.4.3.3 of

this thesis.

6.4.2 Development of the CSM for indeterminate structures

A new design approach that combines features of the traditional plastic design method

and the CSM has been developed to determine the collapse loads of indeterminate

steel structures, with due allowance for the influence of strain-hardening. For a given

collapse mechanism, the critical plastic hinge is first identified as the one that

undergoes the greatest rotation relative to the deformation capacity of the cross-

section at that location. The demands at other plastic hinge locations, i, are then

assigned in proportion to the ratio of the plastic hinge rotations in the mechanism, as

shown in Fig. 6.12, ensuring that, if variable section sizes are used, that the

deformation demand εhingei/εy is reduced in proportion to hiθi/h1θ1 and remains below


192

the deformation capacity at that location εLBi/εy. Based on the resulting deformations,

the corresponding bending moment diagram at collapse is determined.

The key design steps, applied for illustration purposes to the two-span continuous

beam, shown in Fig. 6.12, are summarised below:

(1) Identify the locations of the plastic hinges in a similar manner to traditional

plastic design – see Fig. 6.12.

(2) Based on cross-section slenderness (Eq. (3.7)), calculate the corresponding

cross-section deformation capacity εLB1/εy at hinge 1 (Eq. (5.4)).

(3) Determine kinematically the deformation demands (εhingei/εy) at each plastic

hinge location, i, on the basis of the aforementioned assumptions and Eqs (6.7)

and (6.8), where θ1 > θi.

y

1LB

y

1hinge

εε

εε

= (6.7)

y

LBi

y

1LB

11

ii

y

hingei

εε

εε

θhθh

εε

≤= (6.8)

(4) Calculate the corresponding bending moments at the plastic hinges, Mhingei,

from Eq. (5.10) or (5.11), to yield the collapse bending moment diagram, as

shown in Fig. 6.15. Note that the two hinges forming in the spans undergo the

same rotation, and the moments at these locations are equal and have both

been referred to as Mhinge2 in Fig. 6.15.

(5) Using virtual work, determine the final collapse load by equating the external

work done by the loads to the internal work resulting from rotation of the

plastic hinges, given for the continuous beam shown in Fig. 6.12 by Eq. (6.9).

22hinge11hinge θM2θMδN2 += (6.9)


193

Fig. 6.15: Collapse bending moment diagram (CSM)

Satisfaction of the three conditions of equilibrium, compatibility and plasticity

remains a strict requirement in defining the unique plastic collapse load of a structure

within the continuous strength method. The key diversion from traditional plastic

analysis is in the plasticity condition, where the moment capacity obtained for each

hinge from the CSM is used in place of Mpl. Comparisons of predicted collapse loads

from traditional plastic analysis, representing the Eurocode 3 approach, and the CSM

with those obtained from the reported continuous beam tests and full-scale frame tests

are made and presented later in this chapter.

6.4.3 Parametric studies

6.4.3.1 Validation of kinematic assumption for determining deformation demand

As described above, when the collapse load of a structure is determined by the CSM,

the deformation demands at plastic hinges are required and are assumed to be in

proportion to the ratio of the plastic hinge rotations determined kinematically on the

basis of the chosen collapse mechanism. It is important to verify that this kinematic

assumption accords with the actual physical response of structures.

The two different configurations of continuous beam tests, which were reported in

Chapter 3, were chosen as the basis for this investigation. For each test configuration,

the deformation demand at collapse determined by the kinematic assumption, and the

curvature distribution obtained from ABAQUS, are plotted simultaneously in Figs

6.16 and 6.17 (ABAQUS, 2007).

In both cases, the results show that the actual deformation demands (obtained

numerically) accord closely with the assumed deformation demands determined

Mhinge1

Mhinge2 Mhinge2


194

kinematically. The kinematic assumption for determining deformation demand is

therefore considered to be validated.

Fig. 6.16: Deformation demands for continuous beam configuration 1

Fig. 6.17: Deformation demands for continuous beam configuration 2

θ1 N δ

L/2

N

L/2

θ2

L/2 L/2

Curvature ku1

ku2

θ1 = θ2 according to kinematic assumption

ku1 ≈ ku2 obtained from ABAQUS model

(a) Kinematic assumption

(b) ABAQUS model

θ2

N θ1

N δ

2L/3 L/3 L/3 2L/3

ku1

ku2

Curvature

θ1 = 2θ2 according to kinematic assumption

ku1 ≈ 2ku2 obtained from ABAQUS model

(a) Kinematic assumption

(b) ABAQUS model


195

6.4.3.2 Partial moment redistribution

The cross-section classification system given in Eurocode 3 states that Class 2

sections can reach Mpl but no moment redistribution is allowed (i.e. failure based on

first hinge) while for Class 1 sections full moment redistribution (i.e. failure based on

plastic collapse mechanism). In the CSM, when εLB/εy = 1, a value of which

corresponds to the Class 3 limit, the local buckling moment MLB = Mel; when εLB/εy =

3, a value of which corresponds to the Class 1 limit, MLB = Mpl. However, once εLB/εy

> 1, the structure is inelastic, so for indeterminate structures, the elastic distribution of

forces and moments will change. In this section, theoretical and numerical analyses

are performed to investigate moment redistribution in indeterminate structures

comprising cross-section with εLB/εy = 1 to 3.

For a given collapse mechanism, the CSM assumes that the deformations at the hinges

are proportional to the kinematic hinge rotations, determined on the basis of the

structure’s geometry, loading and boundary conditions. The critical hinge (i.e. the first

hinge to form) is assumed to reach εLB at peak load, while other hinges are at the same

or lower values of εLB defined by these kinematic hinge rotations. Thus, it is not

allowed that all hinges are at εLB (unless all hinges form at the same time, e.g. a

configuration such as that shown in Fig. 6.16), since this would imply that some

hinges would have to go beyond their peak moments while others had yet to attain

theirs.

To investigate the applicability of the above approach, a numerical study has been

carried out to examine moment redistribution in indeterminate structures comprising

cross-sections with εLB/εy between 1 and 3. The elements chosen for the numerical

model were 2-noded linear beam elements, designated as B21 in the ABAQUS

element library. The analysed configuration is shown in Fig. 6.17.

Figure 6.18(a) records the development of the relationship between total load and the

average displacement at the load points. Figures 6.18 (b) and (c) depict the moment-

curvature relationships at the central support and loading points respectively,

throughout the loading process. In order to demonstrate that, during the moment

redistribution, no cross-sections undergo deformation beyond εLB, discrete points on


196

the three curves shown in Figs 6.18 (a-c) were extracted. Points with the same

symbols (circular, square or triangular) and numerical markers, but in different figures,

denote points recorded at the same load levels. The three point types indicate the

stages of loading based on the first hinge: elastic (ε/εy ≤ 1), elastic-plastic (1 < ε/εy ≤ 3)

and strain-hardening (ε/εy ≥ 3), according to the CSM moment predictive model

shown in Fig. 5.24.

In Fig. 6.18, point ‘□1’ indicates first yield. Beyond this point, the structure no longer

behaves elastically and moment redistribution commences. Load increases until

plastic collapse occurs, defined as the point at which the first hinge reaches εLB,

marked ‘О6’ in Fig. 6.18. From Fig. 6.18 (b), the first hinge (at the central support)

may be seen to be at its peak moment, while the hinges at the loading points are yet to

reach εLB and their peak moments – see Fig. 6.18 (c). Therefore, at no points are the

deformation capacities of the cross-sections exceeded. This finding has been

confirmed numerically for a range of configurations, and verifies the safe applicability

of partial moment redistribution beyond εLB = 1, provided deformation demands at the

hinges are derived in proportion to their kinematic rotations.

(a) Load–vertical displacement relationship

0

20

40

60

80

100

0 5 10 15 20 Displacement (mm)

Tota

l Loa

d (k

N)

Elastic Elastic-plastic Strain-hardening

1

2 3

4 5 6

2

3

4

5

6

1

2

4 1

Plastic collapse

First yield


197

(b) Moment–curvature relationship at central support

(c) Moment–curvature relationship at loading points

Fig. 6.18: Numerical study of partial moment redistribution

6.4.3.3 Second order effects

As discussed previously, frames, unlike continuous beams may be sensitive to second

order effects, depending on their geometry and loading. The key parameter αcr, which

was introduced in Section 6.4.1.1 is employed herein as the measure of the sensitivity

of a frame to second order effects. The verified beam element based model was used

0

1

2

3

4

5

6

7

8

0 0.0004 0.0008 0.0012 0.0016 0.002

Curvature kc (m-1)

Mom

ent (

kNm

)

1

2

3

4

5

6 1

2 4 3

1 2 3 4 5 6

Plastic collapse

First yield Elastic Elastic-plastic Strain-hardening

0

1

2

3

4

5

6

7

0 0.0004 0.0008 0.0012 0.0016 0.002

Curvature kc (m-1)

Mom

ent (

kNm

)

1 2 3 4 5 6 1

2 3 4 1

2

3 4 5 6

Plastic collapse

First yield

Elastic Elastic-plastic Strain-hardening


198

to provide accurate estimations of αcr values and load factors at failure αf for a variety

of frames.

Choice of loading and frames

The single bay frame shown in Fig. 6.19 was considered. By varying the height of the

frame, varying values of αcr were obtained.

Fig. 6.19: Frame to be analysed in numerical study

Two load combinations (LC) were considered, derived from Eq. (6.10) of EN 1990

(2002):

LC1 1.35 Dead + 1.5 Imposed + NHL

LC2 1.35 Dead + 1.5 Imposed + 0.75 Wind + NHL

The notional horizontal load (NHL) was applied at the top of the storey and was taken

as 1/200 of the factored total vertical loads, the basic value suggested in EN 1993-1-1

(2005), to allow for frame imperfections.

The following unfactored values of loading were applied to the frames:

Dead Load: 1.0 kN/m

Imposed Load: 5.0 kN/m

Wind Load: 3.6 kN/m

Imposed load

Wind load

Notional horizontal load

Dead load

L

h


199

The properties of the members employed in the frames were the same as those used in

the Charlton test frame (Charlton, 1960), referred to as ‘I-section 1’ in Table 6.2. The

configurations of frames analysed are shown in Tables 6.6 and 6.7.

Table 6.6: Frames analysed under LC1

Frame number

Height h (mm)

Width L (mm) L/h Section used αcr

1a 1000 5000 5.0 I-section 1 18.07 1b 2000 5000 2.5 I-section 1 8.18 1c 3000 5000 1.7 I-section 1 4.47 1d 4000 5000 1.3 I-section 1 2.81 1e 5000 5000 1.0 I-section 1 1.93 1f 6000 5000 0.8 I-section 1 1.41 1g 8000 5000 0.6 I-section 1 0.85

Table 6.7: Frames analysed under LC2

Frame number

Height h (mm)

Width L (mm) L/h Section used αcr

2a 1000 5000 5.0 I-section 1 18.35 2b 2000 5000 2.5 I-section 1 8.93 2c 2500 5000 2.0 I-section 1 6.78 2d 3000 5000 1.7 I-section 1 5.42 2e 4000 5000 1.3 I-section 1 3.88 2f 5000 5000 1.0 I-section 1 3.08 2g 6000 5000 0.8 I-section 1 2.61

Code consideration of second order effects

In addition to the limiting value on the necessity to consider second order effects

given by EN 1993-1-1 (2005), more specific requirements provided by National

Annex (NA) to BS EN 1993-1-1 (2005) rules that consideration of second order

effects is not required if αcr ≥ 5 for load combination 1 (LC1) and αcr ≥ 10 for load

combination 2 (LC2).


200

Merchant-Rankine formula

Merchant (Merchant, 1954; Merchant et al. 1958) proposed an important approximate

approach of estimating the reduction factor from the first-order plastic collapse load

factor to the second order plastic load factor, based on the value of αcr, as given in Eq.

(6.10).

cr

cr

MRpl

f 1α

−α=

αα (6.10)

Owing to it being analogous to the ‘Rankine’ equation and recognising Merchant’s

work in this field, the above formula is generally known as the ‘Merchant-Rankine’

equation. The formula was initially proposed on a purely empirical basis but was later

found by Horne (1963) to have a theoretical basis if the lowest buckling mode and the

plastic collapse mechanism had a similar deflected pattern. If the shapes were

dissimilar, the formula might give conservative results. The formula is included in EN

1993-1-1 (2005).

Analysis and results

Numerical results

For each frame, two analyses were conducted: an eigenvalue (elastic buckling)

analysis in order to estimate αcr and a static Riks analysis allowing for both strain-

hardening and weakening to obtain values of ultimate load factors αf. Second order

effects were taken into account by enabling nonlinear geometry. Traditional plastic

analysis and the CSM were used to calculate the plastic load factor αpl and the CSM

collapse load factor (with strain-hardening) αpl,CSM at failure, respectively, values of

which are tabulated in Tables 6.8 and 6.9.


201

Table 6.8: Ultimate results of frames under LC1

Frame number αcr αf αpl

αf αpl,CSM

αpl,CSM αpl

1a 18.07 1.07 0.95 1.12 1b 8.18 1.05 0.94 1.12 1c 4.47 1.02 0.91 1.12 1d 2.81 1.00 0.90 1.12 1e 1.93 0.97 0.87 1.12 1f 1.41 0.92 0.82 1.12 1g 0.85 0.71 0.64 1.12

Table 6.9: Ultimate results of frames under LC2

Frame number αcr αf αpl

αf αpl,CSM

αpl,CSM αpl

2a 18.35 1.06 0.95 1.12 2b 8.93 1.02 0.91 1.12 2c 6.78 0.99 0.88 1.12 2d 5.42 0.95 0.85 1.12 2e 3.88 0.85 0.76 1.12 2f 3.08 0.75 0.67 1.12 2g 2.61 0.66 0.59 1.12

Presentation of results

The numerically generated results for load combination 1 (LC1) are presented in Fig.

6.20 by plotting the load factor at failure αf normalised by the first order plastic

collapse load factor αpl on the vertical axis against the elastic buckling load factor αcr

on the horizontal axis. Actual frame test data, including the three reported in Table 6.3

and a further 34 tests on multi-storey frames reported by Davies and Brown (1996)

which was originally presented by Low (1959) are also given in Fig. 6.20. The side

loads applied to the frames tested by Low (1959) were all lower than approximately

1/50 times the vertical loads. These horizontal forces are similar in magnitude to the

notional horizontal loads (equal to 1/200 of the vertical loads) employed in load case

1 (LC1). The results are therefore considered in parallel. Design loads according to

the UK NA to BS EN 1993-1-1 (2005), including the Merchant-Rankine reduction

formula, are also presented in Fig. 6.20.


202

Similar to Fig. 6.20, Fig. 6.21 presents the results generated from the numerical

models for load combination 2 (LC2), together with the two frame tests reported in

Table 6.3 and further 20 results on single storey frames generated in other numerical

studies (Lim et al., 2005). The design load according to the UK NA to BS EN 1993-1-

1 (2005) has again been presented.

Fig. 6.20: Comparisons of results against simple plastic analysis for LC1

Fig. 6.21: Comparisons of results against simple plastic analysis for LC2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20

Second order effects neglected

Numerical results Merchant-Rankine Frame test data

UK NA to BS EN 1993-1-1 (2005)

Merchant-Rankine formula

αcr

αf/α

pl

Strain-hardening balances second order effects

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25 30 35

Numerical results

UK NA to BS EN 1993-1-1 (2005)

Lim et al. numerical results (2005) Frame test data

Merchant-Rankine Merchant-Rankine formula

Strain-hardening balances second order effects

αcr

αf/α

pl


203

Analysis of results

The general trend that the higher the value of αcr the higher the normalised load

carrying capacity αf/αpl of the frames may be clearly observed from both the test and

numerical results in Figs 6.20 and 6.21.

The limiting values of αcr = 5 for LC1 and αcr = 10 for LC2, given in the UK NA to

BS EN 1993-1-1 (2005), below which second order effects must be considered may

be seen to be general agreement with the presented test and numerical data. The

numerical results suggest that strain-hardening balances the second order effects at

approximately αcr = 3 for LC1 and αcr = 7 for LC2, and thereafter strain-hardening is

dominant and enables capacities beyond the plastic collapse load to be attained.

For the test and FE results where αcr ≥ 5 for LC1 and αcr ≥ 10 for LC2, increases in

load carrying capacities up to 15% beyond the simple plastic collapse loads are

attained and may be attributed to strain-hardening effects, as discussed earlier in this

chapter. It is therefore required to consider strain-hardening on a rational basis in

order to achieve accurate predictions of ultimate collapse loads in stocky frames.

Note that the FE model employed in the present study was validated against the single

storey frame test carried out by Charlton (1960), and that all parametric studies have

considered single storey frames only. As shown in Figs 6.20 and 6.21, the results of

the FE model agree well with the numerical results obtained by Lim et al. (2005) on

single storey frames, but over-predict the test data on multi-storey frames obtained by

Low (1959). The proposed design method will therefore be restricted to single storey

frames, such as industrial portal frames, pending further investigation.

The Continuous Strength Method

The development of the CSM for indeterminate (single storey) frames is described in

this sub-section.

A normalised value of αcr,CSM was calculated from the equation:


204

CSM,pl

crCSM,cr α

α=α (6.11)

where αcr is the elastic critical load factor obtained from an eigenvalue analysis

relative to the first order plastic collapse load factor from CSM, αpl,CSM.

Figs 6.22 and 6.23 show, for load cases 1 and 2 (LC1 and LC2), respectively, the

frame collapse load factor αf normalised by the CSM collapse load factor αpl,CSM on

the vertical axis against the elastic buckling load factor αcr,CSM on the horizontal axis.

The Merchant-Rankine curve is also shown on both figures.

Fig. 6.22: Comparisons of results against CSM for LC1

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 αcr,CSM

αf/α

pl,C

SM

CSM design model


αcr, CSM = 5

Merchant-Rankine reduction to αpl,CSM

FE model Merchant-Rankine

Frame test data

CSM


205

Fig. 6.23: Comparisons of results against CSM for LC2

The comparisons show that all test results but one (in load case 2) are safely predicted

by the CSM. It is proposed that, as in the UK NA to EN 1993-1-1, explicit account for

second order effects be made by means of the Merchant-Rankine formula for αcr,CSM

below 5 for load case 1 and below 10 for load case 2 The test result that is over-

predicted by the CSM was obtained from Driscoll (1957); a continuous beam test

result from the same source is also over-predicted by the CSM as discussed in the

following section. The presented FE results generally lie below the CSM design

model by about 5%, but were also conservative by a similar margin when compared to

the frame test results of Charlton (1960). Further experimental results for frames with

a range of αcr from approximately 5 to 20, both for direct comparison with the CSM

and for the detailed validation of numerical models would be desirable, and are indeed

planned as part of future studies in this area.

6.5 ASSESSMENT OF THE CSM

6.5.1 Comparison of continuous beam test results with design models

A total of 12 two-span continuous beam tests on steel SHS and RHS were conducted

as part of the present study; two configurations were considered – in Configuration 1,

load was applied centrally between the supports (See Fig. 3.10), while in

Configuration 2, loads were applied closer to the central support, as shown in Fig.

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25 30 35

CSM design model

αcr,CSM

αf/α

pl,C

SM

αcr,CSM = 10


Merchant-Rankine reduction to αpl,CSM

FE model Merchant-Rankine

Frame test data

CSM


206

3.11. Comparisons of the results of the 12 continuous beam tests together with 7

further continuous beam test results collected from the literature (see Section 6.2.1)

with those obtained from traditional plastic analysis and the CSM are shown in Table

6.10. For each test, Table 6.10 contains slenderness pλ , calculated according to EN

1993-1-5 (2006), test collapse load and predicted collapse loads according to simple

plastic design and the CSM. An asterisk signifies that fu was not reported and hence

the strain-hardening modulus was based on nominal values for S275 steel with fu/fy =

430/275. The results are also depicted in Fig. 6.24 and compared with the Eurocode 3

design model, where Ncoll denotes plastic collapse load. FE results for hot-rolled

sections based on RHS 60×40×4-HR1 and for cold-formed sections based on RHS

60×40×4-CF1 section are also presented.

Table 6.10: Comparison of continuous beam test results with design methods

Continuous beam specimen pλ Nu

(kN) NEC3 (kN)

NCSM (kN)

Nu NEC3

Nu NCSM

NCSM NEC3

RHS 60×40×4-CF1a 0.17 83.4 58.3 70.5 1.43 1.18 1.21 RHS 60×40×4-CF2a 0.17 83.3 57.9 70.0 1.44 1.19 1.21 SHS 40×40×3-CF1a 0.26 34.2 26.4 30.0 1.30 1.14 1.14 SHS 40×40×4-HR1a 0.18 44.6 37.9 38.5 1.18 1.16 1.01 SHS 40×40×3-HR1a 0.25 38.1 32.1 32.6 1.18 1.17 1.01 SHS 40×40×4-CF1a 0.17 40.6 31.0 33.8 1.31 1.20 1.09 RHS 60×40×4-HR1a 0.18 78.1 67.8 68.9 1.15 1.13 1.02 SHS 40×40×4-HR2a 0.18 55.2 47.9 48.5 1.15 1.14 1.01 SHS 40×40×3-HR2a 0.25 49.0 40.5 41.0 1.21 1.19 1.01 SHS 40×40×3-CF2a 0.26 42.3 33.4 37.5 1.27 1.13 1.12 SHS 40×40×4-CF2a 0.17 51.5 38.7 42.0 1.33 1.23 1.09 RHS 60×40×4-HR2a 0.19 98.4 84.2 85.4 1.17 1.15 1.01 Beam 1b* 0.14 117.0 91.5 103.1 1.28 1.13 1.13 Beam 2b* 0.14 109.9 89.4 100.8 1.23 1.09 1.13 Beam 3b* 0.13 110.3 93.4 104.8 1.18 1.05 1.12 Beam 4b* 0.14 109.0 91.7 103.3 1.19 1.05 1.13 Beam 5b* 0.14 161.9 141.8 156.2 1.14 1.04 1.10 Beam 6c 0.39 500.8 473.7 507.5 1.06 0.99 1.07 Beam 7d 0.33 1051.7 1042.9 1191.9 1.01 0.88 1.14

Mean 1.22 1.12 1.09 COV 0.09 0.08 -

a Gardner et al. (2010); b Popov and Willis (1957); c Yang et al. (1952); d Driscoll et al. (1957)


207

Fig. 6.24: Continuous beam test and FE data compared with Eurocode 3 design model

From Table 6.10, the continuous strength method may be seen to provide a more

accurate prediction of the test behaviour, with a reduction in scatter and an average

increase in capacity of 9% over traditional plastic methods. This finding is illustrated

in Fig. 6.25, where test collapse loads have been normalised by those calculated using

the CSM.

Fig. 6.25: Continuous beam test and FE data compared with the CSM design model

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

Nu/N

coll

Tests – hot-rolled Tests – cold-formed

FE – cold-formed EC3 design model

FE – hot-rolled

Slenderness

Class 1/2 limit Class 2/3 limit

Class 3/4 limit

p λ

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

Nu/N

CSM

Slenderness p λ

Tests – hot-rolled Tests – cold-formed

FE – cold-formed CSM design model

FE – hot-rolled


208

As discussed in the previous sub-section, there is one continuous beam test reported

by Driscoll et al. (1957) whose resistance is over-predicted by the CSM. Another

frame test from the same resource (Driscoll et al., 1957) is also over-predicted by the

CSM. There is no clear explanation for this over-prediction, but a possible reason may

be related to the reported yield stress of the material.

In addition, the numerically generated on the more slender cross-sections also

dropped below the CSM design model. Since the FE models were validated primarily

on stocky sections, there are clearly uncertainties in extrapolating the model to more

slender sections, and further investigation is required in this area.

6.5.2 Comparison of full-scale frame test results with design models

As stated earlier, experimental results on full-scale steel frames consisting of

members with stocky cross-sections are relatively scarce, though 4 such tests have

been reported by Charlton (1960), Baker and Eickhoff (1955), Driscoll et al. (1957)

and Ruzek et al. (1954). The collapse loads of the frames were predicted using

traditional plastic analysis and the CSM. The collapse loads, as given in Table 6.11,

indicate that the CSM provides a more accurate prediction of the test response, with a

8% average increase in capacity over traditional plastic analysis. This improvement is

summarised in Fig. 6.26, where only test data having αcr,CSM larger than 5 for LC1 and

10 for LC2 are presented, whilst the details of the comparison of the frame tests with

Eurocode 3 are given in Figs 6.20-6.23.

Table 6.11: Comparison of frame test result with design methods

Frame reference

αcr,CS

M Nu

(kN) NEC3 (kN)

NCSM (kN)

Nu NEC3

Nu NCSM

NCSM NEC3

Frame 1a 12.3 107.1 94.7 102.5 1.13 1.05 1.08 Frame 2b 33.2 62.8 57.9 62.0 1.08 1.01 1.07 Frame 4c 28.9 402.1 391.1 427.5 1.03 0.94 1.09 Frame 5d 17.2 462.1 421.1 451.2 1.10 1.02 1.07

Mean 1.09 1.01 1.08

COV 0.04 0.05 - a Charlton (1960); b Baker and Eickhoff (1955); c Driscoll et al. (1957); d Ruzek et al. (1954)


209

Fig. 6.26: Frame test data compared with the CSM design model

6.6 DISCUSSION AND CONCLUDING REMARKS

In this chapter, it has been shown through the analysis of the experimental results

generated in this thesis and from the collection of existing test data, that the collapse

loads predicted by traditional plastic analysis of indeterminate structures are

conservative in the case of stocky sections due to the influence of strain-hardening. In

addition to the developments of the continuous strength method (CSM) presented in

Chapter 5, which offer a rational exploitation of strain-hardening in steel design,

extension of the method to cover indeterminate structures, following the principles of

traditional plastic analysis but allowing bending moments in excess of the plastic

moment capacity, has been proposed. Numerical modelling of the structural response

of continuous beams and frames was carried out to investigate issues that arose during

the development of the method for indeterminate structures. Comparisons have been

made against test results on continuous beams and full-scale frames. These

comparisons show that the CSM provides a more accurate prediction of test response

and enhanced structural load carrying capacity over current design methods. Areas

requiring further investigation in future studies have also been highlighted.

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25 30 35

αf/α

pl,C

SM

Frame test data CSM design model

αcr,CSM

Chapter 7 Conclusions and suggestions for further work

210

CHAPTER 7

CONCLUSIONS AND SUGGESTIONS

FOR FURTHER WORK

7.1 CONCLUSIONS

This section summarises the key findings from the present study and draws overall

conclusions. More detailed concluding remarks are given at the end of each individual

chapter.

Many of the principal concepts that underpin current structural steel design methods

were developed on the basis of bi-linear (elastic, perfectly-plastic) material behaviour;

such material behaviour lends itself to the concept of section classification. This

method is advantageous in terms of simplicity, but fails to reflect accurately the true

continuous nature of the relationship between section slenderness and resistance and

does not effectively utilise strain-hardening. Therefore, the primary objective of this

project has been to develop a more efficient structural steel design method, whilst,


211

where possible, achieving a minimum increase in complexity and maintaining

consistency with current design practice.

The development process of a proposed steel design method included: 1) reviewing

existing design methods; 2) gathering existing experimental data and supplementing

this with further data generated in the laboratory testing programme described in

Chapter 3; 3) developing a material model that allows for strain-hardening; 4)

developing design methods for both determinate and indeterminate structures and 5)

validating the new proposed method against test data plus supplementary finite

element results. The literature review presented in Chapter 2 provided a general

overview of previous work that is relevant to the present study.

An experimental programme, conducted as part of the present study on both hot-

rolled and cold-formed sections, was described in Chapter 3. Tensile coupon tests

were carried out to obtain the basic stress-strain responses of the tested cross-sections.

The marked enhancements of strength in the corner regions of the cold-formed

sections have been compared with the AISI predictive model and revised coefficients

have been proposed. A total of 6 simple beam tests and 12 continuous beam tests

(with two configurations) were performed on the hot-rolled and cold-formed steel

square and rectangular hollow sections to assess the influence of the two different

forming routes on the material and structural behaviour. Evaluation of the current

slenderness limits given in Eurocode 3 was carried out on the basis of the test results

with the results showing that the limits are generally acceptable. The experiments

were carefully conducted and reported and represent a useful contribution to the pool

of laboratory test data on steel structures.

Chapter 4 sets out to develop a material model that incorporates strain-hardening. A

literature review of the key factors influencing material properties suggested that the

ratio of fu/fy was important in determining the level of strain-hardening; this was

supported by the collected experimental stress-strain data. Following an appraisal of

existing material models, an elastic, linear-hardening model was chosen, since this

format provides not only a good representation of strain-hardening but also a

minimum increase of complexity. Existing material tensile coupon tests were

collected and, on the basis of these tests, a quantitative model to determine the strain-


212

hardening modulus was proposed. This model was subsequently employed in the

proposed design method.

The proposed steel design method for determinate structures was developed in

Chapter 5. Some shortcomings of the concepts of the current cross-section

classification system have been highlighted through experimentation and analysis of

existing test data. The limiting resistances adopted in present design practice - the

yield load (Ny) in compression and the plastic moment capacity (Mpl) in bending - are

conservative in the case of stocky sections, due to the influence of strain-hardening.

As an alternative treatment, the continuous strength method (CSM) has been

introduced. The continuous strength method employs a ‘base curve’ to define a

relationship between the slenderness and the deformation capacity of a cross-section,

as limited by local buckling, and incorporates the proposed material model to

rationally exploit strain-hardening.

Numerical comparisons of the results obtained from the CSM and Eurocode 3 were

made with test results for both compression and bending to verify the method. The

level of enhancement in resistance provided by the CSM over traditional design

approaches has been found to be approximately 5% for cross-section compression

resistance and 8% for in-plane bending strength, and there is also a reduction in the

scatter of the predictions. An additional benefit of the proposed approach is that cross-

section deformation capacity is explicitly determined in the calculations, thus

enabling a more sophisticated and informed assessment of ductility supply and

demand.

Chapter 6 described the development of the CSM for indeterminate structures.

Extension of the method to cover indeterminate structures involved following the

principles of traditional plastic analysis but allowing bending moments in excess of

the plastic moment capacity. Various aspects of the structural behaviour of

indeterminate steel structures have been investigated numerically, using the finite

element software ABAQUS. Both shell- and beam-element based numerical models

were employed, with local buckling being artificially simulated in the latter using

moment–curvature data extracted from the former. Following successful replication of


213

experimental results, parametric studies were conducted to address issues raised

during the extension of the CSM to indeterminate structures.

In order to verify the proposed method for indeterminate structures, experimental and

numerical results from both the present study and from existing published sources

were compared with the predicted results. Comparison was also made between the

CSM and with the current steel design method given in EN 1993-1-1 (2005). The

results revealed that EN 1993-1-1 (2005) underestimated, on average, 22% of the test

failure load of continuous beams with a coefficient of variation (COV) of 0.09, and

9% of the test failure load of full-scale frames with a COV of 0.04. By contrast, the

proposed method reduced the underestimation to 12% for continuous beams with a

COV of 0.08 and 1% overestimation on full-scale frames with a slightly increased

COV of 0.05.

Overall, the objective of creating a more efficient steel design method has been

accomplished. The method maintains good consistency with current design methods

and with a similar volume of calculations. It is envisaged that the CSM will be

considered for future incorporation into Eurocode 3 and other international structural

steel design standards.

7.2 SUGGESTIONS FOR FURTHER WORK

The global transition towards the reduced environmental impact and sustainable

development, together with the recent cost increases of steel, justifies efforts for

improving efficiency of material usage in structural design. The design approach

developed in this thesis meets the above changes, but further research is required.

Suggested future areas of investigation are given below.

7.2.1 Member buckling

The scope of the CSM is currently restricted to determination of cross-section

resistance only. Clearly member buckling resistance will also benefit from

consideration of strain-hardening, where the enhancement in capacity will reduce with

increasing member slenderness. Extension of the CSM to member level for both


214

columns (flexural buckling) and beams (lateral torsional buckling) is an area for

future development.

7.2.2 Fire design

Structural deformations experienced in fires are higher than those that can be tolerated

under normal service conditions. Full benefit can therefore be taken of the strain-

hardening capacity of steel at elevated temperatures. Extension of the CSM to design

of steel structures at elevated temperatures is an area for future work.

7.2.3 Other metallic materials

Metallic materials with high alloy content tend to exhibit more rounded stress-strain

characteristics with significant strain-hardening – two such examples are aluminium

alloys and stainless steels. These materials are likely to benefit to a greater extent

from consideration of strain-hardening through the CSM, and indeed, in the case of

stainless steel, this has already been demonstrated to be true. Further research in this

area, including harmonisation between the treatments of different materials, is

required.

7.2.4 Composite construction

Recent studies (Chung, 2010) have shown that strain-hardening can offer enhanced

capacity in composite construction. Extension of the deformation-based design

concept, where deformation capacity will be limited not necessary by local buckling

of the steel beam but perhaps by concrete crushing, to composite construction should

also be investigated.

7.2.5 Experiments on frames

Experimental results for frames that are representative of modern construction

practice with a range of αcr from approximately 5 to 20 are desirable, for both

comparison with the CSM and validation of numerical models on frames.

Furthermore, multi-storey frames are a potential application area for the CSM and


215

further development is required to extend the method which is currently restricted to

single storey frames.

References

216

REFERENCES

ABAQUS (2007). Analysis User’s Manual I-V. Version 6.7-1 ABAQUS, Inc.,

Dassault Systèmes. USA.

Adams, P. F. and Galambos, T. V. (1969). Material considerations in plastic design.

IABSE publications. 29:2, 1-18.

American Iron and Steel Institute (AISI). (1996). Specification for the design of cold-

formed steel structural members.

AISI 360 (2005). Specification for Structural Steel Buildings. American Iron and

Steel Institute.

Akiyama, H., Kuwamura, H., Yamada, S. and Chiu, J.-C. (1992) Influences of

manufacturing processes on the ultimate behaviour of box-section members,

Proceedings of the Third Pacific Structural Steel Conference. Tokyo, Japan. 313-320.

Allen, H. G. and Bulson, P. S. (1980). Background to Buckling. Maidenhead,

McGraw-Hill.

References

217

Alpsten, G. (1968). Thermal residual stresses in hot-rolled steel members. Fritz Eng.

Lab. Report No. 337.3, December.

Alpsten, G. (1972). Variations in mechanical and cross-sectional properties of steel.

Planning and Design of Tall Buildings. 755-803.

American Society of Civil Engineers (ASCE). (1972). Plastic design in steel – a guide

and commentrary. 2nd edition.

AS 4100 (1998). Steel structures. Standards Australia.

American Society for Testing and Materials (ASTM). (2000). Standard Specification

for High-Yield Strength, Quenched and Tempered Alloy Steel Plate, Suitable for

Welding (A514/A514M00a). West conshohocken, PA. USA.

Ashraf, M., Gardner, L. and Nethercot, D. A. (2005). Strength enhancement of the

corner regions of stainless steel cross-sections. Journal of Constructional Steel

Research. 61:1, 37-52.

Ashraf, M., Gardner, L. and Nethercot, D. A. (2006a). Compression strength of

stainless steel cross-sections. Journal of Constructional Steel Research. 62:1-2, 105-

115.

Ashraf, M., Gardner, L. and Nethercot, D. A. (2006b). Finite element modelling of

structural stainless steel cross-sections. Thin-Walled Structures. 44:10, 1048-1062.

Baker, J. F. and Eickhoff, K. G. (1955). The behavior of saw-tooth portal frames.

Proceeding of the Conference on the correlation between calculated and observed

stress and displacements in structures. Institution of Civil Engineers.

References

218

Bambach, M. R. and Rasmussen, K. J. R. (2004a). Effective widths of unstiffened

elements with stress gradient. Journal of Structural Engineering, ASCE. 130:10,

1611-1619.

Bambach, M. R. and Rasmussen, K. J. R. (2004b). Design provisions for sections

containing unstiffened elements. Journal of Structural Engineering, ASCE. 130:10,

1620-1628.

Bjorhovde, R., Brozzetti, J., Alpsten, G. and Tall, L. (1972). Residual stresses in thick

welded plates. Journal of AWS Welding. 51:8, 392–406.

Bredenkamp, P. J., Van Den Berg, G. J. and Van Der Merwe, P. (1992). Residual

stresses and the strength of stainless steel I-section column. The Structural Stability

Research Council. Pittsburgh, PA. USA. 205-223.

Brockenbrough, R. L. and Merritt, F. S. (1999). Structural steel designer's handbook.

3rd Edition. McGraw-Hill. London.

Brown, L. (2010). Strain hardening and second order effects in steel frames.

Unpublished M.Eng thesis, Imperial College London, UK.

Bruneau, M., Uang, C.-M. and Whittaker, A. (1998). Ductile design of steel structures.

New York, McGraw-Hill.

BS 449 (1932). British standard specification No 449 for the use of structural steel in

building. British Standards Institution.

BS 4-1 (2005). Structural steel sections. Specification for hot-rolled sections. British

Standards Institution.

BS 5950 (2000). Structural use of steelwork in building. British Standards Institution.

References

219

Byfield, M. P. and Nethercot, D. A. (1997). Material and geometric properties of

structural steel for use in design. The Structural Engineer. 75:21, 363-367.

Byfield, M. P. and Nethercot, D. A.(1998). An analysis of the true bending strength of

steel beams. Proceedings of the Institution of Civil Engineers-Structures and

Buildings. 128:2, 188-197.

Byfield, M. P. and Dhanalakshmi, M. (2002). Analysis of strain hardening in steel

beams using mill tests. Proceedings of the Third International Conference on

Advances in Steel Structure. Edited by Chan, S. L., Teng, J. G. and Chung, K. F.

Hong Kong, China. 139-146.

Chacón, R., Mirambell, E. and Real, E. (2009). Hybrid steel plate girders subjected to

patch loading, Part 1: Numerical study. Journal of Constructional Steel Research.

66:5, 695-708.

Chan, T. M. and Gardner, L. (2008a) Compressive resistance of hot-rolled elliptical

hollow sections. Engineering Structures. 30:2, 522-532.

Chan, T. M. and Gardner, L. (2008b). Bending strength of hot-rolled elliptical hollow

sections. Journal of Constructional Steel Research. 64:9, 971-986.

Charlton, T. M. (1960). A test on a pitched roof portal structure with short stanchions.

British Welding Journal. 7, 679-685.

Chou, S. M., Chai, G. B. and Ling, L. (2000). Finite element technique for design of

stub columns. Thin-Walled Structures. 37:2, 97-112.

Cruise, R. B. and Gardner, L. (2008). Strength enhancements induced during cold

forming of stainless steel sections. Journal of Constructional Steel Research. 64:11,

1310-1316.

References

220

Davies, J. M. (1966). Frame instability and strain hardening in plastic theory. Journal

of the Structural Division, ASCE. 92:ST3, 1-16.

Davies, J. M. and Brown, B. A. (1996). Plastic design to BS 5950. London, Blackwell

Science.

Davies, J. M. (2002). Second-order elastic-plastic analysis of plane frames. Journal of

Constructional Steel Research. 58:10, 1315-1330.

Davies, J. M. (2006). Strain hardening, local buckling and lateral-torsional buckling in

plastic hinges. Journal of Constructional Steel Research. 62:1-2, 27-34.

Dawson, R. G. and Walker, A. C. (1972). Post-buckling of geometrically imperfect

plates. Journal of the Structural Division, ASCE. 98:1, 75-94.

Doane, J. F. (1969). Inelastic instability of wide-flange steel beams. MSc thesis, the

University of Texas, USA.

Douglas, C. M., Elizabeth, A. P. and Geoffrey, G. V. (2001) Introduction to Linear

Regression Analysis (3rd ed.). New York, Wiley.

Driscoll, G. C., Jr. and Beedle, L. S. (1957). The plastic behavior of structural

members and frames. The Welding Journal. 36:6, 275-286.

Dubina D. and Ungureanu V. (2002). Effect of imperfections on numerical simulation

of instability behaviour of cold-formed steel members. Thin-Walled Structures. 40:3,

239-262.

EN 10002-1 (1990). Metallic materials – tensile testing, part 1: method of test at

ambient temperature. CEN.

References

221

EN 10210-1 (2006). Hot-finished structural hollow sections of non-alloy and fine

grain steels. CEN.

EN 10219-1 (2006). Cold-formed welded structural hollow sections of non-alloy and

fine grain steels. CEN.

EN 10025-1 (2004). Hot rolled products of structural steels - Part 1: General technical

delivery conditions. CEN.

EN 10326 (2004). Continuously hot-dip coated strip and sheet of structural steels.

CEN

EN 1990 (2002). Eurocode: Basis of structural design. CEN.

EN 1993-1-1 (2005). Eurocode 3: Design of steel structures - Part 1.1: General rules

and rules for buildings. CEN.

EN 1993-1-5 (2006). Eurocode 3: Design of steel structures - Part 1.5: Plated

structural elements. CEN.

EN 1999-1-1 (2007). Eurocode 9. Design of aluminium structures. General structural

rules. CEN.

Feng, M., Wang, Y. C. and Davies, J. M. (2003). Structural behaviour of cold-formed

thin-walled short steel channel columns at elevated temperatures. Part 1: experiments.

Thin-Walled Structures. 41:6, 543-570.

Galambos, T. V. and Ravindra, M. K. (1978). Properties of steel for use in LRFD.

Journal of the Structural Division, ASCE. 104:ST9, 1459-1468.

References

222

Gao, L., Sun, H., Jin, F. and Fan, H. (2009). "Load-carrying capacity of high-strength

steel box-sections I: Stub columns." Journal of Constructional Steel Research. 65:4,

918-924.

Gardner, L. (2002). A new approach to structural stainless steel design. PhD thesis.

Imperial College London. London.

Gardner, L. and Nethercot, D. A. (2004). Numerical modeling of stainless steel

structural components - A consistent approach. Journal of Structural Engineering,

ASCE. 130:10, 1586-1601.

Gardner, L. and M. Ashraf. (2006). Structural design for non-linear metallic materials.

Engineering Structures. 28:6, 926-934.

Gardner, L. (2008). The continuous strength method. Proceedings of the Institution of

Civil Engineers-Structures and Buildings. 161:3, 127-133.

Gardner, L. and Theofanous, M. (2008). Discrete and continuous treatment of local

buckling in stainless steel elements. Journal of Constructional Steel Research. 64:11,

1207-1216.

Gardner, L., Saari, N. and Wang, F. (2010). Comparative experimental study of hot-

rolled and cold-formed rectangular hollow sections. Thin-Walled Structures. 48:7,

495-507.

Gardner, L. and Wang, F. (2010). Influence of strain hardening on the behaviour and

design of steel structures. International Journal of Structural Stability and Dynamics.

(Submitted).

Ge, H. and Usami, T. (1992). Strength of concrete-filled thin-walled steel box

columns: Experiment. Journal of Structural Engineering, ASCE. 118:11, 3036-3054.

References

223

Gehring, A. and H. Saal. (2008). Robust finite-element analysis of light gauge cold-

formed sections. Proceedings of the Fifth International Conference on Thin-Walled

Structures. Edited by Mahendran, M. Brisbane, Australia. 297-304.

Guo Y.-J., Zhu A.-Z., Pi Y.-L. and Tin-Loi F. (2007). Experimental study on

compressive strengths of thick-walled cold-formed sections. Journal of


Haaijer, G. (1957). Plate buckling in the strain-hardening range. Journal of the

Engineering Mechanics Division, ASCE. 83:EM2, 1-47.

Han, L., Tao, Z., Huang, H. and Zhao, X. (2004). Concrete-filled double skin (SHS

outer and CHS inner) steel tubular beam-columns. Thin-Walled Structures. 42:9,

1329-1355.

Hancock, G. J. and Zhao, X. L. (1992). Research into the Strength of Cold-Formed

Tubular Sections. Journal of Constructional Steel Research. 23:1-3, 55-72.

Hasan, S. W. and Hancock, G. J. (1989). Plastic bending tests of cold-formed

rectangular hollow. Steel Construction. 23:4, 2-19.

Hill, H. N. (1944). Determination of stress-strain relations from the offset yield

strength values. Technical Note No. 927. National Advisory Committee for

Aeronautics. Washington, D. C.

Horne M. R. (1960). Instability and the plastic theory of structures. Trans Eng Inst

Canada. 4:2, 31-43.

Horne, M. R. (1963). Elastic-plastic failure loads of plane frames. Proceedings of the

Royal Society A. 274, 343-364.

Horne, M. R. (1981). Plastic design of low-rise frames. Granada Publishing Ltd. UK.

References

224

Juhas, P. (2007). Elastic-plastic local stability and load-carrying capacity of steel

members. Proceedings of the Third International Conference on Structural

Engineering, Mechanics and Computation. Edited by Zingoni, A. Cape Town, South

Africa. 964-965.

Kalyanaraman, V., Winter, G. and Pekoz, T. (1977). Unstiffened compression

elements. Journal of the Structural Division, ASCE. 103:9, 1833–1848.

Karren, K. W. (1967). Corner properties of cold-formed steel shapes. Journal of the

Structural Division, ASCE. 93:1, 401-432.

Kato, B. and Aoki, H. (1970). Deformation capacity of steel plate elements. IABSE

publications. 30:1, 93-112.

Kato, B. (1990). Deformation Capacity of Steel Structures. Journal of Constructional

Steel Research. 17:1-2, 33-94.

Kemp, A. R., Byfield, M. P. and Nethercot, D. A. (2002). Effect of strain hardening

on flexural properties of steel beams. The Structural Engineer. 80:8, 188-197.

Key, P. W., Hasan, S. W. and Hancock, G. J. (1988). Column behavior of cold-

formed hollow sections. Journal of structural Engineering, ASCE. 114:2, 390-407.

Key, P. W. and Hancock, G. J. (1993). A theoretical investigation of the column

behaviour of cold-formed square hollow sections. Thin-Walled Structures. 16:1-4, 31-

64.

Kim, Y. (2000). Behavior and design of laterally supported doubly symmetric I-

shaped extruded alumininum sections. MSc thesis, Cornell University, Ithaca, New

York.

References

225

Kim, Y. and Peköz, T. (2008). Ultimate flexural strength of aluminium members.

Proceedings of the Fifth International Conference on Coupled Instabilities in Metal

Structures. Edited by Rasmussen, K. and Wilkinson, T. Sydney, Australia. 81-93.

Knobloch, M. and Fontana, M. (2006). Strain-based approach to local buckling of

steel sections subjected to fire. Journal of Constructional Steel Research. 62:1-2, 44-

67.

Kuhlmann, U. (1989). Definition of flange slenderness limits on the basis of rotation

capacity values. Journal of Constructional Steel Research. 14:1, 21-40.

Kurobane, Y., Ogawa, K. and Ochi, K. (1989). Recent Research Developments in the

Design of Tubular Structures. Journal of Constructional Steel Research. 13:2-3, 169-

188.

Law, A. (2010). Elliptical hollow section test programme. PhD thesis. Structures

Section, Department of Civil and Environmental Engineering, Imperial College

London.

Lay, M. G. and Smith, P. D. (1965). Role of strain-hardening in plastic design.

Journal of structural Engineering, ASCE. 91:ST4, 23-24.

Lay M. G., Galambos T. V. (1965). Inelastic steel beams under uniform moment.

Journal of the Structural Division, ASCE. 91:6, 67–93.

Lay, M. G. and Ward, R. (1969). Residual stresses in steel sections. Journal of the

Australian Institute of Steel Construction. 3:3, 2-21.

Lechner, A., Kettler, M. R., Greiner, R., Boissonnade, N., Jaspart, J.-P. and Weynand,

K. (2008). Plastic capacity of semi-compact steel sections. Proceedings of the Fifth

International Conference on Coupled Instabilities in Metal Structures. Edited by

Rasmussen, K. and Wilkinson, T. Sydney, Australia. 77-84.

References

226

Lim, J. B. P., King, C. M., Rathbone, A. J., Davies, J. M. and Edmondson, V. (2005).

Eurocode 3 and the in-plane stability of portal frames. The Structural Engineer. 83:21,

43-49.

Liu, M., Zhao, J. and Jin, M. (2010). An experimental study of the mechanical

behaviour of steel planar tubular trusses in a fire. Journal of constructional steel

research. 66, 504-511.

Low, M. W. (1959). Some model tests on multi-storey rigid steel frames. Proceedings

of the Institution of Civil Engineers. 13:3, 287-298.

Mirambell, E. and Real, E. (2000). On the calculation of deflections in structural

stainless steel beams: an experimental and numerical investigation. Journal of


Macdonald, M., Taylor, G. T. and Rhodes, J. (1997). The effect of cold forming on

the yield strength of thin gauge steel - Hardness test approach. Thin-Walled Structures.

29:1-4, 243-256.

Machacek, J. (1988). Study on imperfections of stiffened plating. Acta Technica

CSAV. 33:5, 582-606.

Madugula, M. K. S., Haidur, R., Monforton, G. R. and Marshall, D. G. (1997).

Additional residual stress and yield stress tests on hot-rolled angles. Proceedings of

the Structural Stability Research Council Annual Technical Session. Toronto, Canada.

55-68.

Mazzolani, F. M. (1995). Aluminium alloy structures. 2nd ed. London(UK): Chapman

& Hall.

Merchant, W. (1954). Failure of rigid jointed frame works as influenced by stability.

The Structural Engineer. 32:7, 185-190.

References

227

Merchant, W., Rashid, C. A., Bolton, A. and Salem, A. (1958). The behabiour of

unclad frames. Proceedings of the 15th Anniversary Conference, IStructE. London.

McDermott, J. F. (1969). Local Plastic Buckling of A514 Steel Members. Journal of

Structural Divison, ASCE. 95:ST9, 1837-1850.

NA to BS EN 1993-1-1 (2005). UK National Annex to Eurocode 3. Design of steel

structures. General rules and rules for buildings. British Standards Institution.

Nethercot, D. A. (1974). Residual stresses and their influence upon the lateral

buckling of rolled steel beams. The Structural Engineer. 52:3, 89-96.

Nishino, F and Tall, L. (1969). Residual stress and local buckling strength of steel

columns. Proceeding of Japan Society of Civil Engineer. 172:12, 79-96.

Packer, J. A., Chiew, S. P., Tremblay, R. and Martinez-Saucedo, G. (2010). Influence

of Material Properties on Hollow Section Performance. Proceedings of the Institution

of Civil Engineers-Structures and Buildings. (Accepted).

Popov, E. P. and Willis, J. A. (1957). Plastic design of cover plated continuous beams.

Journal of Engineering Mechanics, ASCE. 84:EM1, 1495-1515.

Ramanto, W. H. (2009). Stress-strain report (material modelling). Structures Section.

Department of Civil and Environmental Engineering. Imperial College London.

London.

Ramberg, W. and Osgood, W. R. (1943). Description of Stress-strain Curves by Three

parameters. NACA Technical Note 902, National Bureau of Standard.

Rasmussen, K. J. R. and Hancock, G. J. (1992). Plate Slenderness Limits for High-

Strength Steel Sections. Journal of Constructional Steel Research. 23:1-3, 73-96.

References

228

Rathbone, A. J. (2002). Second-order effect - who needs them?. The Structural

Engineer. 80:21, 19-21.

Roderick, J. W. (1954). Behaviour of rolled-steel joists in the plastic range. British

Welding Journal. 1, 261-275.

Rogers, N.A. (1976). Theoretical Prediction of the Behaviour of Plain Flat Outstands

in Compression. CUED/C-Struct/TR 51 Report. University of Cambridge.

Ruzek, J., Knudsen, K. E., Johnston, E. R. and Beedle, L. S. (1954). Welded portal

frames tested to collapse. The Welding Journal. 33:9, 469-480.

Sawyer, H. A. (1961). Post-elastic behaviour of wide-flange steel beams. Journal of

the Structural Division, ASCE. 87:ST8, 43–71.

Sedlacek, G., Dahl, W., Stranghoner, N., Kalinowski, B., Rondal, J., Boreaeve, Ph.

(1998). Investigation of the rotation behaviour of hollow section beams. EUR 17994

EN, European Commission.

Schafer B. W. and Peköz T. (1998). Computational modeling of cold-formed steel:

characterizing geometric imperfections and residual stresses. Journal of


Schafer, B. W. (2008). Review: The Direct Strength Method of cold-formed steel

member design. Journal of Constructional Steel Research. 64:7-8, 766-778.

Schafer B. W., Li, Z. and Moen, C. D. (2010). Computational modeling of cold-

formed steel. Thin-Walled Structures. 48:10-11, 752-762.

Tao, Z., Han, L. and Wang, Z. (2004). Experimental behaviour of stiffened concrete-

filled thin-walled hollow steel structural (HSS) stub columns. Journal of


References

229

Theofanous, M. and Gardner, L. (2010). Experimental and numerical studies of lean

duplex strainless steel beams. Journal of Constructional Steel Research. 66:6, 816-

825.

Trahair, N. S., Bradford, M. A., Nethercot, D. A. and Gardner, L. (2008). The

behaviour and design of steel structures to EC3. London, Taylor & Francis.

Uy, B. (1998). Local and post-local buckling of concrete filled steel welded box

columns. Journal of Constructional Steel Research. 47:1-2, 47-72.

Van Vlack, L. H. (1989). Elements of materials science and engineering. 6th Edition.

Addison-Wesley. Reading, Mass.

Wadee, M. A. (2000). Effects of periodic and localized imperfections on struts on

nonlinear foundations and compression sandwich panels. International Journal of

Solids and Structures. 37:8, 1191-1209.

Wilkinson, T. and Hancock, G. J. (1997). Tests for the compact web slenderness of

cold-formed rectangular hollow sections. Report R744. University of Sydney,

Australia.

Wilkinson, T. and Hancock, G. J. (1998). Tests to examine compact web slenderness

of cold-formed RHS. Journal of Structural Engineering, ASCE. 124:10, 1166-1174.

Wood, R. H. (1958). The stability of tall buildings. Proceedings of the Institution of

Civil Engineers. 11:1, 69-102.

Yang, C. H., Beedle, L. S. and Johnston, B. G. (1952). Residual stress and the yield

strength of steel beams. The Welding Journal. 31:4, 205-229.

References

230

Young, B. and Hancock, G. J. (2002). Tests of Channels Subjected to Combined

Bending and Web Crippling. Journal of Structural Engineering, ASCE. 128:3, 300-

308.

Zhao, X. L. and Hancock, G. J. (1991). Tests to determine plate slenderness limits for

cold-formed rectangular hollow sections of grade C450. Journal of the Australian

Institute of Steel Construction. 25:4, 2-16.

Zhao, X. L. and Hancock, G. J. (1992). Square and Rectangular Hollow Sections

Subject to Combined Actions. Journal of Structural Engineering, ASCE. 118:3, 648-

668.

a deformation based approach to structural steel design€¦ · statically determinate and...

Documents