i
COMBINED FLEXURAL AND CABLE?LIKE BEHAVIOR OF
DUCTILE STEEL BEAMS
by
Yunus Alp
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
December 18, 2009
Keywords: steel beams, progressive collapse, cable action,
catenary action, ductile behavior
Copyright 2009 by Yunus Alp
Approved by
Hassan H. Abbas, Chair, Assistant Professor of Civil Engineering
James S. Davidson, Associate Professor of Civil Engineering
Mary L. Hughes, Instructor of Civil Engineering
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ABSTRACT
In addressing the propensity of steel building structures to experience progressive
collapse due to extreme loading conditions (e.g., blast), current design guidelines propose
the use of a threat-independent approach that is commonly referred to in the literature as
?the missing column scenario?. Under this scenario, a column from a given story is
assumed to be removed and the resulting structure is analyzed to determine if it could
sustain the loads by activating one or more alternate load carrying mechanisms, with the
idea of mitigating the potential for progressive structural collapse. This study specifically
focuses on the ability of ductile steel beams to carry loads by transitioning from flexural
behavior to cable-like behavior. Theoretical fundamentals of this behavior are described
for rectangular and W-shaped steel beams with idealized boundary conditions and
presumed fully ductile behavior. Two theoretical analysis approaches are used to model
the beam behavior: rigid-plastic analysis and cable analysis. The main factors affecting
the behavior, such as material and geometric properties as well as boundary conditions
are described and corroborating nonlinear finite element (FE) analyses are presented and
compared to the theoretical results. Open System for Earthquake Engineering Simulation,
OpenSees was used in the FE analysis studies. Based upon theoretical and FE analysis
results, a set of equations are proposed that can be used to predict the deflection at the
onset of pure cable behavior.
Additionally, the effect of elastic boundary restraints on the beam behavior was
studied using FE analysis. An approach to evaluate the boundary restraints offered by the
iii
surrounding members in a given frame is also presented. It is shown that axial restraints
have a much more significant effect on the behavior than rotational restraints.
The theory presented in this thesis can serve as basis for designing ductile steel
beams undergoing transition from flexural to cable-like behavior.
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ACKNOWLEDGMENTS
I would first like to express my gratitude to Dr. Hassan H. Abbas for giving me
the chance to work with him. I am thankful for his continuous support, invaluable
guidance, inspiration and patience throughout the course of my studies at Auburn
University. I would like to thank my committee members Dr. Mary L. Hughes and Dr.
James S. Davidson for their time and support. I would also like to thank other Structures
Faculty members for providing valuable coursework during my study.
I take this opportunity to convey my additional thanks to all my family members
for their endless love and support.
Finally, my special thanks go to all my friends who made my stay in Auburn to be
an unforgettable experience.
v
Style manual used The Chicago Manual of Style
Computer Software Used Microsoft Word, Microsoft Excel, Microsoft PowerPoint,
AutoCAD 2008, OpenSees
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TABLE OF CONTENTS
LIST OF TABLES............................................................................................................. xi
LIST OF FIGURES .......................................................................................................... xii
Chapter 1. INTRODUCTION ...................................................................................... 1
1.1 Overview............................................................................................................. 1
1.2 Motivation........................................................................................................... 4
1.3 Research Objectives and Scope .......................................................................... 4
1.4 Approach............................................................................................................. 5
1.5 Organization of Thesis........................................................................................ 5
1.6 Notation............................................................................................................... 6
Chapter 2. BACKGROUND ........................................................................................ 7
2.1 Introduction......................................................................................................... 7
2.2 Existing Methodologies for Design against Progressive Collapse ..................... 8
2.2.1 Indirect Method........................................................................................... 9
2.2.2 Direct Methods.......................................................................................... 15
2.2.2.1 Specific Local Resistance (SLR) .......................................................... 15
2.2.2.2 Alternate Path Method (APM).............................................................. 16
2.3 Recent Relevant Research................................................................................. 21
2.4 Static Plastic Behavior of Simple Beams.......................................................... 25
2.4.1 Rigid?Plastic Behavior ............................................................................. 25
vii
2.4.2 Effect of Finite Displacements.................................................................. 26
2.4.3 M?N Interaction at Cross-Sectional Level................................................ 27
2.4.4 Equilibrium Equations .............................................................................. 30
2.4.5 Deflected Shape Configuration................................................................. 31
2.4.6 Load?Deflection ....................................................................................... 32
Chapter 3. THEORETICAL MODELS ..................................................................... 39
3.1 Overview and Scope ......................................................................................... 39
3.2 Rigid?Plastic Analysis...................................................................................... 39
3.2.1 Simply Supported Beam with a Concentrated Load at Midspan.............. 43
3.2.2 Fully Fixed Beam with a Concentrated Load at Midspan ........................ 51
3.2.3 Fully Fixed Beam with Uniformly Distributed Load along the Length ... 59
3.2.3.1 Beam with Rectangular Cross?Section................................................. 61
3.2.3.1.1 Theory I........................................................................................... 61
3.2.3.1.2 Theory II ......................................................................................... 65
3.2.3.2 Beam with W?Shaped Cross?Section .................................................. 74
3.2.3.2.1 Theory I........................................................................................... 74
3.2.3.2.2 Theory II ......................................................................................... 76
3.2.4 Summary................................................................................................... 81
3.3 Cable Analysis .................................................................................................. 82
3.3.1 Concentrated Load Case ........................................................................... 83
3.3.2 Uniformly Distributed Load Case............................................................. 85
3.3.3 Summary................................................................................................... 88
3.4 Conclusion and Expected Behavior.................................................................. 89
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Chapter 4. ANALYTICAL MODELS ....................................................................... 92
4.1 Introduction....................................................................................................... 92
4.2 Modeling Concepts ........................................................................................... 92
4.2.1 Fiber Section Discretization...................................................................... 92
4.2.2 Corotational Transformation..................................................................... 93
4.2.3 Selection of Beam Element....................................................................... 94
4.2.4 Material..................................................................................................... 94
4.2.5 Loading and Analysis ............................................................................... 95
4.3 Preliminary FE Analyses .................................................................................. 96
4.3.1 Approach................................................................................................... 96
4.3.2 Description of Preliminary FE Models..................................................... 98
4.3.3 Comparison of Results with Rigid?Plastic Theory................................... 99
4.3.3.1 Concentrated Load Case ..................................................................... 100
4.3.3.1.1 Beam with Rectangular Cross?Section......................................... 100
4.3.3.1.2 Beam with W?Shape Cross?Section ............................................ 103
4.3.3.2 Distributed Load Case......................................................................... 107
4.3.3.2.1 Beam with Rectangular Cross?Section......................................... 107
4.3.3.2.2 Beam with W?Shape Cross?Section ............................................ 112
4.3.4 Comparison of Results with Cable Theory............................................. 117
4.3.4.1 Concentrated Load Case ..................................................................... 118
4.3.4.2 Distributed Load Case......................................................................... 121
4.3.5 Summary and Conclusion....................................................................... 125
4.4 Parametric Study............................................................................................. 126
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4.4.1 Approach................................................................................................. 127
4.4.2 Description of Beam Models .................................................................. 129
4.4.3 Concentrated Load Case ......................................................................... 129
4.4.3.1 Constant ?p .......................................................................................... 129
4.4.3.2 Constant ?p / d..................................................................................... 132
4.4.3.3 Constant ?p / 2L ................................................................................... 142
4.4.3.4 Constant L / d...................................................................................... 149
4.4.4 Distributed Load Case............................................................................. 156
4.4.5 Summary and Conclusions ..................................................................... 162
Chapter 5. EQUATIONS FOR THE ONSET OF PURE CABLE BEHAVIOR ..... 164
5.1 Overview......................................................................................................... 164
5.2 Concentrated Load Case ................................................................................. 165
5.2.1 Formulation I .......................................................................................... 165
5.2.2 Formulation II ......................................................................................... 169
5.2.3 Formulation III........................................................................................ 172
5.3 Distributed Load Case..................................................................................... 175
5.3.1 Formulation I .......................................................................................... 175
5.3.2 Formulation II ......................................................................................... 179
5.3.3 Formulation III........................................................................................ 180
5.4 Discussion....................................................................................................... 182
5.5 Conclusion ...................................................................................................... 187
Chapter 6. EFFECT OF ELASTIC BOUNDARY CONDITIONS ......................... 189
6.1 Overview......................................................................................................... 189
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6.2 Effect of Elastic Restraint ............................................................................... 191
6.2.1 Effect of Translational Restraint............................................................. 191
6.2.2 Effect of Rotational Restraint ................................................................. 197
6.2.3 Combined Effect of Translational and Rotational Restraints ................. 206
6.3 Relationship between Beam Models and Frame Models................................ 211
6.3.1 Quantification of Elastic Spring Constants............................................. 212
6.3.2 Comparison between Frame and Beam Models ..................................... 214
6.4 Summary and Conclusions ............................................................................. 220
Chapter 7. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR
FUTURE STUDY........................................................................................................... 221
7.1 Summary......................................................................................................... 221
7.2 Conclusions..................................................................................................... 222
7.3 Recommendations........................................................................................... 223
REFERENCES ............................................................................................................... 224
APPENDICES ................................................................................................................ 228
APPENDIX A: Notation................................................................................................. 229
APPENDIX B: Typical Input Files in OpenSees ........................................................... 232
APPENDIX B.1: Centrally Loaded Beam Fixed at Supports......................................... 232
APPENDIX B.2: Fixed Beam with Uniformly Distributed load along the Length........ 236
APPENDIX B.1: Beam with Elastic Boundary Conditions ........................................... 240
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LIST OF TABLES
Table 2-1 Strength of Ties (Ellingwood et al. 2007) ........................................................ 13
Table 2-2 Load Combinations for Progressive Collapse Analysis ................................... 20
Table 2-3 Definition of Local Collapse (Ellingwood et al. 2007) .................................... 21
Table 3-1 Wcat predicted by Rigid?Plastic Theory............................................................ 81
Table 4-1 Summary of Model Parameters in Preliminary FE Models ............................. 99
Table 4-2 Elasticity Modulus............................................................................................ 99
Table 4-3 Beam Models Considered with constant ?p.................................................... 130
Table 4-4 FE Models with Rectangular Shapes (Constant ?p / d) .................................. 134
Table 4-5 FE Models with W-Shapes (Constant ?p / d).................................................. 134
Table 4-6 FE Models with Rectangular Shapes (Constant ?p / 2L) ................................ 143
Table 4-7 FE Models with W?Shapes (Constant ?p / 2L)............................................... 143
Table 4-8 FE Models with Rectangular Shapes (Constant L / d case)............................ 149
Table 4-9 FE Models with W?Shapes (Constant L / d case) .......................................... 150
Table 4-10 FE Models with Rectangular Sections (Uniform Loading).......................... 156
Table 4-11 FE Models with W-Shapes (Uniform Loading) ........................................... 157
Table 6-1 W14x53 Beam Models with Translational Springs only ............................... 192
Table 6-2 W14x53 Beam Models with Rotational Springs only.................................... 199
Table 6-3 Beam Models with Combined Spring Constant Cases................................... 207
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LIST OF FIGURES
Figure 2-1 Schematic of Tie Forces in a Frame Structure (UFC 2005)............................ 10
Figure 2-2 Calculation of Upper Bound on the Basic Strength (UFC 2005).................... 14
Figure 2-3 Removal of Column (UFC 2005).................................................................... 17
Figure 2-4 Double Span Condition in a Traditional Moment Frame (GSA 2003)........... 18
Figure 2-5 Element Removal Considerations for Framed Structure (GSA 2003)............ 19
Figure 2-6 Behavior Idealizations (Jones 1989) ............................................................... 26
Figure 2-7 Centrally Loaded Simple Beam under Finite Displacements ......................... 27
Figure 2-8 Combination of Axial force and Moment on a Rigid-Perfectly Plastic Beam
with a Rectangular Cross-section (Jones 1989)........................................................ 28
Figure 2-9 Yield Condition Relating M and N required for a Rectangular Cross-Section29
Figure 2-10 Element of a eam subjected to loads which produce finite transverse
displacements (Jones 1989) ...................................................................................... 30
Figure 2-11 Plastic flow of a rigid-perfectly plastic beam with rectangular cross- section
(Jones 1989).............................................................................................................. 34
Figure 2-12 Forces at Midspan (x=0) at Fully Plastic Phase............................................ 37
Figure 2-13 Normalized Load?Deflection curve for simple beam with a rectangular
cross-section (Jones 1989) ........................................................................................ 38
Figure 3-1 Stress distributions in the cross-section based on the location of neutral axis 41
Figure 3-2 M?N interaction for W?Shapes (W30x124) ................................................... 42
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Figure 3-3 W?Shaped Rigid?Plastic Simple Beam with a Concentrated load at Midspan
................................................................................................................................... 43
Figure 3-4 Normalized Axial Force?Deflection Relationship (Midspan) ........................ 49
Figure 3-5 Normalized Moment?Deflection Relationship (Midspan).............................. 49
Figure 3-6 Normalized M?N interaction for W-Shape cross-section (Midspan).............. 50
Figure 3-7 Normalized Load?Deflection Relationship..................................................... 51
Figure 3-8 Centrally loaded, fully fixed beam with W-Shape cross?section ................... 53
Figure 3-9 Normalized Axial Force?Deflection Relationship (Midspan) ........................ 58
Figure 3-10 Normalized Moment?Deflection Relationship (Midspan)............................ 58
Figure 3-11 Normalized Force?Deflection Relationship (Midspan)................................ 59
Figure 3-12 Change in Deflected Shape Configuration Under Uniform Loading............ 60
Figure 3-13 Schematics of Theory I ................................................................................. 62
Figure 3-14 Equilibrium of Forces at Pure Cable State.................................................... 64
Figure 3-15 Schematics of Theory II................................................................................ 66
Figure 3-16 Length of Arc Approximation....................................................................... 69
Figure 3-17 Equilibrium of forces at pure catenary state ................................................. 71
Figure 3-18 Normalized Load?Deflection relationship.................................................... 73
Figure 3-19 Normalized Axial Force?Deflection relationship ......................................... 73
Figure 3-20 Normalized Moment ? Deflection Relationship ........................................... 74
Figure 3-21 Normalized Load?Deflection relationship.................................................... 79
Figure 3-22 Normalized Axial Force?Deflection relationship ......................................... 80
Figure 3-23 Normalized Moment?Deflection relationship............................................... 80
Figure 3-24 Material idealization used in Cable analysis................................................. 82
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Figure 3-25 Free body diagram under concentrated load ................................................. 84
Figure 3-26 Free body diagram under uniform loading.................................................... 86
Figure 3-27 ?? ? relationship......................................................................................... 88
Figure 3-28 Expected behavior representation on M?N interaction ................................. 90
Figure 4-1 Schematic of Fiber Section Discretization of Cross?Sections........................ 93
Figure 4-2 Uniaxial Elastic?perfectly plastic material ..................................................... 95
Figure 4-3 Schematic of preliminary FE beam models .................................................... 98
Figure 4-4 Normalized Load?Deflection Plot ................................................................ 101
Figure 4-5 Normalized Axial Force?Deflection Plot (Midspan).................................... 101
Figure 4-6 Normalized Moment?Deflection Plot (Midspan) ......................................... 102
Figure 4-7 Normalized M?N interaction Plot (Midspan)................................................ 103
Figure 4-8 Normalized Load?Deflection Plot ................................................................ 104
Figure 4-9 Normalized Axial Force?Deflection Plot (Midspan).................................... 105
Figure 4-10 Normalized Moment?Deflection Plot (Midspan) ....................................... 105
Figure 4-11 Normalized M?N interaction Plot (Midspan).............................................. 106
Figure 4-12 Normalized Load?Deflection Plot .............................................................. 108
Figure 4-13 Normalized Axial Force?Deflection Plot (End) ......................................... 109
Figure 4-14 Normalized Moment?Deflection Plot (End)............................................... 109
Figure 4-15 Normalized M?N interaction (End)............................................................. 111
Figure 4-16 Normalized M?N interaction (Midspan) ..................................................... 112
Figure 4-17 Normalized Load?Deflection Plot .............................................................. 114
Figure 4-18 Normalized Axial Force?Deflection Plot (End) ......................................... 114
Figure 4-19 Normalized Moment?Deflection Plot (End)............................................... 115
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Figure 4-20 Normalized M?N interaction (End)............................................................. 116
Figure 4-21 Normalized M?N interaction (Midspan) ..................................................... 117
Figure 4-22 External Load ? Normalized Deflection Plot.............................................. 118
Figure 4-23 Normalized Axial Force ? Deflection Plot (Midspan)................................ 119
Figure 4-24 Normalized M?N interaction (REC Section, Midspan) .............................. 120
Figure 4-25 Normalized M?N interaction (W?Shape, Midspan).................................... 121
Figure 4-26 Normalized Load?Deflection Plot .............................................................. 122
Figure 4-27 Normalized Axial Force?Deflection Plot.................................................... 123
Figure 4-28 Normalized M?N interaction (REC Section, End)...................................... 124
Figure 4-29 Normalized M?N interaction (W-Shape, End)............................................ 125
Figure 4-30 ?p Definition ................................................................................................ 128
Figure 4-31 Normalized Load?Deflection Plot for Constant ?p ..................................... 131
Figure 4-32 Normalized Axial Force?Deflection Plot for Constant ?p (Midspan)......... 131
Figure 4-33 Normalized Moment?Deflection Plot for Constant ?p................................ 131
Figure 4-34 Typical Normalized Axial Force?Deflection plot used to determine Wcat for
................................................................................................................................. 133
Figure 4-35 Wcat / ?p - ?p / d Relationship (Constant ?p / d within each group) .............. 137
Figure 4-36 Wcat / ?p - ?p / 2L Relationship (Constant ?p / d within each group) ............ 137
Figure 4-37 Wcat / ?p ? L / d relationship (Constant ?p / d within each group) ............... 138
Figure 4-38 Wcat / d - ?p / d Relationship (Constant ?p / d within each group) ............... 139
Figure 4-39 Wcat / d - ?p / 2L Relationship (Constant ?p / d within each group) ............. 139
Figure 4-40 Wcat / d ? L / d Relationship (Constant ?p / d within each group)................ 140
Figure 4-41 Wcat / 2L - ?p / d Relationship (Constant ?p / d within each group)............. 141
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Figure 4-42 Wcat / 2L - ?p / 2L Relationship (Constant ?p / d within each group)........... 141
Figure 4-43 Wcat / 2L - L / d Relationship (Constant ?p / d within each group) .............. 142
Figure 4-44 Wcat / ?p - ?p / d Relationship (Constant ?p / 2L within each group) ............ 144
Figure 4-45 Wcat / ?p - ?p / 2L Relationship (Constant ?p / 2L within each group).......... 145
Figure 4-46 Wcat / ?p ? L / d Relationship (Constant ?p / 2L within each group) ............ 145
Figure 4-47 Wcat / d ? ?p / d Relationship (Constant ?p / 2L within each group)............. 146
Figure 4-48 Wcat / d ? ?p / 2L Relationship (Constant ?p / 2L within each group) .......... 146
Figure 4-49 Wcat / d ? L / d Relationship (Constant ?p / 2L within each group).............. 147
Figure 4-50 Wcat / 2L ? ?p / d Relationship (Constant ?p / 2L within each group) .......... 147
Figure 4-51 Wcat / 2L ? ?p / 2L Relationship (Constant ?p / 2L within each group) ........ 148
Figure 4-52 Wcat / 2L ? L / d Relationship (Constant ?p / 2L within each group) ........... 148
Figure 4-53 Wcat / ?p - ?p / d Relationship (Constant L / d within each group) ............... 151
Figure 4-54 Wcat / ?p - ?p / 2L Relationship (Constant L / d within each group)............. 152
Figure 4-55 Wcat / ?p ? L / d Relationship (Constant L / d within each group)................ 152
Figure 4-56 Wcat / d - ?p / d Relationship (Constant L / d within each group) ................ 153
Figure 4-57 Wcat / d - ?p / 2L Relationship (Constant L / d within each group) .............. 153
Figure 4-58 Wcat / d ? L / d Relationship (Constant L / d within each group)................. 154
Figure 4-59 Wcat / 2L - ?p / d Relationship (Constant L / d within each group) .............. 154
Figure 4-60 Wcat / 2L - ?p / 2L Relationship (Constant L / d within each group)............ 155
Figure 4-61 Wcat / 2L ? L / d Relationship (Constant L / d within each group) .............. 155
Figure 4-62 Wcat / ?p - ?p / d Relationship (Constant L / d within each group) ............... 158
Figure 4-63 Wcat / ?p - ?p / 2L Relationship (Constant L / d within each group)............. 158
Figure 4-64 Wcat / ?p ? L / d Relationship (Constant L / d within each group)................ 159
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Figure 4-65 Wcat / d - ?p / d Relationship (Constant L / d within each group) ................ 159
Figure 4-66 Wcat / d - ?p / 2L Relationship (Constant L / d within each group) .............. 160
Figure 4-67 Wcat / d ? L / d Relationship (Constant L / d within each group)................. 160
Figure 4-68 Wcat / 2L - ?p / d Relationship (Constant L / d within each group).............. 161
Figure 4-69 Wcat / 2L - ?p / 2L Relationship (Constant L / d within each group)............ 161
Figure 4-70 Wcat / 2L ? L / d Relationship (Constant L / d within each group) .............. 162
Figure 5-1 Comparison with FE results (REC Shapes) .................................................. 167
Figure 5-2 Comparison with FE results (W?Shapes: Constant ?p / d case).................... 168
Figure 5-3 Comparison with FE results (W?Shapes: Constant ?p / 2L case) ................. 168
Figure 5-4 Comparison with FE results (W?Shapes: Constant L / d case)..................... 169
Figure 5-5 Comparison with FE results (Constant ?p / d case)....................................... 171
Figure 5-6 Comparison with FE results (Constant ?p / 2L case)..................................... 171
Figure 5-7 Comparison with FE results (Constant L / d case)........................................ 172
Figure 5-8 Comparison with FE results (Constant ?p / d case)....................................... 174
Figure 5-9 Comparison with FE results (Constant ?p / 2L case)..................................... 174
Figure 5-10 Comparison with FE results (Constant L / d case)...................................... 175
Figure 5-11 Comparison with FE results (REC Shapes) ................................................ 178
Figure 5-12 Comparison with FE results (W Shapes) .................................................... 178
Figure 5-13 Comparison with FE results (Constant L / d within each group)................ 180
Figure 5-14 Comparison with FE results (Constant L / d within each group)................ 182
Figure 5-15 Distribution of the Difference for Rectangular Shaped Beams .................. 184
Figure 5-16 Distribution of the Difference for W-Shaped Beams.................................. 185
Figure 5-17 Distribution of the Difference for Rectangular Shaped Beams .................. 186
xviii
Figure 5-18 Distribution of the Difference for W-Shaped Beams.................................. 187
Figure 6-1 Schematics of Beam Model with Elastic Boundary Conditions ................... 190
Figure 6-2 Beam model with translational springs only................................................. 192
Figure 6-3 Dependency of Wcat / d on Translational Spring Constant, k? ...................... 193
Figure 6-4 Normalized Load?Deflection Plot (Midspan)............................................... 194
Figure 6-5 Normalized Axial Force?Deflection Plot (Midspan).................................... 195
Figure 6-6 Normalized Moment?Deflection Plot (Midspan) ......................................... 195
Figure 6-7 Normalized M?N interaction (Midspan) ....................................................... 196
Figure 6-8 Normalized M?N Interaction (End) .............................................................. 197
Figure 6-9 Beam with Rotational Springs Only ............................................................. 198
Figure 6-10 Dependency of Wcat / d on Rotational Spring Constant, k?......................... 199
Figure 6-11 Normalized Load?Deflection Plot (Midspan)............................................. 201
Figure 6-12 Normalized Axial Force?Deflection Plot (Midspan).................................. 201
Figure 6-13 Normalized Moment?Deflection Plot (Midspan) ....................................... 202
Figure 6-14 Normalized Moment?Deflection Plot (End)............................................... 202
Figure 6-15 Normalized M?N Interaction (Midspan)..................................................... 203
Figure 6-16 Normalized M?N Interaction Plot (End)..................................................... 204
Figure 6-17 Classification of moment?rotation response of fully restrained (FR), partially
restrained (PR) and simple connections (AISC 2005)............................................ 205
Figure 6-18 Moment?Rotation Response of Elastic Springs in FE Beam Models......... 206
Figure 6-19 Normalized Load?Deflection Plot .............................................................. 208
Figure 6-20 Normalized Axial Force?Deflection Plot (Midspan).................................. 209
Figure 6-21 Normalized Moment?Deflection Plot (Midspan) ....................................... 209
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Figure 6-22 Normalized M?N Interaction (Midspan)..................................................... 210
Figure 6-23 Normalized M?N Interaction (End) ............................................................ 211
Figure 6-24 Reference Frame ......................................................................................... 212
Figure 6-25 Schematic of Frame Analysis...................................................................... 213
Figure 6-26 Schematic for the Quantification of Linear?Elastic Spring Constants ....... 214
Figure 6-27 Frame and Beam Models ............................................................................ 215
Figure 6-28 Normalized Load?Deflection Plot (Beam Midspan) .................................. 216
Figure 6-29 Normalized Moment?Deflection Plot (Beam Midspan)............................. 217
Figure 6-30 Normalized Axial Force?Deflection Plot (Beam Midspan)........................ 217
Figure 6-31 Normalized M?N Interaction (Beam Midspan)........................................... 218
Figure 6-32 Normalized M?N Interaction (Beam End) .................................................. 219
1
Chapter 1. INTRODUCTION
1.1 Overview
In May 16, 1968, 23-story Ronan Point Apartments in London, U.K partially collapsed
due to a gas explosion at the 18th story. The loss of load?bearing precast concrete walls
caused the upper floors to collapse and eventually led to progressive collapse of the entire
corner of the building (Figure 1-1).
Figure 1-1 Ronan Point Apartments collapse (Copyright Daily Telegraph 1968:
http://apps.newham.gov.uk/History_canningtown/pic47.htm)
2
Progressive collapse can be defined as an initial local failure triggering a spread
of failure in a disproportionately large part of the structure. There has been a considerable
effort to develop design guidelines and criteria to decrease or eliminate the vulnerability
of buildings to this kind of failure (Nair 2004). These efforts usually have tended to
concentrate on ensuring a level of redundancy, continuity and ductility in buildings to
prevent such a collapse in case of a local failure.
One of the main methods of analysis for design against progressive collapse in
current design guidelines involves the so-called ?missing column scenario?. In this
method, a designated column from a given floor is hyphotetically removed and the
damaged system is expected to bridge over the beam above the removed column (Figure
1-2(a)). With this ?immaculate? removal of the column, the structure seeks an alternate
path to re-distribute the loads without failure. One of the key load-resisting mechanisms
that is believed to assist the damaged system to achieve this is often referred to as
?Catenary Action?. As illustrated in Figure 1-2 (b), the beam cannot resist the vertical
loads with flexural action alone and the new equilibrium state is reached by development
of axial catenary forces through a formation of catenary?like mechanism (Figure 1-2 (c))
(Khandelwal and El-Tawil 2007).
In spite of many debates, this formation is widely used in most common
progressive collapse design guidelines. For example, Progressive Collapse Analysis and
Design Guidelines for New Federal Office Buildings and Major Modernization Projects
(GSA 2003) clearly states that the capability of the beam or girder to accommodate the
?double span condition? resulting from the missing column scenario is essential.
3
Column to be
Removed
a
b
c
Figure 1-2 Load-resisting mechanism upon column removal
(Adopted from Hamburger and Whittaker 2004)
The term ?catenary? refers to a theoretical curved shape of a hanging chain or
flexible wire supported at the ends under its own weight (Weisstein 2009). However, it is
evident from Figure 1-2 (c) that the shape is more like a cable subjected to a concentrated
load at midspan. Therefore, ?cable?like action? is considered more rigorous describing
4
this behavior. However, since the terminology ?catenary action? is widely used, the
words ?cable? and ?catenary? are used interchangeably throughout this study.
1.2 Motivation
Current design guidelines provide a number of prescriptive methods to prevent or
limit this failure by progressive collapse. However, the theory behind the key load
resisting mechanism of beams resulting from ?double span condition? remains poorly
understood and has not been rigorously studied.
The study presented in this thesis is intended to develop a rigorous description of
the fundamental behavior of ductile steel beams undergoing a transition from a flexural to
a cable-like behavior, as illustrated in Figure 1-2.
1.3 Research Objectives and Scope
The overall objectives of this study are as follows:
square4 Develop rigorous theoretical models that describe the behavior of ductile steel beams
undergoing a transition from flexural to cable-like behavior.
square4 Develop analytical models that can be used to evaluate and assess the theoretical
models as well as help identify the main parameters affecting the behavior.
square4 Assess the effect of boundary conditions on the behavior.
The scope of this study is limited to ductile steel beams with idealized boundary
conditions. Factors related to the potential for instability, fracture and connection
behavior are beyond the scope of this research. It is believed that a thorough
understanding of the behavior at this fundamental level is essential to understanding the
behavior in the presence of other complex factors.
5
The idealized boundary conditions used include simple, fully fixed and partially
restrained supports. Theoretical models are developed for the first two support types
whereas finite element (FE) analysis is used for all support types. Studies related to the
effect of partially restrained supports are limited to elastic boundary conditions.
1.4 Approach
The beam over which the remaining structural system is desired to bridge upon
removal of a supporting column is isolated and modeled with idealized boundary
conditions to study fundamental behavior. Two theoretical approaches are used to model
the beam behavior: rigid-plastic analysis and cable analysis. Rigid-plastic analysis
assumes that the beam behaves rigidly until a flexural collapse mechanism is reached,
whereas cable analysis assumes that the beam does not offer any flexural resistance.
These two theoretical models will serve as bounds to the actual behavior.
In order to verify the theoretical results from the two approaches mentioned above,
preliminary FE analyses were conducted. Following this study, another parametric FE
analysis was conducted in order to identify the main factors affecting the behavior.
OpenSees (McKenna et al. 2000), Open System for Earthquake Engineering
Simulation software was used in the FE analyses.
1.5 Organization of Thesis
Chapter 2 provides background information including: current methodologies
used for design against progressive collapse, a description of recent relevant research
studies and a review of rigid-plastic analysis procedure. Chapter 3 presents theoretical
results using rigid-plastic and cable analyses. FE analysis results are presented in Chapter
6
4. In Chapter 5, an equation is developed and proposed that can be used to predict the
midspan deflection at the onset of pure cable behavior. In Chapter 6, the effect of elastic
boundary conditions on the beam behavior is investigated. Finally, a summary of the
study, along with conclusions and recommendations for future directions, are provided in
Chapter 7.
1.6 Notation
The notation used in this thesis follows AISC Steel Construction Manual 13th Edition
(2005) and is listed in Appendix A.
7
Chapter 2. BACKGROUND
2.1 Introduction
Progressive collapse is a failure mode that may not only occur unexpectedly in buildings
but can also be seen in building demolitions (Bazant and Verdure 2007). Due to the
relative rareness of the events and situations that cause progressive collapse, it is one of
the least researched areas in structural engineering (Mohamed 2006). ASCE Standard 7
(ASCE 2005) defines progressive collapse as ?The spread of an initial failure from
element to element, eventually resulting in the collapse of entire structure or a
disproportionately large part of it?. The keyword ?disproportionate? is widely adopted
since the resulting collapse is generally out of proportion to the initial failure.
The Ronan Point apartment collapse (Figure 1-1) is a landmark case of this
phenomenon in recent history that instigated code changes. In fact, a reform in British
codes started in early 1970?s and has been intensively referenced in literature produced in
the U. S. (Mohamed 2006). Despite the fact that eager interest hascontinued throughout
the decades, the majority of resources were developed during the first several years after
this event. A second increase in interest was aroused by the attack on the Alfred P.
Murrah Federal Building in 1995. Various reports were produced to investigate the
damage and progressive collapse of the building and design?specific recommendations
were provided. Now, the interest has reached its peak level after 9 / 11 tragedy occurred
in New York in 2001 (Dusenberry 2002).
8
A comprehensive survey of all the efforts mentioned above is beyond the scope of
this study. Therefore, a brief review of commonly used concepts and approach methods
for design against progressive collapse is presented in Section 2.2. Recent relevant
research is reviewed in Section 2.3. Finally, a review of rigid?plastic analysis of beams
undergoing finite displacements as provided by Jones (1989) is presented in Section 2.4.
2.2 Existing Methodologies for Design against Progressive Collapse
It is usually not practical to design a structure to an abnormal loading condition
unless it is a special protective system. Nevertheless, local failure effects and progressive
collapse can be limited or mitigated by taking precautions in the design process (ASCE
2005).
There exist a considerable amount of reference documents that address the problem
of progressive collapse. However, most common public sector codes and standards
largely regard general structural integrity and progressive collapse in a qualitative
manner, whereas governmental documents explicitly address the issue (Dusenberry
2002). In fact, a good engineering judgment, design and construction practices in order
for increased structural robustness and integrity are addressed in a number of standards
(Ellingwood et al. 2007). The most commonly used approaches for design against
progressive collapse are indirect and direct design methods. In general, the indirect
design method is a prescriptive method that requires a minimum level of connectivity
between the structural members. Direct design, on the other hand, largely depends on
structural analysis to ensure that the structure resists an abnormal event (Ellingwood et al.
2007).
9
The following sections provide a review of these two methods with an emphasis on
steel structures for the purpose of this study.
2.2.1 Indirect Method
Indirect design refers to implicit considerations to be taken during the design
process for resistance to progressive collapse by means of requiring minimum levels of
strength, continuity and ductility (ASCE 2005). The main purpose of those provisions is
to provide an alternate path within the structure to re-distribute the loads in case of an
abnormal loading condition. ASCE 7-05 gives the following key concepts to be
considered for improving general structural integrity:
square4 Good plan layout
square4 Provide integrated ties among the principal elements within the structures
square4 Returns on walls
square4 Changing directions of floor spans
square4 Usage of load-bearing interior partitions
square4 Catenary action of floor framing (slab)
square4 Beam action of walls
square4 Ductile detailing
square4 Consideration of load reversals
square4 Compartmentalized construction
While most of practices listed above may be considered as examples of
precautions to be considered in the design process, the second concept, which is
commonly referred to as the ?Tie Force Method? was introduced in British codes after
10
Ronan Point collapse and then was adopted by some of the governmental bodies in the U.
S. In this approach, both vertical and horizontal members in buildings are desired to be
tied together at each principal floor level as illustrated in Figure 2-1. Furthermore, these
tensile tie forces are typically provided by the elements and their connections within the
structure that are designed conventionally (UFC 2005). The types of ties are peripheral,
internal, horizontal and vertical ties (Figure 2-1).
Figure 2-1 Schematic of Tie Forces in a Frame Structure (UFC 2005)
The only standard that explicitly utilizes this approach in the U. S. is the
Department of Defense document: Unified Facilities Criteria (UFC) Design of Buildings
to Resist Progressive Collapse (2005). It is noted that the tie force requirements are
11
almost identical to British Standards, and it is assumed that they are applicable to U. S.
construction. Tie force requirements in various design guidelines are summarized in
Table 2-1; the loads generally need not be considered as additive to other loads (e. g.
dead and live loads).
Even though the tie approach does not incorporate intensive calculations of the
structural response to extreme load conditions, it provides a reserve capacity to members
and connections within reinforced concrete and steel?framed structures through either
flexure or membrane action. This event-independent approach is relatively easy to apply
and convenient for all projects (Ellingwood et al. 2007).
The mechanics behind this approach are not well explained. In fact, the
fundamental assumption for the efficiency of this method is that the structural members
and their connections have sufficient rotational ductility to develop axial capacity in the
form of catenary action under large deflections (Marchand and Alfawakhiri 2005). For
the purpose of this study, horizontal tie force requirements which pertain to beams and
girders will be thoroughly presented. According to Ellingwood et al. (2007), the form of
horizontal steel tie force requirement given in UFC (2005) and British Code (BS 5950-I
2000) as detailed in Table 2-1 can simply be obtained with use of the one?half beam
shown in Figure 2-2 in the following way: in case of a column removal, the span length is
now doubled and the slab hangs in a catenary shape with a maximum sag of a (denoted
by S in Figure 2-2). In addition to vertical reactions at the ends, horizontal
reaction force F must also be developed. Therefore, the moment equilibrium requires:
8
)2( 2LwaF ?=? [2.1]
12
where w is load per unit length, L is the bay length. The magnitude of the horizontal
force reactions that must be provided by ties, with transverse spacing s, can be expressed
as:
)/(2 La
LsqF
?
??= [2.2]
where q is the floor load (i.e. swq = )
UFC (2005) points out that the theory behind this concept is based on research
conducted by Burnett (1975), who discussed the logic and theoretical background used to
develop the British tie force requirements. However, the details are limited to reinforced
concrete structures and similar descriptions were not revealed for steel design in the
development of UFC (2005). An illustration of this discussion is given in Figure 2-2.
FT in Table 2-1 and Figure 2-2 is defined as ?Basic Strength? and according to
UFC (2005), the upper limit (60 kN / m) can be obtained from two cases. The first
method is to consider FT as the internal member force developed by catenary action of the
floor in case a vertical load bearing element is removed to produce a presumed transverse
deflection of 10% of the span length (Figure 2-2). The second method for determining FT
is to consider the forces applied to a typical wall panel under a static pressure of 34
kN/m2 (5 psi), which is thought to be the overpressure that occurred in the Ronan Point
explosion. The first approach (catenary action) is the mechanism that the tie forces are
intended to withstand based on the debates with British engineers (UFC 2005). It should
be noted that F provided to obtain the form of steel tie force requirements (Equation 2.2)
has units of force and should not be confused with FT (force per unit length) in the
context of calculations for reinforced concrete as given in Figure 2-2.
13
Table 2-1 Strength of Ties (Ellingwood et al. 2007)
14
Figure 2-2 Calculation of Upper Bound on the Basic Strength (UFC 2005)
There has been same discussion of the approximate deflection level to use in the
calculation of the tie force requirements; beam and slab tests conducted by Creasy (1972)
suggested that the sag of the double span over the missed support should not exceed 20%
15
of single span length. However, Breen (1980) stated that the British calculations are
based on a / L of 0.15. For steel buildings, on the other hand, Marchand and Alfawakhiri
(2005) claim that the required tie forces specified by design guidelines are ?minimum?
catenary forces that develop under a sag of 10 percent of ?double span?.
2.2.2 Direct Methods
Direct design is an explicit consideration of a structure?s resistance to progressive
collapse and damage absorption ability during the design phase. It consists of two
approaches: Specific Local Resistance (SLR) and Alternate Path Method (APM). The
first refers to providing adequate strength to resist an extreme loading condition and
requires that the triggering event be identified so the local resistance can be related to a
particular limit state (Krauthammer et al. 2002). Alternate Path Method, on the other
hand, allows for local failure but requires the availability of alternate load paths to re-
distribute the loads within the remaining structure (ASCE 2005).
2.2.2.1 Specific Local Resistance (SLR)
SLR approach implies the design of critical vertical load bearing elements to a
specific threat. These critical elements are often referred as ?key elements? and are
explicitly designed to withstand abnormal load conditions (e.g., blast pressure).
Therefore, this approach is threat ? dependent (Ellingwood et al. 2007).
ASCE Standard 7 (ASCE 2005) provides the following load combinations to
check the capacity of a structure or structural element to resist an abnormal event:
square4 1.2D + Ak + (0.5L or 0.2S + 0.2W) [2.3]
square4 (0.9 or 1.2)D + Ak + 0.2W [2.4]
16
where Ak represents extreme load condition such as blast pressure and D, L, S, W
represents dead load, live load, snow load, wind load, respectively.
In spite of many debates regarding this method because of the difficulties in
defining the magnitude of the extreme event, this approach is relatively less expensive in
many cases (Mohamed 2006). In fact, this approach is often considered to be more
practical for retrofitting an existing structure because the cost might be significant to have
the structure meet the requirements of other approaches (Ellingwood et al. 2007).
2.2.2.2 Alternate Path Method (APM)
The Alternate Path Method (APM) is based on supplying an alternate load path in
order to limit the local damage and prevent major collapse in case of a local failure
(Khandelwal and El-Tawil 2007). This threat ? independent method addresses the
performance of the structure after some elements are compromised. It is performed by
assuming that the primary structural elements, sequentially one element at a time, are
rendered ineffective and investigating the consequent structural behavior (Dusenberry
2002). The APM relies on continuity and ductility to redistribute the forces in case of a
local damage. Therefore, APM attracts people because the limit state considered is
explicitly related to overall structural performance and, in contrast to SLR method, the
triggering event does not need to be identified specifically (Krauthammer et al. 2002)
In this methodology, key structural elements (typically a column or wall), are
hypothetically removed and the structure is analyzed to evaluate its capacity to bridge
over that removed member (Marchand and Alfawakhiri 2004). Therefore, it is important
that beams and floor girders are able to at least accommodate the double span condition.
17
This necessitates not only beam-to-beam continuity but also requires that girders and
beams deflect further than their elastic limit in flexure without undergoing structural
collapse across the removed element (GSA 2003).
Even though it is not likely to occur in an actual event, the vertical load bearing
elements are notionally removed without degradation of the abilities of the joint above
the removed member. This so called ?immaculate removal? is not necessarily intended to
represent an actual event (UFC 2005). The removal procedure, as well as the response of
the framing scheme, after loss of primary column support in a traditional moment frame
is illustrated in Figures 2-3 and 2-4.
Figure 2-3 Removal of Column (UFC 2005)
18
Figure 2-4 Double Span Condition in a Traditional Moment Frame (GSA 2003)
A theoretical damage state as depicted in Figure 2-3 is assumed in the application
of APM and all other damages that may occur as a result of loss of vertical support are
ignored. The transition is assumed to be instantaneous and dynamic effects are
considered depending on the analysis procedure used (Ellingwood et al. 2007). The
following analysis procedures can be conducted:
square4 Linear Static
square4 Nonlinear Static
square4 Linear Dynamic
square4 Nonlinear Dynamic
19
Vertical member removal considerations are generally common. Interior and
exterior considerations are illustrated in Figure 2-5 for framed structures. A similar but
more detailed set of considerations are also available for shear / load bearing structures.
Figure 2-5 Element Removal Considerations for Framed Structure (GSA 2003)
Load combinations used for this method in different guidelines are summarized in
Table 2-2. It can be seen that a factor 2.0 is used in DoD UFC 4-023-03 (2005) and GSA
(2003) when a linear-static analysis is performed. Powell (2005) states that the maximum
deflection for a linear structure is twice the static deflection, therefore, an ?impact? or
?load amplification? factor is commonly used in the existing design guidelines. There has
been a significant amount of effort on progressive collapse assessment of both RC and
20
steel?framed buildings using these different analysis procedures. Advantages and
disadvantages of each are discussed in detail by Powell (2005).
Table 2-2 Load Combinations for Progressive Collapse Analysis
(Ellingwood et al. 2007)
Two governmental documents in the U. S. explicitly use this methodology.
Department of Defense?s; Unified Facilities Criteria (UFC) Design of Buildings to Resist
Progressive Collapse (2005) and Progressive Collapse Analysis and Design Guidelines
for New Federal Office Buildings and Major Modernization Projects developed by the
General Services Administration (GSA) (2003). However, acceptance criteria used in the
process vary in each. UFC (2005) uses Load and Resistance Factor Design (LRFD)
21
methodology and refers to material specific codes whereas GSA (2003) provides its own
procedure.
In addition to the acceptance criteria for structural members and their connections,
regardless of the analytical method used, it is required that the designer quantify
structural damage during and at the end of the analysis. Table 2-3 provides the damage
definitions and limits stipulated by various design guidelines.
Table 2-3 Definition of Local Collapse (Ellingwood et al. 2007)
2.3 Recent Relevant Research
?Structural integrity? is a trendy term that prompted important arguments
regarding inclusion of provisions into design codes for the purpose of enhanced structural
22
robustness (Gustafson 2009). After the September 11th attacks, research interest in
progressive collapse intensified. Because of the lack of design?specific provisions in
existing codes, a majority of recent considerations have more likely been intended to find
quick answers for structural design practice.
For design of structural steel buildings, AISC has published ?Fact for Steel
Buildings 2: Blast and Progressive Collapse? by Marchand and Alfawakhiri (2005). The
need for careful research is highlighted in regard to assessing weather or not beams and
their connections as currently designed have adequate robustness to develop imperative
plastic rotations and large tensile forces during catenary action, which is considered to be
promising for steel structure design (Hamburger and Whittaker, 2004).
Foley et al. (2007) investigated the level of robustness that structural steel
buildings naturally possess. An application of the Alternate Path Method for steel
moment resisting frames with varying number of stories is presented; the inherent
robustness is evaluated by means of assessing internal force demands in members and
their connections after loss of an exterior column. In addition, the study goes beyond
framework analysis, and the effect of catenary and membrane action in floor systems is
investigated for more accurate quantification of robustness. Gravity load analyses of
various floor sub-assemblages are performed assuming full fixity at the perimeter of the
panels. Recommendations are made for detailing considerations to provide inherent
robustness enhancement for structural systems.
The question of the reliability of catenary action to redistribute forces resulting
from column damage has been studied on particular structural systems. A case study of a
steel?framed building was performed by Byfield and Paramasivam (2007) with regard to
23
the ability of beams to exhibit catenary behavior. The results showed that industry
standard simple beam?column connections have inadequate ductility to exhibit large
floor displacements during catenary action. Furthermore, the absence of rotation capacity
in the tie force method is emphasized and the effect of prying action is also underlined in
regard to the connection performance.
Due to the natural toughness of earthquake resistant construction, which
contributes to alternate load paths and distribution, a widespread notion has arisen that
earthquake resistant design will improve collapse resistance (Khandelwal and El-Tawil
2007). To that end, a research study was carried out by El-Tawil and his co-workers to
investigate the collapse behavior of seismically-designed steel moment resisting frames
and connections. Sub- assemblages of an eight?story special moment resisting perimeter
frame were taken under investigation, which is widely used in the U.S. west coast. The
effect of a number of key design variables on catenary action formation was investigated
and the results indicated that connection ductility and strength are adversely affected by
an increase in beam depth and YUSR (yield to ultimate strength ratio). A set of practical
implications are also provided based on numerical results. However, it should be
remarked that the boundary conditions considered in the sub?assemblages involved
seismic behavior characteristics which may not necessarily reflect actual behavior. The
ensuing effort of this group focused on the development of computationally adequate, so-
called ?macro models? to investigate progressive collapse resistance of seismically
designed moment frame buildings including RC and steel?framed structures (Khandelwal
et al. 2008).
24
As far as recent demand to incorporate design-specific provisions into building
codes, the upcoming International Building Code (IBC) will include new section for
structural integrity. Gustafson (2009) presents these new provisions with an emphasis on
structural steel requirements. According to Gustafson (2009), minimum connection
requirements for beams are as follows:
??End connections of all beams and girders shall have a minimum nominal axial
tensile strength equal to the required vertical shear strength for Allowable Strength
Design (ASD) or 2/3 of the required shear strength for Load and Resistance Factor
Design (LRFD) but not less than 10 kips (45 kN). For the purpose of this section,
the shear force and the axial tensile force need not be considered to act
simultaneously.??
It is emphasized that based on the last sentence of the section provided above, the
proposed procedure is to design the beam end connection conventionally and then to
check for the horizontal force required. Moreover, this force is not intended to be the
force that is required to be redistributed within the structure (Gustafson, 2009). Instead, it
can be considered as a minimum level of structural integrity within the structure.
Finally, the behavior of axially-restrained beams subjected to extreme transverse
loading was studied by Izzuddin (2005), focusing on static response under ambient and
elevated temperature in which beams exhibit large displacements and develop axial
forces. However, the model developed in this study used a linear idealization of the yield
condition under combined bending moment and axial force. Thus, the behavior of W?
shaped beams could not be described rigorously.
25
2.4 Static Plastic Behavior of Simple Beams
Static plastic behavior of a centrally-loaded simple beam with a rectangular cross-
section experiencing finite, but not necessarily large, displacements was presented by
Jones (1989). Due to its relevance, for the purposes of the study presented in this thesis,
the procedure employed by Jones (1989) is reviewed in the remainder of this chapter.
2.4.1 Rigid?Plastic Behavior
Elastic-perfectly plastic and rigid perfectly plastic idealizations on a material
(stress versus strain) level are depicted in Figure 2-6 (a). The corresponding idealizations
on the cross-sectional level (moment versus curvature) are shown in Figure 2-6 (b). Jones
(1989) used the rigid-perfectly plastic idealization on a cross-sectional level for his
analyses, so the effect of elastic deformations is fully ignored. In other words, no cross-
sectional deformations occur until a plastic hinge forms (e.g., when the moment at any
section reaches the plastic moment, Mp for the case shown in Figure 2-7). Once a flexural
collapse mechanism is reached, the beam starts to deflect. As transverse deflection
increases, the beam starts to develop axial force, N as discussed next.
26
?y
?
?
Rigid?Perfectly
Plastic Elastic?Perfectly Plastic
(a)
Elastic?Perfectly Plastic
Rigid?Perfectly
Plastic Bilinear ApproximationM
?
Mp
My (b)
Figure 2-6 Behavior Idealizations (Jones 1989)
2.4.2 Effect of Finite Displacements
Conventional beam theory is developed assuming small displacements, and hence
it is possible to write the equilibrium equations for the undeformed original shape. For
the case illustrated in Figure 2-7, the line through the longitudinal axis of the beam needs
to be longer in the deflected shape configuration due to finite transverse displacements,
W. This extension results in development of an axial strain and an associated axial force,
27
N (Jones 1989). As a result, the load carrying capacity increases beyond the plastic
collapse load as will be shown later. To describe this behavior, a yield condition for
combined bending moment, M and axial force, N needs to be obtained, as presented in the
following section.
P
W
? w
x
L L
L? L?
Figure 2-7 Centrally Loaded Simple Beam under Finite Displacements
2.4.3 M?N Interaction at Cross-Sectional Level
For combined bending moment, M, and axial force, N on a rectangular cross-
section, with the aid of Figure 2-8, the following M-N interaction equation that defines
the yield condition can be derived:
1
2
=
???
?
???
?+
pp N
N
M
M [2.5]
where pN is the plastic axial force and pM is the plastic bending moment.
A graphical representation of this equation is shown in Figure 2-9.
28
z
M N d
b
-?y
?y
_
+ +
?y
-?y
?d
_
+
_
+
d - ?d
d - ?d
+
d (2 ? ? 1) ? d ? d/2
(a) (b) (c)
(d) (f)(e) (g)
?y
?y
-?y
?
?
.
.
?y
(a) Cross-Section
(b) Stress distribution associated under pure plastic bending moment pM
(c) Stress distribution associated under pure plastic axial force pN
(d) Stress distribution associated with combined M and N
(e)-(f) Stress distribution associated with M and N respectively
(g) Strain distribution across the depth due to the stress distribution in (d)
Figure 2-8 Combination of Axial force and Moment on a Rigid-Perfectly Plastic Beam
with a Rectangular Cross-section (Jones 1989)
29
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25
M / M p
N / N p
1
2
=
???
?
???
?+
pp N
N
M
M
Figure 2-9 Yield Condition Relating M and N required for a Rectangular Cross-
Section
When a flexural mechanism forms, transverse deflections result in an axial force,
N and yielding is now controlled by combined M and N, as indicated in Figure 2-8. As
transverse deflection increases, the plastic neutral axis keeps migrating upwards across
the depth and yielding follows the path shown in Figure 2-9 until it is dominated purely
by N.
30
2.4.4 Equilibrium Equations
Under finite displacements, governing equilibrium equations need to be written
on the deformed configuration. Jones (1989) illustrates the infinitesimal free body of a
beam under generalized transverse loads as shown in Figure 2-10.
Horizontal equilibrium requires:
0=dxdN [2.6]
which means that membrane force N is constant throughout the beam
Moment equilibrium requires:
QdxdM = [2.7]
Transverse equilibrium requires:
0/)/( =++ qdxdxNdwddxdQ [2.8]
dxx
w w + dw
d
M Q
N
Q + dQ
M + dM
N + dN
q
Figure 2-10 Element of a eam subjected to loads which produce finite transverse
displacements (Jones 1989)
31
By careful inspection, it can be shown the equations above as derived by Jones
(1989) assume the following:
square4 0sin =? , 1cos =? for horizontal and moment equilibrium
square4 ?? =sin 1cos =? for vertical equilibrium
2.4.5 Deflected Shape Configuration
The deflected shape (Figure 2-7) can be represented as a function of x using:
?????? ?= LxWw 1 [2.9]
The change in length in the longitudinal axis of the beam, ?L is given as:
( )[ ]LWLL ?+?=? 2/1222 [2.10]
The corresponding axial strain, ? over a single plastic hinge of length,l at midspan is
given as:
lLWL /112
2/1
2
2
?
?
?
?
?
?
?
?
???
?
?
???
? +=? [2.11]
Expanding the Equation 2.11 using the binomial theorem and neglecting powers
of
2
?????? LW greater than two, results in:
????????????? lLLW
2
? [2.12]
Then, the axial strain rate,
??
, the first derivative of Equation 2.12 with respect to time, is
given as:
32
lL
WW
?=
?
? 2?
[2.13]
Secondly, the change in angle over a midspan plastic hinge length of l, the
associated curvature rate,
??
is given as:
lL
W
?=
?
? 2?
[2.14]
From the Equations 2-13 and 2.14 it can be shown that
W=
?? ??
/ [2.15]
2.4.6 Load?Deflection
With the use of the previously presented concepts and associated derivations,
expressions for internal forces and external load versus deflection of the beam shown in
Figure 2-7 can be derived. A stepwise presentation of the procedure and resulting
graphical illustration of behavior will also be provided.
Expressions for internal forces (M and N) and then external force (P) as a function
of transverse deflection, W will be derived. Once the expressions for M and N are
obtained, substitution of these equations into the yield condition defined in Equation 2.5
results in normalized load carrying capacity (P / Pc) as a function of W, where Pc is the
plastic collapse load.
square4 Expressions for axial force, N:
Axial forces are derived with use of the deflected shape configuration (Figure 2-7)
as well as a stress / strain profile illustrated over the depth of rectangular cross-section
(Figure 2-8). It is evident from Figure 2-8(f):
33
12 ?= ?
pN
N [2.16]
where,
dbN yp ??=? [2.17]
Figure 2-8(g) suggests:
( )2/dd ?= ?? ??? or ( ) 212/ d?=?? ??? [2.18]
where
??
, is the axial strain rate at the centroidal axis and associated
??
is the curvature
rate.
Substituting Equation 2.16 into 2.18 yields:
))(2(/
pN
Nd=?? ?? [2.19]
which, when combined with Equation 2.15, gives:
d
W
N
N
p
2= for
20
dW ?? [2.20]
Equation 2.20 indicates that at zero deflection, the axial force N is zero. As the
beam deflects, plastic flow begins at Point A (M=Mp) shown in Figure 2-11 and goes
through the yield curve until it reaches to the fully plastic phase where pNN = at a
deflection of 2/dW = (Point C). Therefore:
1=
pN
N for
2
dW ? [2.21]
The solid arrows shown in Figure 2-11 represent the generalized strain rate vector
which is normal to the yield surface due to the normality requirement of plasticity. At
34
point C, Equation 2-15 still controls the behavior and the generalized strain vector rotates
toward the N-axis as W increases beyond d / 2 (Jones 1989).
square4 Expressions for bending Moment, M:
Bending moment is derived with use of the equilibrium equations presented in
Section 2.4.4. The governing equation for the beam shown in Figure 2-7 is obtained by
substituting Equation 2.7 into 2.8 with 0=q :
02
2
2
2
=?+?
?
??
?
???
?
??
?
?+
dx
wdN
dx
dw
dx
dN
dx
Md [2.22]
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25
M / M p
N / N p
C
A
?
?
.
.
?
.
?
.
1
2
=
???
?
???
?+
pp N
N
M
M
Figure 2-11 Plastic flow of a rigid-perfectly plastic beam with rectangular cross-
section (Jones 1989)
35
Recall, Equation 2.9 suggested that the deflected shape expression is a linear
function of x, it follows that the second derivative of w with respect to x is zero. In
addition, Equation 2.6 suggested that N is constant along the length. Thus, Equation 2.22
becomes:
02
2
=dxMd [2.23]
By successive integration of Equation 2.23, it follows that:
BAxxM +=)( [2.24]
where A and B are integration constants. Considering only half of the beam in Figure 2-7
(i.e. Lx ??0 ) and since 0=M at the supports, then:
)()( LxAxM ??= [2.25]
Now, considering vertical equilibrium at midspan (at x = 0), the vertical load P is
resisted by the vertical components of N and Q:
)sin()cos(2 dxdwNdxdwQP ????= [2.26]
For ( ) ?? =sin , ( ) 1cos =?
dx
dwN
dx
dMP ????
2 [2.27]
Equations 2.9 and 2.25 predict that LWdxdw ?= and AdxdM = respectively, thus:
2
P
L
NWA ?= [2.28]
Then, the moment equation becomes:
))(2()( LxPLNWxM ??= [2.29]
36
At 0=x ,
LLNWPM )2( ?= [2.30]
Normalizing Equation 2.30 by the plastic moment, pM , and eliminating N with Equation
2.20, gives:
p
p
pp Md
WN
M
LP
M
M
?
????= 22
2 for 20
dW ?? [2.31]
square4 External Load, P:
Under combined bending and axial load, the external load, P is simply derived
with substitution of the previously obtained M and N equations as given in Equations
2.20 and 2.31, respectively, into yield condition (Equation 2.5). Thus:
2
24
1 dWPP
c
+= ( 20 dW ?? ) [2.32]
where LMP pc 2=
For transverse deflections W beyond d / 2, Equation 2.20 suggests that the beam
has reached the fully plastic phase ( pNN = ) and the applied load is resisted by only
plastic axial loads as illustrated in Figure 2-12. Therefore, vertical equilibrium ? = 0yF
requires:
0sin2 =??? ?pNP [2.33]
For LW== ??sin , then,
L
WNP p ??= 2 . [2.34]
37
Normalizing by cP and rearranging yields:
d
W
P
P
c
?= 4 (
2
dW ? ) [2.35]
Np Np
P
?
x
Figure 2-12 Forces at Midspan (x=0) at Fully Plastic Phase
In conclusion, as can be seen in Figure 2-13, the beam reaches the fully plastic
axial state at a deflection of one-half of the beam thickness at an external load of twice
the plastic collapse load, Pc (Jones 1989).
38
0
1
2
3
4
0 0.25 0.5 0.75 1 1.25
W / d
P / P c
Eq. 2.32
Eq. 2.35
A
C
Figure 2-13 Normalized Load?Deflection curve for simple beam with a rectangular
cross-section (Jones 1989)
A similar procedure was conducted for a fully fixed beam subjected to a
concentrated load at midspan by Haythornthwaite (1959) and results showed that:
2
2
1 dWPP
c
+= ( dW ??0 ) [2.36]
d
W
P
P
c
?= 2 ( dW ? ) [2.37]
where LMP pc 4=
39
Chapter 3. THEORETICAL MODELS
3.1 Overview and Scope
In this chapter, the fundamental behavior of ductile steel beams with idealized
boundary conditions due to finite displacements is presented. The ability of a beam to
carry loads by transitioning from a flexural mechanism to a cable?like mechanism will be
described. Two different theoretical models are developed: rigid?plastic beam model,
which is developed in a manner similar to that described by Jones (1989) as reviewed in
Section 2.4, and cable analysis, which reflects a special case of resisting the loads only by
means of axial forces developed on the member, like a cable.
Two load cases are considered: a concentrated load at midspan and uniformly
distributed load along the length. The theories are presented in such a way that load?
deflection characteristics of each case are described in a graphical manner, which is
obtained with use of a common flow of derivations.
Fully ductile behavior is assumed and stability issues are not considered throughout
this section.
3.2 Rigid?Plastic Analysis
Rigid?perfectly plastic behavior of beams was presented in section 2.4 along with
other important concepts used in the procedure. Therefore, all assumptions made therein
are valid in the development of this section.
40
Theoretical results for W?shaped beams will be presented. Therefore, an
interaction relationship between bending moment, M and axial force, N on the cross?
sectional level for a W?shape needs to be introduced. A similar procedure to that used for
rectangular sections (as presented in Chapter 2) is given. In this case, the M?N interaction
relationship depends upon the plastic neutral axis (PNA) location as illustrated in Figure
3-1 for positive M and N. The interaction equations are given as (Horne 1979):
square4 When the plastic neutral axis is in the web,
xwpp Zt
A
N
N
M
M
41
22
???
?
???
??= 0, ?NM [3.1]
square4 When the plastic neutral axis is in the flange,
xfppp Z
Ad
db
A
N
N
N
N
M
M
22111 ????
?
?
?
?
?
???
?
???
? ???
???
?
???
? ?= 0, ?NM [3.2]
where Mp and Np are the plastic moment and plastic axial force, respectively. Other
cross?sectional properties can be found in Appendix A.
Equations 3.1 and 3.2 are graphically represented in Figure 3-2 for a W30x124. In
this figure, pfM represents the plastic bending moment calculated using the flanges only
(about major axis) and pwN represents the plastic axial force calculated using the web
only. Equation 3.1 is valid between points A and B. Equation 3.2 is valid between points
B and C. At Point A, the PNA is at the centroid of the cross-section. At Point B, the PNA
is at the interface between web and the flange. At Point C, the PNA can be at or beyond
the top of the cross-section.
41
a
a
?y
?y
-?y
?
?
(a) (b) (c) (d)
a
a
?y
?y ?y
-?y
d
(d-2a) / 2
(d-2a) / 2
(e) (f) (g) (h)
tw
dw
d
bf
tf
- PNA in the Web
- PNA in the Flange
.
.
?
?
.
.
PNA
PNA
M N
Centroidal
Axis
Centroidal
Axis
-?y
(a), (e) Cross-Section (AISC notations are used and fillets neglected)
(b), (f) Stress distributions associated with N
(c), (g) Stress distributions associated with M
(d), (h) Strain profile when the neutral axis is in the web and flange respectively
y? is the yield stress and a is the distance between PNA and centroidal axis
Figure 3-1 Stress distributions in the cross-section based on the location of neutral axis
(Horne, 1979)
42
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25
M / M p
N / N p
???
?
???
?
p
pw
p
pf
N
N
M
M ,
(1,0)
(0,1)
C
A
B
PNA
PNA
PNA
PNA
Figure 3-2 M?N interaction for W?Shapes (W30x124)
43
3.2.1 Simply Supported Beam with a Concentrated Load at Midspan
The load?deflection characteristics of a rigid?perfectly plastic simple beam with
W?shape cross?section (Figure 3-3) can be derived with use of a procedure similar to that
described in section 2.4. Expressions for internal forces and external loads throughout the
deformation are derived, and then a graphical representation of overall behavior is
provided.
P
x
W
? w
2?
L L
l??? ,
Figure 3-3 W?Shaped Rigid?Plastic Simple Beam with a Concentrated load at Midspan
square4 Axial Force, N:
Once the mechanism condition is reached as shown in Figure 3-3; finite
transverse deflection, W results in development of axial forces, N. It is evident from
44
Figure 3-2(d) and (h) that regardless of the PNA location, the ratio of generalized
deformation rates is:
a=
?? ??
/ [3.3]
When the neutral axis is in the web, it can be seen from Figure 3-2(b) that:
ywtaN ????= 2 [3.4]
Thus,
A
ta
N
N w
p
??= 2 [3.5]
where AN yp ?= ? .
It is noted that the development of the ratio
?? ??
/ in Section 2.4 was independent of
the cross?section considered. Thus, Equation 2.15 is still valid, therefore Equation 3.3
becomes:
Wa = [3.6]
Substituting a into Equation 3.5 gives:
A
tW
N
N w
p
??= 2 [3.7]
For the special case when the PNA is at the interface between the web and the flange:
wypw ANN ?== ? [3.8]
Substituting Equation 3.8 into Equation 3.7 and solving for W gives:
22
w
w
w d
t
AW =
?= [3.9]
45
Equation 3.9 suggests that the PNA moves into the flange for deflections beyond 2wd . As
a result, Equation 3.9 is valid for 20 wdW ?? .
When the PNA is in the flange, it can be seen from Figure 3-3(f):
( )[ ] yfbadAN ??????= 2 [3.10]
and it follows that:
[ ]
f
p
bA adNN ????= 21 . [3.11]
Substitution of Equation 3.6 into Equation 3.11 gives:
[ ]
f
p
bA WdNN ????= 21 . [3.12]
Equation 3.12 suggests that the fully plastic cable state (i.e., pNN = ) is reached
when the second term is zero so that 2dW = . As a result, Equation 3.12 is valid in the
range of 22 dWdw ?? .
For a pure cable state, Equation 3.12 also suggests that the beam sustains Np for
deflections beyond 2d .Thus, the axial force expression can simply be given as:
1=
pN
N for
2
1?
d
W [3.13]
A plot of N / Np versus W / d is provided in Figure 3-4.
square4 Bending moment, M:
46
Bending moment as a function of transverse deflection, W, can simply be obtained
by substituting the axial force expressions obtained for the two different locations of the
PNA indicated earlier into the interaction equations. Thus, when the PNA is in the web,
substituting Equation 3.7 into 3.1 gives:
x
w
p Z
tW
M
M ??= 21 for
20
wdW ?? [3.14]
and when the PNA is in the flange, substituting Equation 3.12 into 3.2 gives:
f
xp
bZ WdMM ??= 4 )4(
22
for 22 dWdw ?? [3.15]
and finally for pure cable state:
0=
pM
M for
2
dW ? [3.16]
Similarly, a plot of M / Mp versus W / d is provided in Figure 3-5.
square4 External Force, P:
External load derivation was carried out with substitution of internal forces M and
N into the corresponding yield condition, as was reviewed in Section 2.4. Bending
moment, M was obtained based on vertical equilibrium at midspan (Equations 2.23
through 2.31). A similar procedure is used to relate bending moment, M to external load,
P and axial force, N, to finally obtain the external load, P as a function of W. Thus, when
the PNA is in the web, substituting Equation 3.7 into Equation 2.29 at x=0 gives:
LLA NtWPM pw )22(
2
?
????= [3.17]
Now, substituting Equation 3.17 and 3.7 into 3.1, the external load expression is given as:
47
x
w
c Z
tW
P
P 21+= for
20
wdW ?? [3.18]
where LMP pc ?= 2 .
When the PNA is in the flange, substituting Equation 3.12 into Equation 2.30
gives:
( )
???
?
???
? ?
??
?
??
? ??????= W
A
NbWdNLPM pf
p
2
2 [3.19]
Similarly, substituting Equation 3.19 and 3.12 into the yield condition given in Equation
3.2, gives:
???
?
???
? +?
?????? ?= AWbWdZPP f
xc
2
2
21 for
22
dWdw ?? [3.20]
where again LMP pc ?= 2 .
For the pure cable state, with use of vertical equilibrium of forces shown in Figure
2-12, it can be shown that:
xc Z
WA
P
P ?= for
2
dW ? [3.21]
with LMP pc ?= 2 .
The theoretical progression of behavior of a simply supported, centrally loaded,
rigid?perfectly plastic beam having a W-Shape cross-section is depicted in Figures 3-4
through 3-7. Figure 3-4 displays a plot of normalized axial force versus normalized
deflection at midspan. Figures 3-5 shows normalized bending moment with respect to
48
midspan deflection, W. Figure 3-6 represents the corresponding M-N interaction at the
midspan section and finally the relationship of the external load, P with respect to
transverse deflection is given in Figure 3-7. Three characteristic points are indicated in
the figures: A, B and C. Point A represents the onset of the formation of a flexural
mechanism (N=0 and M=Mp at midspan). As the beam deflects, plastic flow follows the
path as shown in Figure 3-6 and the generalized strain vector remains perpendicular to
the yield condition. Due to finite displacements, the beam develops axial forces and the
PNA starts to migrate upwards away from the centroid as indicated in Figure 3-6. Then,
axial force, N, increases linearly and bending moment, M, drops quadratically with W
until Point B is reached, which represents the special level of deflection of W = dw / 2, as
shown in Figures 3-4 and 3-5, respectively. At this point, the PNA is at the interface of
the web and the flange as shown in Figure 3-6. Beyond Point B, N continues to increase
linearly whereas M decreases quadratically with W at a much faster rate than before Point
B. Eventually, the pure cable state is reached (Point C) when the deflection is half as
much as the nominal depth of the beam, 2d (N=Np and M=0). Beyond this point, external
load, P, is resisted by plastic axial force, pN , alone with a cable?like action thus the
generalized strain vector rotates towards the N-axis as the PNA moves away from the
cross-section, as shown in Figure 3-6.
49
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1W / d
N / N p
A
B
C
Eq. 3.7
Eq. 3.12 Eq. 3.13
Npw / Np
W / dw= 0.5
Figure 3-4 Normalized Axial Force?Deflection Relationship (Midspan)
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1
W / d
M / M p
A
B
C
Eq. 3.14
Eq. 3.15
Eq. 3.16
Mpf / Mp
W / dw= 0.5
Figure 3-5 Normalized Moment?Deflection Relationship (Midspan)
50
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25
M / M p
N / N p
C
.?
A
B
?
?
.
.
?
PNA
PNA
PNA
PNA
.
PNA
Figure 3-6 Normalized M?N interaction for W-Shape cross-section (Midspan)
51
0
1
2
3
4
0 0.5 1 1.5 2
W / d
P / P c
A
W / dw=0.5
B
CEq. 3.18
Eq. 3.20
Eq. 3.21
Figure 3-7 Normalized Load?Deflection Relationship
3.2.2 Fully Fixed Beam with a Concentrated Load at Midspan
Similar theoretical steps to those described for a simply supported beam can be
carried out for a fully fixed beam (Figure 3-8 (a)) that has a W-Shape cross-section
subjected to a concentrated load at midspan. Figure 3-8 (b) illustrates the deflected shape
configuration. In this case, three plastic hinges form simultaneously, one at midspan and
one at each support once the flexural mechanism condition is reached. As a result, the
average strain over midspan plastic hinge length, l, is given as:
lLWL 2/112
2/1
2
2
?
?
?
?
?
?
?
?
???
?
?
???
? +=? [3.22]
52
With the aid of binomial series expression, and neglecting powers of
2
?????? LW greater than
two, gives:
????????????? lLLW 2
2
? [3.23]
Thus, the strain rate becomes:
lL
WW
?=
?
??
[3.24]
x
2,
l???
l??? 2,M(0)
x
M(0)
M(0)
? ?
+
P
W
? w
2?
L L
2,
l???
(a)
(b)
(c)
(a) Beam
(b) Deflected shape configuration
53
(c) Statically admissible moment diagram
Figure 3-8 Centrally loaded, fully fixed beam with W-Shape cross?section
On the other hand, the change in angle over the midspan plastic hinge length, l,
gives curvature as:
lL
W
?=
2? [3.25]
thus the curvature rate becomes:
lL
WW
?=
?
? 2?
. [3.26]
From Equations 3.24 and 3.26, it follows that:
2/
W=?? ?? . [3.27]
Now, internal load ? deflection characteristics can be identified.
square4 Axial Force, N:
Equation 3.3 suggests that 2/ Wa ==
?? ??
regardless of the location of the PNA.
Thus, Equations 3.5 and 3.11 give the following expressions respectively, when the PNA
is in the web, and when it is in the flange:
A
tW
N
N w
p
?= [3.28]
[ ]
f
p
bAWdNN ???= 1 [3.29]
For the special case when the PNA is at the interface between web and the flange:
ywpw ANN ??== . [3.30]
54
Substituting Equation 3.30 into Equation 3.28 and solving for W gives:
wdW =
Solving Equation 3.29 for pNN = (i.e. pure cable state):
dW =
As a result, Equation 3.28 is valid for wdW ??0 , whereas Equation 3.29 is valid for
dWdw ?? and suggests that the beam reaches a pure cable state when dW = .
Eventually at the pure cable state, the beam sustains Np, and therefore:
1=
pN
N for dW ? [3.31]
square4 Bending moment, M:
Moment?deflection equations can be obtained by substitution of axial force
expressions into their respective yield condition depending on the location of the PNA.
Thus, when the PNA is in the web, substituting Equation 3.28 into Equation 3.1 gives:
x
w
p Z
tW
M
M
41
2 ?
?= for wdW ??0 [3.32]
and when the PNA is in the flange, substituting Equation 3.29 into Equation 3.2 gives:
f
xp
bZWdMM ??= 4 )(
22
for dWdw ?? [3.33]
For pure cable state:
0=
pM
M for dW ? [3.34]
square4 External Force, P:
55
To use the procedure presented in Section 2.4 (Equations 2-23 through 2-31), a
bending moment expression is needed to obtain the external force as a function of W.
First, it is evident from Figure 3-8(c) that bending moment can be expressed as
(considering one-half of the beam due to symmetry):
BAxxM +=)( . [3.35]
Using the boundary values to obtain the constants A and B, at :0=x
BM =)0( [3.36]
and at :Lx =
BLAM +?=? )0( [3.37]
Solving Equations 3.36 and 3.37 simultaneously, gives:
L
MA )0(2?=
Thus, the resulting moment expression as a function of x is given as:
??
?
??
? +??= 12)0()(
L
xMxM [3.38]
and
L
M
dx
xdM )0(2)( ?= . [3.39]
Therefore, the equilibrium Equation 2.27, developed at midspan, becomes:
L
WN
L
MP ?+= )0(2
2 . [3.40]
Solving for )0(M gives:
24)0(
WNLPM ???= . [3.41]
56
Now, substituting )0(M into Equation 3.38 gives the moment expression as:
??
?
??
? +??
??
?
??
? ???= 12
24)( L
xWNLPxM . [3.42]
Now, external load expressions can be obtained depending on the location of the
PNA as follows:
? When the PNA is in the web, substituting Equation 3.42 (at x = 0) and
Equation 3.28 into Equation 3.1 gives:
x
w
c Z
tW
P
P
?
?+=
41
2
for wdW ??0 . [3.43]
? When the PNA is in the flange; substituting Equations 3.42 (at x = 0) and 3.29
into 3.2 gives:
???
?
???
? +?
?????? ?= 221
2 AW
bWdZPP f
xc
for dWdw ?? . [3.44]
? For the pure cable state, using the vertical equilibrium of forces shown in
Figure 2-12 gives:
xc Z
WA
P
P
2
?= for dW ? [3.45]
where LMP pc ?= 4 in this case.
We can see that the fixed-fixed case resulted in the same form of equations as the
simple beam case. However, twice as much deflection (W=d) is needed to form the cable
mechanism in this case.
This progression of events throughout the deflection of the beam is indicated in
Figures 3-9 through 3-11. Figures 3-9 and 3-10 display, respectively, the plot of
57
normalized axial force, N, and bending moment, M, versus normalized deflection at
midspan. Figure 3-11 displays the corresponding plot of normalized external load, P,
versus normalized deflection at midspan. A rigid?perfectly plastic beam starts to deflect
once three plastic hinges form at a load of cP (indicated as Point A, the onset of formation
of a flexural mechanism). When the transverse displacement W increases, the beam starts
to develop axial forces N and the plastic neutral axis starts to move upwards along the
web as depicted in Figure 3-6. As a result, M starts to drop (Figure 3-10) and yielding is
now controlled by combined bending moment and axial force. When the PNA gets into
the flange (Point B), moment starts to drop faster (Figure 3-10) and consequently, the
axial force increases rapidly (Figure 3-9) until it purely dominates the behavior (Point C).
This behavior between points B and C can be attributed to the fact that more area (wide
flange) attracts more axial force during the distribution of yielding across the cross?
section. Eventually, the beam reaches a cable state (Point C, where pNN = ).
58
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25 1.5
W / d
N / N p
A
B
C
Eq. 3.28
Eq. 3.29
Eq. 3.31
Npw / Np
W / dw= 1
Figure 3-9 Normalized Axial Force?Deflection Relationship (Midspan)
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25 1.5
W / d
M / M p
A
B
C
Eq. 3.32
Eq. 3.33
Eq. 3.34
Mpf / Mp
W / dw= 1
Figure 3-10 Normalized Moment?Deflection Relationship (Midspan)
59
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2
W / d
P / P c
W / dw= 1
B
C
Eq. 3.43
Eq. 3.44
Eq. 3.45
A
Figure 3-11 Normalized Force?Deflection Relationship (Midspan)
3.2.3 Fully Fixed Beam with Uniformly Distributed Load along the Length
Rigid?plastic behavior suggests that there is no deflection until the beam reaches
a flexural mechanism condition. In this case, plastic hinges (at midspan and end sections)
do not form simultaneously. As illustrated in Figure 3-12, it is evident from the theory of
plasticity that two plastic hinges form at the supports first (Figure 3-12(a)), and the beam
acts like a simple beam. Then, with formation of a midspan plastic hinge, the flexural
mechanism condition is reached (Figure 3-12(b)). As the beam deflects further, the
deflected shape can be anticipated to transition to a catenary?like shape towards the
formation of pure cable action (Figure 3-12(c)).
60
q
(a)
(b)
(c)
L L
Figure 3-12 Change in Deflected Shape Configuration Under Uniform Loading
The rigid?plastic analysis procedure presented in this study thus far has assumed a
consistent deflected shape throughout the deformation. However, incorporating this
deflected shape transition (as shown in Figure (3-12)) in the analysis may not be as
straightforward. To implement this phenomenon, two different theories will be presented:
Theory I, in which a triangular deflected shape is considered and Theory II, in which a
parabolic deflected shape is considered throughout the deformation.
A similar procedure to that used for the concentrated load cases can be carried
out. Theories for rectangular and W?Shaped beams are developed and presented next.
61
3.2.3.1 Beam with Rectangular Cross?Section
Using the yield condition provided under combined bending and axial force for
rectangular a cross?section, a rigid?plastic analysis procedure is conducted for each
theory.
3.2.3.1.1 Theory I
Figure 3-13 illustrates the schematics of Theory I, including the deflected shape
configuration and a statically admissible moment diagram.
With the aid of the general equilibrium equations presented in Section 2.4,
moment equilibrium in this case is given as:
qdxMd ?=2
2
[3.46]
which allows us to express moment as follows:
BAxqxxM ++?= 2)(
2
[3.47]
where A and B are integration constants.
To solve for the constants A and B, boundary values are used:
square4 At BMx =?= )0(0
62
x
q
? ?
+
2,
l???
l??? 2,
W w
2?
L L
2,
l????
M(0) M(0)
M(0)
(a)
(b)
(c)
Figure 3-13 Schematics of Theory I
square4 At )0(2)0(
2
MALqLMLx ++?=??=
Thus:
L
MqLA )0(2
2 ?= [3.48]
Then, the moment expression is given as:
)0()0(222)(
2
MxLMqLqxxM +?????? ?+?= [3.49]
Now, with use of the expression for vertical equilibrium at midspan given in Equation
2.27 for P=0, it can be shown that:
63
24)0(
2 NWqL
M ?= [3.50]
Having determined the moment expression, the load?deflection characteristics of
this case can be derived using the same analysis procedure as before.
square4 Axial Force, N:
It can be shown that use of a triangular deflected shape results in the same form of
equations as those of the concentrated load case. For example, the ratio of deformation
rates,
?? ??
/ , is independent of the cross?section considered. Thus:
2/
W=?? ?? [3.51]
and:
d
W
N
N
p
= for dW ? . [3.52]
Equation 3.52 suggests that a pure cable state is reached at a deflection of cross-
sectional depth d. Beyond this deflection, the beam resists the loads with the plastic axial
force, pN , where the expression is simply given as:
1=
pN
N for dW ? . [3.53]
square4 Bending moment, M:
Substitution of Equation 3.52 into the yield condition given in Equation 2.5 gives:
2
2
1 dWMM
p
?= for dW ? . [3.54]
Equation 3.53 suggests:
64
0=
pM
M for dW ? [3.55]
for a pure cable state.
square4 External Load, q:
Having determined internal force equations in Equations 3.50 and 3.52 for M and
N, respectively, we can substitute them into the yield condition for a rectangular cross?
section given in Equation 2.5. Thus, the normalized external load expression is given as:
2
2
1 dWqq
c
+= for dW ? [3.56]
where 24LMq pc = .
It should be remarked that Theory I predicts an equation that is identical to the
form predicted by Haythornthwaite (1959) for beams with a rectangular cross?section
under concentrated loading, as given in Equation 2.36.
At the pure cable state, vertical equilibrium of forces in the free body diagram
illustrated in Figure 3-14 can be constructed as:
Np Np
q
W
L L
?
Figure 3-14 Equilibrium of Forces at Pure Cable State
65
? =????= 0sin.220 ?py NLqF , from each it follows that:
d
W
q
q
c
= for dW ? [3.57]
where again, 24LMq pc = .
By inspection, Equations 3.56 and 3.57 do not give the same load at the deflection
d, at which the beam reaches pN . This discrepancy will be discussed in the context of
Theory II later.
For comparison purposes, the overall load?deflection characteristics of the beam
shown in Figure 3-13 associated with Theory I will be graphically illustrated and
discussed after the presentation of Theory II in the following section.
3.2.3.1.2 Theory II
As illustrated in Figure 3-15(b), a parabolic deflected shape configuration is
considered in development of this theory. The deflected shape can be expressed as:
???
?
???
? ?=
2
2
1 LxWw [3.58]
Taking the derivative of Equation 3.58 twice with respect to x gives:
22LWxdxdw ?= [3.59]
22
2 2
L
W
dx
wd ?= [3.60]
Therefore, the governing vertical equilibrium equation presented in Section 2-4
changes with use of this geometry. Substituting Equation 3.60 into Equation 2.8 gives:
66
0222
2
=+?
?
??
?
? ??+ q
L
WN
dx
Md and [3.61]
qLNWdxMd ?= 22
2 2
. [3.62]
x
q
? ?
+
2,
l???
l??? 2,
W w
2?
L L
2,
l???
?
M(0) M(0)
M(0)
(a)
(b)
(c)
W
Figure 3-15 Schematics of Theory II
By integrating Equation 3.62 twice, bending moment as a function of x becomes:
BAxxqxLNWxM ++?= 222 2)( . [3.63]
Solving for the constants A and B by introducing the boundary values considering
the moment diagram presented in Figure 3-15(c):
67
square4 At BALqLNWMLMLx +??=?=???= 2)0()(
2
( I )
square4 At BALqLNWMLMLx ++?=?=?= 2)0()(
2
( II )
Solving (I) and (II) simultaneously yields:
0=A and )0(2
2
MNWqLB ??=
With aid of the third boundary value, )0(M at midspan, where x=0, gives:
24)0(
2 NWqL
M ?= [3.64]
This deflected shape configuration requires derivation of the change in length of
the beam which relates the axial strain and associated axial force definitions. The actual
length of the deflected shape shown in Figure 3-15 (b) can be obtained with use of the arc
length formulation given as:
[ ] dxxfL b
a
? ?+=? 2)(1 [3.65]
in which )(xf ? is the first derivative of the function of the shape of arc. Due to symmetry,
only one?half of the beam is considered. Substitution of Equation 3.59 into Equation 3.65
gives the actual length as:
dxLWxL
L
? ???????+=?
0
2
2
21 . [3.66]
For convenience, let LW=? and Lx=? then,
??
?
??
? +?=? ? ??? dLL 1
0
2241 [3.67]
68
Evaluation of this integral was performed using Mathcad version 14 (2007) and the
solution was obtained to be a sign function of ? as follows:
( )
?
?
?
?
?
?
?
? ++??++?=?
?
??????
4
14)sgn(2ln)sgn(
2
14)sgn( 222 cccLL [3.68]
where csgn(? )sign function for LW=? .
By inspection, this complex expression may be approximated using a relatively simple
equation in the form:
( )2)(1 ?? ?+?=? LL [3.69]
In Figure 3-16, the normalized actual length and the approximation (Equation
3.69) is plotted versus ? with 707.0=? . It can be seen that a good correlation was
obtained in representing the actual length of the deflected shape with Equation 3.69.
Since the change in length is:
LLL 22 ??=? [3.70]
Then, substituting Equation 3.69 into Equation 3.70, the change in length becomes:
?
?
?
?
?
?
?
?
???
?
?
???
?
??
?
??
? ?+?=? L
L
WLL 212 ? [3.71]
Setting 707.0=? Equation 3.71 reduces to:
2
??
?
??
??=?
L
WLL [3.72]
69
0.75
1
1.25
1.5
0 0.25 0.5 0.75 1
?
L' / L
Actual Length
Approximation
Figure 3-16 Length of Arc Approximation
Then, the change in length over a total plastic hinge length l2 as shown in Figure 3-15(b)
gives an axial strain, ? , of:
2
22 ??
?
??
??=?=
L
W
l
L
l
L? . [3.73]
Thus, the axial strain rate,
??
, is given as:
lL
WW
?
?=
?
??
[3.74]
On the other hand, the total change in angle over the plastic hinge at the end
section, which has a length of 2l , the curvature rate in this case becomes:
lL
W
?=
?
? 4?
. [3.75]
70
Thus, the ratio of
?? ??
/ , is given as:
4/
W=?? ?? . [3.76]
Load?deflection characteristics according to this theory can be developed using
the rigid?plastic analysis procedure.
square4 Axial Force, N:
This ratio of
?? ??
/ was obtained in Equation 2.19 for a beam of rectangular cross?
section. Thus, a combination of this equation with Equation 3.76 gives the axial force
expression as:
d
W
N
N
p 2
= . [3.77]
Equation 3.77 suggests that twice as much deflection, d2 is needed for the beam
to reach the pure cable state ( pNN = ) as in Theory I. Therefore, it is valid for a
transverse deflection range of: dW 20 ?? .
Then for the pure cable state,
1=
pN
N for dW 2? . [3.78]
square4 Bending moment, M:
Again, using the yield condition defined for a rectangular cross?section given in
Equation 2.5, moment equations can be shown to be:
2
2
41 d
W
M
M
p
?= for dW 2? [3.79]
and Equation 3.77 suggests that:
71
0=
pM
M for dW 2? [3.80]
at the pure catenary state
square4 External Load, q:
Having determined internal force equations in equations 3.64 and 3.77 for M and
N respectively, we can substitute them into the yield condition for a rectangular cross?
section (Equation 2.5). Thus, the normalized external load expression in this case
becomes:
2
2
4
31
d
W
q
q
c
+= for dW 2? [3.81]
where 24LMq pc = .
In a similar manner, the derivations can be extended for the deflections
beyond d2 . Figure 3-17 illustrates the free body diagram after the beam reaches a pure
cable state. Vertical equilibrium requires:
? =????= 0sin.220 ?py NLqF [3.82]
W
W
LL
NpNp
q
?
Figure 3-17 Equilibrium of forces at pure catenary state
72
For LW2sin == ?? , it can be shown that:
d
W
q
q
c
2= For dW 2? [3.83]
where 24LMq pc = .
It is evident that Equations 3.81 and 3.83 result in the same load level at the point
of a pure cable state ( dW 2= ).
Figures 3.18 through 3.20 illustrate the overall load?deflection characteristics of
fully clamped, rectangular shaped beams under uniform loading as predicted by Theory I
and II. Characteristic points A and C represent the point of mechanism condition
formation and point of pure cable state, respectively. The subscripts I and II refer to
Theory I and II, respectively.
Since Theory I did not accurately predict the behavior upon formation of the pure
cable state, it is not represented in Figures 3-18, 3-19, and 3-20 beyond point CI.
However, it should be noted that the extension line of Theory I (Figure 3-18) that passes
through the origin coincides with point CII, which represents the point of pure cable state
predicted by Theory II. This agreement can be attributed to the transition in deflected
shape configuration presented in Figure 3-12.
As a concluding remark, different theories based on different deflected shape
considerations provide boundaries to the real behavior, which is discussed in the context
of the finite element analyses presented in Chapter 4.
73
0.0
1.0
2.0
3.0
4.0
5.0
0 0.5 1 1.5 2 2.5
W / d
q / q c
Theory I Theory II
CII
AI,II
CIEq. 3.56 Eq. 3.81
Eq. 3.83
Figure 3-18 Normalized Load?Deflection relationship
0
0.25
0.5
0.75
1
1.25
0 0.5 1 1.5 2 2.5
W / d
N / N p
Theory I Theory II
CII
AI,II
CI
Eq. 3.52
Eq. 3.77
Eq. 3.78
Figure 3-19 Normalized Axial Force?Deflection relationship
74
0
0.25
0.5
0.75
1
1.25
0 0.5 1 1.5 2 2.5
W / d
M / M p
Theory I Theory II
AI,II
CIICI
Eq. 3.54
Eq. 3.79
Figure 3-20 Normalized Moment ? Deflection Relationship
3.2.3.2 Beam with W?Shaped Cross?Section
Two theories based on triangular and parabolic deflected shapes can be developed
for beams with W?Shapes. Characteristics, such as the mechanism condition and
statically admissible moment diagram for each theory as presented in Figure 3-13 and
Figure 3-15, respectively, are independent of the cross?section. The difference here is the
dependency of the yield condition on the location of the PNA across the beam depth, as
given in Equations 3-1 and 3-2. Load?deflection characteristics of W?Shaped beams
predicted by each theory can be obtained in a similar manner as before.
3.2.3.2.1 Theory I
75
It was observed that results using Theory I predicted the identical form of
equation given by Haythornthwaite (1959) for the beam with a rectangular cross?section
under concentrated load case. Therefore, with similar anticipation, the rigid?plastic
analysis procedure was conducted, and the results were shown to be analogous to the
load?deflection characteristics given for W?Shaped beams under concentrated loading as
presented in Section 3.2.2. Depending on the location of the PNA, predictions of Theory I
are given as follows:
square4 When the PNA is in the web, where wdW ??0
A
tW
N
N w
p
?= , [3.84]
x
w
p Z
tW
M
M
?
??=
41
2
,and [3.85]
x
w
c Z
tW
q
q
?
?+=
41
2
, [3.86]
Where 24LMq pc = .
square4 When the PNA is in the flange, where dWdw ??
[ ]
f
p
bAWdNN ???= 1 , [3.87]
( )
x
f
p Z
bWd
M
M
?
??=
4
22
, and [3.88]
?
?
?
?
?
?
?
? +?
?????? ?= 221
2 AW
bWdZqq f
xc
, [3.89]
76
where 24LMq pc = .
As shown in the procedure presented for beams with rectangular cross?section,
Theory I did not accurately predict the behavior beyond pure cable formation (i.e. for
dW ? ) with use of the vertical equilibrium presented in Figure 3-14. Thus, equations
predicted by Theory I beyond the pure cable state are not presented. In similar manner,
for comparison purposes, graphical representation of results given herein will be
presented after presentation of Theory II results.
3.2.3.2.2 Theory II
Schematics of Theory II were given in Figure 3-15 earlier. A similar rigid?plastic
analysis procedure as that presented in Section 3.2.3.1.2 was implemented for W?Shaped
beams with introduction of corresponding yield condition. Therefore, load?deflection
characteristics of the behavior predicted by Theory II can be obtained as follows:
square4 Axial Force, N:
? When the PNA is in the web:
Coupling Equation 3.3 with Equation 3.76, then substituting the result into
Equation 3.5, the normal force expression yields:
A
tW
N
N w
p 2
?= . [3.90]
Applicability of Equation 3.90 can similarly be investigated for the special
case where the PNA is at the interface between web and flange. Solving for W at
pwNN = gives:
wdW 2= . [3.91]
77
Thus, Equation 3.90 is valid for the range of wdW 20 ?? .
? When the PNA is in the flange:
Coupling equations 3.3 and 3.76, then substituting the result into Equation
3.11, the normal force expression is given as:
A
bWd
N
N f
p
??????? ?
?= 21 . [3.92]
Equation 3.92 suggests that the beam reaches a pure cable state at a
deflection of twice the beam depth d2 . Thus it is valid in a range of
dWdw 22 ?? .
? Beyond the pure cable state:
1=
pN
N for dW 2? . [3.93]
square4 Bending moment, M:
Substitution of the previously defined normal force equations into their
corresponding yield condition gives:
? When the PNA is in the web:
x
w
p Z
tW
M
M
?
??=
161
2
wdW 20 ?? . [3.94]
? When the PNA is in the flange:
x
f
p Z
bWd
M
M
42
2
2 ?
???
?
???
?
???????= for dWdw 22 ?? . [3.95]
? Beyond the pure cable state:
78
0=
pM
M for dW 2? . [3.96]
square4 External Load, q:
Substituting the axial force equations for each location of the PNA, along with the
moment equation derived from equilibrium equations (given in Equation 3.64), into the
respective yield condition, gives:
? When the PNA is in the web:
x
w
c Z
tW
q
q
16
31 2+= For
wdW 20 ?? [3.97]
where 24LMq pc = .
? When the PNA is in the flange:
??
?
??
? +?
?
??
?
? ???
?
??
?
? ?= AWbWdWd
Zq
q
f
xc
223241 . [3.98]
? Beyond the pure cable state:
With use of vertical equilibrium of forces in the free body diagram as
given in Figure 3-17, it can be shown that:
xc Z
AW
q
q
?= 2 for dW 2? [3.99]
where 24LMq pc = .
Finally, progression of events in a graphical manner is indicated in the Figures 3-
21, 3-22 and 3-23. Load?deflection characteristics predicted by the two theories for a W?
Shaped beam under uniform loading are presented. Figure 3-21 displays a plot of
79
normalized external load versus normalized deflection, W, at midspan. Figures 3-22
shows the normalized axial force, N, with respect to normalized midspan deflection, W,
and finally the relationship of the bending moment, M with respect to transverse
deflection is given in Figure 3-23. In these figures, characteristic points A, B and C are
again indicated with respective subscripts representing each theory presented herein. It
can be seen that the overall progression of events are common for the two theories except
they occur at a different level of transverse deflection for each theory. Theory I is again
presented with a dashed line beyond the pure cable state (CI), as it was shown that it does
not accurately represent the behavior in this region.
0
1
2
3
4
0 1 2 3 4
W / d
q / q c
Theory I Theory II
CII
AI,II Eq. 3.89
Eq. 3.98
Eq. 3.99
BICI BII
Eq. 3.97
Eq. 3.86
Figure 3-21 Normalized Load?Deflection relationship
80
0
0.25
0.5
0.75
1
1.25
0 1 2 3
W / d
N / N p
Theory I Theory II
CII
AI,II
Eq. 3.87
Eq. 3.92
Eq. 3.93
BI
CI
BII
Eq. 3.90
Eq. 3.84
Figure 3-22 Normalized Axial Force?Deflection relationship
0
0.25
0.5
0.75
1
1.25
0 1 2 3
W / d
M / M p
Theory I Theory II
CII
AI,II
Eq. 3.85
Eq. 3.95
Eq. 3.94
BI
CI
BII
Eq. 3.88
Eq. 3.96
Figure 3-23 Normalized Moment?Deflection relationship
81
3.2.4 Summary
The rigid?plastic behavior of ductile steel beams with idealized boundary
conditions was presented. Load?deflection characteristics for each loading, boundary
conditions and cross?section types were described throughout the deformation of the
beam. Results showed that the load?deflection characteristics for each cross?section type
are predominantly affected by the corresponding yield condition under combined bending
and axial forces. It was also seen that a similar form of equations for normalized load,
axial force and bending moment as a function of transverse deflection W and cross?
sectional properties, described the behavior at each case. The amount of transverse
deflection at the onset of pure cable behavior, defined as Wcat, was predicted in order of
nominal depth, d of the cross?section independent of the cross-section type. Table 3-1
summarizes the Wcat predictions using the rigid?plastic theory for each case presented
herein.
Table 3-1 Wcat predicted by Rigid?Plastic Theory
82
3.3 Cable Analysis
In this section, a theoretical analysis procedure is developed for a special type of
behavior, commonly referred to as ?cable action?. It is assumed that the member has no
resistance to bending; therefore, external loads are resisted only by means of axial forces
that develop in the member as it deflects. With use of the elastic?perfectly plastic
material idealization shown in Figure 3-24, deformations remain elastic until the plastic
axial force, Np, is reached (Point C). Beyond this point, the beam member sustains Np and
yielding occurs along the length. Theoretically, this type of ?pure cable? behavior can be
anticipated from infinitely long beams as discussed later in Chapter 4.
?
?
?y
E
C
?y
Figure 3-24 Material idealization used in Cable analysis
It can be anticipated that the load?deflection characteristics of this type of
behavior are independent of the boundary conditions associated with the rotational degree
83
of freedom at the supports, since there is no resistance to bending. Therefore, assuming
full axial fixity at the supports, an analysis procedure was developed to describe the load?
deflection characteristics under a midspan concentrated load, and under a uniformly
distributed load is presented next.
3.3.1 Concentrated Load Case
An analysis procedure is presented to derive internal and external loads as a
function of midspan deflection, W for a beam under a concentrated load at midspan.
square4 Axial Force, N:
Axial strain associated with axial forces can be determined by deriving the change
in length of the beam as shown in Figure 3-25.
LWLLLL ?+=??=? 22 [3.100]
and
L
L?=? . [3.101]
Substituting Equation 3-100 into Equation 3-101 gives:
11
2/1
2
2
???
?
?
???
? +=
L
W? [3.102]
For small LW , with the aid of a binomial expression, Equation 3.102 can be reduced to the
form:
2
2
1
??
?
??
??=
L
W? . [3.103]
Thus, the associated axial force can be determined as:
84
AEAN ??=?= ?? . [3.104]
Substituting Equation 3.103 into Equation 3.104 yields:
2
2 ??
?
??
???=
L
WAEN . [3.105]
Thus, Equation 3.105 predicts that the axial force, N, is a quadratic function of
transverse deflection, W.
In the special case when the beam reaches the pure plastic cable state
(i.e. ANN yp ?== ? ), Equation 3.105 gives:
EL
W ycat ??= 2 . [3.106]
It can be seen that, Equation 3.106 is independent of the cross?section of the
beam.
W
x
N N
P
?
L L
L? L?
Figure 3-25 Free body diagram under concentrated load
square4 External Load, P:
Considering vertical equilibrium of the forces given in Figure 3.25:
85
? =????= 0sin20 ?NPFy
and assuming that LW== ??sin , it follows that:
L
WNP ??= 2 . [3.107]
Substituting Equation 3.105 into Equation 3.107, gives:
3
??
?
??
???=
L
WAEP . [3.108]
Thus, Equation 3.108 gives the external load, P, as a cubic function of transverse
deflection, W.
3.3.2 Uniformly Distributed Load Case
The deflected shape associated with this load case is represented with a parabolic
configuration as illustrated in Figure 3-26. The rigid?plastic analysis procedure presented
previously indicated that this configuration appropriately represents the behavior at the
pure cable state.
To obtain load?deflection characteristics of this case, a similar procedure is
carried out to that for the concentrated load case presented earlier.
square4 Axial Force, N:
The axial strain based on a parabolic configuration was derived in the context of
rigid?plastic analysis. Recall, considering one?half of the beam that the actual length was
approximated in terms of a parameter,? . For convenience, derivations herein will be
given as a function of? and then, an acceptable value used to represent the actual length
86
in the deflected shape associated with this type of behavior is provided. Therefore, with
use of Equation 3.71, change in length is given as:
2
22 ?
?
?
??
???=?
L
WLL ? [3.109]
Thus axial the strain over the length of the beam member:
2
2
2 ??
?
??
??=?=
L
W
L
L ?? [3.110]
W
W
LL
NN
q
?
Figure 3-26 Free body diagram under uniform loading
Substituting 3.110 into Equation 3.104, gives:
2
2 ?
?
?
??
????=
L
WAEN ? [3.111]
Equation 3.111 predicts that the axial force, N, is a quadratic function of
transverse deflection, W.
For the special case, when the beam reaches the pure plastic cable state
(i.e. ANN yp ?== ? ) it can be deduced from Equation 3.111 that:
87
EL
W ycat ?
? ?=
1 [3.112]
It can be seen that Equation 3.112 is also independent of the cross-section.
square4 External Load, q:
Considering vertical equilibrium of forces given in Figure 3.26:
? =????= 0sin.220 ?NLqFy . [3.113]
It follows that:
2
2
L
NWq = . [3.114]
Substituting Equation 3.111 into Equation 3.114 gives:
322
??
?
??
??=
L
W
L
EAq ? . [3.115]
Thus, Equation 3.115 gives external load, q, as a cubic function of transverse
deflection, W.
To represent the actual length of the deflected shape shown in Figure 3-26, the arc
length approximation given in Equation 3.69 is utilized. In similar manner, an infinitely
long beam assumption will predict an infinitely small LW . In Figure 3-27, the ?
parameter (determined according to Equation 3.69) is plotted versus LW=? . By
inspection, it is found that ? approaches 0.816 when? goes to zero. As a result, under
uniform loading, the deflected shape associated with this special case of ?cable action?
can be approximated by Equation 3.69 with 816.0=? .
88
0
0.2
0.4
0.6
0.8
1
0 0.25 0.5 0.75 1?
?
Figure 3-27 ?? ? relationship
3.3.3 Summary
Load?deflection characteristics of ductile steel beams under different load
conditions predicted by cable behavior were presented. It was seen that the procedure is
predominantly affected by the deflected shape configuration and is independent of cross?
section used. In fact, the deflection at the onset of pure cable behavior, Wcat was shown to
be a function of material properties and span length. By rearranging Equations 3.106 and
3.112, Wcat predictions by cable theory can be given as:
square4 LEW ycat ??= ?2 for a concentrated midspan loading, and
square4 LEW ycat ??= ??1 for uniformly distributed loading,
89
where ? is a parameter used to determine the actual length of the parabolic deflected
shape, and can be taken as 0.816. As remarked earlier, these predictions are independent
of the boundary conditions, therefore, they can be used for simple beams as well as for
beams fixed at both ends.
3.4 Conclusion and Expected Behavior
In this chapter, ductile behavior of steel beams under finite displacements was
described using two different theories; rigid?plastic and cable. It can be recognized that
each of these models represents special cases of beam behavior. Therefore, a conclusion
can be driven with regards to how each theory relates to the actual, elastic?perfectly
plastic behavior which is commonly used in practice. As illustrated in Figure 3-28, which
graphically describe the overall behavior on M?N interaction at a cross?sectional level,
rigid?plastic and cable theories can be regarded as theoretical lower and upper bounds of
the actual behavior, respectively. The paths on the figure represent the behavior depicted
by each case throughout the deformation of the beam. Brief descriptions of these paths
are given as follows:
square4 Path OAC: Represents rigid?plastic behavior. No deflection occurs until the
mechanism condition is reached ( pMM = at all possible plastic hinge locations).
Once the condition is satisfied (Point A), the beam starts to deflect, and develops
axial forces, N. Form then on, yielding occurs with a combination of M and N and
follows the path shown until the beam reaches the pure cable state (Point C).
90
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25
M / M p
N / N p
Ca
ble
Rigid?Plastic
A
C
Figure 3-28 Expected behavior representation on M?N interaction
square4 Path O-dashed lines-C: Represents the actual behavior, depending upon the flexibility
of the elastic?plastic beam member. For very flexible, i.e. long beams, the behavior
may be represented with steeper dashed lines and approaches cable theory. For very
short, relatively rigid beams, the behavior is represented with shallower dashed lines,
approaching rigid-plastic theory. In the case of the dashed lines, the beam may
encounter finite displacements in the elastic regime and develop axial forces. When
91
the axial forces become large enough, plastic hinges form under a combination of M
and N (before reaching full pM ).
square4 Path OC: Represents the cable behavior. The beam has no bending resistance and
carries the loads only by means of axial force throughout the deflection until reaching
a pure cable state (i.e. N=Np). As noted earlier, infinitely long beams may exhibit this
special upper bound behavior.
To summarize, the actual behavior of ductile steel beams will fall somewhere in
between the bounding theories developed in this chapter. Ensuing efforts presented in the
following chapters will focus on how to represent the characteristics of actual behavior
with use of these unique theories.
92
Chapter 4. ANALYTICAL MODELS
4.1 Introduction
In this chapter, Finite Element (FE) analyses of steel beams with idealized
boundary conditions, including the effect of geometric and material nonlinearity, are
described. First, a set of preliminary FE analyses were executed for comparison with the
theoretical findings presented in Chapter 3. Then, a parametric study was conducted to
identify the main geometric factors affecting the actual behavior.
Both loading and cross?sectional cases considered for steel beams, as presented in
Chapter 3 were studied, focusing on beams with fixed ends. FE models were developed
and analyzed using Open System for Earthquake Engineering Simulations (OpenSees)
software (McKenna et al. 2000).
4.2 Modeling Concepts
Modeling tools available in OpenSees (McKenna et al. 2000) are presented next
and were used in this study.
4.2.1 Fiber Section Discretization
In order to keep track of the propagation of yielding through the cross-section, the
fiber section modeling object available in OpenSees was used.
93
Rectangular
tfiber
W - Shape
Subdivisions (Fibers)
Figure 4-1 Schematic of Fiber Section Discretization of Cross?Sections
As illustrated in Figure 4-1, sections considered in this study were subdivided into
smaller regions (fibers) for which the material stress?strain response is integrated to give
the resultant behavior.
Thickness of the fibers, tfiber was consistently taken as 5x10-3 inches for all the FE
models presented herein. For W?Shapes, fillets are neglected and the required number of
fibers was calculated, and then assigned to the web and the flange.
4.2.2 Corotational Transformation
In order to account for geometric nonlinearity, the geometric transformation
object that is available in OpenSees, the so-called ?corotational transformation? was
used. In this procedure, the beam element stiffness and internal forces are transformed
from the basic (reference) system to a global coordinate system. According to De Sousa
94
(2000), geometrically nonlinear systems can be analyzed using this formulation in which
rigid?body displacements are set apart from element deformations by attaching a
reference coordinate system that rotates and translates with the element throughout the
deformation.
4.2.3 Selection of Beam Element
Among the variety of element types available in the OpenSees library, a force-
based ?Nonlinear Beam?Column Element? was selected for use in this study. This
element, which was proposed by Neuenhofer and Filippou (1998) utilizes force
interpolation functions for varying internal forces due to transverse displacements and
explicitly satisfies equilibrium in the deformed shape. The spread of plasticity is
considered along the length of the element. Considering both geometric and material
nonlinearity in the problem, the use of this element type, in conjunction with fiber section
and corotational transformation objects in OpenSees, has been proposed by Scott et al
(2008).
The number of elements along the length of the beam models was consistently
taken as one hundred (100) throughout the FE analyses presented herein.
4.2.4 Material
A uniaxial elastic?perfectly plastic material model, as shown in Figure 4-2 was
used for all the FE models.
95
$E - tangent
$epsyP - strain or deformation at which
material reaches plastic state in
tension
$epsyN - strain at which material reaches
plastic state in compression
$eps0 - initial strain (optional)
Figure 4-2 Uniaxial Elastic?perfectly plastic material
(http://opensees.berkeley.edu/OpenSees/manuals/usermanual/index.html)
4.2.5 Loading and Analysis
The static ?Displacement Control? analysis object available in OpenSees was
used throughout the FE analyses presented herein. In this type of analysis, an appropriate
midspan transverse displacement increment is given as an input. The applied external
load is also inputted as the corresponding plastic collapse load for each load case
considered ( cP or cq ). For the standard case considered here (fully fixed beams spanning
2L), these plastic collapse loads are calculated with use of the theory of plasticity and
given as:
square4 LMP pc 4= for beams with a concentrated load at midspan
96
square4 24LMq pc = for beams with a uniformly distributed load along the length.
where pM was calculated neglecting the fillets.
As a result, at each step, using a given displacement increment, OpenSees
provides the external load as a fraction of the corresponding plastic collapse load inputted
(
cP
P or
cq
q ).
Typical OpenSees input files for the concentrated load case and the distributed
load case are provided in Appendices B1 and B2, respectively.
4.3 Preliminary FE Analyses
The theoretical fundamentals of rigid?plastic and cable behavior for ductile steel
beams with idealized boundary conditions were presented in Chapter 3. Here in Chapter
4, a set of preliminary nonlinear FE analyses on steel beams with similar load and
geometric configuration is described, with an emphasis on the fixed-fixed beams. The
primary objectives of this section are as follows:
square4 Verify the theoretical behavior predicted by each theory.
square4 Generate FE analysis results using elastic-perfectly plastic material properties and
compare to theoretical results.
4.3.1 Approach
The approach taken to accomplish the objectives of this section is given next.
square4 Comparison with Rigid?plastic Theory:
97
In order to enable a direct comparison of FE analysis results with the rigid-plastic
theory, a simple approach that involves the use of various values of Young?s Modulus of
Elasticity, E, was used, with a value of 29,000 ksi for steel as the benchmark case.
square4 Comparison with Cable Theory:
As noted earlier, theoretically pure cable behavior may be anticipated for
infinitely long beams. In order to enable a comparison of FE analysis results with the
cable theory, rather large span lengths were assigned.
square4 Presentation of results:
The preliminary FE analysis results are presented graphically in comparison with
theoretical results. For comparison with the rigid-plastic theory, the following plots are
shown: normalized external load,
cP
P (or
cq
q ), versus normalized deflection,
dW ,
normalized axial force,
pN
N , versus normalized deflection, dW , normalized moment,
pM
M , versus normalized deflection, dW , and normalized M?N interaction
(
pN
N versus
pM
M ). For comparison with cable theory, the following plots are shown:
external load, P (or q) versus normalized deflection, LW , normalized axial force, ,
versus normalized deflection, LW and normalized M ? N interaction (
pN
N versus
pM
M ). M and N are typically given at midspan, unless noted otherwise.
98
4.3.2 Description of Preliminary FE Models
Figure 4-3 shows a schematic of the beam models considered in the preliminary
FE analyses. The beam which is fixed at its ends is modeled with 100 nonlinear beam-
column elements along the length.
P
L L
L L
q
(a)
(b)
Figure 4-3 Schematic of preliminary FE beam models
The cross?section of the beam was chosen as either W-Shaped or rectangular.
When a W-Shaped cross-section was used, a first floor W30x124beam section was used
as typical (Khandelwal and El-Tawil 2007), whereas a rectangular cross?section was
selected in such a way that it gave approximately the same plastic section modulus, Zx, as
a W30x124. As a result, the rectangular section chosen had a unit width of 1 inch and a
nominal depth, d, of 40 inches. In addition, a W24x192 was used in some distributed load
cases as noted in the relevant sections. Cross?sections were discretized as fiber sections
as illustrated in Figure 4-1 with a typical fiber thickness, fibert , of 3105 ?x inches.
99
As noted earlier, a uniaxial elastic?perfectly plastic steel material model was used
in all cases. The various material yield stresses, y? , and span length, 2L, of the beams
used in this study are given in Table 4-1. In addition, the elastic moduli used in the FE
analyses for comparison with the rigid?plastic theory are given in Table 4-2. In these
tables, N/A implies that results for the case given are not available.
Table 4-1 Summary of Model Parameters in Preliminary FE Models
Table 4-2 Elasticity Modulus
4.3.3 Comparison of Results with Rigid?Plastic Theory
The beam shown in Figure 4-3 was modeled and analyzed in OpenSees with the
parameters given in Table 4-1 for different elastic moduli as specified in Table 4-2. The
presentation of results is broken into the two load cases considered with rectangular and
W?Shapes.
100
4.3.3.1 Concentrated Load Case
Analysis results for the beam shown in Figure 4-3(a) are given in comparison
with rigid?plastic theoretical results for each cross?section considered.
4.3.3.1.1 Beam with Rectangular Cross?Section
FE analysis results for the beam shown in Figure 4-3(a) with a rectangular cross-
section are presented in comparison with theoretical results in Figures 4-4, 4-5 and 4-6.
The onset of pure cable behavior is referred to in these figures as Point ?C? for the
benchmark case and the theoretical cases. Figures 4-4 through 4-6 show that there is a
significant difference between theory and FE analysis in predicting the onset of pure
cable behavior. According to rigid-plastic theory, pure cable behavior is reached when
W=d, whereas the FE analysis results for the benchmark case predict that pure cable
behavior is reached when W=1.34 d. When large values of E are used, it can be seen that
the point of pure cable behavior approaches that predicted by theory.
101
Note: C is the onset of point of pure cable behavior
Figure 4-4 Normalized Load?Deflection Plot
Note: C is the onset of point of pure cable behavior
Figure 4-5 Normalized Axial Force?Deflection Plot (Midspan)
102
Note: C is the onset of point of pure cable behavior
Figure 4-6 Normalized Moment?Deflection Plot (Midspan)
Another comparison can be made on the normalized M?N interaction plot shown
in Figure 4-7. The theoretical point for the onset of the flexural mechanism condition is
indicated as Point ?A? whereas Point ?C? represents the point of pure cable behavior. It
can be seen that the beam models with large E almost follow the path predicted by the
rigid?plastic theory. On the other hand, the benchmark case, E reaches Point A before
developing full Mp. The reason is that the axial force developed in the elastic regime
causes a plastic hinge under combined M and N. As a result, the benchmark case
representing the actual behavior reaches pure cable behavior at a significantly larger
transverse deflection than predicted by the rigid-plastic theory.
103
Figure 4-7 Normalized M?N interaction Plot (Midspan)
4.3.3.1.2 Beam with W?Shape Cross?Section
FE analysis results for the beam shown in Figure 4-3(a) with a W30x124 cross-
section are presented in Figures 4-8 through 4-11. The point of pure cable behavior is
referred to in these figures as Point ?C?. In addition, the theoretical Point B where the
PNA is at the interface between the web and flange is also shown in the normalized axial
force versus the normalized deflection and normalized moment versus normalized
104
deflection plots shown in Figures 4-9 and 4-10, respectively. In general, similar
comparison statements can be made as for the rectangular cross?section results. As can
be seen in these figures, the primary difference between the theoretical and benchmark
cases is that E is again observed to greatly affect the prediction of the point of pure cable
state (Point C). According to theory, pure cable behavior is reached when W=d for W-
Shapes, whereas the FE analysis results for the benchmark case predict that pure cable
behavior is reached when W=1.68 d. When large values of E are used though, it can be
seen that the point of pure cable behavior predicted by the FE models approaches that
predicted by theory.
Note: C is the onset of point of pure cable behavior
Figure 4-8 Normalized Load?Deflection Plot
105
Note: C is the onset of point of pure cable behavior
Figure 4-9 Normalized Axial Force?Deflection Plot (Midspan)
Note: C is the onset of point of pure cable behavior
Figure 4-10 Normalized Moment?Deflection Plot (Midspan)
106
In addition, it is evident from the M?N interaction relationship at midspan shown
in Figure 4-11 that all cases eventually fall on the yield curve and follow the same path.
For the benchmark case, in which E=29,000 ksi, the beam again exhibits substantial
transverse deflections in elastic regime and develops axial forces. Thus, the mechanism
condition forms under combined M and N, as opposed to the prediction by the rigid?
plastic theory, where for Point A, M=Mp.
Figure 4-11 Normalized M?N interaction Plot (Midspan)
107
4.3.3.2 Distributed Load Case
Preliminary FE analysis results for the beam shown in Figure 4-3(b) are compared
to results obtained using the rigid?plastic theory for each of the cross?sections and elastic
moduli as given in Tables 4-1 and 4-2, respectively.
4.3.3.2.1 Beam with Rectangular Cross?Section
FE analysis results for the beam shown in Figure 4-3(b) are presented in
comparison with result obtained using rigid-plastic theory in Figures 4-12, 4-13 and 4-14.
The onset of pure cable behavior is indicated in these figures as points CI and CII, for
Theory I and Theory II, respectively, as presented in Chapter 3. According to Theory I,
pure cable behavior is reached when W=d, whereas according to Theory II, pure cable
behavior is reached when W=2d. The FE analysis results for the benchmark case predict
that pure cable behavior is reached when W=2.32 d. For cases when large E values are
used, it can be seen that the point of pure cable behavior prediction approaches that
predicted by Theory II.
Moreover, Figures 4-13 and 4-14 show that the behavior exhibited by Ex100 and
Ex200 is in general bounded by the Theory I and Theory II predictions. In particular, it
can be noted that Theory I at low W / d values is tangent to the FE analysis results, which
gradually converge to Theory II towards the point of pure cable behavior (CII). This can
be attributed to the deflected shape transitioning from a triangular shape to a parabolic
configuration as discussed in Section 3.2.3. Indeed, Theory I, which was developed based
on a triangular deflected shape predicts the behavior well in the early stages, whereas
108
Theory II which was developed based on a parabolic deflected shape, predicts the
behavior well near the point of a pure cable state (CII).
Figure 4-12 Normalized Load?Deflection Plot
109
Figure 4-13 Normalized Axial Force?Deflection Plot (End)
Figure 4-14 Normalized Moment?Deflection Plot (End)
110
Normalized M?N interaction curves are given in Figures 4-15 and 4-16 for an end
section and for midspan, respectively. As seen in Figure 4-15, all FE models perfectly
follow the yield curve once they reach a flexural mechanism condition (Point A). On the
other hand, it is evident from Figure 4-16 that the midspan section indeed develops a
plastic hinge once it reaches the mechanism condition (Point A), but does not necessarily
follow the yield curve later on.
Recall that, besides the different deflected shape considered, both theories under
uniform loading predicted that a midspan plastic hinge is sustained throughout the
deformation of the beam. Therefore, neither of theories perfectly represents the actual
rigid?plastic behavior predicted by stiffer FE models throughout the deformation.
111
Figure 4-15 Normalized M?N interaction (End)
112
Figure 4-16 Normalized M?N interaction (Midspan)
4.3.3.2.2 Beam with W?Shape Cross?Section
Under uniform loading, preliminary FE analysis results presented for a beam with
rectangular cross?section provided a great deal of understanding as to how the FE models
compare with theoretical results. FE analysis results for the beam shown in Figure 4-3(b)
with a W24x192 cross-section are presented in Figures 4-17 through 4-21. The point of
pure cable behavior is indicated in these figures as Points CI and CII for Theory I and
113
Theory II, respectively. In addition, the theoretical Point B where the PNA is at the
interface between the web and flange, is also shown in the normalized axial force versus
normalized deflection and normalized moment versus normalized deflection plots shown
in Figures 4-18 and 4-19, respectively. It can be seen in these figures that the difference
between theoretical results and those for benchmark case, in which E=29,000 ksi, is
again observed to play a significant role in predicting the onset of a pure cable state
(Point C). According to Theory I, pure cable behavior is reached when W=d for W-
Shapes, whereas according to Theory II, pure cable behavior is reached when W=2d. On
the other hand, FE analysis results for the benchmark case predict that pure cable
behavior is reached when W=2.74 d. When large values of E are used, it can be seen that
the point of pure cable behavior prediction by FE models somewhat approach that
predicted by Theory II.
In addition, as can be seen in Figures 4-18 and 4-19, theoretical prediction of
behavior can be considered as the boundaries to the FE models for the Ex100 and Ex200
cases. In fact, it can be noted that the results obtained by Theory I at low values of W / d
are tangent to the FE analysis results, which converge to Theory II towards the point of
pure cable behavior (CII). This result is similar to that described in the discussion of
comparison of FE results with theoretical results for rectangular sections in Section
4.3.3.2.1.
114
Figure 4-17 Normalized Load?Deflection Plot
Figure 4-18 Normalized Axial Force?Deflection Plot (End)
115
Figure 4-19 Normalized Moment?Deflection Plot (End)
Normalized M?N interaction curves are given in Figures 4-20 and 4-21 for an end
section and at midspan, respectively. As can be seen in Figure 4-20, all FE models
perfectly follow the yield curve once they reach the mechanism condition (Point A),
whereas the midspan section indeed develops a plastic hinge once it reaches the
mechanism condition (Point A), but does not necessarily follow the yield curve later on,
as seen in Figure 4-21.
116
Figure 4-20 Normalized M?N interaction (End)
117
Figure 4-21 Normalized M?N interaction (Midspan)
4.3.4 Comparison of Results with Cable Theory
The beam shown in Figure 4-3 was modeled and analyzed in OpenSees using the
rather large span lengths given in Table 4-1. Results are presented in comparison with
theoretical results for the two load cases considered separately. Even though cable theory
predicts that the load?deflection characteristics, as well as the midspan deflection at the
118
onset of pure cable state (Wcat) are independent of cross?section, FE results for
rectangular and W-Shaped cross?sections given in Table 4-1 are considered.
4.3.4.1 Concentrated Load Case
It can be seen from the external load?deflection plot given in Figure 4-22 that the
theoretical results are in excellent agreement with FE model results for both cross?section
types, in spite of the existence of a small flexural resistance as recognized from the M?N
interaction curves shown in Figures 4-24 and 4-25 for the midspan section of a
rectangular beam and a W-Shaped beam, respectively. In fact, the point of pure cable
behavior (C) occurs at the exact same deflection as suggested by the theoretical results
given by Equation 3-106.
0
50
100
150
200
250
0 0.01 0.02 0.03 0.04 0.05 0.06
W / L
P
(k)
FE - REC
FE - W Shape
Theory - REC
Theory - W Shape
C
C
Figure 4-22 External Load ? Normalized Deflection Plot
119
It should be noted that the slight difference in the load?deflection curves shown in
Figure 4-22 for the different cross?section types is due to the difference in cross?
sectional area of the beam models as given in Table 4-1. Recall, an area term exists in
Equation 3-108 which relates external load P to normalized transverse deflection, W / L.
On the other hand, all of the curves resulting from FE models and from theory are in a
perfect agreement once the axial force, N is normalized by Np, as suggested by Equation
3-105 (Figure 4-23)
0
0.25
0.5
0.75
1
1.25
0 0.01 0.02 0.03 0.04 0.05 0.06
W / L
N
/ N
p
REC
W Shape
Theory - REC
Theory - W
Shape
C
Figure 4-23 Normalized Axial Force ? Deflection Plot (Midspan)
120
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25
M / M p
N
/ N
p
FE - REC
R - P Theory
Cable Theory
Figure 4-24 Normalized M?N interaction (REC Section, Midspan)
121
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25M / M
p
N
/ N
p
FE - W Shape
R - PTheory
Cable Theory
Figure 4-25 Normalized M?N interaction (W?Shape, Midspan)
4.3.4.2 Distributed Load Case
Similar to the concentrated load case, FE results were found to be in excellent
agreement with the theoretical results. Independence of the onset point of a pure cable
state with cross-sectional properties was suggested for this case by Equation 3-112.
Indeed, FE results for Point C perfectly coincide with theoretical results for both beam
models as shown in Figures 4-26 and 4-27. It should be noted that the theoretical curves
122
are based on the proposed value of 816.0=? to represent the deflected shape
configuration in this case, as given in Equation 3-116. Similar to results for concentrated
load, a slight difference in cross?sectional area gives two different load?deflection curves
corresponding to the different cross-section shapes, as seen in Figure 4-26, whereas
normalized load?deflection plot again compares perfectly with theory, as shown in Figure
4-27.
0.E+00
1.E-05
2.E-05
3.E-05
4.E-05
5.E-05
6.E-05
7.E-05
0 0.01 0.02 0.03 0.04 0.05 0.06
W / L
q (
kip
/ i
n)
FE - REC
FE - W Shape
Theory - REC
Theory - W Shape
C
C
Figure 4-26 Normalized Load?Deflection Plot
123
0
0.25
0.5
0.75
1
1.25
0 0.01 0.02 0.03 0.04 0.05 0.06
W / L
N
/ N
p
REC
W Shape
Theory - REC
Theory - W
Shape
C
Figure 4-27 Normalized Axial Force?Deflection Plot
Normalized M?N interaction curves for rectangular and W?Shaped beams, shown
in Figure 4-28 and 4-29, respectively, predict a slight resistance in bending as they hit the
yield curve under predominantly axial load, and a slight nonzero bending moment before
reaching the pure cable state.
124
0
0.25
0.5
0.75
1
1.25
-1.25 -1 -0.75 -0.5 -0.25 0
M / M p
N
/ N
p
FE - REC
R - P Theory
Cable Theory
Figure 4-28 Normalized M?N interaction (REC Section, End)
125
0
0.25
0.5
0.75
1
1.25
-1.25 -1 -0.75 -0.5 -0.25 0
M / M p
N
/ N
p
FE - W Shape
R - P Theory
Cable Theory
Figure 4-29 Normalized M?N interaction (W-Shape, End)
4.3.5 Summary and Conclusion
FE models of rectangular and W?Shaped steel beams, fixed at both ends, were
developed and analyzed in OpenSees and the results were compared and discussed with
the results from rigid-plastic and cable theories. In general, the results showed that both
rigid?plastic and cable theories can be replicated numerically with appropriate FE
126
modeling. In particular, the transverse deflection at the onset point of pure cable
behavior, Wcat predicted by each theory was discussed. Following additional conclusions
are made for this study:
square4 For the concentrated load case, both rigid?plastic and cable theory predictions were
explicitly verified using FE modeling. Therefore, the theoretical prediction for the
onset point of pure cable behavior presented in Chapter 3 can be used.
square4 For the distributed load case, FE analysis results showed that Theory I and Theory II
set boundaries for the actual behavior, but neither one represents the behavior
throughout the deformation. Nevertheless, the point of interest, Wcat suggested by
Theory II was verified to be appropriate. FE and theoretical results in regard to the
cable theory were in excellent agreement; thus, the theoretical prediction for the onset
point of pure cable behavior given in Chapter 3 can also be used.
square4 Benchmark FE models representing the actual elastic?perfectly plastic beam behavior
revealed the substantial effect of elastic deformations with regard to the onset point of
pure cable behavior. It was observed that the elastic deformations caused a delay in
forming pure cable action compared to rigid?plastic theory prediction.
4.4 Parametric Study
Preliminary FE analysis results proved the adequacy of the theoretical models
developed in Chapter 3. It was seen that the actual elastic?perfectly plastic beam behavior
was generally in closer agreement with rigid?plastic behavior. In fact, the effect of elastic
deformations was to cause a delay in formation of a pure cable state beyond the one
predicted by rigid?plastic theory. Based on this important observation, a parametric study
127
was conducted with FE analyses in order to identify and study the parameters affecting
the formation of pure cable behavior for elastic?perfectly plastic beams.
A beam fixed at both ends was analyzed for the cross-sections and loading
conditions considered in Chapter 3.
4.4.1 Approach
Based on the results of the preliminary FE analyses, the onset point of pure cable
behavior for elastic? perfectly plastic beams seemed to be predominantly affected by the
elastic deformations of the member. In particular, transverse deflections that a beam
exhibits in the elastic range result in a lag in Wcat beyond that predicted by rigid?plastic
theory.
Rigid?plastic behavior assumes that no deformations take place until the flexural
mechanism condition is reached. An elastic?perfectly plastic beam however, deflects
prior to reaching a mechanism. As illustrated in Figure 4-30, a beam with full end fixity
reaches a flexural mechanism at Point A. Deflection at the plastic collapse load (Pc or qc)
may be associated with the following well known maximum elastic midspan deflection
equations for fixed-fixed beams spanning 2L:
xIE
LP
??
?=
24
3
max? under central concentrated load [4.1]
xIE
Lq
??
?=
24
4
max? under uniform loading along the length [4.2]
For simplicity, the load-deflection relationship for the distributed load case is idealized as
illustrated by the dotted line in Figure 4-30(b). Therefore, substituting the plastic collapse
128
loads, LMP pc 4= and 24LMq pc = for concentrated and uniform loading into Equations 4.1
and 4.2, respectively, gives the same expression for ?p as:
P
W
Pc
?p
q
W
qc
?p
(a) Concentrated Load (b) Uniformly Distributed Load
?p
?p
A A
Figure 4-30 ?p Definition
x
p
p EI
LM
6
2?
=? or
x
xy
p EI
LZ
6
2?
? = [4.3]
where xyp ZM ?= ? .
Equation 4-3 suggests that the effect of elastic deformations consists of geometric and
material properties.
The following parameters are considered in the parametric FE analyses: ?p, ?p / d,
?p / 2L and L / d. The effect of these parameters on the behavior is described next.
129
4.4.2 Description of Beam Models
The beams are modeled and analyzed using OpenSees with the concepts
presented in Section 4-2. A description of the beam models considered in this section is
given below. The other parameters are given for each case separately.
square4 Elastic ? perfectly plastic steel material with 50=y? ksi and 29000=E ksi is used in
all cases.
square4 The cross?sections are discretized using fiber sections with a constant fiber thickness,
tfiber 3105 ?= x inches
square4 The number of elements along the length of the beam members is constantly taken as
100.
4.4.3 Concentrated Load Case
4.4.3.1 Constant ?p
Several beam geometries with W?Shaped cross sections were modeled and
analyzed in OpenSees. First, a 60 ft (2L) beam with W30x124 cross-section was selected
and the corresponding ?p was calculated using Equation 4.3. The other beam sections
were chosen among different W-Shape groups listed in the AISC Steel Construction
Manual (2005) in such a way that they produce same ?p as that for a W30x124 by
changing the span length. Table 4-3 lists the beam cross?sections considered with other
parameters of interest, and theresultant deflection at the onset of pure cable behavior.
130
Table 4-3 Beam Models Considered with constant ?p
It can be seen from Table 4-3 that for deeper members, a larger deflection is
required to reach the pure cable state. However, when Wcat is normalized by nominal
depth, d, an opposite trend is observed in the Wcat / d parameter.
Figures 4-31, 4-32 and 4-33 give normalized load deflection relationships. It can
be seen from these figures that when ?p is constant, the point of pure cable state (C) is
close for different cross-sections. In particular, Wcat / d varies from 1.65 to 1.86, which is
a difference, approximately of 13% with respect to the smallest value of Wcat / d.
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2W / d
P
/ P
c
W30x124 W18x130 W21x122
W24x131 W27x129
C
131
Figure 4-31 Normalized Load?Deflection Plot for Constant ?p
0
0.25
0.5
0.75
1
1.25
0 0.5 1 1.5 2W / d
N
/ N
p
W30x124 W18x130 W21x122
W24x131 W27x129 C
Figure 4-32 Normalized Axial Force?Deflection Plot for Constant ?p (Midspan)
0
0.25
0.5
0.75
1
1.25
0 0.5 1 1.5 2
W / d
M
/ M
p
W30x124 W18x130 W21x122
W24x131 W27x129
C
Figure 4-33 Normalized Moment?Deflection Plot for Constant ?p
132
To summarize, the results showed that the onset point of pure cable behavior,
Wcat is not only sensitive to ?p . In similar manner, with the idea of eliminating the
nominal depth, d, from the problem, FE analyses for different beam geometries with
constant ?p / d were conducted and are presented next.
4.4.3.2 Constant ?p / d
In this section, eight groups of beam models with constant ?p / d were analyzed,
each including three beams with varying nominal depth, d. W?Shaped cross?sections
within each group were chosen from the AISC Steel Construction Manual (2005) to cover
a wide range of d. From the cross?sectional properties, the required length (2L) for ?p / d
is obtained using Equation 4.3 and was assigned to each beam model. Rectangular cross?
sections were chosen to have approximately the same nominal depth and cross?sectional
area as the corresponding W?Shapes. ?p / d varied between groups, from 0.01 to 2.00 to
cover a wide range of behavior.
All cases were analyzed using OpenSees and Wcat was determined by inspection
of the output transverse deflection at midspan at the point when 0.1/ ?pNN is reached.
However, unlike the W?Shape results, rectangular cross-section cases typically exhibit a
plateau at 0.1?
pN
N . This can be seen in Figure 4-34 which shows a typical normalized
axial force?deflection plot for rectangular shape cases that is used in this process. Wcat
was determined as the deflection level at the onset of the apparent plateau which
consistently corresponded to 99.0?
pN
N . Having observed such a trend from FE analysis
133
results prompted the idea that the beam models reach the pure cable state and sustain Np
as predicted by the theory.
The parameters used and the results obtained are given in Tables 4-4 and 4-5 for
the beams with rectangular and W?shaped cross-sections, respectively. The tables also
provide the resultant Wcat in its normalized forms with respect to d, ?p and 2L.
0
0.25
0.5
0.75
1
1.25
W / d
N / N p
Wcat
The end of the
analysis: N = Np
N /Np 0.99 Np? Wcat?
Figure 4-34 Typical Normalized Axial Force?Deflection plot used to determine Wcat for
Rectangular Shape Cases
134
Table 4-4 FE Models with Rectangular Shapes (Constant ?p / d)
Table 4-5 FE Models with W-Shapes (Constant ?p / d)
135
The following are observations from Tables 4-4 and 4-5:
square4 For each group, constant ?p / d resulted in identical L / d and ?p / 2L values for
rectangular section cases (Table 4-4) but a slight difference in those values for W?
Shaped beams (Table 4-5). This can be explained as follows. If the moment of inertia
about the major axis, 2dSI xx ?= is substituted into Equation 4.3, it follows that:
dSE
LZ
x
xy
p ???
??=
3
2?
? . [4.4]
Introducing the shape factor, f , which is defined as:
x
x
S
Zf = [4.5]
into Equation 4.4, gives:
2
3 ??
?
??
??
?
?=
d
L
E
f
d
yp ?? [4.6]
The shape factor is constant ( 5.1=f ) for rectangular cross-sections. Therefore,
for a constant ?p / d, L / d must also be the same within each group in Table 4-4. The
slight difference in shape factor among W?Shapes ( 15.1?f ) results in the slight
difference in L / d within each group in Table 4-5.
In a similar manner, re-writing Equation 4.6 as follows shows why ?p / 2L also
follows the same trend as L / d:
??
?
??
??
?
?=
d
L
E
f
L
yp
62
?? . [4.7]
square4 Rigid?plastic theory suggests that regardless of the cross?section, centrally loaded,
fully fixed beam reaches the onset point of pure cable state when the deflection equals
136
the depth of the beam (i.e., 0.1/ =dWcat ). Indeed, as can be seen from both tables,
dWcat / approaches 1.0 as ?p / d approaches zero (relatively rigid cases).
square4 The output parameters Wcat / d, Wcat / ?p and Wcat / 2L are almost identical within each
group for rectangular shape cases (Table 4-4) but there exists slight difference for the
W?Shape cases (Table 4-5), especially in relatively more rigid groups.
A graphical presentation of the relationships between these variables that seem to
follow a pattern is provided next. The output variables: Wcat / ?p, Wcat / d and Wcat / 2L are
plotted versus the input variables: ?p / d, ?p / 2L and L / d, respectively. Therefore, in
total, nine plots will be shown. In these plots, each point represents one beam model.
Plots of Wcat / ?p versus ?p / d, ?p / 2L and L / d, are given in Figures 4-35, 4-36
and 4-37 respectively. It can be seen that the points representing the rectangular beams
within each group are coincident. However, the points for W?Shaped beams within each
group do not necessarily coincide. In particular, Figure 4-35 shows that the points for the
W-Shapes within each group do nearly coincide except for the very rigid cases. Figure 4-
35 is the only figure in that group of three figures that uses an independent variable of ?p /
d which happens to be constant with each group of three beams. The reason for these
observations will be explored in detail in Chapter 5. The figures also show that agreement
for each group of beams with the same cross-section type has the tendency to improve as
?p / d increases, or as beam flexibility increases. In addition, the figures show that Wcat /
?p decays with increasing ?p / d, ?p / 2L and L / d.
137
0
25
50
75
100
125
0.00 0.50 1.00 1.50 2.00 2.50?
p / d
W
ca
t /
?p
W Shapes REC
Figure 4-35 Wcat / ?p - ?p / d Relationship (Constant ?p / d within each group)
0
25
50
75
100
125
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
?p
W Shapes REC
Figure 4-36 Wcat / ?p - ?p / 2L Relationship (Constant ?p / d within each group)
138
0
25
50
75
100
125
0 10 20 30 40 50 60
L / d
W
ca
t /
?p
W Shapes REC
Figure 4-37 Wcat / ?p ? L / d relationship (Constant ?p / d within each group)
Plots of Wcat / d versus ?p / d, ?p / 2L and L / d are given in Figures 4-38, 4-39 and
4-40, respectively. Again, it can be seen that the points representing the rectangular cross-
section cases within each group are coincident, but the points for the W-Shapes within
each group do not necessarily coincide. The agreement for the W-Shapes is best in Figure
4-38, which is the only figure in that group of figures that uses an independent variable of
?p / d. The figures also show that Wcat / d increases with increasing ?p / d, ?p / 2L and L /
d.
139
0.0
1.0
2.0
3.0
4.0
5.0
0.00 0.50 1.00 1.50 2.00 2.50
?p / d
W
ca
t /
d
W Shapes REC
Figure 4-38 Wcat / d - ?p / d Relationship (Constant ?p / d within each group)
0.0
1.0
2.0
3.0
4.0
5.0
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
d
W Shapes REC
Figure 4-39 Wcat / d - ?p / 2L Relationship (Constant ?p / d within each group)
140
0.0
1.0
2.0
3.0
4.0
5.0
0 10 20 30 40 50 60
L / d
W
ca
t /
d
W Shapes REC
Figure 4-40 Wcat / d ? L / d Relationship (Constant ?p / d within each group)
Plots of Wcat / 2L versus ?p / d, ?p / 2L and L / d are given in Figures 4-41, 4-42
and 4-43, respectively. Similar observations as before can be made. The points for the
rectangular cross-section cases within each group are coincident, but the points for the
W-Shapes within each group do not necessarily coincide. The agreement for the W-
Shapes is best in Figure 4-41, which is the only figure that uses an independent variable
of ?p / d. The figures also show that Wcat / 2L decreases with increasing ?p / d, ?p / 2L and
L / d.
141
0.00
0.04
0.08
0.12
0.16
0.00 0.50 1.00 1.50 2.00 2.50
?p / d
W
ca
t /
2L
W Shapes REC
Figure 4-41 Wcat / 2L - ?p / d Relationship (Constant ?p / d within each group)
0.00
0.04
0.08
0.12
0.16
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
2L
W Shapes REC
Figure 4-42 Wcat / 2L - ?p / 2L Relationship (Constant ?p / d within each group)
142
0.00
0.04
0.08
0.12
0.16
0 10 20 30 40 50 60
L / d
W
ca
t /
2L
W Shapes REC
Figure 4-43 Wcat / 2L - L / d Relationship (Constant ?p / d within each group)
4.4.3.3 Constant ?p / 2L
A similar procedure as before is used. For beams with rectangular cross sections,
the results are the same as shown previously in Table 4-4. This table is rearranged and
presented as Table 4-6. For beams with W-Shapes, the eight groups indicated in Table 4-
5 were re-analyzed. The required span length for each beam model and associated ?p / 2L
values were determined using Equation 4-7.
143
Table 4-6 FE Models with Rectangular Shapes (Constant ?p / 2L)
Table 4-7 FE Models with W?Shapes (Constant ?p / 2L)
144
Plots of Wcat / ?p versus ?p / d, ?p / 2L and L / d are given in Figures 4-44, 4-45 and
4-46, respectively. Plots of Wcat / d versus ?p / d, ?p / 2L and L / d are given in Figures 4-
47, 4-48 and 4-49, respectively. Plots of Wcat / 2L versus ?p / d ?p / 2L and L / d, are given
in Figures 4-50, 4-51 and 4-52, respectively.
Very similar observations as before can be made by comparing Figures 4-44
through 4-52 with Figures 4-35 and 4-43. However, there is one exception. In the current
case, the points for the W-Shapes within each group show the best agreement for cases
that use an independent variable of ?p / 2L, which happens to be constant for each group
of three beams. This can be seen in Figures 4-45, 4-48 and 4-51.
0
25
50
75
100
125
0.0 0.5 1.0 1.5 2.0 2.5 3.0
?p / d
W
ca
t /
?
p
W Shape REC
Figure 4-44 Wcat / ?p - ?p / d Relationship (Constant ?p / 2L within each group)
145
0
25
50
75
100
125
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
?
p
W Shape REC
Figure 4-45 Wcat / ?p - ?p / 2L Relationship (Constant ?p / 2L within each group)
0
25
50
75
100
125
0 10 20 30 40 50 60 70L / d
W
ca
t /
?p
W Shape REC
Figure 4-46 Wcat / ?p ? L / d Relationship (Constant ?p / 2L within each group)
146
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
? p / d
W
ca
t /
d
W Shape REC
Figure 4-47 Wcat / d ? ?p / d Relationship (Constant ?p / 2L within each group)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0.000 0.005 0.010 0.015 0.020 0.025
?p / 2L
W
ca
t /
d
W Shape REC
Figure 4-48 Wcat / d ? ?p / 2L Relationship (Constant ?p / 2L within each group)
147
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0 10 20 30 40 50 60 70L / d
W
ca
t /
d
W Shape REC
Figure 4-49 Wcat / d ? L / d Relationship (Constant ?p / 2L within each group)
0.00
0.04
0.08
0.12
0.16
0.0 0.5 1.0 1.5 2.0 2.5 3.0?p / d
W
ca
t /
2L
W Shape REC
Figure 4-50 Wcat / 2L ? ?p / d Relationship (Constant ?p / 2L within each group)
148
0.00
0.04
0.08
0.12
0.16
0.000 0.005 0.010 0.015 0.020 0.025
?p / 2L
W
ca
t /
2L
W Shape REC
Figure 4-51 Wcat / 2L ? ?p / 2L Relationship (Constant ?p / 2L within each group)
0.00
0.04
0.08
0.12
0.16
0 10 20 30 40 50 60 70L / d
W
ca
t /
2L
W Shape REC
Figure 4-52 Wcat / 2L ? L / d Relationship (Constant ?p / 2L within each group)
149
4.4.3.4 Constant L / d
Again, a similar procedure as before is used. For beams with rectangular cross
sections, the results are the same as those shown in Table 4-4. This table is rearranged
and is presented in Table 4-8. For the beams with W-Shapes, the required span length for
each beam model and associated span-to-depth ratio, L / d, was determined using
Equation 4-7. The parameters used and results obtained for beams with W-shapes are
given in Table 4-9.
Table 4-8 FE Models with Rectangular Shapes (Constant L / d case)
150
Table 4-9 FE Models with W?Shapes (Constant L / d case)
Plots of Wcat / ?p versus ?p / d, ?p / 2L and L / d are given in Figures 4-53, 4-54 and
4-55, respectively. Plots of Wcat / d versus ?p / d, ?p / 2L and L / d are given in Figures 4-
56, 4-57 and 4-58, respectively. Plots of Wcat / 2L versus ?p / d, ?p / 2L and L / d are given
in Figures 4-59, 4-60 and 4-61, respectively.
Very similar observations as before can be made by comparing Figures 4-53
through 4-61 with Figures 4-35 through 4-43, respectively, for the constant ?p / d case,
and with Figures 4-44 through 4-52, respectively, for the constant ?p / 2L case. However,
there is one exception. In the current case, the points for the W-Shapes within each group
show the best agreement for cases that use an independent variable of L / d, which
151
happens to be constant for each group of three beams. This can be seen in Figures 4-55,
4-58, and 4-61.
0
40
80
120
160
0.00 0.50 1.00 1.50 2.00 2.50?
p / d
W
ca
t /
?p
REC W Shape
Figure 4-53 Wcat / ?p - ?p / d Relationship (Constant L / d within each group)
152
0
40
80
120
160
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
?p
REC W Shape
Figure 4-54 Wcat / ?p - ?p / 2L Relationship (Constant L / d within each group)
0
40
80
120
160
0 10 20 30 40 50L / d
W
ca
t /
?p
REC W Shape
Figure 4-55 Wcat / ?p ? L / d Relationship (Constant L / d within each group)
153
0.0
1.0
2.0
3.0
4.0
5.0
0.0 0.5 1.0 1.5 2.0 2.5
?p / d
W
ca
t /
d
REC W Shape
Figure 4-56 Wcat / d - ?p / d Relationship (Constant L / d within each group)
0.0
1.0
2.0
3.0
4.0
5.0
0.000 0.005 0.010 0.015 0.020 0.025?p / 2L
W
ca
t /
d
REC W Shape
Figure 4-57 Wcat / d - ?p / 2L Relationship (Constant L / d within each group)
154
0.0
1.0
2.0
3.0
4.0
5.0
0 10 20 30 40 50
L / d
W
ca
t /
d
REC W Shape
Figure 4-58 Wcat / d ? L / d Relationship (Constant L / d within each group)
0.00
0.03
0.06
0.09
0.12
0.15
0.18
0.00 0.50 1.00 1.50 2.00 2.50?
p / d
W
ca
t /
2L
REC W Shape
Figure 4-59 Wcat / 2L - ?p / d Relationship (Constant L / d within each group)
155
0.00
0.03
0.06
0.09
0.12
0.15
0.18
0.000 0.005 0.010 0.015 0.020 0.025
?p / 2L
W
ca
t /
2L
REC W Shape
Figure 4-60 Wcat / 2L - ?p / 2L Relationship (Constant L / d within each group)
0.00
0.03
0.06
0.09
0.12
0.15
0.18
0 10 20 30 40 50L / d
W
ca
t /
2L
REC W Shape
Figure 4-61 Wcat / 2L ? L / d Relationship (Constant L / d within each group)
156
4.4.4 Distributed Load Case
A similar study was conducted for fully fixed beams under uniform loading, as
shown in Figure 4-3(b).
Only the case of constant L / d is presented herein. Similar observations as for
fully fixed beams under a concentrated load at midspan were made. The parameters used
and results obtained are given in Tables 4-10 and 4-11 for beams with rectangular and W-
Shapes, respectively.
Table 4-10 FE Models with Rectangular Sections (Uniform Loading)
157
Table 4-11 FE Models with W-Shapes (Uniform Loading)
Plots of Wcat / ?p versus ?p / d, ?p / 2L and L / d are given in Figures 4-62, 4-63 and
4-64, respectively. Plots of Wcat / d versus ?p / d, ?p / 2L and L / d are given in Figures 4-
65, 4-66 and 4-67, respectively. Plots of Wcat / 2L versus ?p / d, ?p / 2L and L / d are given
in Figures 4-68, 4-69 and 4-70, respectively.
158
0
50
100
150
200
250
0.0 0.5 1.0 1.5 2.0 2.5?
p / d
W
ca
t /
?
p
REC W Shape
Figure 4-62 Wcat / ?p - ?p / d Relationship (Constant L / d within each group)
0
50
100
150
200
250
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
?p
REC W Shape
Figure 4-63 Wcat / ?p - ?p / 2L Relationship (Constant L / d within each group)
159
0
50
100
150
200
250
0 10 20 30 40 50L / d
W
ca
t /
?p
REC W Shape
Figure 4-64 Wcat / ?p ? L / d Relationship (Constant L / d within each group)
0.0
1.0
2.0
3.0
4.0
5.0
0.0 0.5 1.0 1.5 2.0 2.5?
p / d
W
ca
t /
d
REC W Shape
Figure 4-65 Wcat / d - ?p / d Relationship (Constant L / d within each group)
160
0.0
1.0
2.0
3.0
4.0
5.0
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
d
REC W Shape
Figure 4-66 Wcat / d - ?p / 2L Relationship (Constant L / d within each group)
0.0
1.0
2.0
3.0
4.0
5.0
0 10 20 30 40 50
L / d
W
ca
t /
d
REC W Shape
Figure 4-67 Wcat / d ? L / d Relationship (Constant L / d within each group)
161
0.00
0.10
0.20
0.30
0.0 0.5 1.0 1.5 2.0 2.5? p / d
W
ca
t /
2L
REC W Shape
Figure 4-68 Wcat / 2L - ?p / d Relationship (Constant L / d within each group)
0.00
0.10
0.20
0.30
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
2L
REC W Shape
Figure 4-69 Wcat / 2L - ?p / 2L Relationship (Constant L / d within each group)
162
0.00
0.10
0.20
0.30
0 10 20 30 40 50L / d
W
ca
t /
2L
REC W Shape
Figure 4-70 Wcat / 2L ? L / d Relationship (Constant L / d within each group)
4.4.5 Summary and Conclusions
A parametric FE analysis study of elastic?perfectly plastic steel beams with full
end fixity was conducted. Through preliminary FE analysis results, geometric parameters
that had an influence on the onset of pure cable state, Wcat were identified and studied
using beam models for rectangular and W?Shaped cross?sections.
A cursory study revealed that the midspan displacement at the onset of pure cable
behavior, Wcat , is not only influenced by ?p; even for the same ?p, it was observed that
Wcat varied significantly with the cross-section used. However, further investigations
revealed that the point of pure cable behavior is reached at approximately the same ratio
of midspan displacement to depth, Wcat / d, regardless of the cross section used. This
163
initial study led to another parametric study with a wider scope. In particular, the effect of
?p / d, ?p / 2L and L / d on Wcat / ?p, Wcat / d and Wcat / 2L was investigated.
164
Chapter 5. EQUATIONS FOR THE ONSET OF PURE CABLE BEHAVIOR
5.1 Overview
Rigid?plastic and cable theories for fully fixed beams with rectangular and W?
Shaped cross-sections were developed in Chapter 3. Preliminary FE analysis results
presented in Chapter 4 showed that results from both theories can be simulated
numerically. However, they represent an ideal beam behavior. FE parametric studies as
presented in Chapter 4 showed that the elastic?perfectly plastic response can be neither
represented precisely by rigid?plastic nor cable theory, but is expected to fall in between
the two. The studies focused on finding the midspan deflection level at the onset of pure
cable behavior, Wcat. The observed trends for Wcat, with respect to the geometric
parameters studied, prompted an investigation into the possibility of developing an
equation that would predict the onset point of pure cable behavior in terms of the
theoretical predictions presented in Chapter 3.
Trends for Wcat can be expressed in several ways, such as those discussed in
Chapter 4. The following three forms of relationships were selected:
square4 Wcat / ?p versus ?p / 2L (Formulation I)
square4 Wcat / d versus L / d (Formulation II)
square4 Wcat / 2L versus L / d (Formulation III)
165
Theoretical expressions for Wcat in these forms were developed using both rigid-
plastic and cable theories, and were compared to FE analysis results for the concentrated
and uniformly distributed load cases from Chapter 4.
5.2 Concentrated Load Case
5.2.1 Formulation I
square4 Rigid?plastic theory:
The onset point of pure cable behavior for a fully fixed beam is independent of
the cross-section and occurs when the deflection equals the nominal depth of the section,
d (i.e. dWcat = ). Therefore:
pp
cat dW
?? = [5.1]
which can be written as:
??
?
??
?
??
?
??
??=
Ld
L
W
pp
cat
2
11
2
1
?? . [5.2]
Substituting L / d from Equation 4.7 into Equation 5.2 yields:
2
2
1
12
1
??
?
??
?
??=
L
E
fW
p
y
p
cat
?
?
? [5.3]
square4 Cable Theory
It has been shown in Section 3.3.1 that the onset point of pure cable behavior is
independent of cross?section type and can be written in the form:
166
EL
W ycat ?2= [5.4]
which can be rewritten as:
??
?
??
???=
L
E
W
p
y
p
cat
22
12
?
?
? [5.5]
or
??
?
??
??=
L
E
W
p
y
p
cat
2
1
2 ?
?
? [5.6]
Plots of Wcat / ?p versus ?p / 2L as suggested by each theory are given in Figures 5-
1 through 5-4 in comparison with FE analysis results presented in Chapter 4. Since Wcat
according to the rigid-plastic theory is affected by the shape factor ( f ) (Equation 5-3),
the plots are presented for beams with rectangular and W-Shaped cross-sections
independently. Figure 5-1 shows the comparison with FE analysis results for rectangular
shaped beams. Figure 5-2, 5-3, and 5-4 show the comparison with FE results for W?
Shaped beams with constant ?p / d, ?p / 2L, and L / d, respectively. The shape factor, f ,
in Equation 5.3 was taken as an average value of 15.1 for all W?Shapes considered.
For beams with rectangular cross-section, it can be seen from Figure 5-1 that the
FE analysis results tend to approach those for the rigid-plastic theory for relatively stiff
beams and approach cable theory results for relatively flexible beams. Similar
observations can be made for beams with W?Shapes as shown in Figures 5-2, 5-3 and 5-
4. With the purpose of representing the behavior over the entire range, a new curve is
introduced which was obtained by linear superposition of Equations 5.3 and 5.6:
167
??
?
??
??+
??
?
??
?
??=
L
E
L
E
fW
p
y
p
y
p
cat
2
1
2
2
1
12
1
2 ?
?
?
?
? [5.7]
Equation 5.7 is represented by the solid curve in Figures 5-1 through 5-4.
Equation 5.7 approaches the rigid-plastic theory for relatively stiff beams and cable
theory for relatively flexible beams.
It can be seen that Equation 5.7 is in excellent agreement with the FE results for
W?Shapes (Figures 5-2 through 5-4) and in good agreement with the FE results for
rectangular shapes (Figure 5-1). Minor discrepancies that can be seen in the figures will
be discussed in Section 5.4.
0
30
60
90
120
150
0 0.005 0.01 0.015 0.02 0.025?p / 2L
W
ca
t /
?p
Rigid - Plastic Theory
Cable Theory
Superposition
FE - REC
Figure 5-1 Comparison with FE results (REC Shapes)
168
0
30
60
90
120
150
0 0.005 0.01 0.015 0.02 0.025?p / 2L
W
ca
t /
?p
Rigid - Plastic Theory
Cable Theory
Superposition
FE - W Shape
Figure 5-2 Comparison with FE results (W?Shapes: Constant ?p / d case)
0
30
60
90
120
150
0 0.005 0.01 0.015 0.02 0.025?
p / 2L
W
ca
t /
?p
Rigid - Plastic Theory
Cable Theory
Superposition
FE - W Shape
Figure 5-3 Comparison with FE results (W?Shapes: Constant ?p / 2L case)
169
0
30
60
90
120
150
0 0.005 0.01 0.015 0.02 0.025? p / 2L
W
ca
t /
?p
Rigid - Plastic Theory
Cable Theory
Superposition
FE - W Shape
Figure 5-4 Comparison with FE results (W?Shapes: Constant L / d case)
5.2.2 Formulation II
square4 Rigid?plastic theory:
The onset point of pure cable behavior for a fully fixed beam is independent of
the cross-section and occurs when the deflection equals the nominal depth of the section,
d (i.e. dWcat = ). Therefore:
1=dWcat . [5.8]
square4 Cable theory:
It has been shown in Section 3.3.1 that the onset point of pure cable behavior is
independent of cross?section type and can be written in the form:
170
EL
W ycat ?2= [5.9]
which can be rearranged to give:
??
?
??
??=
d
L
Ed
W ycat ?2 . [5.10]
FE analysis results compared with the results from Formulation I shown
previously demonstrated that the linear superposition of the theoretical predictions
appropriately represented the behavior. Accordingly, the superposition equation for
Formulation II is given as:
??
?
??
??+=
d
L
Ed
W ycat ?21 . [5.11]
As can be seen, this formulation is independent of the shape factor. Thus, the
comparison can be presented for both cross?section types together.
Figure 5-5, 5-6, and 5-7 show the comparison of theoretical predictions with FE
results for rectangular and W?Shaped beams with constant ?p / d, ?p / 2L, and L / d,
respectively. Again, it can be seen that FE analysis results are in excellent agreement with
Equation 5.11.
171
0
1
2
3
4
5
0 10 20 30 40 50 60
L / d
W
ca
t /
d
Rigid - Plastic Theory
Cable Theory
Superposition
FE - REC
FE - W Shape
Figure 5-5 Comparison with FE results (Constant ?p / d case)
0
1
2
3
4
5
0 10 20 30 40 50 60 70L / d
W
ca
t /
d
Rigid - Plastic Theory
Cable Theory
Superposition
FE - REC
FE - W Shape
Figure 5-6 Comparison with FE results (Constant ?p / 2L case)
172
0
1
2
3
4
5
0 10 20 30 40 50L / d
W
ca
t /
d
Rigid - Plastic Theory
Cable Theory
Superposition
FE - REC
FE - W Shape
Figure 5-7 Comparison with FE results (Constant L / d case)
5.2.3 Formulation III
square4 Rigid?plastic theory:
The onset point of pure cable behavior for fully fixed beam was predicted to be
equal to the nominal depth, d, regardless of the cross?section type (i.e. dWcat = ).
Therefore:
L
d
L
Wcat
22 = [5.12]
which can be written as:
??
?
??
??=
d
LL
Wcat
2
1
2 [5.13]
square4 Cable theory:
173
It has been shown in Section 3.3.1 that the onset point of pure cable behavior is
independent of cross?section type and can be written in the form:
EL
W ycat
22
?= . [5.14]
Similarly, the superposition equation in this case becomes:
E
d
LL
W ycat
22
1
2
?+
??
?
??
??= . [5.15]
Similar to the second formulation, this representation is also independent of the
shape factor ( f ). Therefore the comparisons with FE analysis results are presented
together for rectangular and W-Shaped beams.
The following figures (5-8 through 5-10) give the plots of Wcat / 2L versus L / d
predicted by each theory, along with the representation with superposition equation
(Equation 5.15) in comparison with FE results for each case considered in the parametric
study in Chapter 4. As can be seen in these figures, the superposition equation as given in
Equation 5.15 indicates excellent agreement with FE results. In particular, W?Shapes are
again in excellent agreement with results from the superposition curve given in Equation
5-15. Typically, rectangular shaped beams always fall below the curve, which will also
be discussed in Section 5.4.
174
0.00
0.05
0.10
0.15
0.20
0 10 20 30 40 50 60
L / d
W
ca
t /
2L
Rigid - Plastic Theory
Cable Theory
Superposition
FE - REC
FE - W Shape
Figure 5-8 Comparison with FE results (Constant ?p / d case)
0.00
0.05
0.10
0.15
0.20
0 10 20 30 40 50 60 70
L / d
W
ca
t /
2L
Rigid - Plastic Theory
Cable Theory
Superposition
FE - REC
FE - W Shape
Figure 5-9 Comparison with FE results (Constant ?p / 2L case)
175
0.00
0.05
0.10
0.15
0.20
0 10 20 30 40 50 60
L / d
W
ca
t /
2L
Rigid - Plastic Theory
Cable Theory
Superposition
FE - REC
FE - W Shape
Figure 5-10 Comparison with FE results (Constant L / d case)
5.3 Distributed Load Case
A similar procedure for the beams with uniformly distributed load along the
length was implemented.
Theoretical expressions for Wcat in these forms were developed using both rigid-
plastic and cable theories, and werecompared to FE analysis results for the concentrated
and uniformly distributed load cases. Since the preliminary FE analysis results as
presented in Section 4.3.3.2 showed that Theory II predicted the deflection at the onset
of pure cable behavior somewhat accurately, Theory II will be used herein.
5.3.1 Formulation I
square4 Rigid?plastic theory:
176
The onset point of pure cable behavior for a fully fixed beam is independent of
the cross-section and occurs when the deflection equals twice the nominal depth of the
section, 2d (i.e. dWcat 2= ). Therefore:
pp
cat dW
??
2= [5.16]
which can be arranged as:
??
?
??
?
??
?
??
?=
Ld
L
W
pp
cat
2
11
?? . [5.17]
Now, substitution of L / d from Equation 4.7 into Equation 5.17 yields:
2
2
1
6
1
??
?
??
?
??=
L
E
fW
p
y
p
cat
?
?
? . [5.18]
square4 Cable Theory:
It has been shown in Section 3.3.2 that the onset point of pure cable behavior
independent of cross?section type and can be written in the form:
EL
W ycat ?
? ?= 2
1
2 [5.19]
where ? was verified to be 0.816.Re?arranging Equation 5.19 as follows and
using 816.0=? , Equation 5.19 becomes:
??
?
??
???=
L
E
W
p
y
p
cat
2
16127.0
?
?
? . [5.20]
Finally, the superposition equation for this case becomes:
177
??
?
??
???+
??
?
??
?
??=
L
E
L
E
fW
p
y
p
y
p
cat
2
16127.0
2
1
6
1
2 ?
?
?
?
? [5.21]
In Figures 5-11 and 5-12, Wcat / ?p is plotted versus ?p / 2L, including the
predictions given in Equations 5.18, 5.20 and 5.21, in comparison with FE analysis
results from the parametric study in Section 4.4.4. Since the rigid?plastic theory
prediction given in Equation 5.18 involves the shape factor term, comparison is made for
each of the cross section types independently. As in the concentrated load case, an
average shape factor is used for W?Shapes ( 15.1=f ) to produce the theoretical
prediction shown in Figure 5-12. As can be seen in these figures, FE results are in good
agreement with the rigid?plastic theory for small ?p / 2L values, whereas cable theory
produces better agreement for relatively large values of ?p / 2L (i.e. for the flexible
regime). The superposition equation given in Equation 5.21 is again in excellent
agreement with the FE results over the entire range.
178
0
50
100
150
200
250
300
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
?p
FE - REC
Superposition
Rigid - Plastic Theory II
Cable Theory
Figure 5-11 Comparison with FE results (REC Shapes)
0
50
100
150
200
250
300
0.000 0.005 0.010 0.015 0.020 0.025?
p / 2L
W
ca
t /
?p
FE - W Shape
Superposition
Rigid - Plastic Theory II
Cable Theory
Figure 5-12 Comparison with FE results (W Shapes)
179
5.3.2 Formulation II
square4 Rigid ? plastic theory:
The onset point of pure cable behavior for a fully fixed beam is independent of
the cross-section and occurs when the deflection equals twice as much the nominal depth
of the section, 2d (i.e. dWcat 2= ). Therefore:
2=dWcat . [5.22]
square4 Cable theory
It has been shown in Section 3.3.2 that the onset point of pure cable behavior is
independent of cross?section type and can be written in the form:
??
?
??
??=
d
L
Ed
W ycat ?
?
1 . [5.23]
Substitution of 816.0=? into Equation 5.23 yields:
??
?
??
???=
d
L
Ed
W ycat ?225.1 . [5.24]
Having determined the relationship based on each theory, the superposition
equation in this formulation can be given as:
??
?
??
???+=
d
L
Ed
W ycat ?225.12 . [5.25]
Figure 5-13 provides the plots of Wcat / d versus L / d predicted by Equations 5.22,
5.24 and 5.25 in comparison with FE analysis results from the parametric study as
presented in Section 4.4.4. It can be seen that superposition equation adequately
represents the trend exhibited by FE analysis results.
180
0.0
1.0
2.0
3.0
4.0
5.0
0 10 20 30 40 50L / d
W
ca
t /
d
FE - REC
FE - W Shape
Superposition
Rigid - Plastic Theory II
Cable Theory
Figure 5-13 Comparison with FE results (Constant L / d within each group)
5.3.3 Formulation III
square4 Rigid ? plastic theory:
The onset point of pure cable behavior for a fully fixed beam is independent of
the cross-section and occurs when the deflection equals twice the nominal depth of the
section, 2d (i.e. dWcat 2= ). Therefore:
L
d
L
Wcat
2
2
2 = [5.26]
which can be rewritten as:
??
?
??
?=
d
LL
Wcat 1
2 . [5.27]
square4 Cable theory:
181
It has been shown in Section 3.3.2 that the onset point of pure cable behavior
independent of cross?section type and can be written in the form:
EL
W ycat ?
? ?= 2
1
2 . [5.28]
Again, substituting 816.0=? into Equation 5.28 yields:
EL
W ycat ??= 6127.0
2 [5.29]
Finally, the linear superposition equation for this formulation becomes:
E
d
LL
W ycat ??+
??
?
??
?= 6127.0
1
2 [5.30]
Wcat / 2L is plotted versus span-to-depth ratio (L / d) in Figure 5-14, including FE
results from the parametric study (Section 4.4.4) in comparison with predictions
produced by Equations 5.27, 5.29 and 5.30. Similar observations are made and it can be
seen that W?Shaped beams, in particular, seem to be in better agreement with the
prediction given by Equation 5.30 over the entire range of L / d values.
182
0.00
0.05
0.10
0.15
0.20
0 10 20 30 40 50L / d
W
ca
t /
2L
FE - REC
FE - W Shape
Superposition
Rigid - Plastic Theory II
Cable Theory
Figure 5-14 Comparison with FE results (Constant L / d within each group)
The adequacy of representation of FE results by linear superposition of the
theoretical predictions is discussed in following section.
5.4 Discussion
Based on the trends observed in the results of FE parametric study (Section 4.4),
theoretical predictions for the onset point of pure cable behavior were formulated and
compared to FE analysis results. The comparisons showed that the FE results approached
results produced by the rigid-plastic and cable theories for stiff and flexible beams,
respectively. In order to cover the entire range of behavior, a new equation is proposed
that is obtained by linear superposition of the rigid-plastic and cable equations.
Comparison of FE analysis results with the new equation showed that it can be used to
183
predict the onset point of pure cable behavior for elastic?perfectly plastic steel beams.
Some discrepancies were observed between the proposed equation and FE analysis
results. The difference between the two, and possible reasons for this discrepancy are
discussed next.
The difference between results from the proposed equation and FE results can be
quantified as:
100(%) xY YYDifference
FE
FELS ?= [5.31]
where YLS and YFE are y?coordinates of the proposed equation and FE analysis results at a
given L / d level, respectively. Span-to-depth ratio (L / d) was chosen for this analysis due
to its common use in practice.
It can be anticipated that the difference will be identical regardless of which
formulation is used in comparison. The reason is that each formulation presented in this
chapter is a different manipulation of the theoretical onset point of pure cable behavior,
Wcat. Thus, the percent differences between proposed equation and FE analysis results for
each cross?section type are calculated using Equation 5.31 and are presented in the range
of L / d considered in the parametric study.
Figures 5-15 and 5-16 show plots of Equation 5.31 versus L / d of the
concentrated load case, for rectangular and W?Shaped beams, respectively. It can be seen
that the difference is largest for relatively stiff beams and becomes smaller for relatively
flexible beams. While this has not been studied in detail in the thesis, it is believed that
the difference is largest for relatively stiff beams because the deflection at the onset of
pure cable behavior is typically large compared to the span length. Since the rigid-plastic
184
theory is developed for finite, but not necessarily large deflections, it is believed that it
may not give accurate results for very stiff beams.
The shaded region shown in Figures 5-15 and 5-16 covers the range of L / d
between 10 and 30 that is typically encountered in practice. Within this range, the
difference between the proposed equation and the FE analysis results falls in a range of
8~15% for rectangular shapes and less than 3% for W?Shapes. Considering that W?
Shapes are widely used in practice, this difference can be considered acceptable. While
the reason that the difference is larger for rectangular shaped beams than for W-Shaped
beams has not been rigorously studied in the thesis, it is believed that it relates to the way
the onset point of pure cable behavior, Wcat, was determined, as indicated in Figure 4-34
and explained in Section 4.4.
0%
5%
10%
15%
20%
0 10 20 30 40 50 60
L / d
Di
ffe
re
nc
e (
%)
Figure 5-15 Distribution of the Difference for Rectangular Shaped Beams
185
(Concentrated Load case)
-5%
0%
5%
10%
15%
0 10 20 30 40 50 60 70L / d
Di
ffe
re
nc
e (
%)
Figure 5-16 Distribution of the Difference for W-Shaped Beams
(Concentrated Load Case)
In a similar manner, Figures 5-17 and 5-18 show plots of Equation 5.31 versus L /
d for the distributed load case for rectangular and W-Shaped beams, respectively. The
difference is again largest for relatively stiff beams but not necessarily much smaller for
relatively flexible beams. However, within the shaded region (L / d of 10 to 30) the
difference ranges from 6% to 10% for rectangular shaped beams and remains within 7%
for W-Shaped beams.
The fact that the difference is slightly larger for W-Shaped beams compared to the
concentrated load case can be attributed to the discussion made within the context of the
preliminary FE analysis (Section 4.3.3.2) that the onset point of pure cable behavior
186
prediction by the rigid?plastic theory for uniform loading (Wcat =2d) is not rigorous but is
approximate.
0%
5%
10%
15%
20%
0 10 20 30 40 50L / d
Di
ffe
re
nc
e (
%)
Figure 5-17 Distribution of the Difference for Rectangular Shaped Beams
(Distributed Load Case)
187
-5%
0%
5%
10%
15%
20%
25%
0 10 20 30 40 50L / d
Di
ffe
re
nc
e (
%)
Figure 5-18 Distribution of the Difference for W-Shaped Beams
(Distributed Load Case)
5.5 Conclusion
An equation is proposed that predicts the midspan deflection at the onset of pure
cable behavior, Wcat for elastic?perfectly plastic steel beams with a span-to-depth ratio, L
/ d, ranging between 10 and 30. It can be shown that all three formulations presented
herein result in the same reduced form of the onset point of pure cable behavior as
follows:
For beams fixed at both ends spanning 2L under a concentrated load at midspan:
LEdW ycat ??+= ?2 [5.32]
and for beams fixed at both ends spanning 2L under uniformly distributed load along the
length:
188
LEdW ycat ??+= ?225.12 [5.33]
Equations 5.32 and 5.33 suggest that Wcat is independent of the cross-section type
and it can be predicted with knowledge of only d, L, y? and E. Although previous FE
analysis results suggested that the shape of the cross-section has an influence on the onset
point of pure cable behavior, the analysis conducted to quantify the difference between
FE analysis results and theoretical linear superposition results indicated that the
difference is sufficiently small for both rectangular and W-Shaped cross-sections.
Therefore, Equations 5.32 and 5.33 can be used independent of the cross-section type.
189
Chapter 6. EFFECT OF ELASTIC BOUNDARY CONDITIONS
6.1 Overview
The behavior of ductile steel beams undergoing a transition from flexural to cable
behavior was described using idealized boundary conditions in the previous chapters. The
end supports were assumed to be either fully fixed or pinned. However, it can be
anticipated that the boundary conditions of a beam within a building frame may be
different from the idealized ones. In particular, conditions at the ends of the beam may
result in partial rotational and axial restraints. Beam end restraints can be attributed to
two main sources: restraining effects due to the surrounding elements framing into the
beam joint and restraining effects provided by the beam-to-column connection itself.
In this chapter, study on the effect of elastic boundary conditions using FE models
is described. The study consisted of two main objectives: (1) to determine the effect of
elastic boundary restraints on beam behavior and (2) compare frame models to that of
beam models.
The beam models used in the analysis are illustrated in Figure 6-1. The beam A?B
is isolated from the frame and beam end restraints are modeled using linear?elastic
springs. k? and k? are translational and rotational spring constants, respectively, as shown
in Figure 6-1. Due to symmetry, k? and k? are the same for both supports.
190
Nonsymmetrical cases are not considered in this study. In addition, the beam is
loaded at midspan as shown in Figure 6-1. The boundary conditions associated with the
midspan joint are idealized to be rigid and continuous.
L
k?
k?
k?
k?
L
A B
L LLL
Removed Column
A B
L LLL
A B
P
Figure 6-1 Schematics of Beam Model with Elastic Boundary Conditions
The beams and frames considered in this chapter were also modeled and analyzed
using OpenSees (2000). Therefore, all modeling and analysis concepts described in
Section 4.2 are valid here as well.
191
6.2 Effect of Elastic Restraint
FE analyses of steel beams with linear?elastic boundary conditions were
conducted. Both material and geometric nonlinearity were considered for the beams.
First, the effects of translational and rotational restraint were studied separately. For each
case, a wide range of spring constant values were used. Then, both were incorporated
simultaneously into the models, as illustrated in Figure 6-1, to study their combined
effect.
The cross-section of the beam used in the analyses was determined from
preliminary design of a two?bay frame, based on the AISC Manual (2005) LRFD
approach, as presented in Section 6.3. A W14x53 was used for all beam models
considered throughout this section. Cross?sections were discretized with fibers as
explained earlier in Section 4.2.1 with a constant fiber thickness, tfiber = 5x10-3 inches. An
elastic?perfectly plastic steel material with ?y = 50 ksi was used in all models. The
number of elements along the beam length was consistently taken as 100.
6.2.1 Effect of Translational Restraint
In order to study the effect of partial translational restraint, the beam was modeled
as shown in Figure 6-2.
192
Figure 6-2 Beam model with translational springs only
A range of spring constant values was considered and three beam models
(W14x53) with varying lengths for each k? were analyzed. Table 6-1 summarizes the
input parameters, as well as the resulting midspan displacement at the onset of pure cable
behavior, Wcat, and Wcat / d.
Table 6-1 W14x53 Beam Models with Translational Springs only
It can be seen in Table 6-1 that Wcat / d increases with decreasing k?. This trend is
also illustrated in Figure 6-3, in which Wcat / d is plotted versus k?. The figure also shows
that for the same k?, Wcat / d is always larger for longer beams than for shorter beams,
which is consistent with the observations and results of Chapter 4.
193
0
10
20
30
40
50
0 400 800 1200 1600
k ? (k/in)
W
ca
t /
d
20ft 60ft 100ft
Figure 6-3 Dependency of Wcat / d on Translational Spring Constant, k?
The results are also given in the form of normalized load versus deflection,
normalized axial force versus deflection, normalized moment versus deflection, and
normalized M?N interaction plots in Figures 6-4, 6-5, 6-6, and 6-7, respectively. The
figures are given for the 60-ft-long beams only. It should be noted that the external load
is normalized by the plastic collapse load, Pc of a fully fixed beam which in this case is:
26.47360 5.425344 =?=?= LMP pc kips
where 5.4253=pM k-in for a W14x53 section (fillets are neglected).
Figure 6-4 shows how the level of axial restraint can significantly affect the
behavior. For small values of translational spring constant, it can be seen that no
194
significant increase in P beyond Pc can be achieved until extremely large deflections take
place.
Figure 6-4 Normalized Load?Deflection Plot (Midspan)
Figure 6-5 shows that for a low level of axial restraint, the beam develops
insignificant axial forces even at fairly large deflections. And as a result, the behavior is
dominated by flexure, as is seen in Figure 6-6. M-N interaction plots are provided in
Figures 6-7 and 6-8 for midspan and end sections, respectively. Once the mechanism
condition is reached, plastic hinges remain on the yield curve under a combination of M
and N until the behavior is purely dominated by cable action (i.e., pNN = ).
195
Figure 6-5 Normalized Axial Force?Deflection Plot (Midspan)
Figure 6-6 Normalized Moment?Deflection Plot (Midspan)
196
Figure 6-7 Normalized M?N interaction (Midspan)
197
Figure 6-8 Normalized M?N Interaction (End)
6.2.2 Effect of Rotational Restraint
In order to study the effect of partial rotational restraint, the beam with a W14x53
is modeled as shown in Figure 6-8.
198
Figure 6-9 Beam with Rotational Springs Only
A range of values for k? ranging from zero (e.g., pinned) to infinity (fully fixed) at
the beam supports was considered and beams with varying lengths were modeled and
analyzed for each k? level.
In Table 6-2, analysis input parameters, as well as the resulting midspan
deflection at the onset point of pure cable behavior, Wcat, and Wcat / d, are given. A close
examination of the results, along with Figure 6-10, in which Wcat / d is plotted versus k?
reveals that the dependency of Wcat / d on the rotational spring constant, k? is not as
straightforward as it was for the beams with translational springs presented earlier. Figure
6-10 shows that up to a certain level of k?, Wcat / d gradually increases and reaches a peak
at a k? level of approximately 5000 k-in / rad. Then, it starts to decrease with increasing k?
and eventually converges to a value which is the one predicted by the fixed-fixed case.
Table 6-2 in comparison Table 6-1 also shows that Wcat / d is relatively less
sensitive to k? than k? presented earlier. For example, for beams with a total span length,
2L, of 60 ft, Wcat / d varies between 2.08 and 2.57 for this case (Table 6-2), but varies
between 26.36 and 2.58 for beams with translational springs only (Table 6-1).
The results are also graphically presented in the form of normalized load versus
deflection, normalized axial force versus deflection, normalized moment versus
199
deflection and M?N interaction plots in Figures 6-11 through 6-16. The figures are given
for the 20ft-long beams only.
Table 6-2 W14x53 Beam Models with Rotational Springs only
0.00
1.00
2.00
3.00
4.00
0 50000 100000 150000 200000
k ? (k-in/rad)
W
ca
t /
d
20ft 60ft 100ft
5000
Figure 6-10 Dependency of Wcat / d on Rotational Spring Constant, k?
200
Again, external force, P in Figure 6-11 is normalized by the plastic collapse load,
Pc for the fixed-fixed case which in this case is:
78.141120 5.425344 =?=?= LMP pc Kips
Figures 6-12 through 6-14, in particular, show the significance of the level of
rotational restraint on the behavior. For models in which k? is not large enough to develop
a plastic hinge (e.g., ROT3 through ROT8), the tendency of the load?deflection curves is
to ?jump? to the curves that corresponds to models with higher rotational restraints at a
certain deflection level. This can be attributed to the fact that those models do not provide
enough restraint for the beam to develop flexural plastic hinges at end supports, as shown
in Figure 6-14. Therefore, for these low k? values, the beam acts like a simple beam. As
seen in Figure 6-16, with the development of axial forces N, plastic hinges eventually
form at the ends with a combination of M and N, and the beam forms a mechanism. As a
result, it tends to behave as fixed-fixed and load?deflection curves seem to suddeny shoot
towards the one predicted by the fully fixed case.
The plot of M-N interaction at midspan (Figure 6-15) shows that a plastic hinge
forms predominantly by flexure first and follows the yield curve under combined M and
N in all beam models, regardless of k? level, as seen in Figure 6-15.
201
Figure 6-11 Normalized Load?Deflection Plot (Midspan)
Figure 6-12 Normalized Axial Force?Deflection Plot (Midspan)
202
Figure 6-13 Normalized Moment?Deflection Plot (Midspan)
Figure 6-14 Normalized Moment?Deflection Plot (End)
203
Figure 6-15 Normalized M?N Interaction (Midspan)
204
Figure 6-16 Normalized M?N Interaction Plot (End)
The effect of rotational restraint can also be related to the connection behavior at
the beam supports. The AISC Manual (2005) classifies connections on a moment?
rotation (M-?) behavior model as fully restrained (FR), partially restrained (PR) and
simple, as illustrated in Figure 6-17. Dashed lines (Ks) set the boundaries for connection
behavior idealizations in structural analysis. Connections with stiffness between these
two limits are considered as partially restrained (PR). In order to determine how the range
205
of spring constant considered herein relates to these limits, the moment?rotation response
of the spring models are shown in Figure 6-18, along with the AISC regions for
connection behavior. It can be seen that a number of representative beam models are
covered in all three regions. In fact, the majority of the models (ROT1 through ROT8)
fall within the ?simple connection? region, which supports the discussion made regarding
the sudden jump observed in the normalized internal force?deflection plots (Figures 6-12,
6-13).
Figure 6-17 Classification of moment?rotation response of fully restrained (FR),
partially restrained (PR) and simple connections (AISC 2005)
206
Figure 6-18 Moment?Rotation Response of Elastic Springs in FE Beam Models
6.2.3 Combined Effect of Translational and Rotational Restraints
Individual effects of translational and rotational springs at beam ends provided a
great deal of insight into their effect on the behavior. In this section, a W14x53 is
modeled as shown in Figure 6-1 to study the combined effect of elastic boundary
restraints.
Spring constant combinations were chosen in such a way as to cover combinations
of low, intermediate and high spring constants considered in the previous sections. A
typical 60-ft-long, elastic?perfectly plastic beam with W14x53 cross-section was
modeled and analyzed using the 16 different combinations of linear?elastic spring
constants listed in Table 6-3. Labels K11 through K44 denote the beam models analyzed.
207
Table 6-3 also gives the resulting deflection at the onset of pure cable behavior,
Wcat and Wcat / d for each case. It can be seen that Wcat / d is rather more sensitive to the
level of axial restraint, k? than to the level of rotational restraint, k?. In fact, despite the
large variation in rotational spring constant, the level of translational spring constant
determines Wcat / d which changes only slightly within each group. Additionally, the
onset point of pure cable behavior is reached at a relatively reasonable deflection level for
the two groups with higher axial restraints as seen in Table 6-3.
Table 6-3 Beam Models with Combined Spring Constant Cases
FE analysis results are also given in the plots of normalized load versus
deflection, normalized axial force versus deflection, normalized moment versus
deflection in Figures 6-19 through 6-21. It can be seen from these figures that the curves
are assembled into four groups, each with the same k?. Additionally, the significance of
the level of axial restraint, rather than the rotational restraint in regard to the onset point
of pure cable behavior can also be seen in Figures 6-20 and 6-21. In particular, all the
208
beams of Group 1 have to exhibit extremely large deflections to reach pure cable
behavior.
Figure 6-19 Normalized Load?Deflection Plot
209
Figure 6-20 Normalized Axial Force?Deflection Plot (Midspan)
Figure 6-21 Normalized Moment?Deflection Plot (Midspan)
210
Figures 6-22 and 6-23 show normalized M?N interaction results for midspan and
end sections, respectively. It can be seen that when rotational restraints at supports are
low (i.e. K11, K21?K41 and K12, K22?K42), the plastic hinge forms with
predominantly axial force, N (Figure 6-23).
Figure 6-22 Normalized M?N Interaction (Midspan)
211
Figure 6-23 Normalized M?N Interaction (End)
6.3 Relationship between Beam Models and Frame Models
In this section, a description is given of a number of simple FE frame analyses
that were conducted using OpenSees to investigate the sources of boundary conditions
available within a frame and to relate the results to the beam model studies presented in
Section 6.2. First, a preliminary design of a two?bay benchmark frame was performed for
gravity loads only. The frame was assumed to be braced. Also, continuous bracing for the
212
beams due to the existence of a slab was assumed. As illustrated in Figure 6-24, a part of
an interior frame in an office building which has symmetry in both directions was
considered (L=30ft). The AISC LRFD (2005) method was used and typical beam
sections were chosen to be W14x53 and W16x100 were chosen for the columns.
L=360''L=360''
H=180''
q DL = 0.7 k/ft
q LL = 1.5 k/ft
Figure 6-24 Reference Frame
6.3.1 Quantification of Elastic Spring Constants
For the purpose of this study, restraining effects at the beam ends within an actual
frame are assumed to be provided by the surrounding members framing into the beam
joints. In other words, beam-to-column connections herein are considered to be fully
restrained (FR). However, it can be anticipated that the frame given in Figure 6-24 would
not provide any significant translational and rotational restraint at the beam supports if
the middle column is notionally removed and the remaining frame analyzed under a
concentrated load at midspan (Figure 6.25). Instead, one additional bay on each side of
the frame was added to the reference frame to study the quantification of boundary
restraints.
213
L=360''L=360''
H=180''Remove
Column
P
A B
A B
Figure 6-25 Schematic of Frame Analysis
As illustrated in Figure 6-26, elastic boundary conditions offered by the side
frames at joints A and B can be calculated as follows: side frames are isolated from the
actual frame and analyzed with arbitrary loads (PA , MA) one at a time in order to obtain
translational and rotational spring constants, respectively. A linear?elastic analysis was
conducted using OpenSees and the resultant spring constants (i.e. slope of the resultant
load?deflection curve) were obtained as:
k? = 11.8 kip / in
k? = 393,700 k-in / rad.
Due to symmetry, spring constants were taken to be the same at A and B.
214
L=360''L=360''L=360'' L=360''
H=180''Beams : W14x53
Columns : W16x100
A B
PA
MA
PB
MB
k?
k?
k?
k?
A B
L=360''L=360''
Figure 6-26 Schematic for the Quantification of Linear?Elastic Spring Constants
6.3.2 Comparison between Frame and Beam Models
Having determined the linear?elastic spring constants, nonlinear static analyses of
the frame and the beam A?B were conducted under a concentrated load at midspan
separately, as illustrated in Figure 6-27. It should be noted that material nonlinearity was
not included in the members other than for the beam A?B in the frame analysis, as noted
in Figure 6-31.
W14x53 beam A?B was again discretized with fibers with a typical thickness of
tfiber=5x10-3 inches and an elastic?perfectly plastic steel material (?y=50 ksi) was used. The
number of elements along the beam length was also taken as 100.
215
L=360''L=360''L=360'' L=360''
H=180''
P
k?
k?
k?
k?
P
A B
A B
L=360''L=360''
Elastic Beam-Column Element
Nonlinear Beam-Column Element (Elastic-Perfectly Plastic, ?y=50 ksi)
Figure 6-27 Frame and Beam Models
FE analysis results are presented in the form of normalized load versus
normalized deflection, normalized axial force versus normalized deflection, normalized
moment versus normalized deflection, and M?N interaction plots in Figures 6-28 through
6-32 for the beam A?B in each case. In Figure 6-28, the external load, P, is normalized
by the plastic collapse load, Pc, of a W14x53 beam with fixed ends spanning 2L=60ft as
typical.
It can be seen in Figures 6-28 through 6-32 that the behavior exhibited by the
beam model with elastic springs is in excellent agreement with the beam behavior within
the frame. In can be seen from the normalized axial force?deflection plot given in Figure
216
6-30 that the beam within the frame exhibits a slight compression in the elastic regime, as
anticipated, and then axial tension forces due to large deflections start to dominate the
behavior. This can also be seen in the normalized M?N interaction curves as both
midspan and end sections, which follow the yield condition perfectly once the flexural
plastic hinge is formed (Figures 6-31 and 6-32, respectively).
The results show that it is possible to model and analyze the beam above a
notionally removed column in a given frame with adequate representation of boundary
conditions. Nevertheless, it should be noted that the results are limited to the frame
considered herein, and are not necessarily to be generalized at this point.
0
1
2
3
4
5
0 5 10 15
W / d
P
/ P
c
Beam in Frame Beam Model
Figure 6-28 Normalized Load?Deflection Plot (Beam Midspan)
217
0
0.25
0.5
0.75
1
1.25
0 5 10 15W / d
M
/ M
p
Beam in Frame Beam Model
Figure 6-29 Normalized Moment?Deflection Plot (Beam Midspan)
-0.25
0
0.25
0.5
0.75
1
1.25
0 5 10 15
W / d
N
/ N
p
Beam in Frame Beam Model
Figure 6-30 Normalized Axial Force?Deflection Plot (Beam Midspan)
218
-0.25
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25
M / M p
N
/ N
p
Beam in Frame Beam Model
Figure 6-31 Normalized M?N Interaction (Beam Midspan)
219
-0.25
0
0.25
0.5
0.75
1
1.25
-1.25 -1 -0.75 -0.5 -0.25 0
M / M p
N
/ N
p
Beam in Frame Beam Model
Figure 6-32 Normalized M?N Interaction (Beam End)
220
6.4 Summary and Conclusions
Studies of the effect of elastic boundary conditions on the beam response, along
with simple frame studies conducted herein provided the following findings:
square4 The dependency of the deflection at the onset of pure cable behavior on axial restraint
is significantly larger than the dependence on rotational restraint. Therefore, if the
beam ends are not sufficiently restrained axially, it is impractical to take advantage of
the cable behavior described in this thesis as a load-resisting mechanism.
square4 Evaluation of linear?elastic boundary conditions that are offered at the beam joints
within a given frame, and comparison with corresponding beam models, showed an
acceptable agreement for use in future frame studies. However, it should be noted that
general statements can not be made at this time due to the limited data produced.
221
Chapter 7. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR
FUTURE STUDY
7.1 Summary
In addressing the propensity of steel building structures to experience progressive
collapse due to extreme loading conditions (e.g., blast), current design guidelines propose
the use of a threat-independent approach that is commonly referred to in the literature as
?the missing column scenario?. Under this scenario, a column from a given story is
assumed to be removed and the resulting structure is analyzed to determine if it could
sustain the loads by activating one or more alternate load-carrying mechanisms, with the
idea of mitigating the potential for progressive structural collapse. This study specifically
focused on the ability of ductile steel beams to carry loads by transitioning from flexural
behavior to cable-like behavior. Theoretical fundamentals of this behavior were described
for rectangular and W-shaped steel beams with idealized boundary conditions and
presumed fully ductile behavior. Two theoretical analysis approaches were used to model
the beam behavior: rigid-plastic analysis and cable analysis. The main factors affecting
the behavior, such as material and geometric properties as well as boundary conditions
were described and corroborating nonlinear finite element (FE) analyses were presented
and compared to the theoretical results. Open System for Earthquake Engineering
Simulation, OpenSees was used in the FE analysis studies. Based upon theoretical and FE
222
analysis results, a set of equations were proposed that can be used to predict the
deflection at the onset of pure cable behavior.
Additionally, the effect of elastic boundary restraints on the beam behavior was
studied using FE analysis. An approach to evaluate the boundary restraints offered by the
surrounding members in a given frame was also presented. It is shown that axial
restraints have a much more significant effect on the behavior than rotational restraints.
7.2 Conclusions
Conclusions for this study can be summarized as follows:
square4 Rigid-plastic and cable theories serve as bounds to the behavior of ductile steel beams
undergoing a transition from a flexural to cable-like behavior.
square4 Comparisons of FE analysis results with the theoretical results revealed that modeling
the behavior using rigid-plastic theory is appropriate for very stiff beams, whereas
modeling with the use of cable theory is appropriate for very flexible beams. In the
general case, neither theory can alone accurately predict the behavior.
square4 FE analysis parametric studies showed that the midspan beam deflection at the onset
of pure cable behavior, Wcat, is especially sensitive to the deflection at the onset of
flexural mechanism formation.
square4 Expressions for Wcat were developed and compared with theory, and equations were
proposed for use in the range of span-to-depth ratio, L/d, from 10 to 30.
square4 Studies on the effect of boundary conditions indicated that the onset point of pure
cable behavior is predominantly affected by translational restraints.
223
7.3 Recommendations
Despite a number of assumptions made in the development of theory in this study,
it is believed that a profound understanding of the behavior established a solid foundation
for future investigations. The following recommendations for future study are made:
square4 It is recommended that beam models be investigated for:
? Evaluation of ductility demands in regards to accommodating respective
displacements.
? Nonlinear behavior incorporation at boundary conditions.
? Material strain hardening effect.
? Nonlinear-dynamic analysis.
square4 Due to the lack of experimental results in the literature, it is also recommended
that medium-scale laboratory testing be performed for idealized boundary
conditions to corroborate the results obtained in this study.
square4 It is recommended that further frame analyses (2D / 3D) be conducted to
understand the global behavior with regard to how the beam behavior described in
this study would interact within a frame.
square4 It is also recommended that all sources of anchorage possibilities such as those
that exist in an actual building structure be investigated due to the significant
dependency of the behavior to the lateral stiffness provided at each floor.
square4 It is recommended that the equations proposed in this study be compared to and
their applicability be investigated for, development of the Tie Force Method used
in current progressive collapse design guidelines.
224
REFERENCES
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Construction, Chicago, IL
ASCE (2005) ASCE / SEI 7-05: Minimum Design Loads for Buildings and Other
Structures, American Society of Civil Engineers, Reston, VA
Bazant, Z. P., Verdure, M., (2007) ?Mechanics of Progressive Collapse: Learning from
World Trade Center and Building Demolitions?, ASCE Journal of Engineering
Mechanics, 133(3), 308
Breen, J., (1980), ?Developing Structural Integrity in Bearing Wall Buildings,? PCI
Journal, 25(1), 42-73
British Standard Institution (BSI) (2000), BS5950: Part 1, Structural Use of Steelwork in
Building, London, UK
Burnett, E.F.P., (1975), ?The Avoidance of Progressive Collapse: Regulatory
Approaches to the Problem? Report No. NBS-GCR-75-48, National Bureau of
Standards, Washington, DC
Byfield M., Paramasivam, S. (2007), ?Catenary Action in Steel-Framed Buildings?,
Structures & Buildings, 160(5), 247-257
Creasy, L. R., (1972), ?Stability of Modern Buildings? The Structural Engineer, London,
50(1), 3-6
De Souza, R. M. (2000). ?Force-Based Finite Element for Large Displacement
Inelastic Analysis of Frames? Ph.D. Thesis, Univ. of California, Berkeley, CA
225
Dusenberry D.O. (2002), ?Review of Existing Guidelines and Provisions Related to
Progressive Collapse??, Workshop on Prevention of Progressive Collapse,
National Institute of Building Sciences, Washington D. C
Ellingwood B. R., Smilowitz R., Dusenberry D. O., Duthinh D., Lew H. S., Carino N. J.,
(2007), Best Practices for Reducing the Potential for Progressive Collapse in
Buildings, Report No: NISTIR-7396. National Institute of Standards and
Technology, Technology Administration, US Department of Commerce
Foley C. M., Martin K., Schneeman C., (2007), Robustness in Structural Steel Framing
Systems, Report, American Institute of Steel Construction (AISC), Chicago, IL
General Services Administration (GSA), (2003), ?Progressive Collapse Analysis and
Design Guidelines for New Federal Office Buildings and Major Modernization
Projects? Washington, D.C
Gustafson, K. (2009) ?Meeting the 2009 IBC Structural Integrity Requirements?, AISC
North American Steel Construction Conference (NASCC), Phoenix, AZ,
Conference Proceedings, AISC
Haythornthwaite, R. M. (1957), ?Beams with Full End Fixity? Engineering, 183, 110-12
Hamburger, R., Whittaker, A. (2004) ?Design of Steel Structures for Blast-Related
Progressive Collapse Resistance? North American Steel Construction Conference
(NASCC), Long Beach, CA, Conference Proceedings AISC
Horne, M. R. (1979) ?Plastic Theory of Structures? Pergamon Press, Oxford, England,
66-68
Izzuddin, B. A. (2005) ?A Simplified Model for Axially Restrained Beams Subject to
226
Extreme Loading?? International Journal of Steel Structures, 5, 421?429
Jones N. (1989) ?Structural Impact? Cambridge University Press, Cambridge,
Great Britain, 276-290
Khandelwal, K., and El-Tawil, S. (2007). ?Collapse Behavior of Steel Special Moment
Resisting Frame Connections? ASCE Journal of Structural Engineering, 133(5),
646?655.
Khandelwal K., El-Tawil S., Kunnath S. K., Lew H. S. (2008)?Macromodel-Based
Simulation of Progressive Collapse: Steel Frame Structures? ASCE Journal of
Structural Engineering, 134(7), 1070-1079
Khandelwal K., El-Tawil S; Kunnath S. K., Lew H. S. (2008)?Macromodel-Based
Simulation of Progressive Collapse: RC Frame Structures?? ASCE Journal of
Structural Engineering, 134(7), 1079-182
Krauthammer, T., Hall, R. L., Woodson, S. C., Baylot, J. T., Hayes, J. R., and Sohn, Y.
(2002) ?Development of Progressive Collapse Analysis Procedure and Condition
Assessment for Structures? National Workshop on Prevention of Progressive
Collapse in Rosemont, Ill. Multihazard Mitigation Council of the National
Institute of Building Sciences, Washington, D.C
Marchand, K. A., Alfawakhiri F., (2004) Facts for Steel Buildings 2: Blast and
Progressive Collapse, American Institute of Steel Construction (AISC), Chicago,
IL
McKenna, F., Fenves, G. L., and Scott, M. H. (2000) ?Open System for Earthquake
Engineering Simulation (OpenSees)? Univ. of California, Berkeley,
CA,
227
Mohamed, O. A. (2006), ?Progressive Collapse of Structures: Annotated Bibliography
and Comparison of Codes and Standards?, ASCE Journal of Performance of
Constructed Facilities, 20(4), pp. 418?425
Nair, R. S. (2004) ?Progressive Collapse Basics? Modern Steel Construction, American
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Neuenhofer A., Filippou F. C (1998) ?Geometrically Nonlinear Flexibility-Based Frame
Finite Element? ASCE Journal of Structural Engineering, 124(6), 704?711
Powell, G. (2005) ?Progressive Collapse: Case Studies Using Nonlinear Analysis?
ASCE Structures Congress, New York, NY, American Society of Civil Engineers
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Scott M., H., Fenves G. L., McKenna F., Filippou F. C. (2008) ?Software Patterns for
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228
APPENDICES
229
APPENDIX A: Notation
a Distance between cetroid and neutral axis of cross-section, in
A Cross-sectional area of member, in2
Aw Web area, the overall depth times the web thickness, (d x tw) , in2
b width of rectangular cross-section, in
bf Flange width, in
d Full nominal depth of the section, in
dw Nominal web depth, in
E Modulus of Elasticity of steel, 29000 ksi
? Shape function
Ix Moment of inertia about principal axis, in4
k? Linear-Elastic translational spring constant, kip / in
k? Linear-Elastic rotational spring constant, kip-in / rad
l Plastic hinge length
L Length (e.g., beam, bay), in (ft)
L? Actual length in deformed configuration
M Bending moment, kip-in
Mp Plastic Bending Moment, kip-in
Mpf Plastic Bending Moment calculated using the flanges only (about major axis)
N Axial force, kips
230
Np Plastic Axial Force, kips
Npw Plastic Axial Force calculated using the web only
Pc Plastic collapse load of member, kips
P Point load on member, kips
q Uniformly distributed load along the member
qc Uniformly distributed plastic collapse load of member
Q Shear Force, kips
s Mean transverse spacing, in (ft)
Sx Elastic section modulus taken about the principal axis, in3
tfiber Thickness of fiber
tw Beam web thickness, in
w shape function of deformed shape configuration
W Midspan transverse deflection, in
Wcat Midspan Transverse deflection at pure catenary formation
?
W Transverse deflection rate (with respect to time)
Zx Plastic section modulus about the principal axis, in3
?p Midspan deflection at the onset of flexural mechanism, in
? Axial strain
??
Axial strain rate (with respect to time)
? Curvature
??
Curvature rate (with respect to time)
?y Yield stress, ksi
231
? Depth fraction represents the location of neutral axis from bottom of cross-section
? Rotation angle at beam supports
?L Change in length
232
APPENDIX B: Typical Input Files in OpenSees
B.1: Centrally Loaded Beam Fixed at Supports
wipe all;
#---Units: kip, in---#
#------------------Define the Model Builder--------------------------------------------------------#
model basic -ndm 2 -ndf 3;
#------------------File Directory----------------------------------------------------------------------#
file mkdir W30x124;
#------------------Define nodes-----------------------------------------------------------------------#
for {set i 0} {$i<101} {incr i} {
set nodeTag [expr $i+1]
set xdim [expr $i*7.2]
node $nodeTag $xdim 0
}
#-----------------Define Boundary Conditions-----------------------------------------------------#
fix 1 1 1 1;
fix 101 1 1 1;
#-----------------Geometric Transformation--------------------------------------------------------#
geomTransf Corotational 1;
#-----------------Define Materials-------------------------------------------------------------------#
uniaxialMaterial ElasticPP 1 29000 0.00172413;
233
#-----------------Define Fiber Section--------------------------------------------------------------#
#Section W30x124#
set SecTag 1
set d 30.2;
set tw 0.585;
set bf 10.5;
set tf 0.93;
set nfdw 5688;
set nftw 1;
set nfbf 1;
set nftf 186;
set dw [expr $d-2*$tf]
set y1 [expr -$d/2]
set y2 [expr -$dw/2]
set y3 [expr $dw/2]
set y4 [expr $d/2]
set z1 [expr -$bf/2]
set z2 [expr -$tw/2]
set z3 [expr $tw/2]
set z4 [expr $bf/2]
section fiberSec $SecTag {
# nfIJ nfJK yI zI yJ zJ yK zK yL zL
patch quadr 1 $nfbf $nftf $y1 $z4 $y1 $z1 $y2 $z1 $y2 $z4
234
patch quadr 1 $nftw $nfdw $y2 $z3 $y2 $z2 $y3 $z2 $y3 $z3
patch quadr 1 $nfbf $nftf $y3 $z4 $y3 $z1 $y4 $z1 $y4 $z4
}
#----------------Define Elements---------------------------------------------------------------------#
for {set k 0} {$k<100} {incr k} {
set eltag [expr $k+1]
set inode [expr $k+1]
set jnode [expr $k+2]
element nonlinearBeamColumn $eltag $inode $jnode 3 1 1
}
#----------------Define Recorders--------------------------------------------------------------------#
recorder Node -file W30x124/Node51.out -time -node 51 -dof 2 disp;
recorder Node -file W30x124/Node1.out -time -node 1 -dof 3 disp;
recorder Element -file W30x124/ForceE51-S1.out -time -ele 51 section 1 force;
recorder Element -file W30x124/DefE51-S1.out -time -ele 51 section 1 deformation;
recorder Element -file W30x124/ForceE1-S1.out -time -ele 1 section 1 force;
recorder Element -file W30x124/DefE1-S1.out -time -ele 1 section 1 deformation;
#----------------Define Load Case-------------------------------------------------------------------#
pattern Plain 1 Linear {
load 51 0 -230.4474 0
}
#----------------Define Analysis Objects-----------------------------------------------------------#
constraints Plain;
235
numberer Plain;
system BandGeneral;
test NormDispIncr 1.0e-3 10;
algorithm Newton;
integrator DisplacementControl 51 2 -0.02;
analysis Static
analyze 5000;
loadConst -time 0.0;
puts "Done!"
236
B.2: Fixed Beam with Uniformly Distributed Load along the Length
wipe all;
#---Units: kip, in---#
#------------------Define the Model Builder--------------------------------------------------------#
model basic -ndm 2 -ndf 3;
#------------------File Directory----------------------------------------------------------------------#
file mkdir W30x124_DistL;
#------------------Define nodes-----------------------------------------------------------------------#
for {set i 0} {$i<101} {incr i} {
set nodeTag [expr $i+1]
set xdim [expr $i*7.2]
node $nodeTag $xdim 0
}
#-----------------Define Boundary Conditions-----------------------------------------------------#
fix 1 1 1 1;
fix 101 1 1 1;
#-----------------Geometric Transformation--------------------------------------------------------#
geomTransf Corotational 1;
#-----------------Define Materials-------------------------------------------------------------------#
uniaxialMaterial ElasticPP 1 29000 0.00172413;
#-----------------Define Fiber Section---------------------------------------------------------------#
237
set SecTag 1
set d 30.2;
set tw 0.585;
set bf 10.5;
set tf 0.93;
set nfdw 5688;
set nftw 1;
set nfbf 1;
set nftf 186;
set dw [expr $d-2*$tf]
set y1 [expr -$d/2]
set y2 [expr -$dw/2]
set y3 [expr $dw/2]
set y4 [expr $d/2]
set z1 [expr -$bf/2]
set z2 [expr -$tw/2]
set z3 [expr $tw/2]
set z4 [expr $bf/2]
section fiberSec $SecTag {
# nfIJ nfJK yI zI yJ zJ yK zK yL zL
patch quadr 1 $nfbf $nftf $y1 $z4 $y1 $z1 $y2 $z1 $y2 $z4
patch quadr 1 $nftw $nfdw $y2 $z3 $y2 $z2 $y3 $z2 $y3 $z3
patch quadr 1 $nfbf $nftf $y3 $z4 $y3 $z1 $y4 $z1 $y4 $z4
238
}
#----------------Define Elements---------------------------------------------------------------------#
for {set k 0} {$k<100} {incr k} {
set eltag [expr $k+1]
set inode [expr $k+1]
set jnode [expr $k+2]
element nonlinearBeamColumn $eltag $inode $jnode 3 1 1
}
#----------------Define Recorders--------------------------------------------------------------------#
recorder Node -file W30x124_DistL/Node51.out -time -node 51 -dof 2 disp;
recorder Node -file W30x124_DistL/Node1.out -time -node 1 -dof 3 disp;
recorder Element -file W30x124_DistL/ForceE51-S1.out -time -ele 51 section 1 force;
recorder Element -file W30x124_DistL/DefE51-S1.out -time -ele 51 section 1
deformation;
recorder Element -file W30x124_DistL/ForceE1-S1.out -time -ele 1 section 1 force;
recorder Element -file W30x124_DistL/DefE1-S1.out -time -ele 1 section 1 deformation;
#----------------Define Load Case-------------------------------------------------------------------#
pattern Plain 1 Linear {
load 1 0 -2.24 0
for {set m 1} {$m<100} {incr m} {
set Nodetag [expr $m+1]
load $Nodetag 0 -4.48 0
239
}
load 101 0 -2.24 0
}
#----------------Define Analysis Objects-----------------------------------------------------------#
constraints Plain;
numberer Plain;
system BandGeneral;
test NormDispIncr 1.0e-1 10;
algorithm Newton;
integrator DisplacementControl 51 2 -0.01;
analysis Static
analyze 10000;
loadConst -time 0.0;
puts "Done!"
240
B.3: Beam with Linear-Elastic Boundary Conditions
wipe all;
#---Units: kip, in---#
#------------------Define the Model Builder--------------------------------------------------------#
model basic -ndm 2 -ndf 3;
#------------------File Directory----------------------------------------------------------------------#
file mkdir W30x124.wSPrings;
#------------------Define nodes-----------------------------------------------------------------------#
for {set i 0} {$i<101} {incr i} {
set nodeTag [expr $i+1]
set xdim [expr $i*7.2]
node $nodeTag $xdim 0
}
node 102 0 0;
node 103 720 0;
equalDOF 1 102 2;
equalDOF 101 103 2;
#-----------------Define Boundary Conditions-----------------------------------------------------#
fix 102 1 1 1;
fix 103 1 1 1;
#-----------------Geometric Transformation--------------------------------------------------------#
241
geomTransf Corotational 1;
#-----------------Define Materials-------------------------------------------------------------------#
uniaxialMaterial ElasticPP 1 29000 0.00172413;
uniaxialMaterial Elastic 2 320;
uniaxialMaterial Elastic 3 4451000;
#-----------------Define Fiber Section---------------------------------------------------------------#
#Section W30x124#
set d 30.2;
set tw 0.585;
set bf 10.5;
set tf 0.93;
set nfdw 5668;
set nftw 1;
set nfbf 1;
set nftf 186;
set dw [expr $d-2*$tf]
set y1 [expr -$d/2]
set y2 [expr -$dw/2]
set y3 [expr $dw/2]
set y4 [expr $d/2]
set z1 [expr -$bf/2]
set z2 [expr -$tw/2]
set z3 [expr $tw/2]
242
set z4 [expr $bf/2]
section fiberSec 1 {
# nfIJ nfJK yI zI yJ zJ yK zK yL zL
patch quadr 1 $nfbf $nftf $y1 $z4 $y1 $z1 $y2 $z1 $y2 $z4
patch quadr 1 $nftw $nfdw $y2 $z3 $y2 $z2 $y3 $z2 $y3 $z3
patch quadr 1 $nfbf $nftf $y3 $z4 $y3 $z1 $y4 $z1 $y4 $z4
}
#----------------Define Elements---------------------------------------------------------------------#
for {set k 0} {$k<100} {incr k} {
set eltag [expr $k+1]
set inode [expr $k+1]
set jnode [expr $k+2]
element nonlinearBeamColumn $eltag $inode $jnode 3 1 1
}
element zeroLength 101 102 1 -mat 2 3 -dir 1 6;
element zeroLength 102 103 101 -mat 2 3 -dir 1 6;
#----------------Define Recorders--------------------------------------------------------------------#
recorder Node -file W30x124.wSP/Node51.out -time -node 51 -dof 2 disp;
recorder Node -file W30x124.wSP/Node1.out -time -node 1 -dof 1 3 disp;
recorder Node -file W30x124.wSP/Node101.out -time -node 101 -dof 1 3 disp;
recorder Element -file W30x124.wSP/TRSpringForce101.out -time -ele 101 force;
recorder Element -file W30x124.wSP/TRSpringDEF101.out -time -ele 101 deformation;
recorder Element -file W30x124.wSP/TRSpringForce102.out -time -ele 102 force;
243
recorder Element -file W30x124.wSP/TRSpringDEF102.out -time -ele 102 deformation;
recorder Element -file W30x124.wSP/ForceE1.out -time -ele 1 globalForce;
recorder Element -file W30x124.wSP/ForceE100.out -time -ele 100 globalForce;
recorder Element -file W30x124.wSP/ForceE51-S1.out -time -ele 51 section 1 force;
recorder Element -file W30x124.wSP/ForceE1-S1.out -time -ele 1 section 1 force;
recorder Element -file W30x124.wSP/DefE1-S1.out -time -ele 1 section 1 deformation;
recorder Element ?file W30x124.wSP/DefE51-S1.out -time -ele 51 section 1
deformation;
#----------------Define Load Case-------------------------------------------------------------------#
pattern Plain 1 Linear {
load 51 0 -224.046 0
}
#----------------Define Analysis Objects-----------------------------------------------------------#
constraints Lagrange;
numberer Plain;
system BandGeneral;
test EnergyIncr 1.0e-3 10;
algorithm Newton;
integrator DisplacementControl 51 2 -0.01;
analysis Static
analyze 10000;
loadConst -time 0.0;
puts "Done!"