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 
 
 
 
 
 
 
 
ii 
 
 
 
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. 
 
iv 
 
 
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           
vi 
 
 
 
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 
viii 
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 
ix 
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 
x 
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
xi 
 
 
 
 
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 
xii 
 
 
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 
xiii 
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 
xiv 
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 
xv 
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 
xvi 
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 
xvii 
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 
xix 
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 
AISC (2005) Steel Construction Manual 13th Edition, American Institute of Steel  
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 <http://www.aisc.org/> 
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 
<http://www.aisc.org/> 
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, <http://opensees.berkeley.edu> 
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  
Institute of Steel Construction (AISC), 44(3), 37? 42 
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 
(CD-ROM) 
Scott M., H., Fenves G. L., McKenna F., Filippou F. C. (2008) ?Software Patterns for  
Nonlinear Beam-Column Models? ASCE Journal of Structural Engineering, 
134(4), 562?571 
Weisstein, E. W. (2009), ?Catenary? Retrieved from MathWorld, a Wolfram Web  
Resource, June 25: http://mathworld.wolfram.com/Catenary.html  
Unified Facilities Criteria (UFC) (2005) ?Design of Buildings to Resist Progressive  
Collapse?, Department of Defense, Washington, D.C 
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!"