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
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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
2.1 General Remarks 4
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
3.1 General Remarks 19
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
6.4.1 Doubly Symmetric I-Beams 114
6.4.2 Singly Symmetric Channel Sections 115
6.5 Discussion of Experimental and Theoretical Results 115
6.5.1 Doubly Symmetric I-Beams 115
6.5.2 Singly Symmetric Channel Sections 117
6.6 Conclusions 117
v i
7. CONCLUSIONS AND SUMMARY
7.1 General Remarks 154
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
3.3 Mechanical Properties of Test Beam 16-T3-I-HT 28
3.4 Mechanical Properties of Test Beam 24-14-I-HT 29
3.5 Mechanical Properties of Test Beam 24-15-I-HT 30
3.6 Mechanical Properties of Test Beam 32-16-I-HT 31
3.7 Mechanical Properties of Test Beam 40-77-1-1-IT 32
3.8 Mechanical Properties of Test Beam 40-T8-I-HT 33
3.9 Mechanical Properties of Test Beam 40-18-I-HT 34
3.10 Mechanical Properties of Test Beam 40-T8-I-HT 35
3.11 Mechanical Properties of Test Beam 48-T9-I-HT 36
3.12 Mechanical Properties of Test Beam 56-T10-I-HT 37
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
3.15 Mechanical Property Ratios of Test Beam 16-T3-I-HT 39
3.16 Mechanical Property Ratios of Test Beam 24-T4-I-HT 39
3.17 Mechanical Property Ratios of Test Beam 24-T5-I-HT 40
3.18 Mechanical Property Ratios of Test Beam 32-T6-I-HT 40
3.19 Mechanical Property Ratios of Test Beam 40-T7-I-HT 41
3.20 Mechanical Property Ratios of Test Beam 40-T8-I-HT 41
3.21 Mechanical Property Ratios of Test Beam 40-T8-I-HT 42
3.22 Mechanical Property Ratios of Test Beam 48-T9-I-HT 43
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
3.25 Mechanical Properties of Test Beam 12-T2-C-HT 45
3.26 Mechanical Properties of Test Beam 18-T3-C-HT 46
3.27 Mechanical Properties of Test Beam 24-T4-C-HT 47
3.28 Mechanical Properties of Test Beam 30-T5-C-HT 48
3.29 Mechanical Properties of Test Beam 36-T6-C-HT 49
3.30 Mechanical Properties of Test Beam 42-T7-C-HT 50
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
3.5 Analytical and Experimental Stress-Strain Relationship of
Beam 08-T2-I-FIT 56
3.6 Analytical and Experimental Stress-Strain Relationship of
Beam 16-T3-I-HT 57
, 3.7 Analytical and Experimental Stress-Strain Relationship of
Beam 24-T4-I-HT 57
3.8 Analytical and Experimental Stress-Strain Relationship of
Beam 24-15-I-HT 58
3.9 Analytical and Experimental Stress-Strain Relationship of
Beam 32-T6-I-HT 58
3.10 Analytical and Experimental Stress-Strain Relationship of
Beam 40-T7-I-HT 59
3.11 Analytical and Experimental Stress-Strain Relationship of
Beam 40-T8-I-HT 59
3.12 Analytical and Experimental Stress-Strain Relationship of
Beam 40-T8-I-HT 60
3.13 Analytical and Experimental Stress-Strain Relationship of
Beam 48-19-I-HT 61
3.14 Analytical and Experimental Stress-Strain Relationship of
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
6.18 Moment vs. Flange Tip Strain of Beam 24-T4-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
6.21 Moment vs. Flange Tip Strain of Beam 40-T7-I-HT
6.22 Moment vs. Flange Tip Strain of Beam 40-T8-1-HT
6.23 Moment vs. Flange Tip Strain of Beam 48-T9-I-HT
6.24 Moment vs. Flange Tip Strain of Beam 56-T10-I-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
6.30 Moment vs. Flange Tip Strain of Beam 18-T3-C-HT 147
6.31 Moment vs. Flange Tip Strain of Beam 24-T4-C-HT 147
6.32 Moment vs. Flange Tip Strain of Beam 30-T5-C-HT 148
6.33 Moment vs. Flange Tip Strain of Beam 36-T6-C-HT 148
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
6.36 Moment vs. Lateral Deflection of Compression Flange of
Beam 12-T2-C-HT 150
6.37 Moment vs. Lateral Deflection of Compression Flange of
Beam 18-113-C-FIT 150
6.38 Moment vs. Lateral Deflection of Compression Flange of
Beam 24-T4-C-HT 151
6.39 Moment vs. Lateral Deflection of Compression Flange of
Beam 30-T5-C-HT 151
6.40 Moment vs. Lateral Deflection of Compression Flange of
Beam 36-T6-C-HT 152
6.41 Moment vs. Lateral Deflection of Compression Flange of
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
TABLE 3.6 - MECHANICAL PROPERTIES OF TEST BEAM 32-T6-I-HT
(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
COEFFICIENT OF VARIATION (%)
TEST SPECIMEN
Fl
F2
F3
F4
WI
W2
MEAN
STANDARD DEVIATION
COEFFICIENT OF VARIATION (%)
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
TABLE 3.7 - MECHANICAL PROPERTIES OF TEST BEAM 40-T7-I-HT
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
STANDARD DEVIATION 1,5 3,6 7,2
COEFFICIENT OF VARIATION (%) 0,71 0,99 1,80
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
STANDARD DEVIATION 2,7 17,1 6,1 2,8
COEFFICIENT OF VARIATION (%) 1,36 4,84 1,57 0,54
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
COEFFICIENT OF VARIATION (%)
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
TABLE 3.9 - MECHANICAL PROPERTIES OF TEST BEAM 40-T8-I-HT
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
STANDARD DEVIATION 4,2 21,7 13,3 10,9
COEFFICIENT OF VARIATION (%) 2,12 5,64 3,02 2,01
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
STANDARD DEVIATION 4,1 6,6 10,8
COEFFICIENT OF VARIATION (%) 1,92 1,92 2,79
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
STANDARD DEVIATION 4,4 11,0 16,9 10,4
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
STANDARD DEVIATION 4,8 7,3 4,1
COEFFICIENT OF VARIATION (%) 2,25 2,11 1,05
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
STANDARD DEVIATION 4,2 10,1 3,7 1,77
COEFFICIENT OF VARIATION (%) 2,12 2,87 0,99 0,35
• : , :•: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
COEFFICIENT OF VARIATION (%) 3,79 6,20 0,77 10,48
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
TABLE 3.15 - MECHANICAL PROPERTY RATIO'S OF TEST BEAM 16-T3-I-HT
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
COEFFICIENT OF VARIATION (%) 3,56 5,46 3,42 8,47
TABLE 3.16 - MECHANICAL PROPERTY RATIO'S OF TEST BEAM 24-T4-I-HT
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
STANDARD DEVIATION 0,02 0,01 0,02 1,22
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 3.17 - MECHANICAL PROPERTY RATIO'S OF TEST BEAM 24-T5-I-HT
TABLE 3.18 - MECHANICAL PROPERTY RATIO'S OF TEST BEAM 32-T64-11T
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
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
COMPRESSION FLANGE TIP STRAIN
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
COMPRESSION FLANGE TIP STRAIN
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
COMPRESSION FLANGE TIP STRAIN
FIGURE 6.31 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 24-T4-C-HT
-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
COMPRESSION FLANGE TIP STRAIN
FIGURE 6.32 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 30-T5-C-IIT
COMPRESSION FLANGE TIP STRAIN
FIGURE 6.33 - MOMENT vs. FLANGE TIP STRAIN OF BEAM 36-T6-C-HT
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
COMPRESSION FLANGE TIP STRAIN
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
FIGURE 6.36 - MOMENT vs. LATERAL DEFLECTION OF COMPRESSION FLANGE OF BEAM 12-T2-C-HT
14
12
10
MO
ME
NT
(kN
.m)
4
'
—4
0 4 8 12 16 20 24 28
LATERAL DEFLECTION of COMPRESSION FLANGE (mm)
2
FIGURE 6.37 - MOMENT vs. LATERAL DEFLECTION OF COMPRESSION FLANGE OF BEAM 18-T3-C-HT
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
LATERAL DEFLECTION of COMPRESSION FLANGE (mm)
FIGURE 6.38 - MOMENT vs. LATERAL DEFLECTION OF COMPRESSION FLANGE OF BEAM 24-T4-C-HT
MO
MEN
T (
kN.m
)
-2
0
2 4 6 8
10
12
14
LATERAL DEFLECTION (4 of COMPRESSION FLANGE (mm)
FIGURE 6.39 - MOMENT vs. LATERAL DEFLECTION OF COMPRESSION FLANGE OF BEAM 30-T5-C-HT
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)
FIGURE 6.40 - MOMENT vs. LATERAL DEFLECTION OF COMPRESSION FLANGE OF BEAM 36-T6-C-HT
-2
0
2 4
6
LATERAL DEFLECTION (u) of COMPRESSION FLANGE (mm)
MO
MEN
T -
M (
kN.m
)
FIGURE 6.41 - MOMENT vs. LATERAL DEFLECTION OF COMPRESSION FLANGE OF BEAM 42-T7-C-HT
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
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