Restrained composite columns

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A Thesis Submitted for the Degree of PhD at the University of Warwick

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RESTRAINED COMPOSITE COLUMNS

by

0

IAN MELVILLE MAY

ý` fi ;; t1 ii

A Thesis _{submitted to the Department of Engineering}

of The University _{of Warwick}

for the Degree of Doctor of Philosophy

BEST COPY

AVAILABLE

Poor text in the original

thesis.

Some text bound close to

the spine.

SUMMARY

This Thesis _{describes the development of an analysis for}

inelastic _{columns, with cross-sections composed of one or more}

materials, loaded with axial load and biaxial _moments. The

column can have both rotational _{and directional restraints at}

its _{ends. The analysis has been programmed for a computer and}

subsequently tested _{against published results for steel columns,} reinforced _{concrete columns, and concrete-encased steel composite} columns and shown to give good agreement.

A test _{rig with an axial load capacity of 2MN and capable}

of testing full-scale _{columns of any practical length has been} designed _{and built. Columns with elastic and elastic-plastic}

rotational _{restraints or, pin-ends or any combination can be}

tested _{and column end-moments of up to 50 kNm can be applied}

through _{the beams. One important feature of the test rig is sets}

of crossed knife-edges, _{which give both major and minor axis}

rotational freedom and thus allow accurate _{positioning of the} axial load.

Eight _{elastically restrained concrete-encased steel composite}

no-sway columns have been tested, three _with biaxial _restraints and loadings, _{using the rig.} The results from the tests have been compared with predictions using the computer program and

The behaviour _{of column lengths within rigid-Jointed no-sway} frames with both plastically _{and elastically} designed beams has

been studied. _{For the case of a column with elastic restraints} design _{proposals have been checked and shown to be conservative.}

When the beams are designed plastically it _{is recommended that a}

iv

CONTENTS

page

DEDICATION

I

SUMMARY il

CONTENTS _Iv

LIST OF TABLES

_X1

LIST OF FIGURES

_xii

ACKNOWLEDGEMENTS _xx

NOTATION

_xxi

CHAPTER I REVIEW OF RELEVANT

_RESEARCH

_{AND DESIGN METHODS}

1.1

Introduction

1.2

_{General column behaviour}

₁

1.3

_{Composite columns}

₁₀

1.4 _{Current design methods 17}

1.4.1 _{Steel columns 17}

1.4.2 Reinforced _{concrete columns} 20

1.4.3

_{Composite columns}

₂₃

1.5

Conclusions

₂₅

1.5.1 Analysis 25

1.5.2

Tests

25

1.5.3

_{Design methods}

25

CHAPTER 2. THEORY FOR THE ANALYSIS OF BIAXIALLY RESTRAINED COLUMNS

2.1 Introduction 38

2.2 Moment - axial load-curvature _{relationships} 38

2.2.1 Material _properties 38

2.2.1.1 Concrete _{encased steel composite}

columns 38

V

2.2.2

_{Method of solution}

₄₀

2.3

Analysis

_{using forward integration}

₄₄

2.3.1

_Assumptions

₄₄

2.3.2 _{Uniaxial analysis 44}

2.3.3

_{Extension of analysis to biaxially}

restrained _{sway and no-sway columns} 46

2.4

Verification

_{of computer programs}

_"51

2.4.1

_{The moment curvature relationship}

₅₁

2.4.2

_{The column analysis}

₅₂

2.4.2.1

_{Encased composite column in}

uniaxial bending 52

2.4.2.2

_{Encased composite column in biaxial}

bending

₅₂

2.4.2.3

_{Bare steel column in biaxial}

_{single -}

curvature

bending

52

2.4.2.4 _{Bare steel column in biaxial}

double-curvature

_bending

₅₃

2.4.2.5 _{Steel column with elastic 53}

restraints

2.4.2.6 _{Elastically restrained column free}

to sway

53

2.5

_{Extension of analysis}

₅₄

2.5.1

Production

_{of moment-rotation}

characteristics 55

2.5.2 Dealing _{with symmetrical columns 56}

2.6 Discussion _{of analysis} 57

2.7 Convergence _{of analysis} 58

vi

CHAPTER 3. THE BEHAVIOUR OF THE LIMITED SUBSTITUTE FRAME

3.1

Introduction

83

3.2

_{Behaviour of isolated columns}

83

3.3 Behaviour _{of rotationally restrained columns} 84

3.4

The use of be am-lines to investigate

frame

behaviour 85

3.4.1

End-moment-end-rotation

_{relationships}

85

3.4.2

Addition

_{of beam lines to end-moment-end-}

rotation

relationships

86

3.5

Inclusion

_{of elastic-plastic}

beams

86

3.6

_{Behaviour of the limited substitute}

frame

87

3.6.1

_{Columns restrained}

_{by elastic}

beams

87

3.6.2 _{Columns restrained by elastic-plastic}

beams 87

3.6.3 _{Summary of failure modes} 88

3.7 Out-of-plane failure 88

CHAPTER

4.

THE TEST RIG

4.1

Introduction

101

4.2

Choice of specimens

101

4.3

Axial

load system

102

4.3.1 The crossed knife _edges 102

4.3.2 Column beam boxes 104

4.4 Beam restraint and loading system 105

4.5 Geometry of deformations 107

4.6 Overall stability of rig and specimens 110

vii

CHAPTER 5 CHOICE AND MANUFACTURE OF TEST COLUMNS

5.1 Introduction 131

5.2

_{Choice of test specimens}

131

5.2.1

Practical

_slenderness

_ratios

131

5.2.2

_{Specimens used}

132

5.3

_{Manufacture of specimens}

133

5.3.1

Instrumentation

_{of specimens}

135

5.4 _{Auxiliary tests} 135

5.4.1

_Calibrations

₁₃₅

5.4.2

Materials

_tests

135

5.4.3

_{Cross-section}

_dimensions

₁₃₆

5.4.4 _{Residual stresses 136}

5.4.5 _{Initial deflections} 136

CHAPTER

_{6 TEST RESULTS}

6.1 _{Testing procedure 150}

6.2 Results _{of tests on columns RCI, RC2 and RC3} 151

6.2.1 Typical _behaviour 151

6.2.2 Individual _{column behaviour} 152

6.2.2. E

Column RCI

152

6.2.2.2

Column RC2

153

6.2.2.3 Column RC3 153

6.3. Results _{of tests on columns RC4 and RC5} 154

6.3.1. Column RC4 154

6.3.2 Column RC5 154

6.4 Results _{of biaxial column tests} 155

viii

6.4.2 _{Column 6C2 156}

6.4.3 _{Column 6C3 157}

6.5 Accuracy _{of experimental results} 158

6.5.1 _{Measurement errors} 158

6.5.2 Misalignment _{errors and rig defects} 159

CHAPTER 7 COMPARISON _{OF THEORETICAL PREDICTIONS AND TESTS}

7.1

Introduction

201

7.2

_{Parameters used in theoretical}

_predictions

₂₀₁

7.3 _{Discussion of the behaviour of individual}

tests

202

7.3.1

Column RCI

203

7.3.2 _{Column RC2 203}

7.3.3 Column RC3 204

7.3.4 _{Column RC4} 204

7.3.5 _{Column RC5} 205

'7.3.6 Columns BCI, 6C2 and 6C3 205

7.4 General discussion _{and conclusions of test and}

computer results 206

CHAPTER 8 DESIGN METHODS FOR COMPOSITE COLUMNS

8.1 Introduction 211

8.2 The design _{of columns in frames with plastically}

designed beams 212

8.2.1 Moment-rotation _{relationships} for hinges

in beams 212

8.2.2 Analysis _{of simple} frame with plastic

Ix

8.2.2

Analysis

_{of simple frame with plastic}

beams

213

8.2.3

Analysis

_{of limited}

_substitute

frame

with plastic beams 215

8.2.4 The inclusion _{of beams with elastic-}

plastic-strain hardening characteristics 216

8.2.5

Conclusions

218

8.3

_{The design of columns in rigid-Jointed}

frames with elastic beams 219

8.3.1 Introduction 219

8.3.2 _{Proposed design method} 220

8.3.3 _{Parametric study to check use of}

effective lengths 220

8.3.4 _{Study of method with elastic columns} 223 8.3.5 _{The behaviour of an inelastic column} 225 8.4 _{Parametric study of the use of effective}

lengths and moment distribution

226

8.5 _{The use of "squash load" expressions for the} design _{of elastically restrained composite}

columns 228

8.5.1 Introduction 228

8.5.2 _{A re-assessment of the parametric} surveys to establish squash load

expressions 229

8.5.3 Inclusion _{of biaxial end-moments} 230

X

CHAPTER 9 _{CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK}

9.1 _{Introduction 252}

9.2 Conclusions ₂₅₂

9.2.1 Theoretical _{analysis 252}

9.2.2 _{Tests on composite columns 253}

9.2.3 _{Design methods for composite columns in}

rigid

jointed

frames

253

9.3

_{Suggestions for future work}

₂₅₅

REFERENCES ₂₅₇

APPENDIX Al _{THE NEWTON RAPHSON TECHNIQUE FOR THE DETERMINATION} OF ROOTS OF NON-LINEAR SIMULTANEOUS EQUATIONS

Ai. l _{Method for two equations 266}

AI. 2 _{Extension to 'n' equations and use in structural 267} problems

APPENDIX A2 COLUMN ANALYSIS BY INTEGRATION OF THE SHEAR EQUATION

A2. I Theory ₂₆₉

A2.2 Starting _{analysis 270}

A2.3 _{Comments on analysis 270}

APPENDIX A3 USE OF OVER RELAXATION

_{TO ANALYSE}

_COLUMNS

A3.1 Theory 273

A3.2 _{Comments on the analysis} 274

APPENDIX A4 DERIVATION OF FINITE DIFFERENCE EXPRESSIONS

USING TAYLOR'S SERIES 276

xi

LIST OF TABLES

Table

CHAPTER 5 page

5.1 _{Mix details} 137

5.2 Results _{from tensile tests on coupons} 138

5.3 _{Results of concrete cube tests} 139

5.4 Cross-section-measurements 140

5.5 _{Values of initial deflections} 142

CHAPTER 7

7.1 _{Values of parameters used In analyses} 209

7.2 _{Comparison of theoretical and experimental results} 210

CHAPTER 8

8.1 Results _{of analyses of columns with end restraints} 233

xii

LIST OF FIGURES

Fig. CHAPTER

1.,

1.5

1.6

Assumptions _{for the simplified analysis used by} Roderick

Horne's classification

of the stanchion problem

Spread of plasticity

in a cross-section

Load paths used by Young

page

27

28

29

30

Ideal "beta" chart

31

Cross-sections

_{of columns tested by Talbot and Lord}

32

1.7 _{Assumed deflected shape of column} 33

1.8 _{Form of strut curve} 34

1.9 Limited _{frame for column design} 34

1.10 Magnification _{of bending moments due to axial} load 35 1.11 Yielding _{of column with single curvature} bending

about the major axis 35

1.12 Reduction factor design 36

1.13 Magnification factor _method 36

1.14 Typical interaction _{curve and approximate curve} 37

CHAPTER

2

2.1 Steel _{and concrete stress-strain} curves for composite

cross-sections 62

2.2 Concrete _{stress-strain} curve for reinforced concrete

cross-section 63

2.3 Sub-division of cross-section for calculation of

xiii

2.4 Finite _{difference expressions used in analysis} 65

2.5 _{Column with directional restraints} 66

2.6 Details _{of composite section analysed} 67

2.7 Moment-curvature _curves for minor axis bending 68

2.8

_{Moment-curvature curves for major axis bending}

69

2.9

Details

_{of reinforced}

_{concrete cross-section}

_analysed

70

2.10

Moment-curvature

_relations

0-

00

71

2.11

Moment-curvature

_relations

_0-

150

71

2.12 Moment-curvature _{relations 0-} 300 72

2.13

Moment-curvature

_relations

_0-

450

72

2.14

_{Results of column analysis}

73

2.15 Comparison _{of results for composite column} 74

2.16 Comparison _{of results} for column No. 4, Reference 20 76

2.17

_{Comparison of results}

for column No. 15. Reference

77

20

2.18 Comparison _{of results} for _{specimen A4. of Reference}

25

78

2.19

Analysis

_{of sway frame}

79

2.20 Forces and displacements _{on a column in uniaxial} 80. bending

2.21

Node numbering used in symmetrical cases

81

2.22

Convergence of analysis

82

CHAPTER 3

3.1 Limited _substitute frame 90

3.2 Behaviour _{of pin-ended} column 90

3.3 Behaviour _{of elastically} restrained column 91

XIV

3.5

End-moment-end-rotation

_{relationships}

for an

initially

_straight

_elastic

_column

93

3.6

_{End-moment end-rotation}

_{relationships}

for an

inelastic

_column

93

3.7

_{The use of beam lines}

94

3.8

The inclusion

_{of plastic}

hinges in beams

95

3.9 Behaviour _{of short elastically restrained columns} 96

3.10

Behaviour of slender elastically

_restrained

columns

97

3.11

_{Behaviour of a limited substitute}

frame

98

3.12

_{Behaviour of short elastic-plastically}

_restrained

columns 99

3.13 Behaviour _{of slender elastic-plastically restrained}

columns 100

CHAPTER 4

4.1

Frame used for tests

113

4.2

General view of biaxial

_{column testing}

_rig

114

4.3

Schematic representation

_{of test rig}

115

4.4 _{Loading system - axial} loads 116

4.5

_{Crossed knife edges}

116

4.6

The stub stanchion effect

in column testing

117

4.7 Equivalent frames 118

4.8 Application _{of beam loads} 119

4.9

Detail

_{of major axis loading system}

120

4.10 Details _{of beam load and restraint} system 121

4.11 Minor axis loading and restraint system 122

4.12 Co-ordinate axes used to describe deformations 123

xv

4.14 Application _{of displacements to major and minor axes 125} 4.15 _{Geometry of deformations with offset minor axis 126}

4.16 Stability _{of axial load system 127}

4.17 Details _{of slides and rollers 128}

4.18 _{Degrees of freedom for instrumentation rig} 129

4.19 _{Plan of instrumentation rig showing degrees of}

freedom of bearings

130

CHAPTER 5

5.1 Cross-section _{details 143}

5.2 _{Section through formwork and strongback 144}

5.3 _{Location of strain gauges 146}

5.4 _{Position of coupons for tensile tests 147}

5.5 Location _{of readings for cross-section dimensions} 148

5.6 Residual _{stress-measurements} 149

CHAPTER 6

6.1 Column RCi. End-moment-axial _{load curves} 160

6.2

Column RCi.

_{Comparison of deflections}

_{at quarter}

points. 161

6.3 Column RCI. _{Measured strains at mid-height} 162

6.4 Column RC2. End-moment-axial _{load curves.} 163

6.5 Column RC2. Comparison of deflections _{at quarter}

points 164

6.6 Column RC2. Final deflected _{shape of column.} 165

6.7 Column RC3. End-moment-axial load curves. 166

6.8 Column RC3. Comparison _{of deflections} at quarter

points. 167

xvI

6.11

_{Column RC4 Comparison of minor axis deflections}

_at

mid-height

170

6.12 Column RC4 Comparison of major axis deflections _at 171 mid-height

6.13

Column RC4 End-moment-axial

_{load curves}

172

6.14 Column RC4 Final deflected _{shape viewed about major}

axis 173

6.15

Column RC4 Final deflected

_{shape viewed about}

174

minor axis

6.16 _{Column RC5 Comparison of deflections at mid-height} 175

6.17

_{Column RC5 End-moment-axial}

_{load curves}

176

6.18

_{Column RC5 Final deflected}

_{shape viewed about}

minor axis 177

6.19 _{Column RC5 Extent of cracking near beam-column joint} 178

6.20

Column RC5 Final deflected

_{shape viewed about}

major axis

179

6.21 _{Column BCI Major axis end-moment-axial load curves} 180 6.22 _{Column BCI Minor axis end-moment-axial load curves} 181

6.23

Column BCI Final deflected

_{shape viewed about major}

axis 182

6.24 Column BCI Detail _{of hinge} 183

6.25

_{Column BCI Comparison of major axis deflections}

184

6.26 Column BCI Comparison _{of minor axis} deflections 185 6.27 _{Column 0C2 Major axis end-moment-axial} load curves 186 6.28 _{Column BC2 Minor axis end-moment-axial} load curves 187

6.29

Column BC2 Comparison of major axis deflections

at

xvi I

6.30 _{Column BC2 Comparison of minor axis defloctions} at

mid-height

189

6.31

Column E3C2

Crushing due to restraining

_moments

190

6.32

_{Column 6C2 General view of hinge and crushing}

191

6.33

Column BC2 General view of failure

_{of column}

192

6.34 Typical _{crack pattern after application of both}

major and minor axis moments

193

6.35

Column 8C3 Major axis end-moment-axial

load curves

194

6.36

_{Column 6C3 Minor axis end-moment-axial}

load curves

195

6.37 _{Column 8C3 Comparison of minor axis deflections}

at mid-height

196

6.38

_{Column 6C3 Comparison of major axis deflections}

at mid-height

197

6.39 _{Column 6C3 View about minor axis showing crushing}

at ends 198

6.40 _{Column 0C3 View about major axis} 199

6.41 _{Column BC3 View about major axis showing extensive}

crushing 200

CHAPTER 8

8.1 Moment-rotation _curves for hinges _{in compact composite}

beams 235

8.2 _{Example of plastically} designed frame 236

8.3 Moment-rotation _curves 237

8.4 Symmetric frame loaded to give plastic mechanisms

in all beams 238

8.5 Symmetric frame with patterned loading 238

8.6 _{Beams with elastic-plastic-strain-hardening}

xviii

8.7 _{Elastic critical loads of single columns restrained by}

beams (no sway allowed)

240

8.8 _{Frame used for computer tests 241}

8.9 _{Comparison of minor axis effective lengths 242}

8.10

_{Comparison of major axis effective}

_lengths

₂₄₃

8.11 _{Minor axis moment-curvature relationships 244}

8.12 _{Major axis moment-curvature relationships 245}

8.13

_{Comparison of methods of estimating maximum column}

moment 246

8.14

Errors

_{in methods of estimating maximum column}

moments

247

8.15

Magnification

_factor

_{for maximum stanchion moments}

₂₄₇

8.16

_Variation

_{of moment with increase in axial}

_load

₂₄₈

8.17 _{Distribution of moments at failure 249}

8.18 Comparison _{of load capacity with various effective}

lengths ₂₅₀

8.19 Arrangement _{of live loads to give the highest component}

of single curvature in internal _columns 251

APPENDIX A2.

A2.1 _{Forces on a column 271}

A2.2 _{Nodes along the column} 271

A2.3 _{Nodes near top end of column} 272

A2.4 Discontinuities _{In moment curvature relationships} 272

APPENDIX A3.

A3. I _{Forces on a column} 275

xix

APPENDIX A4

A4.1 Definition _{of nodes and deflections} 277

APPENDIX A5

xx

ACKNOWLEDGEMENTS

I would like to thank both my supervisor Professor R. P. Johnson

and also Dr. R. H. Wood of the Building Research Establishment, for

their _{guidance and helpful criticisms throughout the duration} of the research.

I am very grateful for the assistance given by Mr. A. Redhead, Senior Technician, _{and other members of the staff of the Engineering}

Department _{in the design and building of the test rig and in carrying}

out the tests.

The Building _{Research Establishment is to be thanked} for

financial _{support of the research work.}

xx

NOTATION

A

Cross-sectional

_area

Ac _{Area of concrete}

AiJ

_{Area of (ij)th}

_element

Ar Cross-sectional _{area of longitudinal reinforcement} A5 Cross-sectional _{area of joist}

B _{Breadth of section}

b _{Flange width}

Cm Equivalent _{moment factor}

C1,.... C4 Coefficients

_{in concrete stress-strain}

_curve

c

Stability

function

D _{Depth of section}

d Distance _{from centroid to extreme fibre} Clear _{depth of web}

E Young's _modulus

Error _matrix

Ec Young's _{modulus of concrete} Es Young's _{modulus of steel}

Ex, Ey Elastic _{beams about x and y axes respectively}

e Eccentricity _{of load}

Go, Ho Constants _{in extrapolated} Liebemann t, 4ethod Second moment of area

Iy Second _{moment of area about minor axis}

i _{Number of element along x axis}

Node number

j _{Number of element along y axis}

>Yi

K, Kl, KK Stiffness _{of member}

Concr©to _{strength conversion} factor

Parameters to describe _{axial-load-moment} interaction _curve

Kb Stiffness _{of beam}

Kc Stiffness _{of column}

k Factor _{proportional to stiffness of beam}

L _{Length of a member}

Lc Critical _{length of a member}

t Elemental length

Is Stub-stanchion length

M _Moment

Mcen _{Moment at centre of column}

Mcol _{Moment acting on column}

Md _{Moment due to dead load}

MF _{Fixed end moment}

Pip Plastic _{moment of section}

MULT Ultimate _{moment with N-0}

Mux _{Ultimate moment, about x axis, with axial load} N.

Muy _{Ultimate moment, about y axis, with axial} N

Mx, Mxl,

... Mx4 Major axis moment My, My1,... My4 _{Minor axis moment} M, MI... M4 Miscloses

m Number of elements along x axis

Modular _ratio

(mx)ij Major axis moment on (ij)th _element (my)ij Minor axis moment on (ij)th _element

X(: 1.

NA

_{Axial Io, )d}

Na

Ultimate

_axial

load

Nax Ultimate _{axial load when constrained} to fail _{about the}

x axis.

N

Critical

_axial

load

cr

NE Euler load

NEX Euler _{load about x axis} NEY Euler _{load about y axis} Nf Failur e load

N _{Squash load}

sq

NTH Theoretical _{failure load}

NU Required _{axial load capacity}

Nx _{Uniaxial failure load with major axis bending} Nxy Biaxial _{failure load}

Ny Uniaxial failure _{load with minor axis bending} N.... N Axial load

n Node number

Number of elements along _y axis Nij Axial _{load on (ij)th element}

0Y, 0y Pin-Joints _{about x and y axes, respectively}

p Number of analysis

Px, Py Plastic hinges, _adjacent to column, in beams about x

and y axes, respectively.

R Reduction factor

r radius of gyration

Si Shear force _{at node}

S

Shear about x axis

x

`V

s Stability function

si Directional _{restraint at node} I. t Thickness _{of flange}

Un Required _{deflection at node n.}

u Displacement

ui Displacement _{at node} i

u1J x co-ordinate _{of (iJ)th element} Vn Required _{deflection at node n}

v

Displacement

vi

Displacement at node i

vij

_{y co-ordinate}

_{of (iJ)th}

_element

vreq

Required deflection

W _{Width of steel section}

w Thickness _{of web}

wi Displacement _{of node i} Load on beam i

x Distance _{along column}

Linear _axis

y Deflection

Linear _axis

ym Deflection at mid-height

z Linear _axis

z1. Distance of centroid of (iJ)th element to neutral axis zn Depth of neutral axis

a Constant

/ _`I

an Factor in biaxial interaction formula ß Ratio _{of end-moments}

Constant

Y, Y2

Constants in concrete stress-strain

_curve

6

Sway deflection

_{of column}

Deflection _{at and of beam}

E

Strain

Cf Maximum concrete _strain

CIj Strain _{on (ij)th element}

Cu Strain _{at maximum concrete stress} Cy Yield _{strain of steel}

En

_{Measure of initial}

_{out-of-straightness}

Co _Rotation

Orientation _{of neutral axis}

0A Rotation

0bx End-rotation _{about x axis at bottom of column} 0 End-rotation _{about y axis at bottom of column}

by

0 End-rotation _{of column}

c

0nI`-

4 Hinge rotations 6

n

0

,0 , 0 , of - Rotations used to de scribe hinge characteristics

m m

_p

_p

0 Rotation _{of hinge at commenceme nt of strain-hardening} sh

0

End-rotation

_about

_{x axis at top of column}

tx

0 End-rotation _{about y axis at top of column} ty

0 0

, 0 Rotation about x, y, and z axes respectively

x, y z

a

Stress

as Axial compressive stress

abc Bending compressive stress

xxvI

Qlc Initial _curvature stress aij Stress _{on (ij)th element}

O

PC Permissible axial stress

pbc Permissible bending stress

ß Maximum concrete _stress

u a

us Ultimate tensile strength df steel e Yield _{stress of steel}

y n

yr Yield stress of reinforcement

0 Curvature

Ratio

Ar

yr/As y

0o

Curvature due to initial

imperfections

0

Curvature

ýx

Curvature about x axis

CHAPTER

I.

_{REVIEW OF RELEVANT}

_RESEARCH

_{AND DESIGN METHODS}

1.1

Introduction

Analysis,

_testing

_{and design methods for columns have involved}

much research over the last 200 years.

The object of this chapter

Is to report the major developments _{over this period under three}

major headings,

1.2 General column behaviour

1.3 Composite column behaviour

1.4 Current design methods.

Sections

_{1.2 and 1.3 include the major theoretical}

contributions

and also deal with tests carried out and their influence on the

then current design _{rules. Section 1.4 covers the most recent proposals} for design _{in both the United Kingdom and abroad, and any tests}

used to verify these methods.

1.2 General _{column behaviour}

The analysis _{of members under compression} loading _{has developed}

a long way since Euler(1)(2) first _{proposed his analysis} for the strength _{of elastic columns} in the 18th century. _However, Eulers

formula _{was found to over-estimate the strength}

of short columns,

the error increasing _{as columns became shorter. Lamarle established} that _{the elastic limit was the limit of validity of Euler's} formula. Considere(3) _{and Engesser(4), Independently of each other, solved}

the problem of errors in the load by generalizing the Euler

2

Kärmän(5)(6) was the first

to consider the buckling of

eccentrically

loaded columns as a stability

problem and in 1908

gave a complete and exact analysis

for the problem.

Westergaard

and Osgood('),

in 1928, used Kärm3n's analysis with the assumption

of a part cosine deflected _{shape to analyse} a series of eccentrically

loaded columns. This assumption simplified Kärm. n's analysis without impairing _{the accuracy too much. They also investigated the effect}

of initial _{out-of-straightness on the behaviour of columns.}

Chwalla(8) _{used Kärman's analysis to investigate the stability} of eccentrically loaded columns with various _{cross-sections,}

slenderness _{ratios and eccentricity.} Jezek(9) _{showed that} the

use of a simplified bi-linear stress-strain _{curve for steel} had little _{effect on the results of the analysis. Because of the}

immense amount of labour required _{to produce each solution} for

the inelastic behaviour _{of a column results existed only for a few} special cases.

Baker and Holder(10) carried _{out a series of theoretical}

studies for the Steel Structures Research Committee, (S. S. R. C. ), on the behaviour _{of restrained elastic stanchions.} They were

particularly interested in the distribution _{of moments at the}

stanchion-beam _{connection and so compared the moments obtained} using linear _{elastic theory with those obtained from an elastic analysis}

including _{loss of stiffness due to axial} load. Because they were

producing working load methods and maximum permissible stresses

were low, they found that values of N/NE, where N Is the axial

3

moments was very small. They therefore produced design charts, which also included the effects of initial curvature, based on

moments and stresses determined using linear elastic theory.

Following

the theoretical

_{work for the S. S. R. C. Baker and}

Roderick(h1)

_carried

_{out two series of tests of model columns with}

elastic

restraint

and loading about the minor axis.

The first

series _{was on columns bent in uniform} single curvature. The results showed that the choice of initial _{yield as a failure} criterion by

the S. S. R. C. was grossly conservative, that moment reversal could

occur in slender _{columns, and that redistribution of moments occurred} more rapidly _with the spread of plasticity in the column. They also

showed that in uniform single _curvature bending the load producing collapse is not determined by the condition _{of the column at}

mid-height _alone. Before _{collapse can occur the end sections} must also be incapable _{of resisting} further _moment.

In 1948(11) _{tests on columns in symmetric double-curvature}

were carried out on similar _{model columns.} In these tests failure

always occurred _with "unwrapping" from double curvature into single curvature. As in the single curvature tests the results showed

the failure

_criterion

_{of the attainment}

_{of first}

_yield

_{to be grossly}

conservative.

Both series of tests showed how little effect the beam loading had on the collapse axial load of the column.

Horne, Roderick, _{and Heyman(il)developed} the theory of

restrained elattic-plastic columns including the effects of unloading and analysed some of the columns tested and concluded that the

4

Roderick(Ii)developed

_{an approximate}

_analysis

based on

the assumption of the development of plastic hinges in the column, Fig. I. I. A discontinuity due to hinge _{rotation existed} at the

centre hinge but the other two hinges were assumed to have just formed and thus the column end rotation was equal to the beam

rotation. The column lengths between hinges was assumed to remain elastic. The test results were analysed using this simplified

theoretical _{method and good agreement was obtained.}

In 1956 Horne(12) _{produced his classification chart} for

biaxially _{loaded columns,} Fig. 1.2, _{in which} P indicates that the

adjoining beams have plastic hinges _{at the beam-column joint,} 0

indicates _{pinned joints, and E indicates elastic restraint} from the beams. _{He recognised} that _{the PxEY case would give higher} failure

loads, _{even with small y axis beams, than the PxPy case but stated} that, _{at that} time, _{the. problem was intractable.}

He proposed a PxPy design method which took account of the

possibility of flexural torsional buckling due to moments about the major axis.

He later _{proposed(13)(14) a} Px0y _{method which allowed} the

development _{of plastic} hinges _{at the end of the column.} As in his PXPy method account was taken of lateral instability.

In 1960 research at Lehigh commenced with Galambos and Ketter(15)

producing an analysis based on the numerical integration of the equilibrium equation using Newmark's(16) method. They analysed two cases, ß- _{+1 and} ß-0, where B is the ratio of end

5

analysed ß +0.5, and -1.0 but found some difficulty with the ß- _{-1.0 case since} the calculation predicted a neutrally

stable symmetrically deflected shape since these analyses did not

allow for initial curvature although residual stresses were included.

Rossow, Barney and Lee(18), 1967, investigated the effects

of initial curvature on the buckling loads of columns again using a Newmark type integration _method.

The methods discussed _{up to this point} had been used for

the analysis _{of columns with simple stress-strain} curves. The

axial load, _{moment and curvature relationships} had been expressed algebraically with recognition _{of three} distinct _{phases in the}

spread of plasticity, Fig. 1.3. If more complex non-linear stress- strain relationships, _{strain-hardening and cross-sections} of

irregular _{shape or more than one material were to be analysed,} then numerical _{methods would have to be used.}

Cranstonc19), _{1966, proposed a method to obtain axial} load,

moment and curvature relationships which he subsequently used in his analysis for reinforced _concrete columns. To obtain the

relationship he made the following assumptions: -

(a)

_{moment is applied about an axis at right}

angles to an

axis of symmetry,

(b) _{plane sections remain plane}

(c) longitudinal _stress at a point is a function only of longitudinal _strain

6

I

and (e) strain reversal does not occur.

The cross-section was divided into a number of elements in the plane normal to the applied bending moment. The stress on these elements was then assumed to be uniform over an element.

A strain profile was assumed across the cross-section and hence the axial load and moment were calculated. If the axial load

was close to the specified value then the proposed strain profile was correct; if not the profile _{was modified} and the method

repeated. Cranston _{produced results} for a reinforced concrete tee section and gave proposals for extension _{of the method for the}

analysis _{of prestressed} sections _{and biaxial} bending.

Cranston(20) _{used this routine within an analysis for}

uniaxially bent and restrained _columns. The analysis _estimated

an initial deflected _{shape and then calculated moments, curvatures} and deflections and checked equilibrium and compatability. If

convergence was not achieved a new deflected shape was estimated

and the analysis repeated. Elastic unloading was considered and special idealizations to speed up the analysis of symmetrical

cases were used. The method was also applicable to the analysis of sway columns.

Cranston(21), _{1972, used this analysis} to help in the development

of the clauses for column design in the Code of

Practice

for

Reinforced

Concrete, CPI10(22).

7

based on a trial

and correction

_{method, using a second order method}

similar to the Newton-Raphson technique.

They used a computer program to analyse a series of columns. The analysis _{was also used to check the results of tests carried}

out on pin-ended biaxially loaded columns.

Vinnakota _{and Aoshima(24), 1974, have presented a method of}

analysis for the inelastic behaviour _{of rotationally restrained} columns under biaxial loading. _{Account of residual strains} in

the cross-section _{and warping strains that result from the twisting}

of the section is included _{in the analysis.}

The deflected

_{shape of the column was estimated and section}

properties _{and torsional loads calculated. Using these section} properties _{and considering equilibrium, a new set of deflections}

were calculated. These were compared with the estimated _deflections.

If the required degree of convergence had not been achieved the

routine

is repeated using the new deflections.

The analysis _{has been used to check experimental results of}

columns tested by Birnstiel(25) _{and Gent and Milner(26).}

Young(27)(28) _{1971, used a method based on the correction of an} initially _{estimated deflected shape by block relaxation} for the

analysis of axially loaded columns. For beam-columns, because of the lack of symmetry in the deflected shape, point relaxation

was tried but found to converge only slowly. Therefore, to speed up the rate of convergence, the extrapolated Liebemann over-

8

You6g(29) also examined the moment-curvature relationships

for hot rolled,

_{doubly symmetric, i section column and beam shapes}

containing _{realistic residual stress patterns. He examined both}

major and minor axis uniaxial

bending.

in the column analysis Young considered two load paths, Fig. 1.4,

and took account _{of elastic unloading of plastic sections when Using} load path 2. _{It was shown that choice of load path 2 only had any}

effect if yield had taken place under the application _{of axial}

load only and that even then the effects were small.

Young examined the effect of imperfections,

both initial

curvatures

_{and residual}

_stresses,

_{on the failure}

_{loads of columns.}

He pointed out the importance of stocky columns in design and

produced beam-column design charts.

_{That charts,}

_{Fig. 1.5, which}

he called "beta" charts used the three parameters

_M,

L

and ß

pc

where M is the applied _{moment at the top of the column}

Mp _{is the reduced plastic moment under axial load} L _{is the length of the column}

Lc

_{is the critical}

_{length of an axially}

_{loaded column}

and

ß

is the ratio of the moments at the top and the

bottom of the column, single curvature being

+v e.

The slenderness ratio and axial load have been absorbed into the

single

_parameter

L/Lc.

Thus each point on a line of given

ß

represents

an instability

failure

of a column for a given applied

moment and axial load. He compared these charts with test results

(30)(31) (32)

9

was not allowed to occur, in the case of major axis applied

moments, the charts gave good results. However If out of plane failure _{was allowed to occur then the charts were of little use.} Because of this problem a design method was proposedC33) for

the pin-ended biaxially loaded column. _{Young did not produce}

an analysis to check this design method but did check it against available experimental results.

Young also proposed a method(34) for the design of

elastically restrained _{columns using his "beta"} charts with

moment rotation _curves. The method was based on the assumption that _{the deflected shape of the column can be represented by a}

part _{sine curve.} For out-of-plane buckling Young used the criterion suggested by Gent(35) i. e. that a lower bound to the buckling load for I _{section columns will be given by the consideration of}

the minor axis stability of one flange.

The effects of restraint have been studied by Gent(35) who tested _{models representing a typical} heavily loaded no-sway

universal column subjected to single curvature bending about its major axis and with elastic beams. The loading path chosen was to apply high end moments, equal at top and bottom, and then

to apply axial load to failure. Gent showed that failure occurred

either near the squash load or as a snap-through failure about

the minor axis. Torsional buckling was shown to be of secondary importance In his tests _{of elastic-plastic} columns restrained by

elastic beams. He proposed that minor axis stability could be considered as a deterioration of stiffness and that a lower

10

elastic critical load of the tension flange alone. He showed this to be true _with further _{model tests.}

Gent and Milner(26) extended the work to cover the case of biaxially _{loaded, elastically restrained columns.} They showed

the importance _{of the critical load of one flange} for the biaxially loaded columns. _{They concluded that there Is no combination} of

bending _{and axial load under which the ultimate axial} load

capacity _{would be directly reduced by bending} In accordance with the combined stress concept. _{They produced axial} load slenderness

curves for minor axis buckling _{of elastic-plastic columns with} imperfections _{when the stability is controlled by one flange.}

1.3

_{Composite columns}

The problem of the analysis _{of concrete and steel composite}

columns has not received _{as much attention as that of bare steel} columns.

Bondale(36) _{carried out an extensive survey of work on}

composite _{columns prior} to 1959 and therefore this _review will only briefly _{mention work up to 1959.}

In 1912 Talbot _{and Lord(37) attempted to evaluate the effect}

of the various _parameters in a series of 32 tests on axially loaded columns with a built up steel section, Fig. 1.6.10 bare steel

columns were tested, 12 columns were tested with a concrete core

inside, _{3 were tested with concrete cover as well} and the remainder

were encased columns with diameter wire spirals at either 2" or I#" spacing. From the tests on the bare steel section a rule of

Ä-

_{(I. I)}

where A is the cross-sectional

area of the column

r is the radius of gyration of the steel core

and a and 0 are constants

was proposed. They suggested that for the cored columns and the spirally reinforced _{columns the effect of the concrete} could be

included _{by adding to the strength of the bare steel column an}

additional strength _{equal to 2/3 of the 6" cube strength multiplied} by the area of concrete. _{For the unreinforced encased sections}

the concrete _{was found to only carry a stress equal to about} 1/3

of the 6" cube strength, _{due to premature spalling, and it was} suggested that _{no allowance should be taken for the concrete} in design.

These tests together _{with further tests, carried out between}

1912 and 1936, by Mensch(38), Emperger(39), Burr(40 , Faber(41)

and Stang Whittemore and Parson(42) _all indicated _{an increase} of strength _{of up to six times} for encased columns over the bare

columns.

As a result of the above tests BS449: 1948(43) permitted partial

account _{of the concrete encasement to be taken} in design calculations.

The permissible

load, however, was not to exceed 150 per cent of the bare

steel stanchion load.

In 1956 Faber(4' _proposed design formulae from the results

of tests carried out by him.

12

8 were encased and 3 were bare steel.

The results

compared

well with'the

"law of addition".

Rizk also studied eccentrically

loaded columns in uniaxial

and biaxial

bending and proposed an iterative

method for predicting

failure.

StevensO46; 1959, carried out a series of 35 tests on encased

stanchions, in which the variables _{were slenderness ratio,}

reinforcement _{area and concrete strength.} He used the "law of addition" to predict the strengths _{and concluded} that this _was

safe. _{As a result of Stevens' work BS449: 1959(47) allowed} the value of axial load to be increased to twice the value permitted on the bare steel section.

Bondale(36) _{applied Westergaard and Osgood's(7) method to the}

analysis _{of composite columns.} He used algebraic _expressions for

the calculation _{of moment - curvature-axial} load relationships and

assumed the deflected _{shape to be part of a cosine} curve. He also tested eight encased columns with in-plane bending.

Basu(48) extended Bondale's analysis for use on digital

computers by replacing

his algebraic

_{moment curvature}

relationships

with the iterative method as used by Cranston. (19)

From Fig. _{1.7 It can be shown that} if the deflected _shape

of the column is given by

y= ym cos

[(2x/L)cos-1

(e/Ym)]

(1.2)

where y

is the deflection

measured from the line of action

of the

load

e is the eccentricity of the load

13

x

is the position of a point along the column measured

from the centre of the column

L _{Is the length of the column,}

then the curvature,

at the centre due to bending is

0 [(2/L) _cos-1 (e/ym)]2 _{ym -0o} (1.3)

where 0o is The curvature

due to initial

_{out of straightness.}

The moment at the central section is

M- Nym

(1.4)

where N is the axial load.

From the moment curvature relationships

the value of curvature

corresponding _{to the moment can be obtained}

M

_{Nym = F(+)}

(1.5)

and thus

_{ym = f(4)}

(1.6)

The analysis involves _{the solution of equations} (1.3) _and

C1.6) to give values of ym and 4 for a particular

load N.

Basu compared his analysis with _{exact solutions} for universal

columns, unreinforced _elastic concrete (brittle) columns and with test _results for composite _{columns. He concluded} that this

analysis gave good agreement with exact analyses and test results but that _{more tests were required.}

Basu and Hill(49) produced a more exact method of analysis based on the true equilibrium shape. This method had the

advantage over the previous method of being able to handle all

14

with exact solutions

were made. The previous part cosine curve

solution was also checked and found to compare well.

They concluded that _{the use of the straight} line

interaction formula _{as used in steel design for columns with}

equal end eccentricities:

N _{N. e} N

a

MU

LT

Where _{Na is the ultimate load under pure axial load}

and MULT is the ultimate _{moment when} N-0,

is not generally applicable to the failure

load of composite

columns.

They showed it to be safe for medium and short columns;

but it could be unsafe for slender columns.

Basu and Somerville(SO)

_{used these analyses to develop a}

design _{method for pin-ended composite columns; this method is mentioned} in more detail in Section 1.4.

Roderick _{and Rogers(s' proposed an analysis for columns in}

single

curvature

_uniaxial

bending.

The main steps in the method

are: -

(I)

_{Estimate deflections.}

(2)

Calculate

_moments.

(3)

Calculate

_curvatures

_{using moment-curvature}

_{relationships.}

(4)

Calculate

_rotations,

0.

(5)

Calculate

deflections,

_y.

(6)

_{Compare calculated}

_{and estimated deflections;}

if not

within

required

degree of accuracy then use calculated

deflections

15

The moment-curvature _{relationships were obtained from a}

set of algebraic _expressions. A bi-linear _{stress strain curve} was used for steel _{and a tri-linear curve for concrete.}

Three small scale composite columns were tested and

experimental _{and theoretical results compared.} The results _of

the columns tested _{at the Building} Research Station _{by Steven S(46)}

were also compared with the theory. Reasonable agreement between theory _{and test was noted.}

Shapies(52)

_{proposed an analysis}

_{for uniaxial}

_{bending of}

composite columns in which an initial

estimation

of end rotation,

E), was made. Equilibrium

_{was then established}

_{at each node using}

an iterative _process:

(I) _{Calculate the moment, Mn, at the node ignoring secondary}

moments due to deflection _{of the node.}

(2) _{Obtain the curvature, 0}

n, at the node from the moment- curvature _{relationships.}

(3)

_{Knowing 0}

n-l,

On-1, yn-1 and 0

_n,

0n and yn can

be obtained.

(4)

_Calculate

_{the Moment Mn at the node including secondary}

effects (Nyn).

(5) _{Repeat steps 2,3 and 4 until consecutive values of}

Y are within a stated tolerance.

This process is carried out at each node until the last node is reached, at which overall convergence is also checked, in a

no-sway case the deflection should be zero. If convergence is

16

The analysis was extended to biaxial columns but an approximate failure _{criterion was used to determine the failure load due to}

lateral instability.

Sharpies tested a series of model composite columns to obtain

moment rotation

_{curves which he compared with his analysis.}

Reasonable agreement was found.

His tests and theory also pointed

out the possibility

_{of lateral}

Instability.

Virdi

_{and Dowling(53)}

_{tested nine columns in biaxial}

_single

curvature

bending and proposed a method of analysis for biaxial

bending of composite columns.

The deflected

_{shape of the column}

about each axis was assumed to be represented by a part osine curve.

The test _{results were compared with the analysis and found to give}

reasonable _agreement. The interaction formula _{proposed by Basu} and Somerville _{was also checked and found to give good results}

for short columns and conservative

_results

_{for more slender columns.}

The effects of residual _stresses in the steel _section in an

encased section _{were also} investigated _{and found to give a variation} of ±3 per cent in the failure load compared to the failure load of stress-free sections. The use of initial _{out of straightness}

to represent _all imperfections was investigated and it was found that an _{all-inclusive} lack of straightness of 0.0006L2/D, where D Is the depth of the section, or residual stresses plus a lack of

straightness of L/1000 gave results closest to the test results.