the elastic and inelastic lateral torsional buckling …

THE ELASTIC AND INELASTIC LATERAL TORSIONAL BUCKLING

STRENGTH OF HOT ROLLED TYPE 3CR12•STEEL BEAMS

by

HEIN BARNARD

A Dissertation presented to the Faculty of Engineering

for the partial fulfilment of the degree

MAGISTER INGENERIAE

in

CIVIL ENGINEERING

at the

RAND AFRIKAANS UNIVERSITY

SUPERVISOR: Mr. P J BREDENKAMP

CO-SUPERVISOR: PROF G J VAN DEN BERG

JANUARY 1996

i

ABSTRACT

Type 3CR12 steel is a corrosion resisting steel which is intended to be an alternative

structural steel to replace the use of coated mild steel and low alloy steels in mild

corrosive environments. This necessitate the experimental verification of the structural

behaviour thereof.

The purpose of this dissertation is therefore to compare the experimental structural

bending behaviour regarding elastic and inelastic lateral torsional buckling of doubly

symmetric I-beams and monosymmetric channel sections with the existing theories for

carbon steel beams and to modify or develop new applicable theories if necessary.

From the theoretical and experimental results it is concluded that the behaviour of heat

treated Type 3CR12 beams can be estimated fairly accurate with existing theories and

that the tangent modulus approach should be used for more accurate estimates as well

as for beams that are not heat treated.

11

ACKNOWLEDGMENTS

This dissertation was made possible through the contributions and support of a number

of people who needs to be thanked.

Jacques Bredenkamp, my supervisor, is thanked for his guidance during the execution

of this study.

My co-supervisor, Prof Gert van den Berg, is thanked for his advice and support.

Louis Kriek is thanked for his help with the preparation of the test setup and test

beams.

Columbus Stainless for the sponsorship of this study.

I am especially grateful to my Family, Leon, Ettienne and my Mother, Marina, for

their support during my studies.

i ii

TABLE OF CONTENTS

Abstract

Acknowledgements ii

Contents iii

List of Tables vii

List of Figures ix

List of Symbols xiii

INTRODUCTION

1.1 General Remarks 1

1.2 Purpose of Study 1

1.3 Contents of this Study 2

REVIEW OF LITERATURE


2.2 Type 3CR12 Corrosion Resisting Steel 4

2.3 Mechanical Properties of the Material 4

2.3.1 General Remarks 4

2.3.2 Mechanical Properties of Gradual Yielding Material 5

2.3.3 Analytical Representation of the Stress-Strain Curve 6

2.4 Bending Strength of Beams 9

2.4.1 Introduction 9

2.4.2 Yield and Plastic Moment Resistance 10

2.4.2.1 Yield and Plastic Moment Resistance of Type 3CR12 Beams 12

2.4.3 Lateral Torsional Buckling 12

2.5 Conclusion 14

iv

MECHANICAL AND SECTIONAL PROPERTIES


3.2 Experimental Determination of the Mechanical Properties 19

3.2.1 Preparation of Test Specimens 19

3.2.2 Testing of Tensile and Compression Specimens 20

3.2.3 Experimental Mechanical Properties of the Type 3CR12 Beams 21

3.2.4 Analytic Stress-Strain Relationship 22

3.3 Discussion of Experimental Results 23

3.4 Section Properties of the Test Beams 25

3.5 Conclusions 25

YIELD AND PLASTIC MOMENT RESISTANCE OF THE BEAM

SECTION

4.1 Introduction 63

4.2 Bending Theory 63

4.3 General Bending Theory 65

4.4 Resistance Moment of Type 3CR12 Beams 66

4.4.1 Yield Moment Resistance of a Doubly Symmetric I-Beam 67

4.4.2 Plastic Moment Resistance of a Doubly Symmetric I-Beam 71

4.4,.3 Yield Moment Resistance of a Singly Symmetric Channel

Section 71

4.4.3.1 Yield Moment Resistance of a Rectangular Beam Section 72

4.4.3.2 Yield Resistance Moment of Channel Section 74

4.4.4 Plastic Moment Resistance of Singly Symmetric Channel

Section 76

4.5 Calculated Moment Resistance of the Test Beam Sections 76

4.6 Conclusions 77

5. THEORY OF LATERAL TORSIONAL BUCKLING

V

5.1 Introduction 88

5.2 Elastic Lateral Torsional Buckling 88

5.2.1 Doubly Symmetric Cross Section 88

5.2.2 Asymmetric Cross Section 94

5.3 Inelastic Lateral Torsional Buckling 96

5.4 Design for Lateral Torsional Buckling 98

5.5 Moment Resistance of Lateral Continuous Determinate Beams 100

5.6 Design of Type 3CR12 Beam Sections for Lateral Torsional Buckling 102

5.7 Design of Asymmetrical Beam Sections 103

5.8 Conclusions 103

6. EXPERIMENTAL BEAM TESTS

6.1 Introduction 110

6.2 Preliminary Experimental Planning 110

6.2.1 Doubly Symmetric I-Beams 110

6. 2. 2 Singly Symmetric Channel Sections 111

6.3 Experimental Beam Tests 111

6.3.1 Physical Beam Test Setup 111

6.3,2 Experimental Data Recorded 112

6.4 Experimental Beam Test Results 114


6.4.2 Singly Symmetric Channel Sections 115

6.5 Discussion of Experimental and Theoretical Results 115


6.5.2 Singly Symmetric Channel Sections 117

6.6 Conclusions 117

v i

7. CONCLUSIONS AND SUMMARY


7.2 Summary of Research 154

7.3 Future Investigations 155

REFERENCES 156

LIST OF TABLES

VII

3.1 Mechanical Properties of Test Beam 08-T1-I-NHT 26

3.2 Mechanical Properties of Test Beam 08-T2-I-HT 27


3.4 Mechanical Properties of Test Beam 24-14-I-HT 29



3.7 Mechanical Properties of Test Beam 40-77-1-1-IT 32






3.13 Mechanical Property Ratios of Test Beam 08-T1-I-NHT 38

3.14 Mechanical Property Ratios of Test Beam 08-T2-I-HT 38









3.23 Mechanical Property Ratios of Test Beam 56-110-I-HT 43

3.24 Mechanical Properties of Test Beam 06-T1-C-HT 44







VIII

3.31 Sectional Properties of Doubly Symmetric I-Beams 51

3.32 Sectional Properties of Singly Symmetric Channel Sections 52

3.33 Sectional Properties of Singly Symmetric Channel Sections 53

4.1 Yield and Plastic Moment Resistance of Test Beams 78

6.1 Doubly Symmetric I-Beam Lengths and Slenderness 119

6.2 Classification of Doubly Symmetric I-Test Beam Sections 120

6.3 Singly Symmetric Channel Test Beam Lengths and Slenderness 121

6.4 Classification of Singly Symmetric Channel Test Beam Sections 122

6.5 Experimental and Theoretically Estimated Critical Buckling

123 Moments of Doubly Symmetric I-Test Beams

6.6 Experimental and Theoretically Estimated Critical Buckling

Moments of Singly Symmetric Channel Test Beams 124

LIST OF FIGURES

ix

2.1.a Sham Yielding Stress-Strain Behaviour 15

2.1.b Gradual Yielding Stress-Strain Behaviour 15

2.2 Typical Lateral Torsional Buckling Behaviour of Beams 16

2.3 Local Buckling of Flange 17

2.4 Effect of Load Position Application 18

3.1 Mechanical Test Specimen Location 54

3.2 Sample Location of Mechanical Test Specimens of Beam 40-18-I-LIT 54

3.3 Mechanical Test Specimen Dimensions 55

3.4 Analytical and Experimental Stress-Strain Relationship of

Beam 08-T1-1-NHT 56


Beam 08-T2-I-FIT 56


Beam 16-T3-I-HT 57

, 3.7 Analytical and Experimental Stress-Strain Relationship of

Beam 24-T4-I-HT 57


Beam 24-15-I-HT 58


Beam 32-T6-I-HT 58


Beam 40-T7-I-HT 59


Beam 40-T8-I-HT 59


Beam 40-T8-I-HT 60


Beam 48-19-I-HT 61


x

Beam 56-T10-I-HT 61

3.15 Section Dimension Definition 62

4.1 Elastic Stress and Strain Distribution over Cross Section 79

4.2 Inelastic and Plastic Stress and Strain Distribution over

Cross Section 79

4.3.a General Strain Distribution over Cross Section 80

4.3.b General Stress-Strain Relationship 80

4.3.c Stress Distribution over Cross Section and Resultant Forces 81

4.4 Strain Gauge Location over Cross Section 81

4.5 Strain Distribution over Cross Section: Beam 08-T2-I-HT 82

4.6 Strain Distribution over Cross Section: Beam 08-T1-I-HT 83

4.7 Stress Distribution over Cross Section: Beam 08-T2-1-HT 84

4.8 Stress Distribution over Cross Section: Beam 08-T1-I-HT 85

4.9 Stress and Strain Distribution over 3CR12 Beam Cross Section 86

4.10 Yield Stress and Strain Distribution over Section of Unit Width 86

4.11 Yield Stress and Strain Distribution over 3CR12 Channel Section 87

4.12 Plastic Stress and Strain Distribution over 3CR12 Channel Section 87

5.1 General Load Arrangement 104

5.2 Simplified Load Case 105

5.3 Experimental Load Case 106

5.4.a Before and After Buckling Conditions of a Yielded Cross Section 107

5.4.b Loading and Unloading of a Material Fibre 107

5.5.a Moment - Lateral Deformation Curve in the Inelastic Range 108

5.5.b Buckling Curves in the Elastic and Inelastic Region 109

5.6 Adjacent Beam Segment Cases 109

6.1 Test Beam Setup and Moment Distribution 125

6.2 Support and Restraint Details of Test Beam Setup 126

6.3 Setup of Test Beam 08-TI-I-NHT 127

6.4 Testing of Beam 08-T1-I-NHT in Progress

6.5 Setup of Test Beam 56-T10-I-HT

6.6 Testing of Beam 56-T10-I-NHT in Progress Lateral Buckling of

Compression Flange Visible

6.7 Testing of Beam 56-T10-I-NI-IT in Progress Lateral Torsional

Buckling Visible

6.8 Roller System at Load Application Position

6.9 Roller Bearing Restraint System at Cantilever Arm End Tips

6.10 Testing of Beam 12-T2-C-HT in Progress

6.11 Plan View of Test Beam 12-T2-C-HT. Lateral Buckling of

Compression Flange Visible

6.12 Strain Gauge Positioning

6.13 Experimental Displacement Transducer Setup

6.14.a Geometric Lateral and Vertical Relationships

6.14.b Geometric Lateral and Vertical Relationships

6.14.c Geometric Lateral and Vertical Relationships

6.14.d Geometric Lateral and Vertical Relationships

6.15 Moment vs. Flange Tip Strain of Beam 08-T1-I-NHT

6.16 Moment vs. Flange Tip Strain of Beam 08-T2-I-HT

6.17 Moment vs. Flange Tip Strain of Beam 16-13-I-HT


6.19 Moment vs. Flange Tip Strain of Beam 24-T5-1-HT

6.20 Moment vs. Flange Tip Strain of Beam 32-16-1-HT


6.22 Moment vs. Flange Tip Strain of Beam 40-T8-1-HT



6.25 Moment vs. Lateral Deflection of Doubly Symmetric I-Beams

6.26 Moment vs. Twist of Doubly Symmetric I-Beams

6.27 Experimental and Theoretical Critical Buckling Moments vs.

Effective Lengths of Doubly Symmetric 1-Beams

6.28 Moment vs. Flange Tip Strain of Beam 06-T1-C-HT

xi

127

128

128

129

129

130

131

131

132

133

134

135

136

137

138

138

139

139

140

140

141

141

142

142

143

144

145

146

xi i

6.29 Moment vs. Flange Tip Strain of Beam 12-T2-C-HT 146





6.34 Moment vs. Flange Tip Strain of Beam 42-17-C-HT 149

6.35 Moment vs. Lateral Deflection of Compression Flange of

Beam 06-T I-C-HT 149


Beam 12-T2-C-HT 150


Beam 18-113-C-FIT 150


Beam 24-T4-C-HT 151


Beam 30-T5-C-HT 151


Beam 36-T6-C-HT 152


Beam 42-T7-C-FIT 152

6.42 Experimental and Theoretical Critical Buckling Moments vs.

Effective Lengths of Singly Symmetric Channel Beams 153

LIST OF NOTATIONS

A Cross Section Area

Value of Stress Distribution in Flange

Value of Stress Distribution in Web

Section or Flange Width

Bending Stiffness about xx-axis

B y Bending Stiffness about yy-axis

C Compression Force

St. Venant Torsional Stiffness

C., Warping Constant

C. Warping Stiffness

e Strain or Ductility

E. Initial Modulus of Elasticity

E, Tangent Modulus

E, Secant Modulus

F Stress

F, , Proportional Limit

F. Yield Strength

F, Maximum Strength

G Shear Modulus

Q Initial Shear Modulus

G, Tangent Shear Modulus

Stiffness Ratio

h Section Depth

xiv

I, Moment of inertia about the xx-axis

l y Moment of inertia about the yy-axis

I W Warping Constant

J Torsion Constant

Effective Length Factor, Constant

K, Cross Sectional Constant

Beam Length

m Mass of Beam per Unit Length

M Moment

M„ Elastic Critical Buckling Moment

M e Elastic Moment Resistance, Experimental Moment Resistance

M, Tangent Moment

Mie Theoretical Buckling Moment

Ntic Theoretical Buckling Moment

Nty Yield Moment Resistance

ts,4, Plastic Moment Resistance

n , Constant

Axial Compressive Force

rx Radius of Gyration about the xx-axis

rY Radius of Gyration about the yy-axis

tf Flange Thickness

Web Thickness

T Tensile Force

xv

Lateral Deflection of Shear Centre

Vertical Deflection of Shear Centre

z Distance along Longitudinal axis of Beam

a1,2 Real Roots of Characteristic Equation

a34 Complex Roots of Characteristic Equation

Stiffness of Adjacent Beam Sections

Strain

Poisson Constant

Twist Angle of Cross Section

1

CHAPTER 1

INTRODUCTION

1.1 GENERAL REMARKS

Alloy steels consist of alloy elements with iron as the main basic element.

Alloy elements such as nickel, aluminium, manganese, chromium and

molybdenum are added to steel in different combinations to produce alloy

steels with specific characteristics. The characteristics of an alloy steel that are

required, will be determined by the purpose of use for which the steel is

intended.

Two characteristics that are normally required are corrosion- and heat resisting

properties. These specific properties are exhibited by an alloy steel commonly

known as stainless steel. Stainless steels are high alloy steels, with a minimum

chromium content of ten percent'. The corrosion resisting property of stainless

steel is due to a thin, stable chromium oxide film that protects the steel against

corroding media and exists only if the chromium content exceeds ten percent.

In structural applications a need developed for a corrosion resisting steel as an

alternative to painted mild steel, to be used in mild to severe corrosive

environments. An alloy steel known as Type 3CR12 corrosion resisting steel,

a terrific' stainless steel, with a chromium content of twelve percent, was

developed from AISI Type 409 stainless steel by a South African stainless steel

manufacturing company, Columbus Stainless. Type 3CR12 steel is less

expensive than AISI Type 304 stainless steel, and exhibit improved mechanical

properties and weldability over Type 409 stainless steel.

1.2 PURPOSE OF STUDY

The purpose of this study is to develop design criteria for structural

2

applications and to gather experimental test data on the structural bending

behaviour of hot-rolled Type 3CR12 corrosion resisting steel beams. The

lateral torsional buckling behaviour of singly- and double symmetric beams is

of particular interest. Further also to compare the experimental results with

existing theories, modifying them or developing new theories in order to

describe the structural behaviour theoretically.

1.3 CONTENTS OF THIS STUDY

The following chapters contain the applicable theoretical literature and

experimental results in order to reach realistic conclusions on the structural

behaviour of Type 3CR12 steel beams.

A review of the relevant literature is presented in Chapter 2. The mechanical

properties of Type 3CR12 steel and the analytical modelling of the non linear

stress-strain relationship are presented. The theoretical determination of the

ultimate moment resistance for very short beams and the lateral torsional

buckling of short and slender beams are briefly presented.

Chapter 3 deals with the experimental mechanical properties of the Type

3CR12 steel beams. Uniaxial tensile and compression tests are conducted to

determine the mechanical material properties.

In Chapter 4 an analytic model to determine the ultimate flexural moment

resistance of a Type 3CR12 steel beam is presented. This theoretical model is

different from that used for carbon steel beams and is the result of the

different material behaviour of the two types of steel.

The elastic and inelastic lateral torsional buckling of beams are discussed in

Chapter 5. The relevant theory is derived and current design approaches are

presented.

3

In Chapter 6 the experimental beam tests are discussed and the results are

presented. The results are compared with the theoretical theories presented in

Chapters 2, 4 and 5 in order to establish design procedures for Type 3CR12

corrosion resisting steel beams.

The conclusions drawn from the comparison of theoretical and experimental

work are summarised in Chapter 7 and topics for future study are discussed.

4

CHAPTER 2

REVIEW OF LITERATURE

2.1 GENERAL REMARKS

The literature relevant to this study is presented to develop an understanding

of the subject under consideration. The behaviour of the material of which a

beam is manufactured influences the structural bending behaviour. The

mechanical properties of Type 3CR12 steel are therefore discussed and the

analytical modelling thereof is presented. The bending theory of beams is then

discussed regarding the ultimate moment strength and the lateral torsional

buckling of beams.

2.2 TYPE 3CR12 CORROSION RESISTING STEEL

Type 3CR12 steel° is a modified Type 409 stainless steel. The chemical

composition of this steel is similar to that of Type 409 stainless steel except

for the nickel, manganese and titanium contents. The purpose with the

development of this steel was to create a corrosion resisting steel with

improved mechanical properties and weldability compared to that of Type 409

stainless steel. The carbon and nitrogen contents are also kept low to improve

the toughness of Type 3CR12 steel in both the annealed and welded

conditions. Type 3CR12 is therefore superior to AISI Type 409 stainless steel

with a sufficient chromium content to provide a cost effective level of

corrosion resistance.

2.3 MECHANICAL PROPERTIES OF THE MATERIAL

2.3.1 GENERAL REMARKS

The mechanical properties of Type 3CR12 steel were investigated in depth

5

with research by Van Den Berg and Van Der Merwe 5 . The following

conclusions were made:

The material shows a non-linear relationship between stress and strain.

The material has a low ratio of the proportional limit to yield strength.

The material is of the gradual yielding type compared to carbon and

low alloy steels which shows sharp yielding characteristics.

The material properties are anisotropic regarding the longitudinal

direction (that is the direction parallel to the rolling direction of a

section), and the transverse direction (that is the direction perpendicular

to the rolling direction of a section). The material properties are also

anisotropic regarding the uniaxial tensile and compression behaviour.

These differences in mechanical behaviour of Type 3CR12 steel compared to

structural carbon steel indicate differences in structural behaviour and therefore

necessitate the discussion and determination of the mechanical properties of the

beam material.

2.3.2 MECHANICAL PROPERTIES OF A GRADUAL YIELDING

MATERIAL

Figure 2.1 (a) and (b) show sharp and gradual yielding stress-strain curves.

Mechanical properties are determined from the gradual yielding stress-strain

curve as follows:

The initial modulus of elasticity, Ep, is defined as the constant of

proportionality between stress and strain below the proportional limit.

The initial elastic modulus is therefore the slope of the tangent line to

the stress-strain curve in the origin of the curve.

The tangent modulus, E„ is defined as the tangent to the stress-strain

curve at a specific value of stress.

The proportional limit, F p , and the yield strength, F r are defined

6

as the stress corresponding to the intersection of the stress-strain curve

and a line parallel to the initial modulus of elasticity, offset by 0,01%

and 0,2% strain respectively, from the origin.

The ultimate tensile strength, F„, is defined as the maximum stress

reached during a tensile test, and is obtained by dividing the maximum

force reached during the test by the original cross sectional area of the

test specimen.

The ductility of the material is defined as the extent to which the

material can sustain elastic and plastic deformation without rupture.

The ductility is determined by marking an original gauge length, 50mm

or 203mm, on the tensile test specimen. The two parts of the test

specimen are fitted together after rupturing of the test specimen and the

final gauge length is measured. The ductility is then calculated as the

percentage elongation with respect to the originally marked gauge

length.

2.3.3 ANALYTICAL REPRESENTATION OF THE STRESS-STRAIN

CURVE

The incorporation of the effect of the mechanical behaviour of the beam

material in the theoretical calculations of structural behaviour, necessitates the

representation thereof analytically. An analytical model that represent the

mechanical behaviour needs to accurately describe the stress-strain relationship

of the material. Such an analytical model must obey the following

requirements':

The analytic equation must be simple to use.

The curve represented by the analytic equation must pass through the

origin with a slope equal to the initial modulus of elasticity.

The analytical equation must show the characteristic of representing the

stress-strain relationship of a variety of materials by varying the

parameters of the equation.

7

The analytical equation must lead to a determinate integral when

integrated. (The importance of this property will be demonstrated in

Chapter 4)

The parameters of the analytical equation need to be easily determined.

An analytical equation that comply with the above requirements is known as

the Ramberg-Osgood' equation. The Ramberg-Osgood' equation is a three

parameter equation that ensures the exact representation of three properties of

the stress-strain relationship of the material. The analytical equation is given

in equation 2.1.

e = —F + K ( —E,

) 12 E,

(2.1)

= strain

F = stress

Eo = initial elastic modulus

K = constant

n = constant

Ramberg and Osgood' evaluated the constants K and n for the three

parameters, the initial modulus of elasticity, and two secant stresses F, and F2.

These two stresses are determined as the intersection of the stress-strain curve

and the lines through the origin with slopes equal to the secant modulus of

elasticity taken as 0,7E. and 0,85E 0 respectively.

Hills suggested that the constants K and n be evaluated at two proofstresses as

follows:

proofstress F, at an offset strain e, and

proofstress F2 at an offset strain £2.

The constants K and n are then determined as follows:

8

the stress-strain curve that is represented by equation 2.1 deviates from

the straight line curve that is given in equation 2.2.

e = Eo

This deviation is represented by equation 2.3.

e = K ( —F) n

E 0

(2.2)

(2.3)

By applying the logarithmic laws, it follows that,

log (e) = log (K) + n ) (2.4)EF,

Substitution of F 1 , e„ and F2, e 2 , into equation 2.4 leads to

simultaneous equations that simplifies to,

log (e 2

) n -

(2.5)

log(

From equation 2.3,

el K = K - 2

(L)n f2)n Eo Ea

(2.6)

Substitution of equation 2.6 into equation 2.1 and simplifying leads to

the following,

c = n E0 2 F2

(2.7.1)

Or

(2.7.2)

9

A logical choice of the proofstress, F 2 , at an offset strain, e, = 0,002, is the

yield strength, F. Hill° suggested an offset strain, e, = 0,001, at which to

determine the proofstress F,. Van Den Berg' and Van Der Merweg concluded

from extensive research that the proofstress, F,, equal to the proportional

limit, Fp , at an offset strain, e, = 0,0001, resulted in the best analytical

representation of the experimental gradual yielding stress-strain relationships

for materials such as stainless steels.

The modified Ramberg-Osgood' equation is therefore used to represent the

stress-strain relationship for stainless steels and Type 3CR12 steel analytically

and is as follows:

e = —F

+ 0,002 (±')/' Ec, FY (2.8)

where

n -

0 log ( 002 ' 0 , 0001

(2.9)

log( —Y )

2.4 BENDING STRENGTH OF BEAMS

2.4.1 INTRODUCTION

Beams are structural elements that resist external forces, applied perpendicular

to the longitudinal axis of the beam, through the development of internal

bending moments and shear forces. The moment resistance of a beam is

controlled by local buckling of the beam section elements or lateral torsional

buckling. The bending behaviour of beams is therefore divided in three

regions' as shown in Figure 2.2. These region are as follows:

10

Plastic region. Very short beams fail in this region and the moment

resistance is determined by local buckling of the section elements, the

material yield strength and the plastic sectional properties.

Inelastic region. Short to intermediate beams fail in this region and the

moment resistance is determined by inelastic lateral torsional buckling

which is controlled by inelastic stress distributions and slenderness of

the short beam.

Elastic region. Long or slender beams fail in this region and the

moment resistance is determined by elastic lateral torsional buckling

which is controlled by the beam slenderness with no yielding of the

material at buckling.

2.4.2 YIELD AND PLASTIC MOMENT RESISTANCE

The yield and plastic moment resistance are only reached by beams of very

short lengths with small slenderness where the resistance is controlled by

material failure.

The ability of a beam section to reach the yield or plastic moment resistance

depends not only on the material strength and behaviour but also the particular

section profile. An open section such as an I or channel section and closed

profiles such as box sections are typical examples of section profiles composed

of rectangular plate elements. The ability of the composite section profile to

reach the yield or plastic moment resistance depends therefore on the

individual plate elements to reach the material yield strength with or without

considerable deformation. The ability of the plate elements to reach the yield

strength is controlled by local buckling.

Plate elements of the composite profile are subjected to compression stress.

The width to thickness ratio and support condition of the plate elements will

determine if the yield strength is reached or if local buckling occurs at a lower

stress than the yield strength. Local buckling behaviour is shown in Figure

11

2.3. It is clear that for a particular support condition, the larger the width to

thickness ratio the lower the compressive stress at which it will buckle.

Research on local buckling by Holtz and Kulak"• 12 and other researchers led

to limiting width to thickness ratios of plate elements in order to ensure yield

or plastic resistance moment capacities. Classification of section profiles is

therefore determined by the local buckling behaviour of the plate elements as

follows:

Class 1 section:

Class 2 section:

Plastic section - the plastic moment resistance

moment will be reached with more than adequate

rotation capacity to form a plastic hinge before

ultimate failure.

Compact section - the plastic moment resistance

will be reached without enough rotation capacity

to form a plastic hinge before ultimate failure.

Class 3 section: Non Compact section - the yield moment

resistance will be reached before ultimate failure

or local buckling.

Class 4 section:

Slender section: the section will fail at a moment

lower than the yield moment due to local

buckling.

The classification of a section is therefore an indication of the ultimate moment

resistance of a section and can only be achieved with very short beams with

small slenderness.

12

2.4.2.1

YIELD AND PLASTIC MOMENT RESISTANCE OF TYPE 3CR12

STEEL BEAMS

A theoretical procedure for the calculation of the yield and plastic moment

resistance needs to be developed for beams which are classified with adequate

moment resistance capacity according to local buckling theory. This procedure

will be different from the existing theories for carbon steel beams due to the

difference in material behaviour.

These theoretical procedures are discussed in detail in Chapter 4.

2.4.3 LATERAL TORSIONAL BUCKLING

The moment resistance of a beam will decrease as the slenderness increase and

failure will not be controlled by local buckling, but by inelastic and elastic

lateral torsional buckling of the beam.

In-plane bending of a beam causes flexurally induced axial stresses in the

compression flange. As the bending of the beam proceeds the beam may

buckle laterally as the laterally unsupported compression flange becomes

unstable as a result of the axial compression stress. The critical stress or

bending moment at which lateral buckling takes place depend on the following

• criteria:

Section profile: Thin-walled section profiles such as I-sections,

channels, et cetera do not posses a large torsional stiffness as compared

to thick walled or stocky open sections, and are therefore more

susceptible to lateral torsional buckling.

Loading: Different types of loading cause different bending moment

gradients over the beam length which influence the lateral buckling

behaviour. The most critical is a constant bending moment distribution

13

which causes single curvature bending and is usually therefore utilized

in experimental studies.

Position of load application: The position of loads applied to the beam

over the beam length influence the bending moment distribution as

mentioned above. Also critical is the position of loads applied in

relation to the centroid and especially the shear centre of the section

profile. When a load is applied at a position eccentric of the shear

centre it may cause a destabilizing external torsional moment to the

beam. An example is shown in Figure 2.4 where a point load is

applied to the top of the beam at the compression flange. Lateral

torsional buckling of the beam causes the point load to be eccentric in

relation to the shear centre and results in a destabilizing torsional

moment. Applying the point load to the tension flange results in a

stabilizing moment.

Support or boundary conditions: The support conditions influence the

overall stiffness of the beam. A stiff or fixed support will result in

higher critical buckling moments.

Unsupported length of the compression flange: The greater the

unsupported length the more slender the compression flange. The

critical moment or stress at which the flange becomes unstable and

buckles laterally is therefore lower for large unsupported lengths of the

compression flange.

The criteria discussed above influences the lateral torsional buckling behaviour

of a beam. The basic relationship of some of the factors discussed above and

the critical buckling moment for elastic material conditions and doubly

symmetric sections is given in equation 2.10.

14

E C MCI = E Iy G ( 1 +

) G J L 2

(2 . 1 0 )

Equation 2.10 forms the basis for theoretical calculations to estimate the

critical buckling moment. The equation is however based on the assumptions

of elastic stress distributions and a linear relationship between stress and strain

for the beam material which is true for carbon steel.

Modifications are therefore made to equation 2.10 to incorporate effects such

as the inelastic stress distributions, non linear stress-strain relationships, fixed

and partially fixed support conditions, moment gradients over the beam length

and the position of load application to a beam in relation to the shear centre.

Equation 2.10 will be derived in Chapter 5. Modifications and interaction

equations used to estimate the critical buckling moment will also be discussed

in detail.

2.5 CONCLUSIONS

The non linear behaviour of Type 3CR12 steel necessitates the development

of theories to determine the yield and plastic moment resistance of a beam. A

theoretical approach that accounts for the effect of the material behaviour on

the lateral torsional buckling behaviour of beams also needs to be discussed

and experimentally verified to establish a theoretical design approach for

determining the critical buckling moments for Type 3CR12 steel beams.

15

Fp, Fy

E y STRAI N

FIGURE 2.1 a - SHARP YIELDING STRESS-STRAIN BEHAVIOUR

(f) (../) LU

U)

E = 0,0001 E = 0,002 STRAI N

FIGURE 2.1 b - GRADUAL YIELDING STRESS-STRAIN BEHAVIOUR

RE

SIS

TAN

CE

MO

MEN

T

L. INELASTIC REGION ELASTIC REGION

k

PLASTIC REGION

16

SLENDERNESS RATIO

FIGURE 2.2 - TYPICAL LATERAL TORSIONAL BUCKLING BEHAVIOUR

OF BEAMS

17

FIGURE 2.3 - LOCAL BUCKLING OF FLANGE

18

P

fl

n M= Pe M = Pe DESTABILIZING

STABILIZING MOMENT

MOMENT

FIGURE 2.4 - EFFECT OF LOAD POSITION APPLICATION

19

CHAPTER 3

MECHANICAL AND SECTIONAL PROPERTIES

3.1 GENERAL REMARKS

The mechanical properties of Type 3CR12 steel as well as the section

properties of the test beams are presented and discussed. The parameters

obtained in this investigation will be applied in following theoretical

calculations with the aim of predicting the structural behaviour of the test

beams.

3.2 EXPERIMENTAL DETERMINATION OF THE MECHANICAL

PROPERTIES

Type 3CR12 steel shows anisotropic behaviour and therefore are uniaxial

tensile and compression tests performed on the test beam material. The

preparation and testing of the test specimens are performed in accordance to

the requirements of ASTM Standard A370-77' 3 and ASTM Standard E9-70"

for tensile and compression tests, respectively. Stresses that develop through

the bending of beams are in principle longitudinal, that is in the direction of

the longitudinal axis of the beam, which is also the rolling direction of the

beam profile. The tensile and compression test specimens are therefore only

cut in the longitudinal direction to represent the mechanical behaviour of the

beam material in this direction.

3.2.1 PREPARATION OF TEST SPECIMENS

Figure 3.1 shows where the test specimens are cut from the steel sections. The

test specimens are cut at a distance away from the flange end tips as well as

the flange to web junction where local residual stresses exists due to uneven

cooling of the section. The specimens are also cut a distance away from the

20

beam end tips to avoid residual stresses due to uneven cooling of these

exposed end tips. Figure 3.2 shows the sample location of the test specimens

of test beam 40-T8-I-HT. The specimens are cut near the end of the beam and

at position a and b. The reason for this is that the beam had a distinct bend in

the region of position a and b due to handling error at the steel mill and heat

treatment plant, and therefore had to be manually bend straight to fit into the

test setup.

The test specimens are cut with a power- and band saw. Appropriate coolants

were used during the cutting procedure to avoid excessive heating that can

alter the mechanical properties. The test specimens are machined with a

milling machine and thereafter ground with a magnetic metal grinder as the

last machining operation's. The final dimensions and cross sections of the

specimens were within tolerances the dimensions shown in Figure 3.3.

Machining of the compression test specimens were carefully monitored to

ensure that the ends of the specimens at which the compression load is applied,

are parallel to each other and perpendicular to the longitudinal axis of the

specimen to avoid uneven straining of the test specimen as discussed by

Parks'.

3.2.2 TESTING OF TENSILE- AND COMPRESSION SPECIMENS

The stress that develops in a test specimen is obtained by dividing the applied

compression or tensile force by the original cross sectional area of the test

specimen. The average strain is measured with two strain gauges, mounted on

both wide faces of a specimen. The strain gauges are connected in a full

bridge configuration with temperature compensation.

The tests are performed in an Instron 1195 universal testing machine. The

force applied and the strain measured by the strain gauges are recorded with

an Orion Solatron signal compiler and via a computer program written to text

file for data processing. Data recordings are taken at time intervals of 500

•

21

milliseconds.

TENSILE TESTS

The width and depth of the rectangular cross section of the test specimen are

measured accurately to 0,005 mm with a micrometer. Tests are performed at

a strain rate of 0,5 mm/minute until a strain of 1% is reached, after which the

strain rate is increased to 2 mm/minute until fracture of the specimen. The

specimens are marked with an original gauge length of 50mm and fitted

together after fracture to measure the final gauge length accurately to 0,05

MM.

COMPRESSION TESTS

The width and depth of the rectangular cross section of the test specimen is

measured accurately to 0,005 mm with a micrometer. Tests are performed at

a strain rate of 0,5 mm/minute until a strain of 1% is reached, after which

testing is terminated. The test specimens are mounted in a special mechanical

fixture to avoid premature buckling of the specimen about the minor bending

axis.

3.2.3 EXPERIMENTAL MECHANICAL PROPERTIES OF TYPE 3CR12

STEEL BEAMS

A computer program was written to determine the mechanical properties from

the experimental stress-strain data. A linear regression is carried out to

determine the initial modulus of elasticity, after which a spline curve fit of the

experimental data is made and the proportional limit and yield strength are

determined by the offset method.

The mechanical property values are presented in Tables 3.1 to 3.30.

22

The heading of each table indicate the test beam to which the mechanical

material properties belong and the following convention applies:

Example: Beam 08-T1-I-NHT

The first two numerical values indicate the test beam length, example:

0,8 meters.

The next two letters indicate the test number.

The next letter indicates the beam profile, a doubly symmetric I-profile

or a singly symmetric channel.

The last letters indicate if the beam section was heat treated (HT), or

not heat treated (NHT).

3.14 ANALYTIC STRESS-STRAIN RELATIONSHIP

The modified Ramberg-Osgood' equation represents the stress-strain

relationship of stainless steels and Type 3CR12 steel analytically as follows,

e = JE 0,002 (=E) n E, Fy (3 . 1 )

where

n -

0 002 , log ( ' 0,0001

The mean experimental tensile- and compression stress-strain relationship as

well as the analytical representation thereof by the modified Ramberg-Osgood'

equation are shown by Figures 3.4 to 3.14. Figures 3.4 to 3.14 represent the

stress-strain relationship of the doubly symmetric I beams only. The sharp

yielding stress-strain relationship shown by Figure 2.1 (a) represent typically

that of all the singly symmetric channel sections.

23

The mean experimental curves are obtained by fitting spline curves to the

stress-strain data of all compression and tensile tests of each beam and

calculating the mean stress for discrete values of strain.

The analytical curves are obtained with the mean values of E 0, Fp , and Fy as

the parameters of the analytical equation.

3.3 DISCUSSION OF EXPERIMENTAL RESULTS

All the test beams were heat treated to relieve the material of residual stresses,

except beam 08-T1-I-NHT which is a doubly symmetric I-section. This beam

was tested as received from the steel mill to serve as a comparative test to the

heat treated beams.

The following were concluded from the mechanical properties test results:

Research by Van Den Berg and Van Der Merwes led to the conclusion

that Type 3CR12 steel is of the gradual yielding type. This conclusion

followed from extensive research on the mechanical properties of cold

rolled plates. The experimental results of this study shows that:

the mechanical properties of the heat treated singly-symmetric

channels shows sharp yielding characteristics, resulting in the

ratio of yield strength to the proportional limit equal to one.

the mechanical properties of the heat treated doubly-symmetric

I-beams shows gradual yielding characteristics that tends to

sharp yielding characteristics, with a plato at yield strength as

shown by Figures 3.4 to 3.14. The mean ratio of yield strength

to the proportional limit varied from 0,870 to 0,921 for

compression and 0,868 to 0,933 for tension.

the comparative test beam 08-T1-I-NHT that was not heat

treated shows gradual yielding characteristics. The mean ratio

24

of yield strength to the proportional limit is 0,640 for

compression and 0,622 for tension behaviour.

It is thus evident that the mechanical behaviour of Type 3CR12 steel

varies from sham yielding characteristics to gradual yielding

characteristics with low proportional limits depending on the degree of

heat treatment that it was subjected to.

An appropriate approach to model the mechanical behaviour of hot

rolled Type 3CR12 steel would thus be to always assume gradual

yielding behaviour and modelling the behaviour with the modified

Ramberg-Osgood' equation which proof to be realistic as shown by

Figures 3.4 to 3.14.

SABS 0162-PART P', the hot rolled steel design specification stipulate

requirements regarding the ratio of ultimate tensile strength to yield

strength as well as the ductility of special structural steels. These

requirements are stipulated to ensure plastic deformation of the material

to avoid premature brittle fracture at plastic hinges, connections and

are as follows:

the yield strength shall not exceed 700 MPa.

the ratio of the minimum tensile strength to yield strength shall

not be less than 1,2.

the elongation (on a gauge length of 5,56* ✓A,, mm) of a tensile

test shall not be less than 15 %, where A. is the original cross

section area, min 2 .

The mechanical properties of Type 3CR12 steel fulfilled the above

requirements. Yield strengths were below 700 MPa. The mean ratio of

ultimate tensile strength to yield strength of the heat treated steel varied

from 1,23 to 1,66 and is 1,22 for the steel that was not heat treated.

The cross sectional areas of the test specimens varied from 65 mm 2 to

120 mm' which refer to gauge lengths of 44,8 mm to 60,9 mm

according to the above mentioned gauge length to which the elongation

of 15% is applicable. The tensile specimens were marked with a 50

mm gauge length and the mean elongation of the heat treated steel

25

varied from 26,40% to 36,64% and was 21,87% for the steel that was

not heat treated.

3.4 SECTION PROPERTIES OF TEST BEAMS

Singly- and double symmetric beams are tested in this study. The Type 3CR12

steel beams were produced to the standard dimensions of their carbon steel

counterparts currently produced by South African Steel Manufacturing

Company. The two steel sections chosen to be tested in this study consists of

a doubly symmetric I-section (parallel flange) designated as 203*133*25 kg/m

and a singly symmetric channel (DIN taper flange) designated as 100*50*11

kg/m.

The beam section dimensions were taken with a micrometer and a vernier to

0,005 mm and 0,05 mm accuracy, respectively. The section properties of each

individual test beam, measured and calculated, are given in shown by Tables

3.31 to 3.33, and defined as shown in FIGURE 3.15.

3.5 CONCLUSIONS

The mechanical properties of Type 3CR12 steel shows a non linear stress-

strain relationship. The non linearity depends on the extent of heat treatment

applied to the steel. This results in ratios of the yield strength to proportional

limit as low as 0,62 and as high as 1,0 which indicates linear behaviour. The

analytical stress-strain curves shows that the Ramberg-Osgood equation is an

appropriate model of the experimental mechanical behaviour of Type 3CR12

steel.

26

TABLE II - MECHANICAL PROPERTIES OF TEST BEAM 08-TI-I-NIIT

MECHANICAL PROPERTIES :;COMP.RESSION

TEST SPECIMEN Eo (GPa) (MPa) F, (MPa) (MPa)

Fl 198,0 355,0 536,0

F2 221,1 349,5 549,0

F3 205,2 334,4 553,0

F4 195,8 339,5 516,0

WI 216,2 367,0 555,5

W2 221,2 344,2 556,0

MEAN 209,6 348,3 544,3

STANDARD DEVIATION 11,5 11,7 15,7

COEFFICIENT OF VARIATION (%) 5,46 3,36 2,88

MECHANICALPROPERTIES:ETENSILE:::::

TEST SPECIMEN (GPa) (MPa) F, (MPa) F. (MPa)

F1 191,1 288,0 494,5 598,0

F2 187,7 307,0 488,6 597,5

F3 195,8 335,0 506,2 616,6

F4 190,0 322,2 480,5 586,6

WI 205,7 312,0 510,5 629,3

W2 207,4 303,0 522,5 645,1

MEAN 196,4 311,2 500,5 612,2

STANDARD DEVIATION 8,3 16,2 15,5 22,2

COEFFICIENT OF VARIATION (%) 4,21 5,20 3,09 3,63

27

TABLE 3.2 - MECHANICAL PROPERTIES OF TEST BEAM 08-T2-I-HT

MKIiANICALTROPERTIES:: COMPRESSION::

TEST SPECIMEN E„ (GPa) F, (MPa) F, (MPa) F, (MPa)

F1 216,9 367,0 406,8

F2 209,2 379,0 418,0

F3 208,0 414,5 469,8

F4 210,6 423,5 475,4

WI 217,6 389,0 457,5

W2 219,8 358,5 420,8

MEAN 213,7 388,6 441,4



mgc.1.-TANIcAucptiort.ERTIEst .::TgNsiL4

TEST SPECIMEN E„ (GPa) (MPa) F, (MPa) F. (MPa)

Fl 196,7 361,5 383,0 512,6

F2 197,9 364,5 396,0 515,0

F3 196,2 410,0 446,2 541,9

F4 199,7 423,2 453,5 553,7

W I 196,9 395,0 441,0 550,1

W2 202,6 351,5 404,8 526,6

MEAN 198,3 384,3 420,8 533,3



28

TABLE 13 - MECHANICAL PROPERTIES OF TEST BEAM 16-T3-I-HT

TEST SPECIMEN Ep (GPa) Fp (MPa) F, (MPa) F. (MPa)

Fl 208,7 372,8 412,6

F2 212,4 384,2 418,1

F3 218,7 459,0 486,0

F4 223,9 401,0 472,0

W I 217,2 388,8 434,2

W2 215,1 421,4 476,2

MEAN 215,9 404,5 449,9



MECHANICAL : PROPERTIES ;: TENSI E:*

TEST SPECIMEN Ec, (GPa) (MPa) F, (MPa) F. (MPa)

Fl 202,8 370,2 400,2 518,2

F2 193,4 372,2 393,2 515,7

F3 198,2 421,2 450,2 547,9

F4 195,2 410,0 460,0 555,2

WI 199,7 377,4 414,2 530,2

W2 201,6 358,0 441,6 548,0

MEAN 198,5 384,8 426,6 535,9



29

TABLE 3.4 - MECHANICAL PROPERTIES OF TEST BEAM 24-T4-1-HT

TEST SPECIMEN E. (GPa) F,, (MPa) F, (MPa) F. (MPa)

F I 217,8 376,0 432,8

F2 211,8 386,2 424,0

F3 207,6 381,0 429,2

F4 227,7 411,0 458,0

WI 212,7 396,0 450,8

W2 218,7 377,5 444,5

MEAN 216,0 388,0 439,9



frittitAi■tteAt::Oketitittit.:Eltkittm••

TEST SPECIMEN E. (GPa) F,, (MPa) F, (MPa) F. (MPa)

F1 195,9 370,5 394,8 513,6

F2 198,0 370,2 402,8 514,9

F3 202,4 381,0 413,0 526,3

F4 197,5 398,8 419,8 527,4

W I 202,2 384,8 419,4 533,6

W2 199,8 379,5 410,5 531,5

MEAN 199,3 380,8 410,1 524,5



........ . ............. :m c0M -PRESS

30


TEST SPECIMEN E. (GPa) Fp (MPa) F, (MPa) E. (MPa)

F1 206,2 372,5 439,0

F2 208,2 381,5 424,5

F3 208,3 377,3 420,8

F4 205,4 357,6 425,7

WI 211,4 350,0 401,0

W2 211,9 357,8 413,7

MEAN 208,6 366,1 420,8



EIOECILA:$1. 1 6;.LEPROPERTIESt::TEN.$;f..;E:H::.-

TEST SPECIMEN E. (GPa) (MPa) F, (MPa) F. (MPa)

F1 199,3 371,2 434,4 536,8

F2 197,1 388,4 421,7 530,6

F3 197,5 370,8 410,6 523,4

F4 195,8 348,8 407,5 521,6

WI 201,4 331,7 397,5 521,8

W2 200,4 340,5 405,5 525,3

MEAN 198,6 358,6 412,9 526,6



TEST SPECIMEN

F1

F2

F3

F4

WI

W2

MEAN

STANDARD DEVIATION

COEFFICIENT OF VARIATION (%)

TEST SPECIMEN

Fl

F2

F3

F4

W I

W2

MEAN

STANDARD DEVIATION


ED (GPa) FD (MPa)

206,7 358,5

203,4 366,0

208,1 364,4

209,5 382,6

222,9 365,5

221,2 375;4

212,0 368,7

8,1 8,7

3,82 2,36

E. (GPa) (MPa)

197,1 353,0

196,4 334,5

203,1 364,5

199,9 344,2

198,7 346,5

198,8 358,8

199,0 350,3

2,4 10,8

1,20 3,08

F, (MPa)

388,6

399,6

407,2

408,5

398,0

401,6

400,6

7,2

1,80

F, (MPa) F, (MPa)

379,2 504,3

375,0 503,9

386,6 509,6

385,0 514,9

377,5 511,0

385,5 517,8

381,5 510,3

4,9 5,6

1,27 1,09

(MPa)

31


(MPa) F, (MPa)

381,5

369,8

392,0

389,0

372,8

361,0

377,7

11,9

3,16

TEST SPECIMEN

Fl

F2

F3

F4

WI

W2

MEAN

STANDARD DEVIATION


TEST SPECIMEN

Fl

F2

F3

F4

WI

W2

MEAN

STANDARD DEVIATION


E, (GPa) F, (MPa)

209,5 345,5

215,3 346,0

208,7 360,6

216,1 348,2

219,9 350,0

220,0 324,0

214,9 345,7

4,9 12,0

2,28 3,47

Et, (GPO F,, (MPa)

199,8 327,0

201,6 344,0

196,9 362,4

195,3 367,8

206,1 337,5

202,5 310,5

200,4 341,5

3,9 21,5

1,96 6,31

II

F, (MPa) F, (MPa)

367,8 499,5

364,0 504,1

391,6 514,9

402,0 519,4

375,1 509,5

356,0 498,5

376,1 507,7

17,5 8,4

4,65 1,66

32


33

TABLE 18 - MECHANICAL PROPERTIES OF TEST BEAM 40-T8-I-HT

MECHANICAL PROPERTIES ::COMPRESS ION

TEST SPECIMEN En (MPa) Fp (MPa) F, (MPa) (MPa)

Fl 211,1 365,8 409,8

F2 212,4 360,6 398,7

F3 213,0 359,1 397,4

F4 209,4 361,8 397,3

WI 212,7 363,1 399,2

W2 210,0 355,3 387,3

MEAN 211,4 360,9 398,3



MECHANiCA(PROFERTIES. :* . TENSILE; .

TEST SPECIMEN (GPa) Fp (MPa) F, (MPa) F,, (MPa)

Fl 196,5 355,3 396,4 512,2

F2 198,8 373,0 393,8 517,1

F3 200,0 369,2 393,0 514,6

F4 201,5 358,2 392,9 513,0

WI 204,5 338,4 388,5 514,9

W2 201,6 329,3 379,3 508,9

MEAN 200,5 353,9 390,6 513,5



Values presmed by dui, table were obtained from material tee specimens co nay the end of the me beam. Tables 19 and 3.10 present materiel tee data of specimens cut at poddon (a) and (b).

reaPeerhtlY. 'flown in Figure 3 2 The maim being that the beam had to be gag araienened abort mime secdonal axh and was not bat treated afterwards.

:MECHANICALPROIS:BRTI,

F, (MPa) F. (MPa)

454,8

457,0

447,0

450,0

473,9

491,2

F, (MPa) F. (MPa)

433,4

442,0

450,5

439,4

451,0

453,7

16,3

3,59

TEST SPECIMEN

F I

F2

F3

F4

WI

W2

TEST SPECIMEN

F1

F2

F3

F4

WI

W2

MEAN

STANDARD DEVIATION


Eo (GPa) Fp (MPa)

210,4 412,5

211,8 416,3

212,4 397,2

214,5 404,1

219,8 416,0

221,2 441,3

Ep (GPa) Fp (MPa)

211,8 388,2

216,4 401,5

211,3 406,5

212,5 394,8

218,1 411,6

214,6 408,2

3,7 14,2

1,73 3,48

34


Wits manned by this table were &mined from material tea wecimens cut at position (a) on the as barn, town In Figure 3.2.

Values presented by this table were &rained from material ten recimens cu at position (b) cm the tea beam, *own in Figure 3.2.

35

TABLE 3.10 - MECHANICAL PROPERTIES OF TEST BEAM 40-T8-I-FIT

MECHANICAL PROPERTIES T:E.NSI[ E::i

TEST SPECIMEN En (GPa) Fp (MPa) F, (MPa) F„ (MPa)

Fl 199,5 395,8 437,2 541,6

F2 190,1 337,7 437,5 540,3

F3 195,9 397,6 443,8 545,4

F4 198,4 401,3 449,2 543,7

WI 201,4 405,6 464,9 562,5

W2 205,4 410,1 461,5 561,7

$100-14,(Ai E0401E$'17,ENIWE

TEST SPECIMEN (GPa) Fr, (MPa) F, (MPa) F, (MPa)

Fl 195,4 400,7 435,3 533,7

F2 199,7 387,5 429,3 536,21

F3 198,7 386,1 424,4 529,4

F4 199,3 374,8 421,1 530,5

WI 200,2 361,8 444,4 549,8

W2 205,9 363,6 442,3 550,3

MEAN 199,2 385,2 440,9 543,8



Values presented by this able wart drained Iran material in prime's Cy at Nadal (a) on the in beim. shown In Figure 3.2.

Values presented by this table were chained (ran material tea specimens cut a, patinae (b) on the tea beam, limn in Figure 3.2.

36

TABLE 3.11 - MECHANICAL PROPERTIES OF TEST BEAM 48-T9-1-HT

MECHA:NI(At;!!RopERTIES::COS4pRE”ON:: :.

TEST SPECIMEN E„ (GPa) Fr, (MPa) F, (MPa) F. (MPa)

Fl 210,1 343,8 380,0

F2 212,1 336,5 373,2

F3 213,3 347,8 402,0

F4 208,9 345,0 396,0

WI 219,9 345,2 382,0

W2 216,4 357,0 390,8

MEAN 213,5 345,9 387,3



MECHANICALTRiCIPEktIES:::MNSILE::::

TEST SPECIMEN Eo (GPa) F,, (MPa) F, (MPa) F, (MN)

Fl 198,3 351,2 375,0 502,2

F2 195,3 354,0 375,0 504,1

F3 199,1 374,5 411,2 524,1

F4 195,2 352,5 406,4 525,6

W I 203,8 341,0 374,2 505,0

W2 205,8 352,0 383,0 510,6

MEAN 199,6 354,2 387,5 511,9


COEFFICIENT OF VARIATION (%) .2,20 3,10 4,36 2,03

. • . :•:: ••

COMPRESSION::::

37

TABLE 3.12 - MECHANICAL PROPERTIES OF TEST BEAM 56-TI0-I-HT

TEST SPECIMEN Ea (GPa) F. (MPa) F, (MPa) F. (MPa)

Fl 217,0 337,6 384,0 •

F2 221,2 344,8 389,4

F3 208,7 354,0 394,0

F4 209,1 356,4 384,0

WI 216,1 341,0 387,5

W2 213,8 348,0 383,8

MEAN 214,3 347,0 387,1



M EPISAINTP.it;J!R0.

TEST SPECIMEN E. (GPa) F, (MPa) F, (MPa) F. (MPa)

F1 195,2 357,8 379,2 503,0

F2 198,4 353,0 375,3 502,4

F3 202,9 339,0 379,4 505,3

F4 198,9 358,5 374,4 504,9

W I 200,7 335,5 369,6 502,2

W2 191,1 355,6 373,4 506,5

MEAN 197,9 349,9 375,2 504,1



• : , :•:RATTErs.or

TEST SPECIMEN

COMPRESSION TENSILE

PPM, PP/F),

PdP), e

F I 0,66 0,56 1.21 22,65

P2 0,64 0,63 1,22 24,02

F3 0,63 0,66 1,22 23,33

P4 0,66 0, 67 1,22 23,08

WI 0,66 0.61 1,23 18,16

W2 0,62 0.58 1,24 19.96

MEAN 0,64 0. 62 1,22 21,87

STANDARD DEVIATION 0,02 0.04 0,01 2,29


38

TABLE 3.13 - MECHANICAL PROPERTY RATIOS OF TEST BEAM 08-TI-I-NHT

TABLE 3.14 - MECHANICAL PROPERTY RATIO'S OF TEST BEAM 08-T2-I-HT

TEST SPECIMEN COMPRESSION TENSILE

Pp/FY

pp/Ps FiFy % e

PI 0,90 0,94 1.34 29,55

F2 0,91 0.92 130 2834

F3 0,118 0,92 1.21 25,65

P4 0,89 033 1,22 26,39

WI 0.85 0,90 1,25 24.18

W2 0,85 0,87 1,30 26,54

MEAN 0,88 0,91 1,27 26,87

STANDARD DEVIATION 0,02 0,03 0,05 2.03

COEFFICIENT OF VARIATION ( %) 2,78 3,00 3.95 7,55

39


TEST SPECIMEN

COMPRESSION TENSILE

Pp/Py Pp/FY Pulpy S c

PI 0,90 0,93 1.29 29,76

F2 095 1,31 28.83

F3 0,94 094 1.22 25,78

P4 0,85 0,89 1.21 25,70

W I 0,90 0,91 1.28 26,16

W2 0,88 0,81 1,24 23,63

MEAN 0,90 0,90 1,26 26,64

STANDARD DEVIATION 0,03 0,05 0,04 2.26



TEST SPECIMEN

COMPRESSION TENSILE

FOIE r Fo/Py St

F' 0,87 0,94 1,30 29,62

P2 0.91 0,92 1,29 28,57

F3 0.89 0,92 1,27 28,30

1,4 0,90 0,95 1.26 26,44

WI 0,88 0,92 1.27 27,11

W2 0,85 0,92 1,29 26.80

MEAN 0,88 0,93 1,28 27,81


COEFFICIENT OP VARIATION (%) 2.46 1.38 1,26 41,40

TEST SPECIMEN

COMPRESSION 1

TENSILE

:•:•:•:•:•:•:•:• :•: :•:•: :•:.: :•:-:•:•.

PJP y PdPy % e

0,85 1,24 27,05

0,92 1.26 27,07

0,90 1.27 27.79

0,86 138 27,27

0,83 1,31 26,45

0,84 1,30 2735

0.87 1.28 27,23

0,04 0,03 0.50

4.09 2,13 1E4

17 1

n

P3

P4

WI

W2

MEAN -

STANDARD DEVIATION

COEFFICIENT OP VARIATION (%)

PJF r

0,85

0,90

0,90

0,84

0,87

0.87

0,87

0,02

237

40


TABLE 3.18 - MECHANICAL PROPERTY RATIO'S OF TEST BEAM 32-T64-11T

• • •

TEST SPECIMEN

COMPRESSION TENSILE

PA, y 12JP r

PdPy % e

PI 0,92 0,93 1,33 31.65

P2 0,92 0,89 1,34 31,27

P3 0,89 0,94 31.14

P4 0.94 0,89 1,34 28,00

WI 0.92 0,91 1.35 28.66

W2 0.93 0,93 1,34 29,75

MEAN 0,92 0.92 1,34 30,08

STANDARD DEVIATION 0.02 0,02 0.01 1,51

COEFFICIENT OP VARIATION (%) 1,65 2,28 0,93 5,04

COMPRESSION

TEST SPECIMEN II P /F

P r Pu/Py S .

0.89 1,36 29.39

0,95 1,39 32.30

0,93 1.31 30,16

0,91 1.29 29,30

0.90 1.36 27,59

0,87 .40 26,40

0,91 1.35 29,19

0,03 0.04 2.05

2,89 3.03 7,02

STANDARD DEVIATION


MEAN

WI

W2

PI

P2

174

F3

0.94

0,91

0,92

0,90

0.94

0,90

0,92

0,02

2,08

FJP r

TENSILE

: : : : : : : : : : : : : : : • : : : : : : : : : : : :

TEST SPECIMEN

COMPRESSION

PJP r

pp/P1

PI 0,89 0,90

P2 0,90 0.95

P3 0.90 0.94

P4 0.91 0.91

WI 0,91 0,87

WI 0.92 0.87

MEAN 0,91 0,91

STANDARD DEVIATION 0.01 0,03

COEFFICIENT OP VARIATION (S) 0,93 3,70

TENSILE

S t Fu/Py

28,04 1.29

29,20 1.31

1.31 30,37

1,31 28,78

28.14 1,33

27.02 1.34

28.59 1.32

0,U2 1,14

4,00 1.30

41



Vetoes mined by this table sere obtained from mathrIal in thecimcns cut ow the end of the n beam.

42

TABLE 3.21 - MECHANICAL PROPERTY RATIO'S OF TEST BEAM 40-T8-1-1IT

I

TEST SPECIMEN COMPRESSION.

1

TErisrio*

Ppiry Pp/Fy Pp/P, I c

PI 001 0.91 1.24 27,89

F2 0,91 0.77 1,23 26,80

P3 0.89 0,90 1.23 26.90

P4 0.90 0,90 1,21 27,55

WI 0,88 0,87 1.21 25,12

W2 0,90 0,89 1,22 24.83

TEST SPECIMEN COMPRESSION. mime.

Pp/Fy Pp/Py P p/Py I e

Fl 0.90 0.92 1.23 26,85

P2 0.91 0,90 1.25 28,10

P3 0.90 0.91 1 . 25 27.60

P4 0.90 0.89 1.26 28.02

WI 0.91 0.81 1.24 25.40

W2 . 0.82 1.24 25,35

MEAN 0.90 0.91 1.23 26.70

STANDARD DEVIATION 0.01 0,03 0,02 1.22

COEFFICIENT OF VARIATION (%) 1,14 3,70 1.27 4.55

Values presented by this table were &mined from material letcpecimens at at position (a) as then barn, Mown in Figure 3.2.

S

Value, presented by this table were obtained from materiel test Ipecimens on at position (b) as the me beam, shown in Figure 3.2.

. . . . . . . . . . . . . . . . . • .

TEST SPECIMEN

COMPRESSION TENSILE

pp/P r Fp/Py ri/pr S c

Fl 0,91 0.94 1.34 30.39

0,90 0,94 1.34 18.49

P3 0,87 0.91 1,27 25,50

P4 0.87 0,87 1.29 28,91

WI 0,90 0,91 1.35 27,66

W2 0,91 0,92 1,33 30,27

MBA N 0,89 0,92 1.32 28,54


COEFFICIENT OP VARIATION ( 5) 2,24 2,94 2.33 6,38

43


TABLE 3.23 - MECHANICAL PROPERTY RATIO'S OF TEST BEAM 56-TIO-I-HT

ONMECEANICAL:PROtERTIES:

TENSILE COMPRESSION TEST SPECIMEN

Fp/Py S t

Fl 29.85 1,33 0.94 0,88

P2 29,88 1,34 0.94 0,89

F3 29.46

0,93 0,96 1,33

WI 0.88 0.91 1,36

W2 0,91 0.95 1,36

MEAN 0,90 0.93 1,34


COEFFICIENT OF VAR LAnoN (%) 2.12 2,77 0.99

28,14

29,34

29,55

29,37

0,64

2,17

0.90 0,89 1.33

44

TABLE 3.24 - MECHANICAL PROPERTIES OF TEST BEAM 06-TI-C-HT

MECIIANICAL PROPERTIES :: COMP.

TEST SPECIMEN E. (GPa) = F, (MPa) F. (MPa)

Fl 208,1 319,3

F2 206,2 291,8

WI 212,0 320,9

W2 208,6 314,6

MEAN 208,7 311,6

STANDARD DEVIATION 13,5


1,17 4,33

1•:•:•:•: : : :•:•.

TEST SPECIMEN E. (GPa) = F, (MPa) F. (MPa) %e

Fl 197,9 308,4 477,6 1,55 36,25

F2 199,2 304,9 471,6 1,55 34,99

WI 192,4 313,4 468,1 1,49 33,47

W2 193,7 313,1 469,4 1,50 33,93

MEAN 195,8 309,9 471,7 1,52 34,66

STANDARD DEVIATION 3,3 4,1 4,2 0,03 1,24


1,68 1,32 0,89 1,96 3,57

45

TABLE 3.25 - MECHANICAL PROPERTIES OF TEST BEAM 12-T2-C-HT

MF,CHANICAL PROPERTIES: COMPRESSION


F1 208,1 319,3

F2 206,2 291,8

WI 212,0 320,9

W2 208,6 314,6

MEAN 208,7 311,6



1,17 4,33

iOtd4.4-$14;100:•:Eis:kbOtittit,Hitisi•SittH:

TEST SPECIMEN E. (GPa) Fp = F, (MPa) F. (MPa) F./F, %e

F1 197,9 308,4 477,6 1,55 36,25

F2 199,2 304,9 471,6 1,55 34,99

WI 192,4 313,4 468,1 1,49 33,47

W2 193,7 313,1 469,4 1,50 33,93

MEAN 195,8 309,9 471,7 1,52 34,66



1,68 1,32 0,89 1,96 3,57

-MECHANICALi:PROPERTIES:COMPRESSION: ..

E - MECHAN I.CALIEPRQPEITIES;:iTPN$11.4: .: :;.

TEST SPECIMEN (GPa) = F, (MPa) F. (MPa)

F I 202,9 269,2

F2 202,5 262,9

WI 208,9 281,3

W2 204,5 277,4

MEAN 204,7 272,7

STANDARD DEVIATION 2,9 8,3


1,43 3,03

TEST SPECIMEN E. (GPa) = F, (MPa) F. (MPa) F,/F, %e

F1 185,0 251,6 421,2 1,67 37,65

F2 190,7 253,6 428,1 1,69 37,92

W I 179,9 264,0 433,0 1,64 34,70

'W2 186,5 269,3 434,1 1,61 36,27

MEAN 185,5 259,6 429,1 1,65 36,64



2,41 3,26 1,37 2,08 4,04

46

TABLE 3.26 - MECHANICAL PROPERTIES OF TEST BEAM 18-T3-C-I-IT

TEST SPECIMEN F. (MPa) E. (GPa) F, (MPa)

4 EdANICAL:PiOREkTIE :COMPRESSI ON

Fl 212,6 301,8

F2 213,1 303,6

WI 210,2 302,3

W2 209,4 297,9

MEAN 211,3 301,4



0,85 0,82

TEST SPECIMEN E. (GPa) = F, (MPa) F. (MPa) FJF, %e

Fl 199,7 289,2 469,4 1,62 37,80

F2 199,7 288,5 465,0 1,61 36,64

WI 200,8 293,2 463,3 1,58 34,56

'W2 199,5 294,5 459,9 1,56 34,26

MEAN 199,9 291,4 464,4 1,59 35,82


COEFFICIENT OF VARIATION ( %)

0,29 1,02 0,85 1,78 4,73

47


• F1 208,7 325,3

F2 207,0 330,5

WI 208,8 330,1

W2 213,4 338,1

MEAN 209,5 331,0

„„ MECH.ANIC AL:PROP ERTIFS::CO MPRESSIONE::::

MECHANIcAli'POOSIES: . T.4NSILE

48


TEST SPECIMEN E. (MPa) p = F, (MPa) F. (MPa)

STANDARD DEVIATION


2,7 5,3

1,31 1,60

TEST SPECIMEN E. (GPa) F„ = F , (MPa) F. (MPa) %e

Fl 199,5 320,5 482,3 1,51 34,96

F2 196,1 321,4 485,8 1,51 34,23

W I 193,5 323,7 475,0 1,47 33,40

W2 192,6 321,4 470,6 1,46 32,44

MEAN 195,4 321,8 478,4 1,49 33,76



1,57 0,42 1,44 1,65 3,22

•


Fl 199,3 265,6 •

F2 234,4 267,3

WI 203,6 273,6

W2 203,8 277,6

MEAN 202,3 272,3



1,25 2,25

MECHANICAL PROPE

TEST SPECIMEN E. (GPa) Fp = F, (MPa)

Fl 191,1 259,5

F2 188,9 263,4

WI 193,1 266,0

W2 192,9 264,3

MEAN 191,5 263,3



1,03 1,04

F. (MPa) %e

431,4 1,66

437,2 1,66

438,0 1,65

436,3 1,65

435,7 1,66

3,0 0,01

0,68 0,45

49


50

TABLE 3.30 - MECHANICAL PROPERTIES OF TEST BEAM 42-T7-C-IIT

:MRCHAN CAL:P.R:OPOTIES:::COMPRuSSION:::::

TEST SPECIMEN E. (MPa) Fp = F, (MPa) F. (MPa)

F1 202,9 306,9

F2 210,8 316,5

WI 316,0 205,3

W2 204,8 302,9

MEAN 205,9 310,6


COEFFICIENT OF 1,64 2,17 VARIATION (%)

TEST SPECIMEN E. (MPa) F„ = F, (MPa) F. (MPa) F./F, %e

F1 200,0 309,4 474,0 1,53 37,64

F2 460,8 1,07 24,68

WI 198,2 310,9 459,0 1,48 33,10

W2 197,6 309,0 459,9 1,49 33,33

MEAN 198,6 309,8 464,3 1,50 34,69

STANDARD DEVIATION 1,3 l,0 8,4 0,03 2,56


0,64 0,33 1,81 1,96 7,37

51

TABLE 3.31 - SECTIONAL PROPERTIES OF DOUBLY SYMMETRIC I - TEST BEAMS

SECTIONAL PROPERTIES T(13ST Raw .Sd.:• : .

08-71-1-NIT 08-T2-1-HT 16-T3-1-HT 24-74-1-HT 24-T5-1-HT

m (k6/93) 23.60 23,77 23.67 23.49 24,15

A (mm) 204.00 203.93 203,33 203,47 204.20

e r (mm) 132.30 132,50 132.55 132,15 132,85

lw (mm) 5,91 5,80 5.83 5.82 6.00

'r (mm) 7,26 7,32 7,22 7.22 7.45

rl (mm) 7,16 7,07 6,94 7.08 6.95

A (103 mm2) 3,09 3.08 3.06 3,06 3,16

(106 mm4) 22,41 22,42 22.11 22.05 22.97

'An (mm) 85.12 85.27 84.96 84.96 85,22

15,3, (106 mm 4) 2.85 2,86 2.83 2,81 2,93

`5,3, (mm) 30.38 30.46 30,40 30,33 30,46

1 (103 mm4) 56.71 56,09 55,04 54,58 59,77

C,, (109 com 6) 29.78 29,71 29.36 29,07 30,59

SECTIONAL PROPERTIES C' 141:0:48 15i:C . :':".

32-76-1-1IT 40-T7-1-11T 40-11-I-HT 48-19-1-HT 56-710-1-HT

co (k8ico) 23,53 23.71 23,98 23,67 23,55

e (mm) 203.40 203,53 203,75 204.03 203,27

b r (mm) 132.35 132,70 132,20 132,90 132.10

iw . (mm) 5,91 5,82 5,93 5.81 5,115

it (mm) 7,23 7.32 7,42 7,29 7,22

rl (mm) 7,28 7,18 7,79 7.05 7,29

A (103 mm2) 3,08 3.09 3.14 3,08 3,06

lxx (106 mm4

r

22,19 22.35 22,74 22,45 22,06

rxx (mm) 84 • 85 85,09 85,09 65.31 84.86

1YY (106 mra5 2,85 2,87 2.88 : 2,87 2.81

rYY (mm) 30,40 30,48 30,29 30.52 30,30

r 1 (10 mm4) 56,13 57,49 59.16 57,10 55,34

Cy (109 mm6) 29.45 29,66 29,87 29,91 29.09

52

TABLE 3.32 - SECTIONAL PROPERTIES OF SINGLY SYMMETRIC CHANNEL - TEST BEAMS

SECTIONAL PROPERTIES itsr.setat NC( :-'.

06-T1.C-HT I 2..T2-C-11T 18-13-C-HT 24-74-C-HT 30.15-C-IIT

m (kW m) 9,40 9,40 10.24 10,22 9.64

It (mm) 103,35 100.35 103.65 100,80 100.21(

b f (mm) 48,3 48.3 49.00 49,75 47,90

C (mm) 5,60 5.60 6.20 6,08 5.97

I f (mm) 8.03 8.03 8,58 8.50 8.70

cl (mm) 8,04 8.04 8.42 8.61 834

'2 (mm) 2,90 2,90 2.59 1.98 1.70

hi . (mm) 25,00 25,00 23,00 25.00 25.00

I (deg) 94,54 94,54 94,47 94.43 94,73

A (103 mm2) 1,23 1.23 1,34 1.34 1,26

• lc (mm) 28,27 28,27 28,63 29,29 27,80

AY (mm) 14,97 14,97 15,35 15.66 14,72

(106 mm4) 1,99 1.99 2,19 2,23 2,07

rxx (mm) 40.30 40.30 40.45 40.86 40,57

(106 01111 4) 0,26 0,26 0.29 0,30 0,25

57 (ram) 14,43 14,43 14,62 14,89 14,22

• 111• (mm) 7,78 7.78 8,33 8,27 7,88

II (103 osm4) 19,25 19.25 24.47 24,02 20.77

Cwt. (109 mm6) .1 0,46 0,46 0.50 0,53 0,45

Thickness of equavalent pal el flange channel with anal area m miler flute thalami.

Distance to the they centre from the cress sectional centred of an •quavalent parallel flange channel ignoring the caner fillets, and calculated arcording to the following equadcm 18 :

a = b 2 3b 2

a + 2b a + 6b

when a h '11 b Is f - V2

Torsional camant of an equavalent parallel flange channel ignoring the earner fillets, and calculated acconing!. the following equation 18 :

J =E 3 b t3.

where b long Ode of rectangular section element

- thickness of recummtler sectional element

(3.2)

(3.3)

53

TABLE333- SECTIONAL PROPERTIES OF SINGLY SYMMETRIC CHANNEL -TEST BEAMS

SECTIONAL PROPERTIES

36,T6-C-HT 42-TI-C-UT

m (4/8 ) 10,01

b

9,62

(mm) 100,45 100.55

b r (mm) 48,48 47.90

tw (mm) 6,10 5.87

(mm) tr

8,45 8,20

tll (mm) 8,33 8,54

(mm) r2 3,84 1.70

bl (mm) 25,00 25,00

5 (deg) 94.43 94,73

A (103 mm2) 1,31 1,25

• 4C (mm) 28,24 27,81

(mm) 4Y

15,08 14,76

Iu (105 mm4) 2,08 2,08

(mm) 39,81 40.76

1 YY (106 mm 4) 0,27 0,25

57 (mm) 14,44 14,24 ,

• ' ri• (mm) 8,22 7,88

1# (103 mm 4) 23,27 20,50

Cy" (109 mm6) t

0,48 0,45

Meknes equivalent petal el flange channel with equal uea to taper flange channel.

SI Warping cannot of an equivalent parallel flange channel ignoring the comer fillets, and calculated according to the following equarlonle:

Where a b - b f - td2

- t a 2 b 3 2a + 3b) Ch,

12 a + 6b (3.4)

V/I

7

W2

I 1///A ////1 F3 F4

F1

W1

we

F2

- 1600

54

Fl F2

v/A 17/AI J

DOUBLY SYMMETRIC I - SECTION

SINGLY SYMMETRIC CHANNEL SECTION

FIGURE 3.1 - MECHANICAL TEST SPECIMEN LOCATION

TEST BEAM 40-T8-I-HT

POSITION a - POSITION b END OF BEAM'

FIGURE 3.2 - SAMPLE LOCATION OF MECHANICAL TEST SPECIMENS OF BEAM 40-T8-I-HT

70

VARIES CuU)

1 50

150

TENSILE TEST SPECIMEN

VARIES

70

COMPRESSION TEST SPECIMEN

55

FIGURE 3.3 - MECHANICAL TEST SPECIMEN DIMENSIONS

600

500

4-00

a

300

ce

200

100

O

56

Longitudinal Compression - Experimental Longitudinal Compression - Analytical Longitudinal Tension - Experimental Longitudinal Tension - Analytical

---------

Longitudinal Compression - Experimental Longitudinal Compression - Analytical Longitudinal Tension - Experimental Longitudinal Tension - Analytical

0.000

0.001

0.002

0.003

0.004

0.005

STR4IN

FIGURE 3.4 - ANALYTICAL AND EXPERIMENTAL STRESS-STRAIN

RELATIONSHIP OF BEAM O8-TI-I-NHT

0.001

0.005

STRAIN


RELATIONSHIP OF BEAM 08-T2-I-HT

0.005 0.002 0.003 0.004

57

0.000

CL o 300

(11 Ll

200

Longitudinal Compression – Experimental Longitudinal Compression – Analytical Longitudinal Tension – Experimental

-• Longitudinal Tension – Analytical

STRAIN

500

400

500

400

c7 300 a_

u.;

H 200

---- Longitudinal Compression – Experimental ---- Longitudinal Compression – Analytical — Longitudinal Tension – Experimental

Longitudinal . Tension – Analytical

0.001 0.002 0.003 0.004 0.005

STRAIN


RELATIONSHIP OF BEAM 16-T34-11T



58

0.004 0.005 0.002 0.003

500

Longitudinal Compression - Experimental

Longitudinal Compression - Analytical

Longitudinal Tension - Experimental

Longitudinal Tension - Analytical

STRAIN

300 -

2

re K

200 -

0.001 0.005 0.004 0.003 0.002

500

400

o_ 300 2

ct. lj 200

100

0 0.000

Longitudinal Compression - Experimental Longitudinal Compression - Analytical Longitudinal Tension - Experimental Longitudinal Tenzion - Analytical



STRAN



400

300

0 0

co

ce`j 200

VI

.100

0 0.000 0.002 0.001 0.003 0.004 0.005

59

Longitudinal Compression — Experimental Longitudinal Compression — Analytical Longitudinal Tension — Experimental

Longitudinal Tension — Analytical

n.

500

400

300

100

0 0.000 0.001 0.002 0.003 0.004 0.005

Longitudinal Compression — Experimental Longitudinal Compression — Analytical Longitudinal Tension — Experimental Longitudinal Tension — Analytical

STRAN



ST RN N



500

400

rc? 300

2

to

200

100

0 0.000 0.004 0.001 0.002 0.005 0.003

60

Longitudinal Compression — Experimental Longitudinal Compression — Analytical Longitudinal Tension — Experimental

--- Longitudinal Tension — Analytical

STRNN


RELATIONSHIP OF BEAM 40-T8-I-HT: SPECIMENS CUT AT

POSITION (a) AND (b) SHOWN IN FIGURE 3.2

0.001 0.004 0.002 0.003 0.005

.300

400

Longitudinal Compression — Experimental Longitudinal Compression Analytical Longitudinal Tension — Experimental Longitudinal Tension — Analytical

100

ST

RE

SS

(M

Po)

200

400

300

0 7

in tn ce 200

Longitudinal Compression — Experimental Longitudinal Compression — Analytical Longitudinal Tension — Experimental Longitudinal Tension — Analytical

61

100

0 0.000 0.001 0.002 0.003 D.004 0.005

STRNN



STRNN



bf

62

>-

S = SHEAR CENTER C = SECTION SENTROID

w

X x

bf


SINGLY SYMMETRIC CHANNEL SECTION

FIGURE 3.15 - SECTION DIMENSION DEFINITION

63

CHAPTER 4

YIELD AND PLASTIC MOMENT RESISTANCE OF THE BEAM SECTION

4.1 INTRODUCTION

The behaviour of very short beam lengths in the plastic range and the

determination of the yield and plastic moment resistance of the beam section

will be discussed in this chapter. Inelastic and elastic behaviour of short and

slender beams will be discussed in detail in Chapters 5.

4.2 BENDING THEORY

The equations used to calculate the yield and plastic moment resistance are

determined from the principles of statics' as follows:

The position of the neutral axis is determined by static equilibrium of

the forces on the cross sectional area as is given in equation 4.1.

E P), = 0 fA F dA = 0

4 . 1

where

F = stress distributed over the cross section

A = cross sectional area

The moment resistance is the resultant of the moments of the forces on

the cross section about the neutral axis as shown in equation 4.2.

E mz = o ; fA F y dA = 0

4 . 2

where y = distance from the neutral axis to the infinitesimal

area dA

To determine the yield or elastic moment resistance of a beam by the elastic

bending equation 4.3, which satisfy the above equilibrium requirements, the

64

following assumptions must be met:

plane sections through the beam, perpendicular to the longitudinal

beam axis, remain plain after bending, i.e the strain distribution over

the section is linear.

the material obey Hooke's law, i.e stress is proportional to strain below

the yield strength.

the material is isotropic regarding tensile and compression

characteristics.

The elastic moment resistance equation is given in equation 4.3.

F le M — Me

Ye 4 . 3

where = elastic moment resistance

F = stress at extreme fibres of the section and is

smaller or equal to the yield strength

= elastic section modulus

ye = distance from the elastic neutral axis location to

the extreme fibre of the section

The strain and the stress distribution for the case above are shown in Figure

4.1.

To determine the plastic moment resistance of a beam by the plastic moment

equation 4.4, which satisfy the above equilibrium requirements, the following

assumptions must be met:

plane sections through the beaM, perpendicular to the longitudinal

beam axis, remain plain after bending, i.e the strain distribution over

the section is linear.

the material stress-strain relationship is represented by a bi-linear curve

65

such as the sharp yielding curve shown by Figure 2.1 (a).

the material is isotropic regarding tensile and compression

characteristics

The plastic moment resistance is as follows:

M = F Z P Y P 4 . 4

where

Mp = elastic resistance moment

Fy = yield strength

Z„ = moment of equal areas about the neutral axis

The stress-strain distribution of this case is shown in Figure 4.2. The transition

stress-strain distribution from the case of yield moment resistance through the

inelastic case to the plastic moment resistance case is also shown.

The equations above are applicable to structural carbon steel which comply

with the assumptions as stated. The material properties of Type 3CR12 steel

do not comply with these assumptions and the general bending theory must

therefore be applied. The general bending theory will be described briefly

according to Popov' after which a bending theory for Type 3CR12 steel

sections will be formulated.

4.3 GENERAL BENDING THEORY

The static equilibrium requirements are still valid as well as the assumption

that plane sections remain plane, i.e the strain distribution is linear over the

cross section. It will also be assumed that the material is anisotropic, that is

to say the tensile and compression stress-strain relationships differ. The

theoretical process to determine the moment resistance is as follows.

A strain distribution over the section with a known neutral axis

location, where the strain is equal to zero, is assumed as shown in

66

Figure 4.3 (a). The strain distribution over the section is converted to

a stress distribution with the use of a stress-strain relationship obtained

from material tests. A general stress-strain relationship is shown in

Figure 4.3 (b) and the corresponding stress distribution over the section

in Figure 4.3 (c). The stress distribution on the tensile and compression

side is integrated over the cross sectional area to obtain the

compression and tensile force resultants of the stress distributions,

force C and T.

A trial-and-error procedure follows where the process described above

is repeated until equilibrium of the forces is satisfied, i.e the tensile

force is equal to the compression force. The real location of the neutral

axis is known when equilibrium is satisfied and it must be noted that

the location of the neutral axis may not coincide with the centroidal

axis of the cross sectional area. It will only coincide if the cross-section

has two axis of symmetry and the tensile and compression stress-strain

relationships are identical.

With the neutral axis location known as well as the stress distributions it

follows that the line of action of the tensile and compression forces can be

determined as shown in Figure 4.3 (c). The moment resistance is now a couple

and can be determined as given in equation 4.5.

Mr =T ( a + b) =C (a + b) 4 . 5

4.4 RESISTANCE MOMENT OF TYPE 3CR12 STEEL BEAMS

The resistance moment of Type 3CR12 steel beams will be determined by the

general bending theory and the following assumptions:

67

plane sections remain plane after bending, i.e the strain distribution is

linear across the section. This assumption was verified by experimental

strain measurements at midspan across the section depth of two of the

doubly symmetric I-beams, 08-T1-I-NHT and 08-T2-I-HT. The

location of the strain gauges are shown in Figure 4.4 and they were

connected in a quarter Wheatstone bridge configuration. These beams

were chosen to verify the assumption as they are the shortest beam

lengths tested for lateral torsional buckling and theoretically should

reach the yield moment resistance. The experimental strain distribution

versus the section depth, at discrete experimentally applied moments,

is shown in Figures 4.5 and 4.6 for the two beams, respectively.

The stress distribution over the section depth will be represented

analytically by the modified Ramberg-Osgood' equation for the

theoretical calculations. The stress distribution over the section depth

is shown in Figures 4.7 and 4.8. These stress distributions are

determined by correlating the strain values of the strain distributions

shown in Figure 4.5 and 4.6, with stress values determined from the

mean experimental stress-strain curve obtained from the mechanical

property tests of the beam material. The stress distributions over the

cross section shows that a non linear stress distribution over the cross

section develops when the stresses exceed the proportional limit. The

modified Ramberg-Osgood equation, transformed to represent the stress

over the cross section will therefore be valid.

4.4.1 YIELD MOMENT RESISTANCE OF A DOUBLY SYMMETRIC I-

BEAM

Figure 4.9 shows the profile of a doubly symmetric I-section. It is assumed

that yielding occur when the steel fibres at extreme distances from the neutral

axis, which endure the most strain in bending, reach the yield strength. From

the material properties presented in Chapter 3 it is concluded that the tension

yield strength and tension initial modulus of elasticity are always smaller than

68

the compression yield stress and compression initial modulus of elasticity. The

implication of this is that the strain at compression yielding is usually smaller

than that of tension yielding. The strain corresponding to compression yielding

will therefore be taken as constant while the neutral axis and strain at the

tension side are varied to obtain the stress distributions and ultimately

equilibrium. The method to obtain the yield moment resistance of the Type

3CR12 steel 1-beams will therefore be as follows:

The compression yield strain is calculated with the modified Ramberg-

Osgood equation. A location of the neutral axis is chosen at a distance

cc from the outside face of the compression flange as shown in Figure

3.10. The distance from the neutral axis to the outside face of the

tension flange and the strain at this outside face are determined with

equal triangles as follows,

c t = h - cc 3 . 7

C „

t = Cc

ec 3 . 8

The strain, E„ and t ic , at the inside faces of the flanges are also

determined by the same principle.

The stresses that correspond to these strain values are determined

numerically from the modified Ramberg-Osgood equation.

The resultant tensile and compression forces, of the stress distributions,

on the cross section will be determined as the following step. The

calculation will be described for the tension side but applies in the

same manner to the compression side, the only difference being the

variable designations. The calculation is as follows.

The modified Ramberg-Osgood equation is a function g that

69

represents the strain, in terms of the stress i.e e = g(F). This

function is transformed by the principle of equal triangles to a

function f that represents the distance over the cross section, in

terms of stress i.e y = f(F). The function y = f(F) is given in

equation 4.9.

c t F 0 . 0 02 n Cr

( , F

y + u.v02 — --)n e t E, e F t y

4.9

The value of the stress distribution acting on the web and flange

can now be determined as follows:

the area under the curve for the web as shown in Figure

4.9 is determined by integration of equation 4.9 and is

as follows,

11 1 = Fre f (F) dF 4.10 0

The stress distribution over the web can now be

determined as follows,

A, = crt Fit t — A 4.11

The stress distribution over the flange is determined by the

same principle. From Figure 4.9:

= Fr̀ f (F) dF 4.12 Ffc

and the stress distribution in the flange is given in equation

4.13.

Af = c t Ft - (A l + Aw + A") 4.13

The centroid of A' is determined by the following equation,

70

y i = A/ f Fr' 2 [f (F)] 2 dF 4.14

0

and the centroid of the stress distribution in the web by taking

moments about the neutral axis as given in equation 4.15.

•

4.15 Yw

A,

The centroid of A" is determined by the following equation,

Y 1 I FL 1

F — f (F) ) 2 dF

All it 2 4.16

and the centroid of the stress distribution in the flange by taking

moments about the neutral axis,

(A' + A„) cft

2 - y i

Yf

cc (A / + A, A" + = -A'y' - A, y, - A" y"

2 4.17

A f

The resultant tensile force of the stress distribution over the

web and flange on the section is given in equation 4.18.

T = A, t 4.18

The line of action of the tensile force is again determined by

taking moments about the neutral axis is as follows,

A„ y, + A f br

jit A, t„ + Ai br 4.19

The resultant tensile and compression forces of the stress distributions

are determined for every choice of the neutral axis location until

equilibrium of the forces is satisfied.

71

The yield moment resistance is then a couple of the forces and

determined by using equation 4.20.

Mw = T (yt + = C (37t + 57c ) 4.20

The strain and stress at the tension side must always be checked after

equilibrium is satisfied. If the tensile yield stress is exceeded at equilibrium,

the procedure must be repeated with the tensile yield strain as constant and the

strain on the compression side varied to obtain the equilibrium of the forces

and the neutral axis location.

4.4.2 PLASTIC MOMENT RESISTANCE OF A DOUBLY SYMMETRIC I-

BEAM

The classification of a beam section according to local buckling will be

discussed in subsequent chapters, suffice to say that the test beams were

classified as class 3 sections and will therefore not reach the plastic resistance

moment before premature local buckling of the flanges and/or web. The plastic

moment resistance capacity as well as rotation capacity of Type 3CR12 steel

beams with gradual yielding material properties were not the aim of the

current study and will therefore not be discussed for the doubly symmetric !-

sections. The reader is referred to Bredenkamp, Van Den Berg and Van Der

Merwe" for a discussion and experimental results on the subject.

4.4.3 YIELD MOMENT RESISTANCE OF A SINGLY SYMMETRIC

CHANNEL SECTION

The yield moment resistance of a singly symmetric channel section of Type

3CR12 steel can not be determined by the elastic moment resistance equation

4.3 as the material tensile and compression characteristics differ. The general

bending theory approach must therefore be applied and equilibrium of

equations 4.1 and 4.2 satisfied.

72

The mathematics of the theory will first be derived for a section of constant

width equal to unity to illustrate the approach after which derivation of the

theory for channel sections will follow.

4.4.3.1 YIELD MOMENT RESISTANCE OF A RECTANGULAR BEAM

SECTION

It is concluded from the stress-strain relationships presented in chapter 3 that

the tensile yield strength and initial modulus of elasticity are lower than the

compression yield strength and initial modulus of elasticity. The compression

yield strain is therefore usually smaller than the tensile yield strain and it will

be assumed that the section reached the compression yield strain first. The

strain on the tensile side is therefore varied to obtain static equilibrium of the

forces on the cross section. The strain and stress distribution at yielding over

the cross section is shown in Figure 4.10. The location of the neutral axis and

the tensile stress needs to be determined first and are determined by the

following method.

The neutral axis location, at a distance ; from the extreme fibres on

the compression side of the beam section, can be expressed by the

principle of equal triangles :

zs e t 4.21

cc h-ce

Since the material obey Hooke's law i.e stress is proportional to strain

it follows that,

Ft Et

Y„ C

GC

Ec

4.22

Substitution of equation 4.22 into equation 4.21, the resulting equation

simplifies to equation 4.23.

73

F F (—Y ` + c h

E, Et c Ec 4.23

Equilibrium of forces on the cross section required by equation 4.2

leads to equation 4.24,

• 1 2 cc Fye b = —2

(h - cc) Ft b

which simplify to equation 4.25.

cc - ( 1

) h

1 + FY`

Ft

4.24

4.25

The tensile stress at equilibrium is determined by substitution of

equation 4.25 into 4.23 and simplifies to,

Ft = FY` E

t 4.26

The location of the neutral axis is determined by substituting equation

4.26 into 4.25 and simplifying equation 4.27.

Cc -

Et

E, h

Et 1 +

Ec

4.27

The yield moment resistance is determined by taking moments about the

neutral axis and simplifies to the following,

MY = —3

(Fr, ) +h (h - 2 cc) Ft ] 4.28

74

4.4.3.2 YIELD RESISTANCE MOMENT OF CHANNEL SECTION

The strain and stress distribution are shown in Figure 4.11. It is assumed that

the compression yield strength has been reached and the tensile stress is

smaller than the tensile yield strength and needs to be determined along with

the neutral axis location with equilibrium of the forces on the cross sectional

area satisfied. The tensile stress is expiessed in terms of the distance from the

extreme compression fibre to the neutral axis by the principle of equal

triangles and is as follows,

h - c F E F - ( C ) yc t

t cc Ec

4.29

The tensile and compression stress inside the flanges are also determined by

equal triangles and is given in equation 4.30.

cc - tf Ffr = ( ) Fyc

c h - cc t f F

Y c

Et F

c

it ) Ec ct,

4.30

The location of the neutral axis is determined by equating the tensile and

compression forces on the web and flanges, and substitution of F„ F, and F ic

as given in equations 4.29 and 4.30 and simplifies to equation 4.31.

with

-a2 J

I ay - 4 al a 3 cc

2 a l 4.31

a l = (Ec - Et) tw 4.32

a 2 = 2 t if (EC + Et ) (be + 2 Et h t i,, 4.33

a 3 = (Et + EC) (bf - - 4.34 2 h t f Et (bf - h 2

(cc - tp ) 3 tw, + trbf - 3 t fcc + 34) 414 = 3 cc

4.36

75

The tensile stress can now be determined from equation 4.29. The yield

moment resistance is now determined by taking moments of the forces on the

cross section about the neutral axis and simplifies to the following,

My = a4 4.35

with

t rbf (3h 2 - 6hcc - 3htf + 3c,2 + 3 t fcc + a 5 -

3 (h - cc ) (h - cc - t f ) 3 tw.

3 (h - cc )

4.37

The stress on the tensile side of the beam must be smaller than the tensile yield

stress as assumed. If the tensile yield stress is exceeded it follows that the

calculation must be repeated with the assumption that the section first yield on

the tension side with the stress on the compression side smaller than the

compression yield stress at equilibrium. Derivation of the equations to

determine the forces and resistance moment of the section follows that

described above. The stress on the compression side is determined by equation

4.38.

cc Ec Fc - ( ) F t

h cc Et Y 4.38

The location of the neutral axis is determined by equation 4.31 as given above.

The moment resistance moment is determined as follows,

My= "4 Fc + as- 4.39

with the constants as given in equation 4.36 and 4.37.

76

4.4.4

PLASTIC MOMENT RESISTANCE OF SINGLY SYMMETRIC

CHANNEL SECTIONS

The plastic resistance moment can not be determined by the plastic moment

resistance equation presented by paragraph 4.2, the reason being that the

tensile and compression characteristics of the material are not isotropic. The

other assumptions stated are however still applicable. The strain and stress

distribution are shown in Figure 4.12. It is assumed that all the steel fibres

over the cross sectional area has yielded and the location of the neutral axis

must be determined by satisfying equilibrium of the forces on the cross

section. Equating the tensile and compression forces over the web and flanges

leads to the neutral axis location as given in equation 4.40.

t f (t„ - be) (Fyc - Fyt ) + h t, Fyt cc = 4.40

tw (Fy, + Fyt )

The plastic moment resistance is determined by taking moments of the tensile

and compression forces about the neutral axis and simplifies to the following,

MP —

2 (a

6 Fyc + a 7 Fyt )

4.41

with

a 6 = t w c 2 + tf (be - t„) (2cc - t f ) 4.42

a 7 = ch 2 + (b1 - t„) (2h - tf ) + c, t h, ( cc - 2h - 2 tf) - 2 t rbf

4.43

4.5 CALCULATED MOMENT RESISTANCE OF THE TEST BEAM

SECTIONS

The calculated yield moment resistance values for the doubly symmetric I-test

beams are shown in Table 4.1 as well as the yield and plastic moment

77

resistance for the singly symmetric channel sections. Also shown is which side

according to the calculation of stresses will yield first, the tensile or

compression side.

It must be noted that although the beam sections are the same their moment

resistance is different as a result of their different experimental mechanical

properties and the use of each sections own profile properties presented in

Chapter 3. It must further be noted that the moment resistance values are

technically approximations as the corner fillets are ignored and the equivalent

parallel flange properties are used for the moment resistance moment of the

channel sections for simplicity.

4.6 CONCLUSION

The difference in material behaviour between that of carbon steel and Type

3CR12 steel necessitates that the moment resistance of Type 3CR12 beams be

determined by the general bending theory. The modified Ramberg-Osgood

equation which is transformed to represent the stress distribution over the cross

section. The methods described for Type 3CR12 steel beams is based on

experimental observations and provide a theoretical approach to the

determination of the yield and plastic moment resistance of Type 3CR12

beams.

78

TABLE 4.1 - MOMENT RESISTANCE OF TEST BEAMS

DOUBLY SYMMETRIC

I - SECTION

RESISTANCE MOMENT

MY (kN.m) mp (kN.m) Alp / M1

08-TI-1-NIT 123,67*

0842-I-HT 103,961 •

16-11-1-HT 104,801

24-T4-I-HT 102.00

24-11.1-HT 101,40

32-T6-I-HT 93,80

40-T7-I-HT 89.50

40rre-1-irrn 108.20 •

48-19-1-HT 91,71 1

56-110-1-HT 90,231

SINGLY SYMMETRIC

CHANNEL SECTION

RESISTANCE MOMENT

My (kN.m) Mp (kN-m) Mp I MY

06-71-C-HT 11,60 12,61 1,080

12•72-C-HT IL& 12,61 1,080

18-T3-C-HT 10.72 * 11.62 1,084

24-T4-C-HT 12,05.

13,03 1.081

30-73-C-11T 12,40 13,36 1.01

36-16C-HT , 10.56* 11,42 1,081

42-T2-C-HT 11.821 12,70 1,074

Compression side yield

Tension side yield

Moment resistance calculated with mechanical properties tested at location (a) Ind (b) thaw in Figure 3.2.

SECTION

N _C

.0

b

79

STRAIN DISTRIBUTION STRESS DISTRIBUTION

Ma NEUTRAL AXIS

TENSICN SIDE

y CCIeRESSON SIDE

TENSON:SLDE

Fy

Fy

is NEUTRAL AXIS

C

b

SECTION

FIGURE 4.1 - ELASTIC STRESS AND STRAIN DISTRIBUTION OVER CROSS SECTION

NEUTRAL AXIS

b

SECTION

INELASTIC STRAIN DISTRIBUTER' INELASTIC STRESS DISTRIBUTION

PLASTIC STRAIN DISTRIDUT ON

PLASTIC STRESS DISTRISUTIEN

FIGURE 4.2 - INELASTIC AND PLASTIC STRESS AND STRAIN DISTRIBUTION OVER CROSS SECTION

80

COMPRESSION SIDE

NEUTRAL AXIS

TENSION SIDE

.4

SECTION STRAIN DISTRIBUTION

FIGURE 4.3.a - GENERAL STRAIN DISTRIBUTION OVER CROSS SECTION

F (COMPRESSION)

Fl

F2

e t

F3

F4 e 4

F (TENSION)

FIGURE 4.3.b - GENERAL STRESS-STRAIN RELATIONSHIP

Mp o

.

FL

C

F2

F3

T

COMPRESSION SIDE

NEUTRAL AXIS

TENSION SIDE

81

F4

STRESS DISTRIBUTION

FIGURE 4.3.c - STRESS DISTRIBUTION OVER CROSS SECTION AND RESULTANT FORCES


FIGURE 4.4 - STRAIN GAUGE LOCATION OVER CROSS SECTION

OUTSIDE FACE OF FLANGE - TENSION

STRAIN DISTRIBUTION AT MOMENT

1,58 kN.m --- 30,43 kN.m - - - 51,67 kN.m

70,58 kN.m 90,47 kN.m

97,31 kN.m 100,08 kN.m

_ CENTROIDAL AXIS LOCATION

OUTSIDE FACE OF FLANGE - COMPRESSION

SE

CTI

ON

DE

PTH

10

-20

-30

40

50

60

70

80

90

-100

-110

110

100

90

8D

70

60

50

40

30

20

10

0

82

-0.5 -0.4 -0.3

-0.2 -0. 1 0.0

0.0

0.2

0.3

0.4

STRAIN (%)

FIGURE 4.5 - STRAIN DISTRIBUTION OVER CROSS SECTION: BEAM

- C CENTROIDAL AXIS LOCATION

OUTSIDE FACE OF FLANGE

- TENSION

STRAIN DISTRIBUTION AT MOMENT

1,52 kN.m 30,55 kN.m 50,37 kN.m 70,10 kN.m

100,90 kN.m 115,57 kN.m 123,39 kN.m

1

I I I

I N .

I N.N I 1

1

OUTSIDE FACE OF FLANGE

- COMPRESSION

SEC

TION

DE

PT

H (

mm

)

-10

20

-30

40

-50

60

70

80

-90

-100

-110

110

100

90

80

70

60

50

40

30

20

10

0

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.0 0.2 0.3 0.4 05

83

STRAI N (%)

FIGURE 4.6 - STRAIN DISTRIBUTION OVER CROSS SECTION: BEAM 08-T1-I-NHT

SE

CTI

ON

DE

PT

H (

mm

)

100

110

10

20

-30

40

50

60

-70

80

-90

110

100

80

90

70

50

40

20

60

30

10

0


ii

IN

%•.

%. •

\

CENTRO1DAL AXIS LOCATION

STRESS DISTRIBUTION AT MOMENT

1,58 kN.m

--- 30,43 kN.m • - - 51,67 kN.m

70,58 kN.m 90,47 kN.m

97,31 k.N.m 100,10 kN.m

.‘


84

-500 -4-00

-300

-200

-100

0 100

200

300

400

500

STRESS (MPa)

FIGURE 4.7 - STRESS DISTRIBUTION OVER CROSS SECTION: BEAM 08-T2-I-HT

SE

CTI

ON

DE

PTH

(m

m)

50

60

70

80

90

-100

-110

30

10

20

-4D

110

100

90

80

70

40

30

60

50

20

10

0


STRESS DISTRIBUTION AT MOMENT — 1,52 kN.m --- 30,55 kN.m • - - - 5D,37 kN.m 70,10 kN.m 100,90 kN.m

115,57 kN.m 123,39 kN.rn

L

•

CENTROIDAL AXIS LOCATION

\

1 •


85

-500 -400 -300 -200 -100

0

100

200

300

400

500

600

STRESS (MPa)

FIGURE 4.8 - STRESS DISTRIBUTION OVER CROSS SECTION: BEAM 08-TI-I-NHT

et MM _ •f

t IMENgra=1 t .

IENSIDI SIDE

MIUTRAL AXIS

C 0 IMRESSIEN SIDE

If

.c

b

ec COMPRESSION SIDE Fyc

NEUTRAL AXIS

by

86

SEMEN STRAIN DISTRIBUT ION

A'

MSS Aw

AP

EMI A"

TENSION SIDE

ICUTRAL AX IS

CEMPRESSI ON SIDE

STRESS DISTRIBUTION

FIGURE 4.9 - STRESS AND STRAIN DISTRIBUTION OVER 3CR12 BEAM CROSS SECTION

SECTION

STRAIN DISTRIBUTION

STRESS DISTRIBUTION

FIGURE 4.10 - YIELD STRESS AND STRAIN DISTRIBUTION OVER SECTION OF UNIT WIDTH

Fyc

NEUTRAL AXIS

EQUIVALENT CHANNEL SECTION STRAIN DISTRIBUTION STRESS DISTRIBUTION

TENSION SIDE Fyt b

Cs.

87

t„

bF

°c CONPRESSIEN SIX Fyc

FFc

NEUTRAL AXIS

TENSION SIDE F

EQUIVALENT [HAMEL SECTION

STRAIN DISTRIBUTION

STRESS DISTRIBUTION

FIGURE 4.11 - YIELD STRESS AND STRAIN DISTRIBUTION OVER 3CR12 CHANNEL SECTION

FIGURE 4.12 - PLASTIC STRESS AND STRAIN DISTRIBUTION OVER 3CR12 CHANNEL SECTION

88

CHAPTER 5

THEORY OF LATERAL TORSIONAL BUCKLING

5.1 INTRODUCTION

The behaviour of short and slender beams which is controlled by inelastic and

elastic lateral torsional buckling will be discussed in this chapter. The relevant

theory is derived and the South African Design specification approach is

discussed as well as methods by Galambos'n' and Nethercot and Trahair 22 to

determine the critical lateral torsional buckling moments.

5.2 ELASTIC LATERAL TORSIONAL BUCKLING

The critical lateral torsional buckling moment will be derived for a beam with

equal end moments causing single curvature as it constitutes the most critical

loading behaviour. The derivation is according to the theory by Galambos n

with the following assumptions:

the beam material behave elastically,

the beam is geometrically perfect,

deflections are small,

the profile of the section remain in tact, no distortion occur.

5.2.1 DOUBLY SYMMETRIC CROSS SECTION

The behaviour of an open section beam element subject to end bending

moments and a compression force as shown in Figure 5.1, with an assumed

small initial out of plane deformation, can be described by the following three

differential equations,

89

Bx + -4 [ Mby (Mty Mby) ±Px01 =

Mbx ( Mbx)

By U ll + P u - [ Mbx - (Mtx + Mbx ) - P yo ) 1 =

- M + (M + M ) by L (Pity by

and

- (C, v i [ Mbx (Mbx Mtx )

+ P yo - V [ Mby — (Mby + Mby ) + P xo

- L(Pity mby ) - L (frftx fribx ) = 0

where u = deflection of shear centre in the x - direction

v = deflection of shear centre in the y - direction

z = direction along the longitudinal axis of the beam

L = beam length

= angle of rotation of the cross section

M, = bending moments as shown in Figure 5.1

B x , B, = bending stiffness in the x - and y - direction

P = axial force on beam

C, = St. Venant torsional stiffness

C. = warping stiffness

K 1 = cross sectional constant

The bending case of interest is however that shown in Figure 5.2. Equations

5.1, 5.2 and 5.3 simplify to the following,

Bx = - Mx , 5 . 4

B u ~~ + Mx 4 = 0

5 . 5

5 . 1

5.2

5 . 3

+ - (Ct + + - (

m Mu = 0

where

B x = E lx

= E I,

= G 1

C = E

K, = cross sectional constant

It is clear from inspection that equation 5.4 is independent of equation 5.5 and

5.6. Equation 5.4 represent the case of the normal theory of bending in the y -

z plane.

The lateral torsional buckling theory follows from equations 5.5 and 5.6,

which are dependant on each other. Differentiating both equations twice with

respect to z and setting the end moments equal with opposite directions lead

to the following differential equations,

EI u 1111 + Mt, = 0

5 . 7

E 41111 - G J 4" + Mo u ll = 0 5 . 8

The cross section constant K, is equal to zero because the cross section is

doubly symmetric. The boundary conditions for a simply supported beam is

as follows,

lateral deflection boundary conditions:

u(0) = u(L) = 0(0) = u"(L) = 0 5 . 9

warping boundary conditions:

90

5 . 6

91

5(0) = +(L) = 4"(0) = 4 11 (L) = 0 5.10

Integrating equation 5.7 twice and applying the boundary conditions result in

the following expression for the second derivative of the lateral deflection,

u II — —

5.11 E I

Substitution of equation 5.11 into equation 5.8 and simplifying lead to the

following forth order homogenous differential equation,

4) //// _ 4y/ _ A.2 0

where

_ G J 1 E

MZ 1 2

5.12

5.13

E 2 Iy I,,,

The solution of the differential equation is determined by assuming a function

that represents the solution as follows,

= e" 5.14

Equation 5.14 and derivatives of this function are substituted into equation

5.12 and leads to the following characteristic equation.

a 4 - A 1 a 2 - A 2 = 0

4.15

The roots of the equation above are as follows,

1 1 + sP.1 +4 1 2 5.16 c4 1,2 = f 2

where

= real roots

i.e,

= C1 cosh(a 1 z) + C2 sinh(a l z) + C3 cos (a 1 z) + C4 sin(a 3 z)

5.21

92

and

where

\I A l - + 4 1 2 a 3,4 = 2

c4 3.4 = complex roots

5.17

The general solution of the real part of the complete solution of equation 5.12

is given in equation 5.18.

= C1 cosh (a1 z) + C2 sinh (al z) 5.18

where

C„ C, = constants

The general solution of the complex part of the complete solution of equation

5.12 is given in equation 5.19.

432 = C3 cos (17 3 z) + C4 sin(a 3 z) 5.19

where

C„ C, = constants

The complete solution is given in equation 5.20 and 5.21.

= .1 4) 2 5.20

Applying the boundary conditions given in equation 5.10 to equation 5.20

leads to a system of equations with unknowns CI, C2, C 3 , and C4 . These

equations are homogenous and there exist two solutions namely, (i) the trivial

solution and (ii) the non-trivial solution.

93

(i) Trivial solution:

C, = C 2 = C, = C, = 0

This solution has clearly no use as it just assign values equal to zero to and

u.

) Non-Trivial solution:

This solution only exist if the determinant of the system of equations is equal

to zero. The characteristic determinant is set equal to zero and simplifies to

equation 5.22.

a 3 ) sinh(a l L) sin(a 3 L) = 0

5.22

Equation 5.22 will be satisfied if sin(a, L) = 0. Setting the other two parts of

equation 5.22 equal to zero leads to the trivial solution. The roots of the sine

functions are therefore solutions of equation 5.22 and are as follows,

5.23

where

j = 1, 2, 3,

The critical buckling moment will be at the lowest root, i.e when j is equal to

one. Substitution of equation 5.17 into equation 5.23 and simplification leads

to the critical lateral torsional buckling equation.

The elastic critical lateral torsional buckling moment is then determined as

given in equation 5.24.

where

94

E 7r 2 CrM = L „ EI G J (1 +

EC,,,)

G J L 2 5.24

(h - t f) 2 CG,- I" - _Ty

4 (approximate warping constant for symmetric I-sections)-

5.2.2 ASYMMETRIC CROSS SECTION

The critical lateral torsional buckling moment for an asymmetric cross section

is also derived from equations 5.5 and 5.6 which are as follows,

By ull +Mx4 = 0 5.25

M + M ca — (c, K1 ) erh' bfx u" - ( tx L bx) u - o 5.26

and it was shown by Galambosy that the cross sectional constant K, is equal

to the following,

K1 = Mx ct x 5.27

where

f y (x 2 + y 2 ) dA px - 2 yo

Ix 5.28

Differentiating equation 5.25 once and equation5.26 twice with respect to z,

and setting the end moments equal and applied in opposite directions as shown

in Figure 5.3 leads to the following differential equations that describe the

asymmetrical beam section lateral torsional buckling behaviour as follows,

95

E I Li" + Mo 4)" = 0

5.29

E Ch"" — J + Mo px) 4>" mo U " = 0

5.30

Equation 5.29 is the same as equation 5.7. Integration thereof and introducing

the simple boundary conditions of equations 5.9 and 5.10 again results in

equation 5.11. Substitution thereof into equation 5.30 leads to the following

homogenous equation,

_ _ 2 =0

5.31

where

G J + Mo 13 x =

E A

2 -

mo2

5.32

E 2 r y (.)

By the same mathematical solution process previously demonstrated it follows

that the solution is that given in equation 5.33.

CC 3 = L 5.33

where

j = 1, 2, 3, ... and a, as defined by equation 5.17

The critical lateral torsional buckling moment will be the lowest when j is

equal to one. Substitution of a, and simplifying leads to the following

expression for the critical elastic lateral torsional buckling moment for beams

with an asymmetric sectional profile,

2 L 2 Y 1 ± 1+ ( p4x2 EG2 LTE 11_2y - 5.34

E 2 E I 13 x

Equation 5.34 reduce to equation 5.24 for the doubly symmetric case when B

is equal to zero. This special case also accounts for channel section as 13, is

equal to zero for channels sections.

MCI

96

5.3 INELASTIC LATERAL TORSIONAL BUCKLING

The elastic lateral torsional buckling equations are only applicable if the

stresses at all locations in the beam are in the elastic region. This case will

only exist for beams with a high slenderness ratio. Beams with a low

slenderness ratio will buckle when the stress at highly stressed locations has

exceeded the yield strength of the material. When exceeding the yield strength

at a local section of the beam, it results in the decrease of the stiffness

properties B,,, C„ and C. at that location. For the case of sharp yielding steel,

these properties decrease because yielding of the beam material leads to a

smaller core of elastic material that is able to resist the lateral and torsional

buckling. These stiffnesses are determined by considering two cases. The one

case involves the stress distribution condition that exists just before buckling

and the other case that condition that exists just after buckling as discussed by

Galambosn .

These two cases are presented graphically in Figure 5.4. The cross section

positions of a rectangular beam section are shown as well as the stress

distribution over the cross section and the loading and unloading in the stress-

strain curve. For the first case it is assumed that buckling is just about to

occur. The hatched areas indicate that parts of the cross section where the

yield strength are exceeded and therefore the material no longer posses the

ability to provide any resistance to an increase in strain because the modulus

of elasticity is zero above the yield strength. The stiffness of the section is

therefore only determined by the elastic core of material. For the second case

it is assumed that the beam has buckled. Buckling of the section reduces the

strain in some of the yielded material and therefore reduces the stress to below

the yield strength due to elastic unloading of the steel as shown in Figure 5.4.

The stiffnesses that resist buckling therefore increase due to the increase in the

elastic material core.

The latter approach of determining the stiffnesses that resist buckling will

97

therefore yield larger critical buckling moments than that of the first approach.

This led to the tangent modulus approach and the reduced modulus approach

to determining the critical buckling moments. The tangent modulus approach

is based on the assumption that the unloading of the material follows the same

path of loading. The reduced modulus is based on elastic unloading of the

material as discussed above and therefore leads to higher stiffnesses and

critical buckling moments. Shanley' made the following conclusions regarding

the inelastic bifurcation-type buckling problem presented above:

The critical buckling load determined by the tangent modulus approach

is always smaller than that determined by the reduced modulus

approach.

For an initial straight beam there exists no lateral deformation before

the critical tangent modulus buckling moment is exceeded.

An increase in moment above the tangent modulus buckling moment

may lead to lateral deformations and an increased maximum moment

resistance.

The reduced modulus buckling moment is an upper bound to the

tangent modulus moment and the maximum moment resistance.

The conclusions made by Shanley" are illustrated graphically in Figure 5.5

which shows the relationship between the critical moment and the variables,

lateral deflection and twisting of the section. Figure 5.5 also illustrate the two

approaches with the moment versus length of the beam relationship.

The tangent modulus approach provides therefore a lower bound solution to

the critical buckling moment and is therefore preferred because it yields

conservative estimates. The process of determining the critical buckling

moment by the tangent modulus approach for sharp yielding material will

therefore in general be as follows.

The critical buckling moment is determined according to the elastic

98

torsional buckling theory as already discussed. When the material of

the cross section yields it is assumed not to be effective in resisting the

applied moment. The stiffness properties B,,, C„ C., and K, are

determined for the elastic material core taken into account that the

shear centre has also changed position. The differential equations for

lateral torsional buckling are now solved with the new stiffness

properties.

The method is however tedious and complicated because of the presence of

residual stresses. A numerical process must also followed for the case where

the moment varies over the length of the beam.

The approach for beams of material which are of the gradual yielding type is

similar but with the distinct difference that even though the material yield

strength may be exceeded there will still be resistance provided by the so

called yielded material as the stress strain curve gradually increases after

yielding. The approach therefore will be to determine the stress induced by an

applied moment and to determine the tangent modulus which will be equal to

the initial elastic modulus for stresses below the proportional limit. The

reduction of bending and torsional stiffnesses therefore depends on the elastic-

and shear modulus and not on an effective elastic cross sectional core of

material. Subjecting a beam to a moment gradient will however again lead to

a numerical process as yielding of the cross section differ over the length of

the beam.

5.4 DESIGN FOR LATERAL TORSIONAL BUCKLING

The approach of design codes is to determine the critical buckling moment

with the elastic lateral torsional buckling equation for slender beams and

approximating the buckling moment with a transitional equation for stocky

beams and beams of intermediate slenderness which fail by inelastic buckling.

The design equations given by the South African Design Specification" which

99

is based on the Canadian Design Specification" are as follows,

for doubly symmetric

when Ma

Mr = 1,15 cp

when Ma

class 1 and class 2 sections:

> 0,67 M,, (Inelastic buckling)

Mp (1 - 0,28 m

P Mp 5.35 Mcr

< 0,67 Mp (Elastic buckling)

Mr = P Mcr 5.36

for doubly symmetric class 3 section and channel sections:

when M„ > 0,67 M y (Inelastic buckling)

Mr = 1,15 p My (1 - 0,28 MY s 5.37 My Mcr

when M„ < 0,67 My (Elastic buckling)

Mr = P Mcr 5.38

where

n Mcr - 2 E ) ( n 2 5.39

L Ky Y Kz L

and

= 1,75 + 1,05 K + 0,3 le- G 2,5

K = ratio of smaller end moment to the larger end moment, positive

for double curvature and negative for single curvature

K y, K, = effective length factor for lateral bending and torsional

fixity respectively

100

The 0,67 factor incorporate the effect of residual stresses which based on

research" can be as much as one third of the yield strength of the material

without residual stress. The ce, factor incorporate the moment distribution over

the length of the beam and is equal to one for uniform bending. The effective

length factors incorporate the end restraint conditions of the beam segment

under consideration. Effective length factors's." are tabulated for different

support conditions for which the lateral torsional buckling equations can not

be solved in closed form.

5.5 MOMENT RESISTANCE OF LATERAL CONTINUOUS

DETERMINATE BEAMS

The effective length factors for lateral continuous determinate beams are

usually taken as one when designing the beam segments between lateral

supports. A simple hand calculation method according to Galambos' s and

Nethercot, Trahairn can however be used to take the stiffening effect of the

segments adjacent to the critical beam segment into account. The method is as

follows.

Elastic lateral buckling:

Calculate the properties El y, al, EL.

The in plane bending moments are calculated.

The elastic critical buckling moment and load of the beam

segments between lateral supports are determined from the

elastic buckling equation for effective length factors of one.

The critical segment is indicated by the lowest critical buckling

moment.

The effective lateral stiffnesses for the critical beam segment is

determined as given in equation 5.40,

101

= 2 E y 5.40

L

and the effective lateral stiffness factors for the adjacent

sections, a, and a r , are determined as given in equation 5.41.

a n E Y ( 1

)Pm l , r

, r P1, r

5.41

where n = 2 if far end of beam is continuous

n = 3 if far end of beam is pinned

n = 4 if far end of beam is fixed

The different cases for adjacent sections are shown

schematically in Figure 5.6. The term (1-1)./P) takes into

account the reduced stiffness provided by the adjacent sections

because of the moments applied to them as well.

Calculate the stiffness ratios for the adjacent sides as shown in

equation 5.42.

a a= = =n , Gr a1 ar 5.42

Determine the effective length factor k = K , = ICz from the

column nomograph for non-sway columns.

With the effective length factor calculate the critical buckling

moment and load again.

Inelastic lateral buckling:

The approach is similar to the elastic method except that the

critical buckling moment and load are calculated with the

transitional equations as given in paragraph 5.4 above, and the

stiffness of the critical beam segment is calculated as follows,

102

2 E Iy Mr 5.43

L

MCI

and the stiffness of the adjacent sections as given in equation

5.44.

• n E

aer T Y ( 1 l r

Pm ) Mr

131,r Mcr 5.44

where n = 2 if far end of beam is continuous

n = 3 if far end of beam is pinned

n = 4 if far end of beam is fixed

The extra term M r/M„ takes into account the reduced stiffness

as a result of yielding of beam material at highly stressed

regions.

5.6 DESIGN OF TYPE 3CR12 BEAM SECTIONS FOR LATERAL

TORSIONAL BUCKLING

The critical buckling moment will be determined with the elastic critical

buckling equation 5.24 with the exception that the stiffness of the beam will

be reduced according to the tangent modulus and the appropriate shear

modulus.

Tangent Modulus

The tangent modulus is determined by differentiating the modified Ramberg-

Osgood equation with respect to stress and setting the inverse equal to the

tangent modulus as follows,

Er - F E y o

F + 0, 002 n E, (—FF

5.45

103

Shear Modulus

The initial shear modulus is determined from elastic material relationships as

shown by Popov' to be as given in equation 5.46.

G — o 2(1 + v )

S.46

and the shear modulus with the variation of the elastic modulus as follows,

Gt = Gc, Et

5.47

The tangent shear modulus is thus determined by multiplying the initial shear

modulus with a plasticity reduction factor as suggested by Galambos".

5.7 DESIGN OF ASYMMETRICAL BEAM SECTIONS

The approach for designing asymmetrical sections will be similar to those

discussed above with the exception that the elastic critical lateral torsional

buckling equation is that derived for asymmetrical sections and is given in

equation 5.34.

5.8 CONCLUSIONS

The elastic an inelastic lateral torsional buckling of beams are theoretically

investigated. Equations are derived in order to describe the buckling behaviour

and different design approaches are presented and discussed. These critical

buckling moments estimated by these design approaches will be compared with

experimental data in Chapter 6.

104

1 z

FIGURE 5.1 - GENERAL LOAD ARRANGEMENT

105

FIGURE 5.2 — SIMPLIFIED LOAD CASE

106

-J

FIGURE 5.3 - EXPERIMENTAL LOAD CASE

107

Mx'

MATERIAL UNLOADING ELASTICALLY

REGION OF YIELDED MATERIAL

POSITION OF SECTION JUST BEFORE BUCKLING

POSITION OF SECTION\ AFTER BUCKLING

INELASTIC STRESS DISTRIBUTION

= 0 GPa E = 0 GPa F y

E = Eo

ELASTIC UNLOADING PATH

e

FIGURE 5.4.a - BEFORE AND AFTER BUCKLING CONDITIONS OF A

YIELDED CROSS SECTION

LOADING AND UNLOADING PATH IN THE STRESS-STRAIN DIAGRAM

FIGURE 5.4.13 - LOADING AND UNLOADING OF A MATERIAL FIBRE

Mo

REDUCED MODULUS MOMENT

MAXIMUM MOMENT STRENGTH .

TANGENT MODULUS MOMENT

108

Mo, -"1-,\ MO

42 L

u, PHI

FIGURE 5.5.a - MOMENT-LATERAL DEFORMATION CURVE IN THE

INELASTIC RANGE

TANGENT MODULUS MOMENT

MAXIMUM MOMENT STRENGTH

REDUCED MODULUS MOMENT

ELASTIC MOMENT

Mo

Mp

Frc My(1 Fy )

L

109

FIGURE 5.5.b - BUCKLING CURVES IN THE ELASTIC AND INELASTIC

REGION

L,

ADJACENT SEGMENT PROVIDE EQUAL END RESTRAINT

ADJACENT SEGMENT HINGED AT FAR END

ADJACENT SEGMENT FIXED AT FAR END

FIGURE 5.6 - ADJACENT BEAM SEGMENT CASES

110

CHAPTER 6

EXPERIMENTAL BEAM TE TS

6.1 INTRODUCTION

The full scale beam tests are discussed in this chapter. Beams with variable

slenderness values were tested to evaluate their lateral torsional buckling

behaviour and to compare the experimental results with current design

specification equations as well as the tangent modulus approach to estimate the

critical buckling moment.

6.2 PRELIMINARY EXPERIMENTAL PLANNING

The beams of Type 3CR12 corrosion resisting steel were hot-rolled to produce .

sections similar to their standard South African carbon steel counterparts. The

sections tested were the 203x133x25 kg/m I-section and the 100x50 DIN taper

flange channel. The beams were also heat treated to minimize the effect of

residual stress on the experimental results.

6.2.1 DOUBLY SYMMETRIC I-BEAMS

Different beam lengths were chosen to vary the slenderness ratio, defined as

the ratio of the beam length to radius of gyration (L/r,), from 26 to 185. This

range of the slenderness parameter was chosen to obtain experimental data of

the structural bending behaviour in the three regions as shown in Figure 2.2

namely the plastic-, inelastic- and elastic lateral torsional buckling. Table 6.1

shows the exact beam lengths and the discrete values of the slenderness

parameter Lir,.

The width to thickness ratio of the section plate elements of each beam are

shown in Table 6.2. According to these ratios the beam sections are classified

111

as class 3 sections with the capacity to develop the yield moment before local

buckling occurs. The yield moment may however only be reached if the beam

slenderness, support conditions and moment gradient are such that lateral

torsional buckling does not occur prematurely. The test beam 08-T1-1-NHT

falls into class 4 and according to the code limits should not reach the yield

moment before the onset of local buckling.

6.2.2 SINGLY SYMMETRIC CHANNEL SECTIONS

The beam lengths were chosen to vary the slenderness ratio, Ur,,, from 41 to

295. These lengths were also chosen to obtain experimental data of the

structural behaviour of the beams in the plastic-, inelastic- and elastic lateral

torsional buckling regions of this monosymmetric section. Table 6.3 shows the

discrete beam lengths chosen as well as the corresponding slenderness values.

Classification of the sections are also shown in Table 6.4. The channel sections

are classified as class 3. The SA88-0162 1 ' hot-rolled design code does not

permit channels to be classified as a class 1 or class 2 section.

6.3 EXPERIMENTAL BEAM TESTS

6.3.1 PHYSICAL BEAM TEST SETUP

The beam test setup is shown in Figure 6.1 along with the in plane moment

distribution which is the aim of the test setup. The test setup consists of a one

span test beam with cantilevers on both sides when viewed in the vertical

bending plane. The cantilevers serve as moment lever arms in order to subject

the test length to equal and opposite_ moments between the intermediate

supports. A constant moment therefore exist over the test section.

The beam is supported by a pin and roller system in the vertical plane. The

test setup forms a lateral continuous beam with three spans in the horizontal

112

plane i.e when viewed in plan. The in-plan intermediate supports are knife

edged which prevent lateral movement as well as twisting of the beam section.

The cantilever end tips are also restrained from lateral movement and twisting

but are free regarding vertical in-plane movement in order to apply the

concentrated load to the cantilevers. The details are shown in Figure 6.2. The

cantilever lengths are also listed in Table 6.1 and Table 6.3 for the beams

tested.

The load on the cantilever ends is applied by dividing the force of the

hydraulic controlled load cell with a 457x191x67 loading I-beam and

subjecting the test beam to it through a pin and roller system also shown in

Figure 6.1.

Figure 6.3 and 6.4 show the real laboratory beam setup for beam 08-T1-I-

NEIT and testing thereof in progress. Figure 6.5 to 6.7 show the testing of

beam 56-T10-I-HT. These successive figures clearly exhibit the lateral

torsional buckling of the beam. Lateral torsional buckling of the bottom flange

which is in compression and the twisting of the unrestrained section between

the knife edged supports are clearly visible. Figure 6.8 shows the roller

connection at which the force on the cantilever end tip is applied. Figure 6.9

shows the roller bearing system utilized to prevent lateral movement and

twisting of the cantilever end tip but still allow vertical movement to the

applied concentrated force on the cantilever end tip. Figure 6.10 shows the

testing of beam 12-T2-C-HT in progress and Figure 6.11 shows a plan view

of the same beam test. The lateral buckling of the bottom flange which is in

compression is clearly visible when compared to the upper top flange which

is in tension.

6.3.2 EXPERIMENTAL DATA RECORDED

The purpose of this chapter is to experimentally determine the critical lateral

torsional buckling moment of the beams tested. The critical buckling moments

113

were determined by two methods which are as follows:

METHOD A: Strain gauges were attached to the compression flange

tip/tips. The positioning of the strain gauges for the 1-

beams and the channel sections are shown in Figure

6.12. Bending of a beam subject all of the strain gauges

to normal compressive strain/stress. Lateral buckling of

the beam increase the compression strain measured by

the strain gauges positioned on the concave side of the

buckled beam. The strain gauges on the convex side are

subjected to tension strain which lowers the compression

strain which was recorded up to the point where

buckling commenced and a complete or partial strain

reversal occur. This peak strain/stress reversal indicate

the maximum moment of resistance of the test beam

which is taken as the experimental critical buckling

moment.

METHOD B: This method which was used by O'hEachteirn and

Nethercot" on their tests of monosymmetric plate

girders, involves the setup of three displacement

transducers at the midspan of the test beam and

connected to the test beam as shown in Figure 6.13. By

recording the initial starting lengths LI, L2, L3 and the

change thereof with the transducers as the beam test

progress, the vertical and lateral deflection of the

midspan cross section can be determined with simple

geometrical mathematics as shown in Figure 6.14. The

twisting of the cross section when the compression

flange buckles laterally and the cross section rotates can

also be determined. Comparing the applied moment to

lateral movement of the cross section indicates the

114

critical moment at which a distinctive increase in lateral

movement due to buckling of the compression flange

takes place.

6.4 EXPERIMENTAL BEAM TEST RESULTS


Figure 6.15 to Figure 6.24 shows plots of the applied moment versus flange

tip strain for the I-section test beams. Figure 6.25 shows a plot of the applied

moment versus lateral deflection of the section centroid and Figure 6.26 shows

a plot of the applied moment versus the midspan twist of the cross section.

The maximum moments that were experimentally obtained from these data

comparisons are shown in Table 6.5. Table 6.5 also shows the theoretically

estimated buckling moments. The theoretical moments were calculated by

following three approaches as follows:

The South African Design Specification" equations were used in the

method described by Galambos is , Nethercot and Trahairn, which takes

the stiffening effect of the adjacent spans into account.

By calculating the moment resistance with the South African Design

Specification" equations and appropriate effective length factors to

account for the lateral continuity of the beam. The support conditions

at the intermediate supports are approximated as simply supported for

bending i.e K Y = 1,0 and fixed for warping due to the lateral

continuity of the beam i.e Ki = 0,5. The effective length factor

approach is however an approximation because the lateral continuity of

the beam does not provide a full fixity against warping that is assumed.

The tangent modulus approach whereby the bending and warping

stiffnesses are reduced according to the tangent modulus for Type

115

3CR12 steel as well as the corresponding shear modulus.

Figure 6.27 shows the graphical comparison of the theoretically calculated

critical buckling moments and the experimentally obtained moments versus the

effective beam lengths.


Graphical representation of the experimentally applied moments versus

measured strain of the compression flange for the channel sections are shown

by Figures 6.28 to 6.34. Figures 6.35 to 6.41 shows the applied moment

versus lateral deflection of the compression flange.

The moment versus lateral buckling of the compression flange plots indicate

distinctive critical moments at which premature buckling of the beams occur.

The experimental ultimate and critical buckling moments are listed in Table

6.6. The theoretical critical moments were calculated by the first two

approaches as described for the 1-beam tests. A graphical representation of the

critical buckling moments versus the effective beam lengths, is shown in

Figure 6.42.

6.5 DISCUSSION OF EXPERIMENTAL AND THEORETICAL RESULTS


The moment versus flange tip strain in Figures 6.15 to 6.24 shows smooth

curves with the strain reversal process as already discussed above, and the

critical moments at peak strain reversal. The moment versus lateral deflection

in Figure 6.25 also shows the maximum moment of resistance of the beams

at which the curves develop a plato with no increase of strength with the rapid

increase of lateral deflection. It must be noted that these curves were

normalised with zero as the initial deflection. This was done because the initial

116

out of plane straightness of the test beams was unfortunately not measured.

The moment versus twist plots in Figure 6.26 do however show the initial

twist angles and the changing thereof with the onset of lateral torsional

buckling.

All of the curves discussed above indicates agreement of the maximum

experimental moments with no premature buckling behaviour at lower moment

values. The maximum experimental moment resistances are plotted against the

effective beam lengths as shown in Figure 6.27.

The following conclusions follow from the moment versus effective length

plot:

The theoretical approach of the design specification taking the

stiffening effect of the adjacent lateral continuous spans into account is

very conservative, especially at an effective beam length of 2400 mm

and longer.

The theoretical approach with the use of effective length factors agree

well with the experimental data at high slenderness but becomes

unconservative at short beam lengths. This is due to the fact that the

effective length factors were derived for fully fixed supports to restrain

warping. The test beams are laterally continuous and therefore only

partially fixed.

The tangent modulus approach estimate the critical moments accurately

with high slenderness beams and becomes conservative at low

slenderness ratios.

The theoretical and experimental moments agree well within limits. It must

however be noted that the beams tested were heat treated and therefore

behaves structurally well in accordance to the theories developed for sharp

117

yielding carbon steel beams.


The moment versus flange tip strain plots in Figures 6.28 to 6.34 show that

all the test beams reached their yield moment resistance. The strain gauge

positioned at the web to flange junction shows a definite change in strain

behaviour of the material at a lower moment than the maximum reached. The

plots of the applied moment versus lateral deflection of the compression flange

clearly shows premature buckling for beams with high slenderness ratios after

which the moment resistance increase to an maximum equal to the yield

moment resistance. These premature buckling moments were obtained by

straight line curves fitted to the experimental data above and below the

premature buckling moment. The intersection of these straight lines is taken

as the critical buckling moments.

Figure 6.42 shows the experimental buckling moments and theoretical

estimated moments versus the effective beam test lengths from which the

following conclusions are made:

The theoretical approaches using effective length factors and including

the stiffening effect of the adjacent sections tends to over estimate the

experimental critical moments. The theoretical moments resistance

factor of 0,9 required when designing beams in practice will however

provide for a safe design moment.

6.6 CONCLUSIONS

The theoretical estimates of the critical buckling moments agree well with the experimental

moments. From the symmetric I-beam estimates as shown in Figure 6.27 it is clear that the

tangent modulus approach provide a very accurate theoretical method of estimating the critical

buckling moments. The current design specification equations could however still be used

118

because the resistance factors applied when designing beams in practice will still provide safe

and even conservative estimates.

The theoretical estimates using the design specification equations might however not be so

impressive when critical buckling moments are estimated for beams that were not heat

treated. The tangent modulus approach which directly depends on the material stress-strain

behaviour should however provide good estimates for this case, but should be experimentally

verified.

119

TABLE 6.1 - DOUBLY SYMMETRIC I - TEST BEAM LENGTHS AND SLENDERNESS

TEST BEAM

No.

TEST BEAM

LENGTH (mm)

CANTILEVER

MOMENT ARM (mm)

SLENDERNESS

RATIO L / ry

08-T1-I-NHT 800 1630 26,34

08-T2-I-HT 800 1630 26,26

16-T3-I-HT 1600 1390 52,64

24-T4-I-HT 2400 1250 79,13

24-T5-I-HT 2400 1250 78,79

32-T6-I-HT 3200 1160 105,28

40-17-I-HT 4000 1370 131,23

40-T8-I-HT 4000 1370 132,06

48-T9-I-HT 4800 1130 158,42

56-T10-I-HT 5600 960 184,36

120

TABLE 6.2 - CLASSIFICATION OF DOUBLY SYMMETRIC I - TEST BEAM SECTIONS

WIDTH-TO-THICKNESS RATIOS

TEST BEAM

No.

FLANGE WEB

ACTUAL CODE LIMIT ACTUAL CODE LIMIT

1 -, --. - - -- - 1

I linn/JF --w - -w

08-T1-I-NHT 9,11 CLASS 4 32,06 47,15

08-T2-I-HT 9,05 9,52 32,64 52,36

16-T3-I-HT 9,18 9,43 32,40 51,86

24-T4-I-HT 9,15 9,54 32,48 52,45

24-T5-I-HT 8,92 9,75 31,55 53,63

32-T6-I-HT 9,15 9,99 31,97 54,96

40-T7-I-HT 9,06 10,29 32,46 56,60

40-T8-I-HT 8,91 9,39 31,86 51,68

48-T9-I-HT 9,12 10,16 32,61 55,89

56-T10-I-HT 9,15 10,17 32,28 55,91

200/

121

TABLE 6.3 -

SINGLY SYMMETRIC CHANNEL TEST BEAM LENGTHS AND

SLENDERNESS

TEST BEAM

No.

TEST BEAM

LENGTH (mm)

CANTILEVER

MOMENT ARM (mm)

SLENDERNESS

RATIO L / ry

06-T1-C-HT 600 480 41,58

12-T2-C-HT 1200 620 83,16

18-T3-C-HT 1800 840 123,14

24-T4-C-HT 2400 560 161,20

30-T5-C-HT 3000 490 211,05

36-T6-C-HT 3600 420 249,29

42-T7-C-HT 4200 330 294,94

122

TABLE 6.4 - CLASSIFICATION OF SINGLY SYMMETRIC CHANNEL TEST BEAM

SECTIONS

I WIDTH-TO-THICKNESS RATIOS

TEST BEAM

No.

FLANGE WEB

ACTUAL CODE LIMIT ACTUAL CODE LIMIT

1 -. -. I

-- _ ___.. _ y

06-T1-C-HT 6,21 11,33 17,92 62,31

12-T2-C-HT 6,21 11,33 17,92 62,31

18-T3-C-HT 5,89 12,11 16,23 66,61

24-T4-C-HT 6,02 11,52 16,57 63,36

30-T5-C-HT 6,08 11,00 16,78 60,46

36-T6-C-HT 5,90 12,12 16,47 66,66

42-T7-C-HT 6,08 11,35 17,13 63,20

200/JF 1100/JF

123

TABLE 6.5 - EXPERIMENTAL AND THEORETICALLY ESTIMATED CRITICAL

BUCKLING MOMENTS OF DOUBLY SYMMETRIC I - TEST BEAMS

TEST BEAM

No.

EXPERIMENTAL

TEST BEAM

MOMENT

- Me

(kN.m)

THEORETICAL

MOMENT :

EFFECTIVE

LENGTH

APPROACH

- Mte

(kN.m)

THEORETICAL

MOMENT :

SABS CODE -

APPROACH

- Mtc

(kN.m)

THEORETICAL

MOMENT :

TANGENT

MODULUS

APPROACH

- Mt

(kN.m)

18-T1-1-NHT 130,73 119,52 119,52 139,81

08-T2-I-HT 103,18 97,05 97,05 96,47

16-T3-I-HT 94,38 97,84 97,84 91,47

24-T4-I-HT 74,25 95,34 87,18 84,74

24-T5-I-HT 89,63 94,67 87,20 83,76

32-T6-I-HT 78,15 81,34 70,81 77,98

40-T7-I-HT 64,12 70,10 60,13 71,31

40-T8-I-HT 68,06 72,97 62,31 74,58

48-T9-I-HT 63,17 61,15 50,18 63,20

16-TIO-I-HT 50,88 47,91 40,47 48,50

124

UILE 6.6 -

EXPERIMENTAL AND THEORETICALLY ESTIMATED CRITICAL BUCKLING

MOMENTS OF SINGLY SYMMETRIC CHANNEL TEST BEAMS

PEST BEAM

No.

EXPERIMENTAL

ULTIMATE

TEST BEAM

MOMENT

- Mu

(kN.m)

THEORETICAL

MOMENT :

EFFECTIVE

LENGTH

APPROACH

- Mte

(kN.m)

THEORETICAL

MOMENT :

SABS CODE -

APPROACH

- Mtc

(kN.m)

EXPERIMENTAL

CRITICAL

BUCKLING

MOMENT

- Me

(kN.m)

06-TI-C-HT 13,15 12,36 12,36 13,15

12-112-C-HT 13,27 12,36 12,36 13,27

18-T3-C-HT 11,31 11,74 11,37 11,31

24-T4-C-HT 13,22 11,94 11,66 13,22

30-T5-C-HT 13,13 10,13 10,09 9,26

36-T6-C-HT 11,47 8,57 8,61 8,63

42-T7-C-HT 12,10 7,17 7,31 6,35

125 LOAD A

PPLIED BY

HYDRAULIC CON

TROL

LED

LOAD

CEL

L

CANTILEVERED MOM

ENT LEVER ARMS

MO

ME

NT

DIS

TR

IBU

TIO

N O

VE

R T

ES

T B

EA

M

ti

Cl

Li

TE

ST

BE

AM

SE

TU

P

BEAM TES

T LENGTH

at

ti

O

t

0

1N3HOW

FIGURE 6.1 - TEST BEAM SETUP AND MOMENT DISTRIBUTION

.6)

0 0

em ! 0 0'

126

CANTILEVER END TIP RESTRAINT

INTERMEDIATE KNIFE EDGE SUPPORTS

FIGURE 6.2 - SUPPORT AND RESTRAINT DETAILS OF TEST BEAM SETUP

STRAIN GAUGE

l■

1

132

■I

COMPRESSION FLANGE COMPRESSION FLANGE

FIGURE 6.12 - STRAIN GAUGE POSITIONING

FIGURE 6.3 - SETUP OF TEST BEAM 08-T1-I-NHT

FIGURE 6.4 - TESTING OF BEAM 08-TI-I-NHT IN PROGRESS

FIGURE 6.5 - SETUP OF TEST BEAM 56-T10-I-HT

FIGURE 6.6 - TESTING OF BEAM 56-T10-I-HT IN PROGRESS. LATERAL BUCKLING OF COMPRESSION FLANGE VISIBLE

FIGURE 6.7 - TESTING OF BEAM 56-T10-I-HT IN PROGRESS. LATERALTORSIONAL BUCKLING VISIBLE.

FIGURE6.8 - ROLLER SYSTEM AT ,.‘ LOAD APPLICATION POSITION.

- 1, 1

FIGURE 6.9 - ROLLER BEARING RESTRAINT SYSTEM AT CANTILEVERED MOMENT LEVER ARM END TIPS.

FIGURE 6.10 - TESTING OF BEAM 12-T2-C-HT IN PROGRESS

FIGURE 6.11 - PLAN VIEW OF TEST BEAM 12-T2-C-HT. LATERAL BUCKLING OF COMPRESSION FLANGE VISIBLE

COLUMNS OF FRAME SUPPORTING HYDRAULIC CONTROLLED LOAD CELL

STIFF PLATE FITTED WITH SMALL PULLY'S

TEST BEAM

CABLES

WEIGHT

DISPLACEMENT TRANSDUCER "; ,</„‘A,644;4. ,44,417,rAft.41=2:17W4WeiteSttIMM.Y.W,XtleX.,

133

FIGURE 6.13 - EXPERIMENTAL DISPLACEMENT TRANSDUCER SETUP

134

= 12

CO

S 1

90°-

(B

ETA

tLA

mB

DA

) I

111, =

13 S

IN (

BE

TA

)

W =

13

CO

S (

BE

TA

)

r.

LAM

BD

A =

AC

OS

lL B

ETA

= A

CO

S

• A

(1) (,)

( BE

TA+L

Am

BD

A)

SIN

190

° -

(BE

TA

+LA

MB

DA)

= t 1

2 S

IN 19

0° - (B

ETA

-t-LA

MB

DA

)I

FIGURE 6.14 (a) - GEOMETRICAL LATERAL AND VERTICAL RELATIONSHIPS

BC

= L6

CO

T (

PH

I')

IF P

HI"

>

PHI' T

HE

N

U, =

t4

• B

C S

IN (

PHI "

*PH

I')

CO

S (P

HI"

*P

III'

)

01

4-

10

>s

135

FIGURE 6.14 (b) - GEOMETRICAL LATERAL AND VERTICAL RELATIONSHIPS

136

N

< I- <

0_

1±. a_

n 1-...4 1 cL \ -/

V)

0

N _1

4-

=

II

D

n

I' iii ¶ cL NJ \ -/

Z (/)

V) D

N _1 N

_I + +

bn

II II

>e. Da

/Th.

r CL N.-/

Z I—I V)

N —I

+

u

II

> 1+1 V)

Z L.L.I M I—

D

\/

D

Cu D°

V

FIGURE 6.14 (c) - GEOMETRICAL LATERAL AND VERTICAL RELATIONSHIPS

•

=

"

PHI

=P

HI *

PH

I'

EC

• L6 C

OS

EC

(PH

I')

AC

= L7 C

UT

(PH

I')

1.1;

t A

C SIN

(PH

I)

S'

>6

L6

COT

(PH

I')

z

V U,=

U',,

BC

SIN

(PH

I)

n

137

FIGURE 6.14 (d) - GEOMETRICAL LATERAL AND VERTICAL RELATIONSHIPS

MO

MEN

T (

kN.m

)

-20

100

140

120

80

60

40

20

0.5 0.0 0.5 1.0 1.5

CONVEX CONCAVE

138

COMPRESSION FLANGE TIP STRAIN

FIGURE 6.15 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 08-TI-I-NUT


FIGURE 6.16 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 08212-I-HT

1.5 20 -20

-0.5 0.0 0.5 1.0

MO

ME

NT

(kN

.m)

40

20

- 1.0 -0.5 0.0 0.5

1.0

1.5

20

100

80

60

40

20

0


FIGURE 6.17 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 16-T3-1-11T

80

60

0

20


FIGURE 6.18 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 24-T4-I-HT

MO

ME

NT

(kN

.m)

139

CONVEX CONCAVE

100

80

60

MO

ME

NT

(k

N.m

)

40

20

20

100

80

60

MO

ME

NT

(kN

.m)

40

20

0

-20

140

-0.5 0.0 0.5

1.0 1 5


FIGURE 6.19 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 24-T5-1-1-1T

CONVEX

CONCAVE

-0.5

0.0

0.5

1.0

15


FIGURE 6.20 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 32-T6-1-11T

141

-1.0

-0.5

0.0 0.5

1.0

15

70

60

50

40

30

0 n

70

O

10


FIGURE 6.21 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 40-T74-11T

80

60

MO

ME

NT

(kN

.m)

40

20

0

-20 -0.4 -0.2 0.0 0.2 0.4

0.6

08


FIGURE 6.22 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 40-T8-1-1-IT

CONVEX CONCAVE

142

70

60

50

40

30 z

O 20

10

-10 -1.0 -0.5

0.0 0.5

1.0

15


FIGURE 6.23 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 48-T9-I-HT

60

50

40

MO

ME

NT

(kN

.m)

30

20

10

0

-10 -0.6 -0.4 -0.2 0.0 0.2 0.4

0.6

08


FIGURE 6.24 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 56-TI0-I-HT

MO

ME

NT

(kN

.m)

120

100

80

60

40

20

MO

ME

NT

(kN

.m)

100

80

60

40

20

120

0

143

0

5 10 15 20

25

30

LATERAL DEFLECTION OF CENTROID (mm)

—5

0

5 10 15 20

25

30

LATERAL DEFLECTION OF CENTROID (mm)

FIGURE 6.25 - MOMENT vs. LATERAL DEFLECTION OF DOUBLY SYMMETRIC I-BEAMS

08-T2-1- HT

16-T3-1- HT

24 - T5 -I-HT

120

100 -

80 -

60 Ui

0 40 -

20 -

0 -

32-T6 -I-HT

24 -T4 -I-HT

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20

0.25 0.30

MIDSPAN CROSS SECTION TWIST (rad.)

120

100

80

60

40

20

0

z

0

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20

0.25

0.30

MIDSPAN CROSS SECTION TWIST (rad.)

FIGURE 6.26 - MOMENT. vs. TWIST OF DOUBLY SYMMETRIC I-BEAMS

144

120

100

80

60 z

0

145

40

20

— THEORETICAL cuncAL MOMENT — Mte

---- TANGENT MODULUS MOMENT — Mt

EXPERIMENTAL MAXIMUM MOMENT — Me

THEORETICAL CRITICAL MOMENT — Mtc

0 0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

EFFECTIVE BEAM TEST LENGTH (mm)

FIGURE 6.27 - EXPERIMENTAL AND THEORETICAL CRITICAL BUCKLING MOMENTS vs.

EFFECTIVE LENGTH OF DOUBLY SYMMETRIC I-BEAMS

FIGURE 6.29 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 12-T2-C-HT

2 -0.5 2.5 3 0 0.0 0.5 1.0 1.5 2.0

14

12

10

2

2 0 5

2 -0.5 0.0 0.5 1.0 1.5 2 0


FIGURE 6.28 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 06-TI-C-UT

MO

ME

NT

(k

N.m

)

14

12

10

8

6

4

2

0

, COMPRESSION FLANGE TIP STRAIN

146

- STRAIN AT WEB-FLANGE JUNCTION --- STRAIN AT FLANGE TIP - - - STRAIN AT FLANGE TIP

- STRAIN AT WEB-FLANGE JUNCTION --- STRAIN AT FLANGE TIP - - - STRAIN AT FLANGE TIP

- STRAIN AT WEB-FLANGE JUNCTION --- STRAIN AT FLANGE TIP. • - - STRAIN AT FLANGE TIP

14

12

10

B

6

4 MO

MEN

T (

k N.m

)

2

O

2

14

12

1 0

5

6

4 MO

MEN

T (

kN.m

)

2

0

-2

147

rrkin;r-,rtve:=- -a- y

- STRAIN AT WEB-FLANGE JUNCTION --- STRAIN AT FLANGE TIP - - STRAIN AT FLANGE TIP

-0.2 0.0 0.2 0.4 0.6

0.8

1 0


FIGURE 6.30 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 18-T3-C-IIT

-0.5 0.0 0.5 1.0

1.5

20



-0.4 -0.2 0.0 0.2 0.4 0.6 08

14

1 2

10

8 -

6 -

- STRAIN AT WEB-FLANGE JUNCTION --- STRAIN AT FLANGE TIP • - - STRAIN AT FLANGE TIP

4

2

-

2

MO

ME

NT

(kN

.m)

06 2 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

0.5

14

12

10

- STRAIN AT WEB-FLANGE JUNCTION --- STRAIN AT FLANGE TIP - - STRAIN AT FLANGE TIP

2

0

MO

MEN

T (

kN.m

) 8

6

4

148


FIGURE 6.32 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 30-T5-C-IIT



STRAIN AT WEB—FLANGE JUNCTION --- STRAIN AT FLANGE TIP - - STRAIN AT FLANGE TIP

149

14

12

10

8

1— 6 z w

O 4

2

0

2 —0.2 0.0 0.2 0.4

0.6

08


FIGURE 6.34 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 42-17-C-HT

14

12

10

MO

MEN

T (

kN.m

) 8

6

4

2

0

— 2 —5 0 5 10 15 20

25

30

LATERAL DEFLECTION of COMPRESSION FLANGE (mm)

FIGURE 6.35 - MOMENT vs. LATERAL DEFLECTION OF COMPRESSION FLANGE OF BEAM 06-T1-C-HT

150

MO

ME

NT

(kN

.m)

14

12

10

a

6

4

2

0

2 —10 —5 0 5 10 15 20

LATERAL DEFLECTION of COMPRESSION FLANGE ( mm)

25 30


14

12

10

MO

ME

NT

(kN

.m)

4

'

—4

0 4 8 12 16 20 24 28


2


151

-<---- INTERSECTION MOMENT = 9.26 kN.m LATERAL DEFLECTION = 1.60mm

— EXPERIMENTAL DATA - LINEAR REGRESSION CURVE FIT

= 0.113085•M + 0.555653 - - - LINEAR REGRESSION CURVE FIT

u = 1.318403•M - 11.71575

MO

ME

NT

- M

(kN

.m)

- 2

14

12

10

6

14

12

10

8

6

4

2

0

2 -4 0 4 8 12 16 20 24

28



MO

MEN

T (

kN.m

)

-2

0

2 4 6 8

10

12

14

LATERAL DEFLECTION (4 of COMPRESSION FLANGE (mm)


10 -

INTERSECTION MOMENT = 8.63 kN.m LATERAL DEFLECTION = 0.68m m

— EXPERIMENTAL DATA -- LINEAR REGRESSION CURVE FIT

u = 0.167562*M - 0.764149 - - LINEAR REGRESSION CURVE FIT • u = 1.463101.'1%4 - 11.93869

INTERSECTION MOMENT = 6.35kN.m LATERAL DEFLECTION = -0.523mm

— EXPERIMENTAL DATA --- LINEAR REGRESSION CURVE FIT

-0.100436:PM 4- 0.113997 - - LINEAR REGRESSION CURVE FIT

u = 0.18106«M - 1.672404

14

12

10

MO

ME

NT

- M

(kN

.m)

152

14

12 -

0 2 4 6 8 10 12 14

LATERAL DEFLECTION (u) of COMPRESSION FLANGE (mm)


-2

0

2 4

6

LATERAL DEFLECTION (u) of COMPRESSION FLANGE (mm)

MO

MEN

T -

M (

kN.m

)


153

14

• • •

MO

ME

NT

(kN

.m)

12

10

8

6

•

4

THEORETICAL CRITICAL MOMENT — Mte -- MEAN MAXIMUM EXPERIMENTAL MOMENT — Mu

EXPERIMENTAL CRITICAL BUCKING MOMENT — Me o THEORETICAL CRITICAL MOMENT — Mtc

2

0

0 500 1000 1500 2000 2500 3000

3500

4000

4500

EFFECTIVE BEAM TEST LENGTH (mm)

FIGURE 6.42 - EXPERIMENTAL AND THEORETICAL CRITICAL BUCKLING MOMENTS vs.

EFFECTIVE LENGTH OF SINGLY SYMMETRIC CHANNEL BEAMS

155

used to determine the moment resistances which leads to an iterative

procedure. This is necessary because the stress-strain relationship is non linear

and the Material is anisotropic.

The theoretical background on elastic and inelastic lateral torsional buckling

were presented and discussed in Chapter 5. The tangent modulus approach to

determine the critical buckling moments of Type 3CR12 steel beams was also

presented. Theoretical methods to account for the lateral continuity of the test

beams were also discussed.

The beams tests were discussed in Chapter 6. The theoretically and

experimental buckling moments were compared and it was concluded that the

tangent modulus approach to determine the buckling moments is an accurate

method that could be used to determine the buckling moments of Type 3CR12

beams.

7.3 FUTURE INVESTIGATIONS

This investigation led to topics that need further investigation. The following

topics are recommended for further study.

The lateral torsional buckling behaviour of beams that was not heat

treated. Steel beams coming directly from the rolling steel mill are

normally used in construction. These beams contain high residual

stresses and there influence on the bending behaviour needs to be

investigated.

An more comprehensive investigation on the lateral torsional buckling

of monosymmetric sections is needed, especially for plate girder

sections where the top and bottom flange width is different.

154

CHAPTER 7

CONCLUSIONS AND SUMMARY

7.1 GENERAL REMARKS

The purpose of this investigation was to compare experimental data on the

lateral torsional buckling behaviour of Type 3CR12 steel beams and theoretical

critical buckling moments.

It was concluded in Chapter 6 that the theoretical equations of the South

African Design Specification'', with the use of tabulated effective length

factors or the effective length factors determined by the method by Galambos"

and Nethercot and Trahair 22 , provided reasonable estimates of the critical

buckling moments. The tangent modulus moments were however the most

accurate and will in general be the best to use for beams that were not heat

treated.

7.2 SUMMARY OF RESEARCH

A literature investigation was presented in Chapter 2. The mechanical

behaviour of Type 3CR12 steel was investigated and a method derived to

analytically represent the stress-strain relationship. An introduction on the

bending behaviour of beams was also given regarding the plastic, inelastic and

elastic regions.

The experimental mechanical properties were presented in Chapter 3. It was

also shown that the stress-strain relationship was best represented by the . - modified Ramberg-Osgood equation' s ''.

In Chapter 4 methods were derived to determine the yield and plastic moment

resistance of Type 3CR12 steel beams. The general bending theory must be

156

REFERENCES

Van Vlack L H. Materials for Engineering - Concepts and Applications. Addison Wesley Publishing Company, 1982.

Avner S H. Introduction to Physical Metallurgy. McGraw Hill International Book Company, 1974.

Van Den Berg G J; Van Der Merwe P. Stainless Steel in Structural Applications. Rand Afrikaans University. Internal Report No. MD-75, December 1992.

Van Den Berg G J. The Torsional Flexural Buckling Strength of Cold Formed Stainless Steel Columns. D.Eng. Thesis. Rand Afrikaans University, 1988.

Van Den Berg G J; Van Der Merwe P. Design Criteria for Stainless Steel Structural Members. Rand Afrikaans University. internal Report No. MD-70, June 1992.

Osgood W R. Stress Strain Formulas. Journal of the Aeronautical Sciences, January 1946.

Ramberg; Walter; Osgood W R. Description of Stress Strain Curves by Three Parameters. N.A.C.A Technical Note No. 902, 1943.

Hill H N. Determination of Stress Strain Relations from "offset" Yield Strength Values. N.A.C.A Technical Note No. 927, October 1943.

Van Der Merwe P. Development of Design Criteria for Ferritic Stainless Steel Members and Connections. Ph.D Thesis, University of Missourri-Rolla, 1987.

Kulak; Adams; Gilmor. Limit States Design in Structural Steel. Canadian Institute of Steel Construction, Fourth Edition, December 1990.

Kulak E L; Holtz N M. Web Slenderness Limits for Compact Beams. Structural Engineering Report No. 43. Department of Civil Engineering, University of Alberta Canada, March 1973.

Kulak E L; Holtz N M. Web Slenderness Limits for Non-Compact Beams. Structural Engineering Report No. 51. Department of Civil Engineering, University of Alberta Canada, August 1975.

American Society for Testing and Materials. Standard Methods and Definitions for Mechanical Testing of Steel Products. ASTM A370-77, Annual Book of ASTM Standards, 1981.

American Society of Testing and Materials. ASTM E9-70, 1981.

157

Galambos T V. Guide to Stability Design Criteria for Metal Structures. 4th Edition, John Wiley & Sons, 1988.

Parks M B; Yu W W. Design of Automotive Structural Components using High Strength Sheet Steel. Civil Engineering Study 83-3, Second Progress Report, University of Missourri-Rolla, August 1983.

South African Bureau of Standards. The Structural Use of Steel, SABS 0162-93 part 1: Limit States Design of Hot Rolled Steelwork. Pretoria, South Africa.

Yu W W. Cold Formed Steel Design. John Wiley & Sons, 1985.

Popov E P. Mechanics of Materials. Prentice Hall, Second Edition, 1978.

Bredenkamp P J; Van Den Berg G J; Van Der Merwe P. The Plastic Strength of Stainless Steel Beams. Rand Afrikaans University, 1993.

Galambos T V. Structural Members and Frames. Prentice Hall, 1968.

Nethercot D A; Trahair N S. Inelastic Lateral Buckling of Determinate Beams. Journal of the Structural Division, April 1976.

Shanley F R. Inelastic Column Theory. Journal of Aeronautical Science, Volume 14, No. 5, May 1947.

Canadian Standards Association. Steel Structures for Buildings (Limit States Design), CSA-S16.1 M89 84b. Ontario, Canada.

Galambos T V. Inelastic Lateral Buckling of Beams, Journal of the Structural Division. October 1963.

Chen W F; Lui E M. Structural Stability - Theory and Implementation. Elsevier Science Publishing Co. Inc., 1987.

Galambos T V. As Discussed at Rand Afrikaans University. September 1993.

O'hEachteirn P; Nethercot D A. Lateral Buckling Tests on Monosymmetric Plate Girders. Journal of Constructional Steel Research, 1988.

the elastic and inelastic lateral torsional buckling …

Documents