PROGRESSIVE COLLAPSE OF MOMENT RESISTING STEEL...

PROGRESSIVE COLLAPSE OF MOMENT RESISTING STEEL FRAMED BUILDINGS QUANTITATIVE ANALYSIS BASED ENERGY APPROACH

By

STEFAN TADEUSZ SZYNISZEWSKI

A DISSERTAION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2009

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© 2009 Stefan Tadeusz Szyniszewski

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Dla mojego Taty Zdzisława i Mamy Aleksandry oraz Kasi, Oli i mojej kochanej żony Ani, której wsparcie i miłość były dla mnie bezcenne w trakcie pisania tej pracy

To my Father Zdzisław and Mother Aleksandra, sisters Kasia and Ola and my beloved wife Anna, whose support and love made possible the completion of this dissertation

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ACKNOWLEDGMENTS

This study was conducted at the Center for Infrastructure Protection and Physical Security

at the University of Florida under contract with the US Army Corps of Engineers, Engineer

Research and Development Center (ERDC), Vicksburg, MS. The author wishes to acknowledge

the generous support provided by the sponsor.

Foremost, I would like to thank my advisor, Dr. Theodor Krauthammer, for his time,

guidance and insight throughout the different stages of this study. I also want to thank my

committee members: Dr. Joseph Tedesco, Dr. Gary Consolazio and Dr. Rafael Haftka for their

valuable recommendations and comments.

To acknowledge everyone who contributed to this study in some manner is clearly

impossible, but a major debt is owed my teachers at the University of Florida from whom I

learned so much: Mike McVay, Trey Hamilton, Kurt Gurley, Youping Chen, John Lybas and

others. The author is also indebted to his teachers at the Warsaw University of Technology:

Wojciech Radomski, Tomasz Lewiński, Wojciech Gilewski, Stanislaw Jemioło, Aleksander

Szwed and Wawrzyniec Sadkowski, Andrzej Reterski and Kazimierz Cegiełka.

The manuscript was carefully reviewed in its various versions, several times, by Michael

Davidson, Nick Henriquez, Hyun Chang Yim and the University of Florida Editorial Office,

resulting in important consistency and grammatical improvements.

Finally, I want to express my special thanks to my family in Poland, to Aleksandra and

Zdzisław, my parents, for their total support and encouragement on pursuing my education goals;

to my sisters Kasia and Ola, for always being there; and to my dear wife Anna for her enduring

love and sacrifice. To all of those wonderful people, my friends and teachers who have

generously given me their advice and help, I am pleased to express my gratitude.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS .............................................................................................................. 4

LIST OF TABLES .......................................................................................................................... 7

LIST OF FIGURES ........................................................................................................................ 9

ABSTRACT .................................................................................................................................. 19

CHAPTER

1 INTRODUCTION ............................................................................................................... 20

Problem Statement .............................................................................................................. 20 Importance ........................................................................................................................... 20 Research Objectives ............................................................................................................ 20 Scope ................................................................................................................................... 21

2 LITERATURE REVIEW .................................................................................................... 23

Overview ............................................................................................................................. 23 Event Control ...................................................................................................................... 24 Indirect Design .................................................................................................................... 24

Historical Cases 25 Building Code Measures to Prevent Progressive Collapse 30

Direct Design: Finite Element Method ............................................................................... 36 Static Procedures for Progressive Collapse Modeling ........................................................ 38 Energy Concepts ................................................................................................................. 43

Deformation Work (Internal Energy) 45 Kinetic Energy 48

Energy Based Procedures for Progressive Collapse Modeling ........................................... 49 Energy Flow between Members .......................................................................................... 55 Simplified Methods for High Rise Buildings ...................................................................... 59 Summary ............................................................................................................................. 67

3 RESEARCH APPROACH .................................................................................................. 70

Proposed Theory ................................................................................................................. 70 Virtual Experiments ............................................................................................................ 79

Overview 79 Material Modeling 79 Structural Modeling 84

Selected Structures .............................................................................................................. 91 Summary ............................................................................................................................. 97

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4 KINEMATIC RESULTS .................................................................................................... 98

Overview ............................................................................................................................. 98 Virtual Experiments on the Simplified Steel Framed Buildings ......................................... 98 Kinematic Observations .................................................................................................... 100 Validation of Results ......................................................................................................... 118 Steel Framed Building with Shorter Beam Spans ............................................................. 120 Summary ........................................................................................................................... 129

5 ENERGY APPROACH TO THE ANALYSIS OF PROGRESSIVE COLLAPSE ......... 131

Overview ........................................................................................................................... 131 Energy Definitions in LS-DYNA ...................................................................................... 132 Energetic Characteristics of Individual Columns .............................................................. 133

Displacement controlled Buckling 133 Force controlled Buckling 138 Column Buckling Energy 148

Energy Flow and Redistribution ....................................................................................... 149 Energy Propagation through the Building 167 Energy based Column Buckling Criterion (Full Building Analysis) 171 Usefulness of the Energy Buckling Limits 194 Energy based Building Failure Limit 197 Analytical Solution of Elasto-Plastic Column Buckling 201

Verification of Energy Approach on Realistic Steel Building .......................................... 207 Two Columns Removed. CASE 2 209 Three Columns Removed. CASE 3 219

Summary ........................................................................................................................... 232

6 CONCLUSIONS AND RECOMMENDATIONS ............................................................ 236

Summary ........................................................................................................................... 236 Progressive Collapse Conclusions .................................................................................... 236 Energy Conclusions ........................................................................................................... 238 Recommendations ............................................................................................................. 240

APPENDIX

A VERIFICATION OF ENERGY EXTRACTION PROCEDURE .................................... 241

B VERIFICATION OF ENERGY APPROACH TO PROGRESSIVE COLLAPSE .......... 249

LIST OF REFERENCES ............................................................................................................ 285

BIOGRAPHICAL SKETCH ...................................................................................................... 291

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LIST OF TABLES

Table page 2-1 Steel building: selected results from nonlinear dynamic analysis (Powell, 2005) ........... 42

2-2 Calculation of Force Ratios, R1,i in Phase 1 (Dusenberry and Hamburger, 2006) .......... 51

3-1 Lightly reinforced slab modeling ...................................................................................... 87

3-2 Moment resistant beams (designated with “A”) ............................................................... 95

3-3 Column schedules of the typical SAC building ................................................................ 96

5-1 W12x58, 156 [in] column buckling results..................................................................... 145


5-3 Demand Capacity comparison ........................................................................................ 195






5-9 Steel profiles of columns (designations according to AISC, 2006) ................................ 207

5-10 Moment resistant beams (designated with “A”) ............................................................. 208

5-11 Demand capacity (D/C) ratios ........................................................................................ 233

B-1 Removed columns in CASE 1 ........................................................................................ 250


B-3 W14x74, 156 [in] column buckling results..................................................................... 258





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B-8 Failure energy limits for the selected columns ............................................................... 272


B-10 Failure energy limits for the selected columns ............................................................... 280

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LIST OF FIGURES

Figure page

2-1 Single degree of freedom system ...................................................................................... 38

2-2 Loading applied to the SDOF system (relative time to the natural period T) ................... 39

2-3 Displacement response to dynamic loads shown in Figure 2-2 ........................................ 40

2-4 Typical inelastic force-deformation relationship .............................................................. 41

2-5 Simple bar in tension ........................................................................................................ 45

2-6 Stress vs. strain relationship of the rod ............................................................................. 46

2-7 Strain and energy decomposition into plastic and elastic components ............................. 48

2-8 Released mass falling on the spring .................................................................................. 49

2-9 Deflected shape under static gravity forces ...................................................................... 50

2-10 Load-displacement during push-down static analysis ...................................................... 52

2-11 Spring coupled, longitudinally vibrating rods excited by a harmonic point force ............ 58

2-12 Comparison of exact and simplified solutions for the coupled rod system ...................... 58

2-13 Progressive collapse of the World Trade Center towers ................................................... 59

2-14 Typical load-displacement diagram of columns of one story ........................................... 60

2-15 Continuum model for propagation of crushing front ........................................................ 63

2-16 Free body diagram in the crush-down and crush-up phase ............................................... 65

3-1 Energy state in a hypothetical structural system ............................................................... 70

3-2 Energy transformation in a generic system ....................................................................... 71

3-3 Energy transformation between potential, internal and kinetic energies .......................... 72

3-4 Energy based approach to progressive collapse ................................................................ 73

3-5 High rise building collapse, initiating story ...................................................................... 74

3-6 Energy flow- displacement of single column (H = height of story) ................................. 75

3-7 Collapse propagates into consecutive story ...................................................................... 76

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3-8 Energy release with energy absorption by the crushing story itself ................................. 77

3-9 Collapse propagation due to insufficient energy absorption ............................................. 77

3-10 Typical stress-strain relationship for A36 steel ................................................................ 81

3-11 Three-dimensional solid cube buckling modeling ............................................................ 84

3-12 Integration scheme for W section in the Hughes-Liu beam element ................................ 86

3-13 Displacement- force history comparison of brick and Hughes-Liu modeling .................. 88

3-14 Buckling stress- slenderness parameter λc ....................................................................... 89

3-15 Selected three-story steel framed building ........................................................................ 91

3-16 Two-dimensional steel frame selected for the analysis .................................................... 92

3-17 Three-dimensional steel frame selected for the analysis .................................................. 93

3-18 Simplified steel framed building ....................................................................................... 93

3-19 Model of the SAC Modified Boston Building .................................................................. 94

3-20 Framing plan used for SAC three story building .............................................................. 95

3-21 Orientation of columns ..................................................................................................... 96

4-1 Selected three story steel framed building (W14x74 columns) ........................................ 99

4-2 Two dimensional frame in CASE 1 ................................................................................ 101

4-3 Three dimensional frame in CASE 1 .............................................................................. 101

4-4 Displaced shape of the simplified steel framed building in CASE 1 .............................. 102

4-5 Displacement of the building corner (CASE-1) .............................................................. 102

4-6 Beam moment time history in CASE-1: A) static, B) dynamic phase ........................... 103

4-7 Collapse sequence of two-dimensional frame in CASE 3 .............................................. 106

4-8 Collapse sequence of the three-dimensional frame in CASE 3 ...................................... 107

4-9 Building with hardened slabs - Collapse sequence in CASE 3 ...................................... 108

4-10 Building with typical slabs - Collapse sequence in CASE 3 .......................................... 110

4-11 Displacement time history of point A-1 in CASE 3 ....................................................... 112

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4-12 Moment time history of beam A-B at B-2 in CASE 3 .................................................... 113

4-13 Normalized internal forces in A2.1 column: A) hardened slabs, B) typical slabs .......... 114

4-14 Dynamic phase of normalized internal forces in A2.1 column: A) hardened slabs, B) typical slabs ..................................................................................................................... 115

4-15 Normalized internal forces in B2.1 column: A) hardened slabs, B) typical slabs .......... 116

4-16 Dynamic phase of normalized internal forces in B2.1 column: A) hardened slabs, B) typical slabs ..................................................................................................................... 117

4-17 Free fall requirement in CASE 1 .................................................................................... 119

4-18 Adherence to free fall in CASE 3 ................................................................................... 119

4-19 Selected three story steel framed building for buckling onset analysis .......................... 120

4-20 Arrested collapse of three-dimensional building w/ typical slabs in CASE A ............... 121

4-21 Arrested collapse of three-dimensional model w/ typical slabs in CASE B ................... 121

4-22 Collapse sequence of three-dimensional model w/ typical slabs in CASE C ................. 122

4-23 Displacements of A2.1 and B2.1 columns: 1) Complete history, 2) Dynamic, collapse phase. .............................................................................................................................. 124

4-24 Normalized internal forces in A2.1 column: 1) CASE A, 2) CASE C ........................... 125

4-25 Dynamic phase of normalized internal forces in A2.1 column: 1) CASE A, 2) CASE C......................................................................................................................................... 126

4-26 Normalized internal forces in B2.1 column: 1) CASE A, 2) CASE C ........................... 127

4-27 Dynamic phase of normalized internal forces in B2.1 column: 1) CASE A, 2) CASE C......................................................................................................................................... 128

5-1 Simulated buckling mode of W12x58, 156 [in] clamped column (beam elements) ...... 134

5-2 Prescribed column top displacement time history .......................................................... 134

5-3 Reaction force time history ............................................................................................. 135

5-4 Resistance-top displacement function ............................................................................. 136

5-5 Internal energy (deformation work) time history ............................................................ 137

5-6 Internal energy displacement history .............................................................................. 137

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5-7 Acceleration time history at the column top ................................................................... 139

5-8 Acceleration force history at the column top .................................................................. 140

5-9 Velocity force history at the column top ......................................................................... 140

5-10 Vertical displacement at the column top- force history .................................................. 141

5-11 Kinetic energy time history ............................................................................................. 141

5-12 Internal energy time history ............................................................................................ 142

5-13 Internal energy time history in pre-buckling phase ........................................................ 143

5-14 Internal energy rate ......................................................................................................... 144

5-15 Internal energy rate in the pre-buckling phase ................................................................ 144

5-16 W12x58: buckling force for the selected, force controlled, loading rates ...................... 146

5-17 W12x58: Internal buckling energy rate for the selected loading rates ........................... 147

5-18 W12x58: Internal buckling energy for the selected loading rates .................................. 147

5-19 Buckling force for the selected loading rates in W14x74 column .................................. 150

5-20 Internal buckling energy for the selected loading rates in W14x74 column .................. 150

5-21 Final shape of the selected structure after sudden column removal in CASE 1 ............. 151

5-22 Global energy histories (from GLSTAT) ....................................................................... 152

5-23 Dynamic phase of the global energies ............................................................................ 152

5-24 Internal energy time histories for all columns ................................................................ 153

5-25 Kinetic energies for all columns ..................................................................................... 154

5-26 Final deflection of the selected structure in CASE 2 ...................................................... 155

5-27 Global energies (GLSTAT) ............................................................................................ 155

5-28 Global energies in dynamic phase (GLSTAT) ............................................................... 156

5-29 Internal energies in columns ........................................................................................... 157

5-30 Instable columns as inferred from the internal energy results ........................................ 158

5-31 Kinetic energies of all columns ....................................................................................... 158

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5-32 Internal energy in selected columns (initial collapse phase) .......................................... 159

5-33 Identification of column buckling from the kinetic energies .......................................... 159

5-34 Global energies (from GLSTAT) .................................................................................... 161

5-35 Internal energies in columns ........................................................................................... 162

5-36 Internal energies in selected columns with buckling energy threshold .......................... 163

5-37 Verification of energetic results – Collapse sequence assessment ................................. 164

5-38 Energy propagation zones ............................................................................................... 167

5-39 Energy propagation through the inclusive building zones .............................................. 168

5-40 Normalized energy allocation in the building (inclusive zones) .................................... 168

5-41 Energy rates in the columns ............................................................................................ 169

5-42 Energy distribution among members in zone 1 .............................................................. 170

5-43 Global energy redistribution in CASE A ........................................................................ 171

5-44 Internal energies in CASE A........................................................................................... 172

5-45 Energy flow in building with typical slabs in CASE A (arrested collapse): A) Axial force-displacement, B) Internal energy-displacement .............................................................. 173

5-46 Energy rates in the columns in CASE A ......................................................................... 174

5-47 Energy propagation through the inclusive zones in CASE A ......................................... 175

5-48 Energy propagation through the exclusive zones in CASE A ........................................ 175

5-49 Normalized energy allocation in the inclusive zones in CASE A .................................. 176

5-50 Energy split between members in zone 1 (CASE A) ...................................................... 177

5-51 Arrested collapse of three-dimensional model w/ typical slabs in CASE B ................... 177

5-52 Global energies in CASE B ............................................................................................ 178

5-53 Internal column energies in CASE B .............................................................................. 179

5-54 Energy flow in building with typical slabs in CASE B (arrested collapse): A) Axial force-displacement, B) Internal energy-displacement. ............................................................. 180

5-55 Energy rates in columns in CASE B ............................................................................... 181

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5-56 Energy propagation through the inclusive zones in CASE B ......................................... 182

5-57 Energy propagation through the exclusive zones in CASE B ........................................ 182

5-58 Normalized energy allocation in the building in CASE B .............................................. 183

5-59 Energy split between members in zone 1 (CASE B) ...................................................... 183

5-60 Collapse sequence of three-dimensional model w/ typical slabs in CASE C ................. 184

5-61 Global energies in CASE C ............................................................................................ 186

5-62 Internal column energies in CASE C .............................................................................. 187

5-63 Energy flow in building w/ typical slabs: CASE C (total failure): A) Axial force-displacement, B) Internal energy-displacement. ............................................................. 188

5-64 Energy rates in columns in CASE C ............................................................................... 189

5-65 Energy propagation through the exclusive zones in CASE C ........................................ 190

5-66 Normalized energy allocation in the building in CASE C .............................................. 190

5-67 Energy split between members in zone 1 (CASE C) ...................................................... 191

5-68 Elastic component of the absorbed energy in A2.1 column ........................................... 192

5-69 Normalized energy decomposition into elastic and plastic component .......................... 192

5-70 Elastic component of absorbed energy in A1.1 beam y-y .............................................. 193

5-71 Normalized energy absorption in A1.1 beam y-y ........................................................... 194

5-72 Buckling force demand/capacity ratios ........................................................................... 195

5-73 Buckling energy demand/capacity ratios ........................................................................ 195

5-74 Parallel of energy capacity with axial capacity of W12x58, 156 [in] column ................ 198

5-75 Kinematics of column buckling ...................................................................................... 201

5-76 Bending stretches in the hinge ........................................................................................ 202

5-77 Moment equilibrium ....................................................................................................... 203

5-78 Elasto-plastic material model .......................................................................................... 205

5-79 Force displacement of W12x58 column of 156 [in] height ............................................ 206

5-80 Energy displacement of W12x58 column of 156 [in] height .......................................... 206

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5-81 Three story, moment resisting framed building for verification analysis ....................... 207

5-82 Framing plan used for SAC three story building ............................................................ 208

5-83 Deflected, final configuration of the building in CASE 2 .............................................. 209

5-84 Global Energies in CASE 2 ............................................................................................ 209

5-85 Internal column energies in CASE 2 .............................................................................. 210

5-86 Internal forces in CASE 2: A) column A2.1, B) column B2.1. ...................................... 211

5-87 Internal forces in the dynamic phase. CASE 2: A) A2.1 column, B) B2.1 column. ..... 212

5-88 Energy absorption in the selected columns. CASE 2: A) Axial force-displacement, B) Internal energy-displacement. ......................................................................................... 213

5-89 Energy rates in columns in CASE 2 ............................................................................... 214

5-90 Building zones used to trace the energy propagation ..................................................... 215

5-91 Energy propagation through the exclusive zones in CASE 2 ......................................... 215

5-92 Energy propagation through the inclusive zones in CASE 2 .......................................... 216

5-93 Normalized energy allocation in the building in CASE 2 .............................................. 216

5-94 Decomposition of the absorbed energy (deformation work) in A1.1 beam y-y ............. 217

5-95 Decomposition of the absorbed energy (deformation work) in B2.1 column ................ 218

5-96 Energy split between members in zone 2 (CASE 2)....................................................... 218

5-97 Global energies in the building ....................................................................................... 219

5-98 Collapse sequence of the steel building in CASE 3 ........................................................ 220

5-99 Building zones used to trace the energy propagation ..................................................... 221

5-100 Energy propagation through the exclusive zones in CASE 3 ......................................... 222

5-101 Normalized energy allocation in the building in CASE 3 .............................................. 222

5-102 Energy split between members in zone 2 (CASE 3)....................................................... 223

5-103 Decomposition of the absorbed energy (deformation work) in A2.1 beam y-y ............. 224

5-104 Decomposition of the absorbed energy (deformation work) in B1.1 beam x-x ............. 224

5-105 Close-up view of the energy (deformation work) decomposition in B2.1 column ........ 225

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5-106 Energy decomposition of the absorbed energy (deformation work) in B2.1 column ..... 225

5-107 Decomposition of the absorbed energy (deformation work) in C2.1 column ................ 226

5-108 Internal column energies in CASE 3 .............................................................................. 227

5-109 Normalized internal forces in: A) B2.1 column, B) C2.1 column .................................. 228

5-110 Normalized internal forces in dynamic phase: A) B2.1 column, B) C2.1 column ......... 229

5-111 Energy flow in the SAC building: CASE 3 (total failure): A) Axial force-displacement, B) Internal energy-displacement ..................................................................................... 230

5-112 Energy rates in columns in CASE 3 ............................................................................... 231

5-113 Buckling force demand/capacity ratios ........................................................................... 233

5-114 Buckling energy demand/capacity ratios ........................................................................ 234

5-115 Failure energy demand/capacity ratios ........................................................................... 234

A-1 Steel frame used for the energy benchmark test ............................................................. 241

A-2 Final, displaced shape of the 2-D frame used in the energy benchmark test .................. 242

A-3 Global energies reported by LS-DYNA in GLSTAT file ............................................... 242

A-4 Analytical and numerical external work results .............................................................. 243

A-5 Analytical and numerical external work results during the static preloading phase ....... 244

A-6 Global (GLSTAT) and sum of local energies (MATSUM) during the static preloading245

A-7 Comparison of global (GLSTAT) and sum of local energies (MATSUM) .................... 245

A-8 Location of the selected node used for energy- displacement histories .......................... 246

A-9 Internal, kinetic and total energy during the static preloading ........................................ 247

A-10 Internal, kinetic and total energy .................................................................................... 247


B-2 Final configuration of the steel building (CASE 1) ........................................................ 250

B-3 Global energies in CASE 1 ............................................................................................. 250

B-4 Internal column energies in CASE 1 .............................................................................. 251

B-5 Energy rates in columns in CASE 1 ............................................................................... 252

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B-6 Building zones used to trace the energy propagation ..................................................... 253

B-7 Energy propagation through the building in CASE 1 ..................................................... 254

B-8 Normalized energy allocation in the building in CASE 1 .............................................. 254

B-9 Energy split between members in zone 2 (CASE 1)....................................................... 255


B-11 Final configuration of the building in CASE 4 ............................................................... 257




B-15 Building zones used to trace the energy propagation in CASE 4 ................................... 260





B-20 Collapse sequence of the steel building in CASE 5 ........................................................ 263





B-25 Energy propagation through the building in CASE 5 .................................................. 268




B-29 Final configuration of the building in CASE 6 ............................................................... 271


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B-39 Collapse sequence of the steel building in CASE 7 ........................................................ 278



B-42 Internal column energies in CASE 7. Close-up view ..................................................... 281






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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

PROGRESSIVE COLLAPSE OF MOMENT RESISTING STEEL FRAMED BUILDINGS

QUANTITATIVE ANALYSIS BASED ENERGY APPROACH

By

Stefan Tadeusz Szyniszewski

May 2009

Chair: Theodor Krauthammer Major: Civil Engineering

An energy based, quantitative analysis of collapse propagation in steel framed buildings

was developed in this work. Column buckling energy was proposed as a necessary condition to

initiate the collapse (but it is not a sufficient). The column failure energy was introduced and

verified as the sufficient collapse criterion. This study demonstrated that energy based analysis

and energetic failure criteria facilitate understanding of collapse propagation and reveal the

underlying mechanisms during the collapse arrest or propagation.

Simulations of structural response to the sudden removal of key structural member(s) have

been carried out for a number of moment resisting steel frames and steel framed buildings. The

modeling aspects (e.g. retrieving reliable structural information, probabilistic geometric

inaccuracies, and material models) are presented. Kinematic and energetic results with limited

experimental validation are elaborated. An easy to use energy based analysis of collapse

propagation has been developed and clearly explained in this study.

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CHAPTER 1 INTRODUCTION

Problem Statement

“Progressive collapse is a catastrophic structural failure that ensues from local structural

damage that cannot be prevented by the inherent continuity and ductility of the structural

system“(Ellingwood and Dusenberry, 2005). The local damage or failure initiates a chain

reaction of failures that propagates through the structural system, leading to an extensive partial

or total collapse. The resulting damage is disproportionate to the local damage caused by the

initiating event. Such local initiating failures can be caused by abnormal loads not usually

considered in design. Abnormal loads include gas explosions, vehicular collisions, sabotage,

severe fires, extreme environmental effects and human errors in design and construction.

Importance

All buildings are susceptible to progressive collapse in varying degrees because the

combination of their behavioral characteristics and abnormal loading may not be completely

predicted nor eliminated. Progressive collapse usually leads to numerous deaths. For example

2750 people died in the World Trade Center (WTC) collapse (Associated Press, 2007). No

building system can be engineered to be absolutely free from the risk of progressive collapse due

to the presence of numerous uncertainties arising in the building process or from potential

failure-initiating events. The destruction of the WTC on September 11, 2001 and other cases to

be discussed confirm that potential threats cannot be totally eradicated.

Research Objectives

The main objective of this study was to enable the development of a rational energy-based

analysis of progressive collapse of moment resisting steel framed buildings by studying simple

frame systems focusing on the role of energy flow.

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• Objective 1: Examine energy flow and redistribution during progressive collapse,

• Objective 2: Provide expressions for column buckling energy (deformation work) for typical

W-shape columns to facilitate the energy based analysis approaches,

• Objective 3: Provide an energy-based approach for collapse assessment of moment resisting

steel buildings by analyzing the internal energy (deformation work) of structural members in

order to evaluate dynamic alternate paths reformations, competing failures, and structural safety,

Column buckling energy (deformation work) was proposed as a rate independent buckling

criterion in progressive collapse considerations. Moreover, the column failure energy

(deformation work) was proposed and verified as the limit of the global building safety. This

study demonstrated that energy-based analysis and energy buckling and failure criteria facilitate

a better understanding of collapse propagation, and reveal the underlying mechanisms during the

collapse arrest or propagation.

Scope

This research was limited to understanding the characteristics of progressive collapse in

moment resisting steel framed buildings. The initial structural responses to abnormal loadings

(e.g. a plane crash, an impact, an explosion) were not considered. Damage caused by the

conditions cited above was assumed to result in abrupt removal of columns at critical locations.

Structural details, such as walls and partitions, can affect the response of the structure.

Including these secondary elements in the analysis would increase the complexity of the

responses. Therefore, the effects of secondary elements were not considered because the

response of the major load carrying members should be clarified first. The analysis was

conducted using finite element simulations. The level of modeling complexity was gradually

21

enhanced: frames, frames with slabs, and finally frames with slabs including semi-rigid

connections.

Simulations of structural response to the sudden removal of a key column(s) have been

carried out for a number of moment resisting steel frames and steel framed buildings. The

modeling aspects (e.g. retrieving reliable structural information, probabilistic geometric

inaccuracies, and material models) are presented in detail.

The objectives were accomplished within the following scope:

1. The analysis was limited to high fidelity finite element simulations by means of a

commercially available software package (LS-DYNA),

2. Simplified steel framed buildings with seismic detailing were analyzed to assess the

feasibility of energy-based approaches to studying progressive collapse,

3. Calculation procedures for buckling energy were proposed for typical W-shape steel

columns,

4. Collapse of a three story seismic steel building was simulated after the initiation of

numerous localized failures to verify the proposed energy-based analysis procedure.

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CHAPTER 2 LITERATURE REVIEW

Overview

Numerous incidents of progressive collapse attest to its danger. To name just a few:

Ronan Point apartments in the United Kingdom in 1968 (Griffiths, et al., 1968, Pearson and

Delatte, 2005), where a kitchen gas explosion on 18th floor sent a 25 story stack of rooms to the

ground; the 2000 Commonwealth Ave. tower in Boston in 1971, triggered by punching of

insufficiently hardened slab; bombing of the Murrah Federal Building in Oklahoma City, in

1995, where the air blast pressure sufficed to destroy only a few columns and slabs at lower

floors, whereas the upper floors failed by progressive collapse; attack on the Pentagon (Mlakar,

Dusenberry, and Harris, 2002); WTC collapse in 2001; residential building collapse in Italy

(Palmisano, et al., 2007) and others.

A structural system should be designed to withstand local damage without the

development of a general, total structural collapse and thus prevent the aforementioned disasters.

A structure should remain stable to allow for evacuation and emergency operations and permit

temporary support or repair. Successful design must result in a structure which has the capacity

to limit a local failure to the immediate area.

In general, three approaches to prevent progressive collapse have been postulated

(Ellingwood and Dusenberry, 2005):

1. Event control. Elimination and/or protection from incidents that might cause progressive

collapse. This is not practical because abnormal events cannot be completely predicted and,

even identified, threats cannot be entirely eliminated,

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2. Indirect design. Preventing progressive collapse by specifying minimum requirements

with respect to strength, structural continuity, ductility, etc. Accidental load and the

ensuing structural behavior are not explicitly evaluated,

3. Direct design. Considering resistance against progressive collapse and the ability to absorb

damage directly by means of a non-linear finite element method or any available methods

as a part of the design process.

Event Control

Event control aims to eliminate the cause of the progressive collapse by:

1. Isolating the building from possible threats by limiting access to structure, tightening

security checks, etc.

2. Specific Local Resistance. “Hard spots” are designed into the structure, at areas that are

believed to be prone to accidental loads (e.g. exterior columns at risk are designed to

withstand a blast and an impact load).

Unfortunately, event control provides resistance only to specific, perceived hazards that are

usually difficult to quantify reliably. Specific abnormal loads seldom can be designed against,

economically, because they produce bulky, redundant members, which should survive the

accidental load.

Indirect Design

Indirect design aims at implementing good design and construction practices, which boost

structural safety. Indirect design is more qualitative than quantitative in its nature. After each

catastrophic collapse, reasons of a disproportionate collapse are identified and investigators

recommend remedies and fixes for recognized problems. Examination of historical catastrophic

failures and their effects on the design community exposes how the indirect design

recommendations were acquired.

24

Historical Cases

Collapse of the 22-story Ronan Point apartment tower, in May 1968, in London resulted in

provisions of sufficient ductility and continuity so that structures would remain stable under local

damage. The collapse was initiated by a gas explosion on the 18th floor (5 people died and 17

were injured). The force of the explosion knocked out the exterior, corner walls of the

apartment. These walls were the sole supports for the walls directly above. This created a chain

reaction in which floor 19 collapsed, then floor 20 and so on, propagating upwards. The four

floors fell onto level 18, which initiated a second phase of the collapse. This sudden impact

loading on floor 18 caused it to give way, smashing down to floor 17 and progressing until it

reached the ground (Griffiths, et al., 1968, Pearson and Delatte, 2005).

A panel formed to investigate the collapse determined that the precast elements were not

sufficiently tied together (Griffiths, et al., 1968). Building joints between walls and slabs were

constructed by filling voids with mortar and tightening panels by use of lifting rods (Levy and

Salvadori, 1994). These connections heavily relied on friction between precast panels

(especially bottom connections of wall panels with slabs). Poor workmanship aggravated the

inherent structural weakness. Some of the joints had less than fifty percent of the mortar

specified.

The lessons from Ronan Point resulted in recommendations of “tying building elements

together and increasing ductility so that the building elements can better sustain deformations

from the failure of a portion of the building’s structure. Transverse ties create cantilever action

form adjacent walls. Vertical ties provide suspension from panels above, peripheral ties hold

floors together, and longitudinal ties string floor planks – large prestressed panels – together”

(Pearson and Delatte, 2005).

25

The Murrah building bombing resulted in recommendations of designing for the loss of a

column at the building perimeter without progressive collapse and provisions of sufficient

reinforcement for reverse bending in beams and slabs. On April 19, 1995, a truck loaded with an

ammonium nitrate and fuel oil (ANFO) bomb caused the collapse of fully half of the total floor

area of the nine-story, reinforced concrete Murrah Federal Building in Oklahoma city. The

extent of the collapse, resulting from the initial loss of up to three columns due to the original

explosion, extended well beyond the zone of direct structural blast damage (Osteraas, 2006).

The truck bomb, estimated to be 4,000 [lb] (1,800 [kg]) TNT equivalent, was centered

approximately 13 [ft] (4 [m]) from the 1st floor column located in the center of the external

building’s facade. Given its proximity to the blast, it is generally believed that the concrete in

the column was shattered by the blast, leaving only the column’s bundled reinforcing steel. The

short wall segment located only 7 [ft] (2.1 [m]) further into the building was largely intact, as

was the third-floor transfer girder. Absent any alternative load path, loss of that column led to

the loss of four bays over the full height of the building.

As the blast wave expanded, it exerted an upward force on the floor slabs. As the thick

floor slabs were not reinforced to resist upward pressures, differential blast pressures of less than

1 [psi] (7 [kPa]) were sufficient to uplift the floor slabs. Because the floor slabs were cast

monolithically with the transverse floor beams, and the connection between the two was well

reinforced, reactions from the upward pressure on the floor slabs were efficiently transferred into

the floor beams, which also were not designed to resist upward pressures. The results were

reverse bending in the beams with reverse flexural and shear cracking at the columns.

The upward movement of the floor slab also generated large catenary forces, which pulled

the top edge of the transfer girder, causing it to rotate inward. Once the blast wave passed,

26

gravity took over, beam/column connections failed in punching shear, and the floor structure

draped as a catenary. Of a total of 20 bays in the footprint of the building, 10 collapsed over the

full height of the building.

Destruction of the Murrah Building was a combination of the direct blast damage and the

structural configuration that led to the progressive collapse that occurred. The following

conclusions were drawn from the Murrah Building collapse (Osteraas, 2006):

• A complete three-dimensional space frame that interconnects all load path elements

provides better stability than a frame with antiredundant features such as the transfer girder;

• The frame must be robust and ductile to absorb overloads with large deformations while

maintaining continuity;

• Beams and slabs must be sufficiently reinforced to resist not only downward but also

upward bending,

• Lower portions of perimeter columns should be designed, to the greatest extent possible, to

resist the direct effects of blast;

• The frame could be protected with the provision of “mechanical fuses” that allow slabs and

walls to fail without destroying the frame, should an explosion occur.

Interestingly, a robust connection between floor slabs and floor beams contributed to the

final collapse, as opposed to the common belief that strong structural ties unconditionally

improve the building safety against progressive collapse. Much attention has been paid to

adopting seismic detailing for important buildings in non-seismic zones. Protecting the frame

with mechanical fuses is in some ways in conflict with the good seismic design practice of tying

all components together as well as possible. Thus, the catastrophic failure of the Murrah

27

Building revealed that provisions of sufficient ductility and continuity should not be used as a

cure-all against progressive collapse and that only direct collapse analysis can produce a safe

design (Osteraas, 2006).

Collapse of the World Trade Center (WTC) towers in New York in 2001 resulted in

provisions of sufficient fireproofing protection of steel members so that a structure would remain

stable for the time needed for safe evacuation and fire extinguishing. As generally accepted by

the community of specialists in structural mechanics, the failure scenario was as follows

(National Institute of Standards and Technology, 2005; Bazant and Verdure, 2007):

1. About 60% of the 60 columns of the impacted face of framed tube and about 13% of the

total of 287 columns were severed, and many more were significantly deflected. This

caused stress redistribution, which significantly increased the load of some columns,

attaining or nearing the load capacity for some of them,

2. Because a significant amount of steel insulation was stripped, many structural steel

members heated up to 600°C, as confirmed by annealing studies of steel debris. The

structural steel lost about 85% of its yield strength at 600°C;

3. Differential thermal expansion, combined with heat-induced viscoplastic deformation,

caused the floor trusses to sag. The catenary action of the sagging trusses pulled many

perimeter columns inward by about 3[ft] (1[m]). The bowing of these columns served as a

huge imperfection, inducing multistory out-of-plane buckling of framed tube wall. The

lateral deflections of some columns due to aircraft impact, the differential thermal

expansion, and overstress due to load redistribution also diminished buckling strength,

28

4. The combination of the following seven effects finally led to buckling of columns:

a. overstress of some columns due to initial load redistribution,

b. overheating due to loss of steel insulation,

c. drastic lowering of yield limit and creep threshold by heat,

d. lateral deflections of many columns due to thermal strains and sagging floors,

e. weakened lateral support due to reduced in-plane stiffness of sagging floors,

f. multistory bowing of some columns (for which the critical load is an order of

magnitude less than it is for one-story buckling),

g. local plastic buckling of heated column webs.

5. The upper part of the tower fell, with little resistance, through at least one floor height,

impacting the lower part of the tower. This triggered progressive collapse because the

kinetic energy of the falling upper portion exceeded, by an order of magnitude, the energy

that could be absorbed by limited plastic deformations and fracturing in the lower part of

the tower (Bazant and Zhou, 2002).

Although the structural damage inflicted by the aircraft was severe, it was only local.

Without the stripping of a significant portion of the steel insulation during impact, the

subsequent fire would likely not have led to overall collapse (National Institute of Standards and

Technology, 2005). Therefore, the stripping of fireproofing steel insulation proved critical to the

collapse of both towers. However, there are also claims that the structural system adopted for the

Twin-Towers may have been unusually vulnerable to a major fire, irrespective of the plane crash

effects (Usmani, Chung, and Torero, 2003).

Indirect design is a set of provisions and recommendations learned from catastrophic

collapses. It advocates design and construction practices, which eliminate roots and causes of

29

historical failures and thus enhance structural safety. Collapse of Ronan Point in London

resulted in provisions of sufficient ductility and continuity, so that structures would remain stable

under local damage. The Murrah building bombing prompted recommendations of designing for

the loss of a column at the building perimeter without progressive collapse and provisions of

sufficient reinforcement for reverse bending in beams and slabs. Collapse of the World Trade

Center towers resulted in provisions of sufficient fireproofing protection of steel members so that

structures would remain stable for the time needed to put out the fire.

The main flaw of indirect design lies in its tendency to attribute findings based on a single

sample of catastrophic failure to the whole population of structures. Whereas lack of sufficient

ties between precast slabs and walls proved fatal in the Ronan Point building, excessive ties

between slabs and transfer beams in the Murrah Building contributed to the damage of the main

frame and extended the collapse zone. Therefore, only the direct design of a given structure can

fully evaluate its resistance or predisposition to progressive collapse.

Building Code Measures to Prevent Progressive Collapse

Specific direct design approaches to prevent progressive collapse as a result of abnormal

loads have not been standardized in the United States or elsewhere. Building codes and

standards resort to indirect design and invariably treat general structural integrity and progressive

collapse in qualitative rather than quantitative terms. This is due to the current lack of insight

into the nature of this phenomenon.

The ASCE Standard 7-05, Minimum Design Loads for Buildings and Other Structures

(American Society of Civil Engineers (ASCE), 2005), previously known as ANSI Standard

A58.1, first introduced a requirement for progressive collapse due to “local failure caused by

severe overloads” in section 1.3.1 in the 1972 edition - published following the 1968 Ronan

30

Point collapse. ANSI Standard A58.1 published in 1982, section 1.3: General Structural

Integrity, contained a more comprehensive performance statement, and contained a greatly

expanded commentary with references. Recommendations were made to provide a good plan

layout, returns on walls, at least minimal two-way action for floors, load-bearing interior

partitions, catenary action in floor systems, and beam action in walls; these were accompanied by

figures. The 1988, 1993, 1995, and 1998 editions of ASCE Standard 7 (standard and

commentary) were similar, although in the course of time the commentary was shortened by

eliminating the figures and other specific guidance but retaining the discussion of general design

approaches to general structural integrity (Elingwood and Dusenberry, 2005). This gradual

removal of detailed guidance has been caused by fading confidence that the specifics acquired

from particular, historical collapses can be generalized to all types of structures.

In 2005, the provisions in section 1.4 of ASCE Standard 7-05 were revised to reflect more

recent information and to include more specific suggestions for the enhancement of general

structural integrity. A new section 2.5 (and commentary) was added to the load combinations

section of ASCE Standard 7-95, which required a check of strength and stability of structural

systems under low-probability events, where required by the authority having jurisdiction.

Section 1.4 stipulates that “buildings and other structures shall be designed to sustain

local damage with the structural system as a whole remaining stable” and not being damaged to

an extent disproportionate to the original local damage. This shall be achieved through an

arrangement of the structural elements that provides stability to the entire structural system by

transferring loads from any locally damaged region to adjacent regions capable of resisting those

loads without collapse” (American Society of Civil Engineers (ASCE), 2005).

31

Section C2.5 of the commentary recommends that after an element is notionally removed;

the capacity of the remaining structure should be checked using the following load combination:

W 0.2+S)0.2orL(0.5+D1.2) or (0.9 (2-1)

The 0.5 L corresponds to the mean value of maximum live load. The 0.9 load factor is used

when the dead load helps with overall building stability. The load combination in Equation (2-1)

has an annual probability of being exceeded equal to 0.05.

Structural Concrete (ACI 318-02) and Commentary (ACI 318R-02) address general

structural integrity by specifying prescriptive detailing requirements (American Concrete

Institute (ACI), 2004). The commentary to section 7.13 includes the statement: “Experience has

shown that the overall integrity of a structure can be substantially enhanced by minor changes in

detailing of reinforcement.” The code itself addresses detailing of reinforcement and

connections, to effectively tie together the structural members to “improve integrity of the

overall structure.” The requirements address continuation of reinforcement through supports, the

location and nature of splicing, and provisions for hooks at terminations. Special requirements

are cited for pre-cast construction. In section 7.13, the code requires transverse, longitudinal,

and vertical tension ties around the perimeter of the structure. Section 16 describes details for

the required ties, and prohibits use of connections that rely solely on friction from gravity load.

Steel Construction Manual - Load and Resistance Factor Design (LRFD) (American

Institute of Steel Construction (AISC), 2005) does not appear to contain provisions specifically

addressing progressive collapse or general structural integrity.

The Interagency Security Committee (Interagency Security Committee, 2001) issued

specifications on blast resistance or other specialized security measures. Progressive collapse

analysis is handled indirectly, by reference to ASCE Standard 7-95 for specific details on

32

prevention of progressive collapse. It allows progressive collapse analyses for an effective live

load that is as low as 25% of the code prescribed live load, and for enhanced ultimate strengths,

to recognize that material strengths often are higher at strain rates associated with impact than

they are at strain rates common for application of normal environmental loads.

The General Services Administration (GSA), in its progressive collapse analysis and

design guidelines for new federal office buildings and major modernization projects published in

2003, assisted in the reduction of the potential of progressive collapse in new Federal office

buildings. The first step in the process defined by this standard is to evaluate the risk and threat

by evaluating occupancy and vulnerability.

The GSA guidelines also provide a simple means to assess collapse potential through

linear analysis. Although such a simplistic approach can be very appealing to civil engineers, its

ability to model the real collapse behavior is rather doubtful. The structure, with a missing

vertical load-carrying element, is analyzed for the applied load:

( )LDLPC 25.02 += (2-2)

where LPC = load applied for collapse analysis; D = dead load; and L = design-base live load.

The load factor of two on the combined dead and live loads can be thought of as a dynamic

amplification factor to account for the rapid application of the load in an elastic system. The

potential for disproportionate collapse is determined by the calculation of a demand-capacity

ratio (DCR) for each primary and secondary structural element as:

CEUD QQDCR /= (2-3)

QUD = force determined by linear elastic, static analysis in the element or connection QCE = expected ultimate capacity of the component and/or connection.

If the DCR < 2, a concrete structure is deemed safe, and the collapse arrested. A table of

DCRs defines failure criteria based on a variety of potential failure modes for steel elements.

33

Limiting values of the DCR represent correlation factors that might loosely account for the

ductility (energy absorption) inherent in conventional structures (Dusenberry and Hamburger,

2006).

The Department of Defense requires prevention of the progressive collapse for all

inhabited structures of three stories or more. The provisions are similar to those in ASCE

Standard 7-98. The alternate load path procedure requires notional removal of one primary

vertical and one primary horizontal element at key locations in the structure. Connections must

be able to develop the capacity of the weaker connected element, and detailing must conform to

requirements for seismic design. A capability to withstand load reversals must be provided.

Damage must be contained to the stories above and below the location of notional initiating

damage, and horizontal damage must be confined to 750 [ft2] (70 [m2]) or 15% of floor area.

Specific design provisions to control the effect of extraordinary loads and risk of

progressive failure can be developed with a probabilistic basis (Ellingwood and

Leyendecker, 1978; Elingwood and Dusenberry, 2005). One can either attempt to reduce the

likelihood of the extraordinary event or design the structure to withstand or absorb damage from

the event if it occurs. Let F be the structural failure and Hi be the structurally damaging event.

The probability of failure due to Hi (i = 1...n) is:

∑=

⋅⋅=n

iiiif HPHDPDHFPP

1)()|()|(

(2-4)

F = event of structural collapse; P[Hi] = probability of ith hazard Hi; P[D |Hi] = probability of local damage D, given that hazard Hi occurs; P[F |DHi] = probability of collapse, given that hazard Hi and local damage D both occur;

The separation of P(F|Hi) = P(F|DHi) P(D|Hi) and P(Hi) allows one to focus on strategies

for reducing risk. P(Hi) depends on siting, controlling the use of hazardous substances, limiting

34

access, and other actions that are essentially independent of structural design. In contrast,

P(F|Hi) depends on structural design measures ranging from minimum provisions for continuity

to a complete post-damage structural evaluation.

The probability P(Hi) depends on the specific hazard. Limited data for severe fires, gas

explosions, bomb explosions, and vehicular collisions indicate that the event probability depends

on building size, which is measured in dwelling units or square footage, and ranges from about

0.23 x l0-6 [dwelling unit/year] to about 7.8 x l0-6 [dwelling unit/year]. Thus, the probability

that a building structure is affected may depend on the number of dwelling units (or square

footage) in the building.

The probability of structural failure P(F) must be limited to some socially acceptable value

through prudent professional practice and appropriate building regulation. If one were to set the

conditional limit state probability, P(F|Hi) = 0.1~0.2 [1/year], the annual probability of structural

failure from Equation (2-4) would be on the order of 10-7 up to 10-6, placing the risk in the low-

magnitude background. Design requirements corresponding to P(F|Hi) = 0.1~0.2 can be

developed using first-order reliability analysis if the limit state function describing structural

behavior is available. While current building standards ensure that building failures occur only

rarely, no one knows exactly what a socially acceptable building failure rate might be. However,

there is evidence (Elingwood and Dusenberry, 2005) that the risk (measured in terms of

probability) below which society normally does not impose any regulatory guidance, is on the

order of 10-7/year. Accepting this target value requires a sociopolitical decision that is outside

the scope of this work.

35

Direct Design: Finite Element Method

Direct design considers resistance against progressive collapse and the ability to absorb

damage directly by means of non-linear finite element methods or any other available methods.

The most straightforward approach is to use non-linear commercial finite element codes to

simulate dynamic behavior after removal of one or more key members. Finite element codes

such as LS-DYNA, ABAQUS and so on, provide more sophisticated simulation capabilities

every year.

Moment resisting steel frames have been analyzed in recent years using commercially

available finite element package ABAQUS (Krauthammer et al., 2004; Lim and Krauthammer,

2006). Among other structures, a realistic 10-story moment resisting frame was modeled in

Abaqus. One, two or three columns at the ground level were instantaneously removed at the

prescribed time (following the quasi-static application of gravity, dead and live loads).

A numerical estimation of internal forces during collapse can also be retrieved from the

simulation results (Liu, et al., 2005). A time history of tying force between a girder and column

adjacent to the collapse initiation as well as the effect of column removal time on displacements

can be estimated from the computations. Connections play an important role in the overall

structural behavior, and their properties significantly affect the structural behavior of individual

members (Liu, et al., 2005).

Whereas most of the connections can be treated as moment resisting connections (assumed

to transfer moments between members) and shear connections (modeled as pins, i.e. no moment

transfer), the reality is more complicated because most of the connections are capable of partially

transferring moments. Such connections can fail before catenary action is developed in beams

and slabs, thus their performance affects the global structural behavior. In order to incorporate

connection behavior, Lim and Krauthammer (2006) obtained force-displacement and moment-

36

rotation relations using high resolution FEM models (solid elements). These relations were later

used for springs’ properties at connections in simplified models.

High resolution models of perimeter resisting frames were also reported by Khandelwal

and El-Tawil (2005). Columns and girders of the perimeter resisting frame were modeled with

shell elements. Existing general purpose finite element method codes enable modeling of

structural elements, connections, bolts, welds, etc. at very high resolution. This trend is expected

to continue. However, more complicated models require more input information and expensive

computer resources, and produce vast output, which must be parsed and analyzed in order to

answer relevant design questions.

More detailed models require additional parameters than just extra geometrical information

in comparison to simplified representations. To name just a few: the fracture criterion (depends

on element size, geometric details such as beam access hole and welded stiffeners, etc.),

geometric imperfections (to produce more physical local behavior) and others.

37

Static Procedures for Progressive Collapse Modeling

In view of the aforementioned complications and excessive labor involved in high

resolution modeling of progressive collapse; there is a widespread call for simplified procedures

to assess structural resistance to progressive collapse. Complex structural systems do not have

an analytical, closed form solution, which would explicitly reveal the underlying dynamics. On

the contrary, computational mechanics result in vast numerical outputs, which are more difficult

to generalize. Analytical solutions exist for single degree of freedom systems. Explicit solutions

of such problems, in terms of symbolic parameters, provide insight into the investigated

phenomena. Such results cannot, however, be uncritically extrapolated to multi degree of

freedom systems.

The concept of amplified static loads, to account for dynamic effects, directly originates

from the analysis of the dynamic response of a single degree of freedom (SDOF - Figure 2-1)

system to step loading (Figure 2-2).

Figure 2-1. Single degree of freedom system

Response of a SDOF system without damping to a step load (Figure 2-2) can be calculated

by solving the following differential equation (representing equilibrium of forces and/or

conservation of momentum):

.0 constgmFkxxm ===+&& (2-5)

m – mass, k – spring’s stiffness, x(t) – spring’s displacement, g – gravity acceleration

38

Figure 2-2. Loading applied to the SDOF system (relative time to the natural period T)

Assuming the absence of an initial velocity and initial displacement, solution to

Equation(2-5):

Tmt

mktx ω,cos1)( 0 =≡⎟

⎟⎠

⎜⎜⎝

⎟⎟⎠

⎜⎜⎝

⋅−=kkF π2⎞⎛ ⎞⎛

(2-6)

Maximum response (without damping) is reached every T/2 and its magnitude is:

( ) ...,3,2,1,2)cos(1)2

( 00max =⎟

⎠⎞

⎜⎝⎛⋅=⋅−=⋅== n

kF

nkFTntxx π

(2-7)

The system reaches similar displacements under step loading and under the application of a

doubled quasi-static load, but their time histories differ significantly (Figure 2-3). Displacement

shown in Figure 2-3 is relative to static displacement = u dynamic / u static (assuming 5% of critical

damping). Force in the spring under quasi-static doubled load is equal to the maximum dynamic

39

force because the force-displacement spring relation ( ) does not depend on the

loading (static vs. dynamic).

xkFspring ⋅=

Figure 2-3. Displacement response to dynamic loads shown in Figure 2-2

The guidelines for equivalent static approach are essentially as follows (Powell, 2005):

1. Remove column. First remove column(s) and then apply static loads,

2. Loads. The basic gravity load is 1.2D + 0.5L (dead load = D, live load = L). This load is

applied in all bays except those adjacent to the removed column. In these bays an impact

amplification factor of 2.0 is applied, giving a load of 2.4 D + 1.0 L. Lateral wind load

0.2W is also applied,

3. Strength Capacities. Nominal strengths are calculated from the formulas in the ACI,

AISC, etc. design codes. Material over-strength (Ω) factors are specified, typically 1.25 for

reinforced concrete and 1.3 for structural steel. Strength reduction (φ) factors are the ACI

and AISC values (typically 0.9 or 0.85),

40

4. Deformation Capacities. Only flexural behavior. For shear effects, in both concrete and

steel, zero ductility is assumed, and yield is not allowed,

5. Strength Loss. If the deformation capacity is exceeded for any beam, it is removed, and its

loads applied on the beam below, approximating collapse. It is unlikely that this option

will be used in most static analysis cases because it is a complex process.

Tying force results exhibit a cyclic nature. Cyclic loading reduces strength capacity of a

member (Powell, 2005) as shown in Figure 2-4. Therefore, when comparing the simulated

maximum forces to static capacities of designed members, caution and engineering experience

must be exercised.

First yield

Strain hardening

Ultimate strength

Force

Ductile limit

Residual strength

With cycling

Deformation

Initialstress

Figure 2-4. Typical inelastic force-deformation relationship

Whereas it has been reported that the actual dynamic amplification factor is lower than 2

and closer to 1.5 for selected moment resisting frames (Ruth, et al., 2006), its actual values can

well exceed 2 in other structures (Powel, 2005). Kaewkulchai and Williamson (2004) reported

that static, material non-linear analysis produced un-conservative results for different two

dimensional frames as compared to their dynamic, time-history analysis. Their analysis showed

that static loading applied to the damaged structure (e.g. with one column removed) resulted in

41

unsafe results. Whereas static analysis produces only four plastic hinges, dynamic time history

calculations resulted in thirteen plastic hinges. Hence, static analysis underestimated the final

damage to the structure.

A number of frame structures were analyzed by Powel (2005) to illustrate the variation

between results obtained from dynamic and static analysis. A representative comparison of

demand/capacity (D/C) results for one of the models is presented in Table 2-1. The ratio of

dynamic to static results is the actual dynamic amplification factor. However these factors vary

considerably between internal forces, displacements and rotations (even in a single analysis for

the same structure). Additional variation has been observed between various structures (Ruth, et

al., 2006).

Table 2-1. Steel building: selected results from nonlinear dynamic analysis (Powell, 2005) Result Type Maximum Value Dynamic Amplification

Exterior column strength, D/C ratio .28 2.8

Interior column strength, D/C ratio .86 1.7

Floor beam moment strength, D/C ratio .86 2.3

Floor beam connection strength, D/C ratio .97 2.1

High variation between dynamic amplification factors makes the selection of such a

consistent factor in advance for equivalent static analysis almost impossible. Therefore, the

dynamic amplification factor approach proves unreliable for a realistic assessment of structural

resistance to progressive collapse. Unfortunately there is ungrounded belief among practicing

engineers that static analysis with a dynamic amplification factor of 2 is overly conservative

(Marjanishvili, 2004) and its value could even be reduced to 1.5 (Ruth, et al., 2006).

The poor agreement between time-history dynamic analysis and static analysis with

amplified static dead loads to account for inertia arises from the multi-degree of freedom

42

(MDOF) nature of real systems. Whereas a dynamic amplification factor (DAF) of two is an

excellent approximation for a single degree of freedom (SDOF), such a uniform value does not

exist for MDOF systems. Moreover, DAF = 2 is not the upper bound for actual amplification

factors in an MDOF system, which may well exceed two (Table 2-1). Therefore, static methods

with a DAF shall not be used for the design of progressive collapse resistant structures because

they may produce unconservative estimations, and their error is unbounded and unknown to the

designer.

Energy Concepts

The word "energy" derives from Greek ἐνέργεια (energeia), which appeared for the first

time in the work Nicomachean Ethics (Aristotle, translated by Jonathan Barnes, 1984) of

Aristotle in the 4th century BC. Several different forms of energy exist to explain all known

natural phenomena. These forms include (but are not limited to) kinetic, potential, thermal,

gravitational, sound, light, elastic, and electromagnetic energy. Although certain energies can be

transformed to another, the total energy remains the same. This principle, the conservation of

energy, was first postulated in the early 19th century, and applies to any isolated system.

“The concept of energy emerged out of the idea of vis viva, which Leibniz defined as the

product of the mass of an object and its velocity squared; he believed that total vis viva was

conserved. In 1807, Thomas Young was the first to use the term "energy" instead of vis viva, in

its modern sense. Gustave-Gaspard Coriolis described "kinetic energy" in 1829 in its modern

sense, and in 1853, William Rankine coined the term "potential energy." It was argued for some

years whether energy was a substance (the caloric) or merely a physical quantity, such as

momentum. William Thomson (Lord Kelvin) amalgamated all of these laws into the laws of

thermodynamics, which aided in the rapid development of explanations of chemical processes

43

using the concept of energy by Rudolf Clausius, Josiah Willard Gibbs, and Walther Nernst. It

also led to a mathematical formulation of the concept of entropy by Clausius and to the

introduction of laws of radiant energy by Jožef Stefan.” (Smith, 1998).

During a 1961 lecture (Feynman, 1964) for undergraduate students at the California

Institute of Technology, Richard Feynman, a celebrated physics teacher and Nobel Laureate, said

this about the concept of energy: “There is a fact, or if you wish, a law, governing natural

phenomena that are known to date. There is no known exception to this law; it is exact, so far we

know. The law is called conservation of energy; it states that there is a certain quantity, which we

call energy, that does not change in manifold changes which nature undergoes. That is a most

abstract idea, because it is a mathematical principle; it says that there is a numerical quantity,

which does not change when something happens. It is not a description of a mechanism, or

anything concrete; it is just a strange fact that we can calculate some number, and when we finish

watching nature go through her tricks and calculate the number again, it is the same.”

The total energy of a system can be subdivided and classified in various ways. For

example, it is sometimes convenient to distinguish potential energy (which is a function of

coordinates only) from kinetic energy (which is a function of coordinate time derivatives only)

and internal energy (which is a function of body deformations).

The transfer of energy can take various forms; familiar examples include e.g., work and

heat flow. Because energy is strictly conserved, it is important to remember that by definition of

energy the transfer of energy between the "system" and adjacent regions is work. A familiar

example is mechanical work. In simple cases this is written as:

WE =Δ (2-8)

ΔE = the amount of energy transferred W = work done on the system

44

Den Hartog (1961), an MIT professor, said that “energy and work are practically

synonyms; when outside forces do work on a system, the energy of the system is said to have

increased by the amount of the work done. Sometimes this work done (or the energy gained) is

recoverable, as, for example, when a rigid body is raised against its own weight (potential energy

of gravitation) or when the force acting on a rigid body has given it speed (kinetic energy). In

other cases the work is not recoverable; it is then said to have been dissipated, but still the system

has gained an amount of energy equal to the work done upon it, although that energy appears

only in the form of heat.”

Deformation Work (Internal Energy)

A simple bar in tension (Figure 2-5) is discussed herein to introduce the concept of internal

energy (deformation work). This definition is used by LS-DYNA to report energy results. The

bar has cross-sectional area A, length L, volume V and is loaded axially with tensile force P.

L

P

Figure 2-5. Simple bar in tension

The internal energy (deformation work) of the bar is defined as its strain energy:

45

( )dVdE ∫ ∫= εσint (2-9)

For the given bar under uniform tension, the internal energy (deformation work) is:

∫⋅= εσ dALEint (2-10)

An elasto-plastic material is characterized by stress-strain relationship depicted in Figure

2-6. The internal energy (deformation work) of the bar under uniform tension is purely elastic in

phase 0-1 and exhibits elasto-plastic behavior in phase 1-2.

ε

σ

σ

0

1

2

yσ

E

tE

strainE

Figure 2-6. Stress vs. strain relationship of the rod

The incremental stress-strain relationship in elastic phase 0-1:

Ed dσε = (2-11)

ε = strain increment, σd = stress increment and E = elastic modulus. d

Hence, the internal energy (deformation work) in elastic phase 0-1:

EALd

EALdALE

2ˆ 2ˆ

0

10int

σσσεσσ

⋅=⋅=⋅= ∫ ∫− (2-12)

The incremental stress-strain relationship in elasto-plastic phase 1-2:

46

tEdd σε = (2-13)

Thus, the internal energy (deformation work) in elasto-plastic phase 1-2:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⋅=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⋅=⋅= ∫ ∫∫−

t

yy

t EEALd

Ed

EALdALE

y

y

2ˆ

2

222ˆ

0

21int

σσσσσσσεσ

σ

σ

σ

(2-14)

The total strain can be decomposed into elastic (recoverable) and plastic (irrecoverable):

pe ddd εεε += (2-15)

The irreversible, plastic strain component in phase 1-2 is:

σσσεεε dEEE

dEdddd

tt

ep⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=−=

11 (2-16)

The plastic portion of internal energy (deformation work) can be calculated as follows:

2ˆ11

22ˆ

inty

tt

pp

EEALd

EEALdALE

y

σσσσσεσ

σ

σ

−⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅=−⋅=⋅= ∫∫ (2-17)

Elastic energy in elasto-plastic phase 1-2:

2ˆ11

2ˆ

2

22222

intintinty

tt

yype

EEAL

EEALEEE

σσσσσ −⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⋅=−= (2-18)

EAL

EEALE yye

2ˆ

2ˆ

2

2222

intσσσσ⋅=⎟

⎟⎠

⎞⎜⎜⎝

⎛ −+⋅=

(2-19)

Only the elastic portion of internal energy (deformation work) can be considered as stored

energy because it can be potentially retrieved. If a bar is unloaded, only the elastic portion of

bar’s internal energy will be recovered. The plastic portion of deformation is irreversibly

47

dissipated as heat. Figure 2-7 depicts decomposition into elastic and plastic components. Plastic

energy corresponds to the irreversible strains. Only elastic energy is potentially recoverable.

ε

σ

σ

yσ

E

tE

elasticplastic

E

eεpε

Figure 2-7. Strain and energy decomposition into plastic and elastic components

Kinetic Energy

However, not only deformation work (internal energy) results from external work done on

a system. If there is a beam falling down in a rigid motion with velocity v, external work (done

by gravity) results in kinetic energy but no strains and thus no internal energy (deformation

work) is induced in the system. During a collapse there is both strain related energy and velocity

related energy. The external forces (gravity acting on displacements) result not only in strains

but also in motions (velocities). Thus part of the external work is converted into kinetic energy.

The remaining portion results in strains. Kinetic energy of a simple bar is defined as:

dVvEkin ∫= 2

21 ρ (2-20)

ρ = mass density, v = particle velocity.

48

Energy Based Procedures for Progressive Collapse Modeling

Energy based procedures to asses resistance to progressive collapse directly originate from

the analysis of energy flow in a single degree of freedom system (SDOF, Figure 2-8) during

downward motion caused by the gravity.

Figure 2-8. Released mass falling on the spring

If a mass resting on a spring (with stiffness k) is instantaneously released, gravity forces

will produce a step load (m = mass [kg], g = gravitational acceleration [kg m/s2]).

Conservation of energy yields that as the mass falls, the initial gravitational potential energy is

converted into spring internal energy (deformation work) and kinetic energy (motion).

gmF ⋅=0

),0(22 maxxxforkxxmg ∈+=⋅1 2

2xm &

(2-21)

However, when the mass reaches its maximum deflection, its velocity is 0.

0)()( maxmaxmax =⇔= txxtx & (2-22)

Thus at this instance released gravitational energy is equal to absorbed internal spring

energy and kinetic energy vanishes because the velocity is 0.

2maxmax 2

1 kxmgx =

(2-23)

49

Therefore, in order to find maximum displacement, it is sufficient to find the displacement

satisfying Equation (2-23). The maximum displacement calculated by solving Equation (2-23)

and its corresponding force experienced by the spring are:

mgkxFkmgx 22

maxmaxmax ==⇒=

(2-24)

Dusenberry and Hamburger (2006) attempted to generalize the above SDOF solution to

MDOF. If the structure is able to arrest collapse, kinetic energy at its final deflection vanishes.

In other words, the released gravitational energy equals the absorbed strain energy, to satisfy the

conservation of energy. However, the maximum deflected shape corresponding to failure is not

known in advance, unless dynamic, time-history analysis is performed.

To illustrate this static push-down procedure, a grillage structure depicted in Figure 2-9

was analyzed after removal of a central support by Dusenberry (2006).

40 [kip]

Girder

Beam 1.29 [in]

Figure 2-9. Deflected shape under static gravity forces

In order to estimate the final deflected shape, the structure was statically pushed-down in

bays adjacent to the removed column. Push-down was stopped when released gravitational

energy equaled absorbed strain energy. If internal forces at this deformed state were supported

by the structure, it was concluded that the structure was able to survive the considered sudden

column removal. If such a deformed state produced internal forces greater than carrying

50

capacities or released gravitational energy exceeded absorbed stain energy for all admissible

deflections, the structure was doomed to failure.

The ratios of the computed plastic capacities at each critical point to computed forces

under statically applied gravity forces are illustrated in Table 2-2.

Table 2-2. Calculation of Force Ratios, R1,i in Phase 1 (Dusenberry and Hamburger, 2006)

Member Location Plastic capacity FPi [kip-ft]

Elastic Force FE1,i

FPi/FEj,i j = 1 (1st phase)

Girder End supports -225 -137 1.64 Girder Central support 225 137 1.64 Beam End supports -138 -62 2.22 Beam Center support 138 62 2.22

The smallest of these ratios (denoted as βj, where j = stage of loading) occurred

simultaneously at the center and end supports of the girder and has a value of:

64.1)min( ,11 == iRβ (2-25)

The presented procedure used linear elastic analysis software (most common in design

offices) and accounted for plasticity by introducing plastic hinges at connections after internal

forces exceeded the plastic capacities. The initially fixed connections were changed to hinges,

when of static gravity load was applied. The corresponding deflection at

the center of the grillage was (

64.1)min( ,11 == iRβ

Figure 2-10):

][4.5][12.2,11,1 cminiEie === δβδ (2-26)

ie ,1δ = displacement at ith node, corresponding j=1st set of hinge formations,

iE ,1δ = displacement at ith node, corresponding to static gravity load;

The effective forces causing this deformation were:

][8.291][6.65][4064.11,1 kNkipkipgmF iie ==⋅== β (2-27)

51

Figure 2-10. Load-displacement during push-down static analysis

It should be noted that the above force has no physical meaning in relation to progressive

collapse. It was only used to obtain feasible deflected shape, which was assumed to occur during

collapse. The internal energy corresponding to the aforementioned displaced geometry was of

interest. However, it was equal to the work done on the system by an auxiliary force because, in

the quasi-static application, no kinetic energy was produced. Therefore, the increment of the

strain energy was calculated using the area under the force-deformation curve:

( ) ∑∑==

− ⋅=⋅=Δ≡Δi

iEiiEii

gmgmSEEnergyStrain1

,11,1111

10 2)(

2δβδββ

nn211

(2-28)

Since the considered example had only one point of interest (intersection of beams), its

number of members was n = 1, and the summation over structural members reduced to:

][85.7][5.692 1,11110 mkNinkipgmSEEnergyStrain E −=−=⋅=Δ≡Δ − δβ1 2 (2-29)

52

Simultaneous with the energy absorption, potential, gravitational energy was released:

][57.9][7.84

)(

1,111

1,1110

mkNinkipgm

gmPEEnergyPotential

E

n

iiEi

−=−=⋅=

⋅=Δ≡Δ ∑=

−

δβ

δβ (2-30)

Since more energy was released by gravity than absorbed by strain energy, the surplus

resulted in kinetic energy:

∑=

− −⋅⋅=Δ≡Δ

Δ=Δ−Δn

iiEi gmKEEnergyKinetic

EnergyKineticEnergyStrainEnergyPotential

1

1,1110 )

21(

βδβ (2-31)

][72.1][2.15)2

1( 11,11110 mkNinkipgmKE E −=−=−⋅⋅=Δ −

βδβ (2-32)

Kinetic energy is present in the system after the formation of the 1st hinge, and collapse

further progressed. Hinges were placed at locations where internal forces reached plastic

moment capacity. Since constant (assumption) plastic moments resisted downward motion, they

acted in the opposite direction lifting the grillage. Plastic resisting moments were denoted

as , where j = loading stage, and i = location id. Successively, gravity was applied to

produce a realistic displaced shape in the second phase, which was scaled by the β to achieve the

formation of the next plastic hinges. The capacity/applied static gravity force ratio was

consecutively computed for the 2nd stage of loading (after the formation of 1st set of hinges).

ijF ,0 iEF ,2

82.1minmin,1

,012

,

,0 =⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⇒⎟

⎟⎠

⎞⎜⎜⎝

⎛ −=

iE

iPi

iEj

ijPij F

FFF

FFββ

(2-33)

The new effective deformations were calculated as follows:

ijjEjjjej ,0,, δδβδ += (2-34)

][14.7][81.21,2 cmine ==δ (2-35)

53

In phases after the formation of hinges, the energies were:

∑=

−−− −⋅=Δ≡Δn

iijeiejijj gmPEEnergyPotential

1),1(,)1( )( δδ (2-36)

( ) (∑=

−−−− −⋅⋅+=Δ≡Δn

iijeiejijjjj gmSEEnergyStrain

1),1(,)1()1( 2

1 δδββ ) (2-37)

(∑=

−−−− ⎥⎦⎤

⎢⎣⎡ +−⋅−⋅=Δ≡Δ

n

ijjijeiejijj gmKEEnergyKinetic

1)1(),1(,)1( 2

11)( ββδδ ) (2-38)

With the deflected shape at the formation of the second yield hinge defined, potential

energy, strain energy and kinetic energy were estimated:

1,2eδ

][57.029.272.1][29.2][27.1342.585.7][42.5][70.1213.357.9][13.3

2021

2021

2021

mkNKEmkNKEmkNSEmkNSEmkNPEmkNPE

−−=−=Δ−−=Δ−=+=Δ⇒−=Δ−=+=Δ−=Δ

−−

−−

−−

(2-39)

The calculated strain energy accumulated by the system at the end of the second phase

exceeded the change in potential energy, and the calculated kinetic energy was negative. Of

course, kinetic energy cannot be less than zero: this negative value meant that the structure has

arrested collapse prior to formation of the second set of hinges.

The aforementioned approach can be generalized to a multi-degree of freedom

system (MDOF). Mass is assumed to be lumped in each column-slab connection. Such an

MDOF approach would produce analogous formulas with summation throughout time (index j)

and throughout the structure (index i). In this approach, the determination of collapse potential is

resolved by calculating the sum of the kinetic energy changes in each phase, and comparing it to

zero.

( )∑∑= =

−−− ⎥⎦⎤

⎢⎣⎡ +−⋅−⋅=Δ

k

j

n

ijjijeiejik gmKE

1 1)1(),1(,0 2

11)( ββδδ

(2-40)

54

The accuracy of the aforementioned method depends on the correctness of the assumed

deformed shape. It can be either conservative or unconservative. Moreover, its error is

unbounded and unknown, thus confidence in such results cannot be assessed rationally.

Experience with other dynamic methods relying on a guess of an approximated deformed shape

(e.g. the Ritz method for computing 1st natural frequency) shows even those small inaccuracies

in deflected shape can result in significant error.

Energy Flow between Members

Nefske and Sung (1987) proposed a power flow analysis of a dynamic system in noise

control and acoustics problems. The power flow method has been developed for predicting the

vibration response of structural and acoustic systems to high frequencies at which the traditional

finite element method is no longer practical. The formulation of the power flow analysis is

based on a differential, control-volume approach and on a partial differential equation of the

heat conduction type, which can be solved by applying the finite element method.

To begin, one considers an elemental control volume V of a physical system. One

considers the time-averaged vibrational energy per unit volume, w (energy density) in V (control

volume). With this definition, one now applies conservation of energy to V. The governing

partial differential equation is readily shown (Nefske and Sung, 1987) to result in the power

balance:

dissPqtw

−⋅∇−=∂∂

(2-41)

tw ∂Which relates the quasi-steady variation of the energy density within volume V,∂ / , to

the energy flux density q crossing the boundary and to the internal power dissipated per unit

volume, Pdiss. To complete the formulation, it was necessary to assume relationships for the

power dissipated within V and for the energy flux density. These were chosen analogous to

55

those in statistical energy analysis (SEA) for the power dissipated in a subsystem and for the

power conducted between subsystems.

This energy flow hypothesis was further examined by Wohlever and Bernhard (1992) for

both longitudinal vibration in rods and transverse flexural vibrations of beams. The rod was

shown to behave approximately according to the thermal energy flow analogy. However, the

beam solutions behaved significantly differently than predicted by the thermal analogy unless

locally space averaged energy and power were considered.

The heat transfer analogy works well for the vibration of rods. The equation of motion for

a rod, driven by a harmonic point force is:

tjc exxFtxu

tStxu

xSE ωδρ )(),(),( 02

2

2

2

−−∂∂

=∂∂

(2-42)

EcS is the rod stiffness per unit length (S = cross-sectional area), ρS is the density per unit length (ρ = density per unit volume), u(x,t) is the longitudinal displacement, Fδ(x-x0)ejωt is the harmonic point force applied at point x0.

The general solution of Equation (2-42) is:

tjjkxjkx eBeAetxu ω)(),( += − (2-43)

Where k is a complex wave number, and c is the phase speed in the rod (where c2 = E/ρ).

In a rod, where only axial forces are present, the power q, defined locally as the axial force at a

point times the velocity, is written in terms of displacement as:

⎭⎬⎫

⎩⎨⎧∂∂

⎭⎬⎫

⎩⎨⎧∂∂

−= ),(),( txut

txux

ESq

(2-44)

The total energy density at a point is the sum of the strain and kinetic energies:

TVe += (2-45)

56

The displacement solutions can be used to develop expressions for the energy density in

the rod. The strain (V) and kinetic (T) energy densities are:

22

),(21),(

21

⎥⎦⎤

⎢⎣⎡∂∂

+⎥⎦⎤

⎢⎣⎡∂∂

= txut

Stxux

ESe ρ (2-46)

In lightly damped structures, the hysteretic damping coefficient ≡ η << 1, and thus the

imaginary part (describing damping contribution) of the complex wave number is small

compared with the real part, i.e. |k1| >> |k2|, the time averaged expressions for the power and total

energy density were approximately:

xkxk eBeAc

ESq 22 2222

21 −−=

ωω

(2-47)

and

xkxk eBeASe 22 22222

21 −−= ωρ

(2-48)

Where the operator indicates a time averaged quantity.

(2-47) and (2-48) can be used to develop the simple relationship between local values of

time averaged power and energy density in a rod:

edxdcq

ηω−=

(2-49)

The time averaged power is proportional to the gradient of the time averaged density.

Equation (2-49) is characteristic of the energy transmission mechanisms in a rod. To develop an

energy formulation for other types of structures, it is necessary to find a transmission relationship

similar to Equation (2-49). The only approximation made was that hysteretic damping is

small, η<<1.

To develop the governing equations which model the energy density in a rod, it is

necessary to perform an energy balance on a differential rod element. The time rate of change of

57

energy within the control volume must be equal to the net power entering the volume minus the

power dissipated within the volume. The resulting balance can be written as:

dissqx

et

π−∂∂

−=∂∂

(2-50)

Where πdiss is the power dissipated within the differential control volume. The time

derivative of energy density is zero since the power is being considered here for a steady state

condition. To solve the particular problem at hand:

02

22

=−⎟⎟⎠

⎞⎜⎜⎝

⎛disse

dxdc π

ηω (2-51)

To demonstrate the applicability of the proposed mathematical model, the energy flow

relation between two coupled rods (Figure 2-11) was analyzed by Wohlever (1992). Energy

flow results for these coupled rods are shown in Figure 2-12.

Fδ(x-x0)e jωt

x1=0 x2=5 x3=10

1 2

Figure 2-11. Spring coupled, longitudinally vibrating rods excited by a harmonic point force

0.0

0.5

1.0

5.0 10.00.0 0.0 5.0 10.0

2.5

1.25

0.0

Length Length

Pow

er

Ener

gy d

ensi

ty (

x10-3

)

exact - - - simplified

Figure 2-12. Comparison of exact and simplified solutions for the coupled rod system

58

Simplified Methods for High Rise Buildings

Bažant and Verdure (2007) proposed an energy based analysis of progressive collapse of

high-rise structures. Their concept was illustrated using the collapse of the World Trade Center

(WTC) towers in 2001. Collapse was divided into crush-down and crush-up phases (Figure

2-13). When columns at the damaged floor buckle, they yield to the gravity of the upper portion

of the structure and a fracture wave propagates. Once all floors below the upper portion have

collapsed progressively, the upper block impacts the debris and the fracture wave propagates

upward until the whole structure is destroyed (Figure 2-13).

Crush-Down Phase Crush-Up Phase

Figure 2-13. Progressive collapse of the World Trade Center towers

The main focus of the analysis was on energy absorption capacities of columns as the most

critical factor in progressive collapse of high rise structures. The mass of columns was assumed

to be lumped, half and half, into the mass of the upper and lower floors.

To analyze progressive collapse, the complete load-displacement diagram F(u) must be

known (Figure 2-14). It begins by elastic shortening, and, after the peak load F0, the curve F(u)

steeply declines with displacement due to plastic buckling, combined with fracturing. For a

single column buckling, the inelastic deformation localizes into three plastic hinges.

59

F0

mg

u0 uc

Floor displacement, u uf h

λhDeceleration phase

Acceleration phase

Cru

shin

g fo

rce,

F(u

) Fc

Maxwell line

Figure 2-14. Typical load-displacement diagram of columns of one story

When the difference )()( zgmuF ≠ )()( zgmuF − causes deceleration of mass m(z) if

positive and acceleration if negative. The equation of motion of mass m(z) during the crushing

of one story (or one group of stories, in the case of multistory buckling) reads as follows:

)()(

zmuFgu −=&&

(2-52)

The energy loss of the columns, Φ(u) , up to displacement u is:

)()()()())()(()(00

uWuduFwherezgmuWudzgmuFuu

u

u

u

≡′′−=′−′=Φ ∫∫

(2-53)

z = constant = column top coordinate; W(u) = energy dissipated by the columns = area under the load-displacement Figure 2-14; gm(z) u = gravitational potential change causing an increment of kinetic energy of mass m(z).

Note that, since the possibility of unloading can be dismissed, W(u) is path independent

and thus can be regarded, from the thermodynamic viewpoint, as the internal energy, or free

energy, and thus Φ(u) represents the potential energy loss. If F(u) < gm(z) for all u, Φ(u)

continuously decreases. If not, then Φ(u) first increases and then decreases during the collapse

of each story.

60

The Criterion of Arrested Collapse

Collapse will be arrested if and only if the imparted kinetic energy from collapse through a

previous story(ies) does not suffice for reaching the interval of accelerated motion, i.e., the

interval of decreasing Φ(u). So, the crushing of columns within one story will get arrested before

completion if and only if

)()()( zgmuWuE ccK −=Φ< (2-54)

EK = kinetic energy of the impacting mass m(z) Φ(uc) = net energy absorption up to uc during the crushing of one story

This is the criterion for preventing progressive collapse from starting. Graphically, this

criterion means that EK must be smaller than the area under the load-deflection diagram lying

above the horizontal line F = gm(z) (Figure 2-14). If this condition is violated, the next story

will again suffer an impact, and the collapse process will get repeated.

The Criterion of Accelerated Collapse

The next story will be impacted with higher kinetic energy if and only if:

pg WW > (2-55)

fg uzgmW ⋅= )( = loss of gravitational energy when the upper part of the tower is moved down by distance uf = final displacement at full compaction;

∫==fu

fp duuFuWW0

)()( = area under the complete load-displacement curve F(u);

For the WTC, it was estimated that EK ≈ 8.4Wp >> Wp for the story where progressive

collapse initiated (Bazant and Zhou, 2002). As Wg was, for the WTC, greater than Wp by an

order of magnitude, acceleration of collapse from one story to the next was ensured.

The aforementioned representation was homogenized to the global continuum model using

energetically equivalent mean crushing force. A non-softening energetically equivalent

characterization of snap-through in discrete elements was pursued. It corresponds to

61

nonstandard homogenization, in which the aim is not homogenized stiffness but homogenized

energy dissipation.

Energetically Equivalent Mean Crushing Force

For the purpose of continuum smearing of a tower with many stories, the actual load-

displacement diagram F(z) can be replaced by a simple diagram that is story-wise energetically

equivalent, and is represented by the horizontal line F = Fc. Here Fc is the mean crushing force

(or resistance) at level z, such that the dissipated energy per story represented by the rectangular

area under the horizontal line F = Fc, is equal to the total area Wp under the actual load-

displacement curve:

∫==fu

ff

pc duuF

uuW

F0

)(1

(2-56)

The energy-equivalent replacement avoids unstable snap through and is analogous to what,

in physics of phase transitions, is called the Maxwell line. Although the dynamic u(t) history for

the replacement Fc is not the same as for the actual F(u), the final values of displacement u and

velocity v at the end of crushing of a story are exactly the same. So the replacement has no

effect on the overall change of velocity v of the collapsing story from the beginning to the end of

column crushing as long as Fc is not large enough to arrest the downward motion. Fc may also

be regarded as the mean energy dissipated per unit height of the tower, which has the physical

dimension of force.

One-Dimensional continuum model for crushing front propagation is based on the

following simplifying hypotheses: (1) the only displacements are vertical and only the mean of

vertical displacement over the whole floor needs to be considered. (2) Energy is dissipated only

at the crushing front (this implies that the blocks in Figure 2-15 may be treated as rigid, i.e., the

deformations of the blocks away from the crushing front may be neglected). (3) The relation of

62

resisting normal force F (transmitted by all the columns of each floor) to the relative

displacement u between two adjacent floors obeys a known load-displacement diagram (Figure

2-14), terminating with a specified compaction ratio λ (which must be adjusted to take into

account lateral shedding of a certain known fraction of rubble outside the tower perimeter). (4)

The stories are so numerous (Bazant and Verdure, 2007), and the collapse front traverses so

many stories, that a continuum smearing (i.e., homogenization) gives a sufficiently accurate

overall picture.

The one-dimensionally idealized progress of collapse of a tall building (of initial height H)

is shown in Figure 2-15, where: ξ = z(t) = coordinate of the crushing front measured from the

initial tower top; η = y(t) = coordinate of the crushing front measured from the current tower top;

H

z0

s0

z

A

C

B

A

C

z0

s = λs0

ς

y0 = z0

λ(H-z0)

y

r0

B BB B B’

CC y

r = λr0

ηλz0

λH

(a) (b)

(c)(d)

(e)

Crush-Down Phase Crush-Up Phase

dz/dt

Figure 2-15. Continuum model for propagation of crushing front

Firstly, the compaction ratio is introduced. When the upper floor crashes into the lower

one, with a layer of rubble between them, the initial height h of the story is reduced to λh, with λ

denoting the compaction ratio (in finite-strain theory, λ is called the stretch). After that, the load

63

can increase without bounds. In a one-dimensional model pursued here, one may use the

following estimate

( )0

11VV

out ⋅−= κλ

(2-57)

V0 = initial volume of the tower; V1 ≈ volume of the rubble on the ground into which the whole tower mass has been compacted; κout = correction representing mainly the fraction of the rubble that has been ejected during collapse outside the perimeter of the tower and thus does not resist compaction.

The rubble that has not been ejected during collapse but was pushed outside the tower

perimeter only after landing on the heap on the ground should not be counted in κout. The volume

of the rubble found outside the footprint of the tower, which can be measured by surveying the

rubble heap on the ground after the collapse, is an upper bound on V1, but probably much too

high a bound for serving as an estimate (Bazant and Verdure, 2007).

Let μ = μ(ξ) = initial mass density at coordinate ξ = continuously smeared mass of

undisturbed tower per unit height. During crush-down, the ejected mass alters the inertia and

weight of the moving compacted Part B, which requires a correction to m(z), whereas during

crush-up no correction is needed because Part B is not moving:

( ) ξξμξξμκξξμ dzmwherezzforddzmzz

zout

z

∫∫∫ ≡>⋅−+=0

0

0

000

0

)()()(1)()(

(2-58)

The initial location of the first floor crashing into the one below is at ξ = z(t0) = z0 = y0.

The resisting force F and compaction ratio λ are known functions of z. A and C label the lower

and upper undisturbed parts of the tower, respectively. B denotes the zone of crushed stories

compacted from initial thickness s0 to the current thickness, s(t):

( )0

)(

)()()(0

ztzdtstz

z

−⋅== ∫ λξξλ

(2-59)

64

≡− 0)( ztz distance that the crushing front has traversed through the tower up to time t.

The velocity of the upper part of the tower C is:

( ) ( )( ) ( )(1)()()()()( 0 tzztztzt

tstzt

tvC &⋅−=−⋅−∂

)∂=−

∂∂

= λλ

(2-60)

The differential equation for z(t) – crush-down can be obtained from a dynamic free body

diagram (Figure 2-16). In the crush-down phase, the compacted Zone B and the upper Part A of

the tower move together as one rigid body accreting mass, with combined momentum:

zzmtvzm C &)1()()()( λ−⋅=⋅ (2-61)

Crush-Down

Crush-Up

tzΔ&

ς cvm &

mg

cF cF

cvm &

mg

cF cFtyΔ&

Figure 2-16. Free body diagram in the crush-down and crush-up phase

The negative of the derivative of this momentum is the upward inertia force. Additional

vertical forces are weight m(z)g downward, and resistance Fc(z) upward. The condition of

dynamic equilibrium according to the d’Alembert principle yields the following differential

equation for compaction front propagation in the crush-down Phase I of progressive collapse:

)()()()1()( downcrushzFgzmdt

zmdt c

⎭⎬

⎩⎨

dzd−−=−⎫⎧ ⋅−⋅ λ

(2-62)

The initial conditions for the crush-down Phase I are z = z0 and v = 0. Downward

propagation will start if and only if:

65

)()( 00 zFgzm c> (2-63)

The differential equation for y(t) – crush-up phase can be obtained from a dynamic free

body diagram (Figure 2-16). In the crush-up phase, the crushing front at η = y is moving up

(debris is stacking up) with velocity

yzvstacking &)(λ= (2-64)

Thus the downward velocity of Part C is:

( )yzyzyvC &&& )(1)( λλ −=−= (2-65)

It should be noted that part C is decelerating upon contact with stationary rubble stack B.

Thus, using the d’Alembert principle, a force equal to inertia is applied downward to bring the

system into equilibrium. The differential equation of progressive collapse in the crush-up phase:

( ) )()()(1)( upcrushyFgdtdyy

dtdym c −=

⎭⎬⎫

⎩⎨⎧

+⎟⎠⎞

⎜⎝⎛ − λ

(2-66)

The following characteristics of the analytical results were noticed:

1. Varying the building characteristics, particularly the crushing energy Wf per story, made a

large enough difference in response to be easily detectable;

2. For the typical WTC characteristics, the collapse takes about 10.8 [s] which is not much

longer (precisely only 17% longer) than the duration of free fall in vacuum from the tower

top to the ground, which is 9.21 [s]. For all of the wide range of parameter values

considered, the collapse took less than about double the free fall duration.

“If the total energy loss/dissipation during the crushing of one story (representing the

energy dissipated by the complete crushing and compaction of one story, minus the loss of

gravity potential during the crushing of that story) exceeds the kinetic energy impacted to that

story, collapse will continue to the next story. If it is satisfied, there is no way to deny the

66

inevitability of progressive collapse driven by gravity alone (regardless of by how much the

combined strength of columns of one floor may exceed the weight of the part of the tower above

that floor). What matters is energy, not the strength, nor stiffness.” (Bazant and Verdure, 2007).

It is also instructive to consider progressive collapse also from the point of view of an

implosion contractor who regularly demolishes buildings through explosives-induced

progressive failure (Loizeaux and Osborn, 2006). All buildings want to fall down. However,

they are prevented from doing so through their structural columns, walls and transfer girders.

Innumerable ergs of potential energy are just waiting to be released. The implosion contractor

creates a progressive demolition by releasing this energy through the sequential explosive

removal of key structural supports, allowing gravity to do the remaining work, simultaneously

using the minimum amount of explosives, creating the maximum amount of fragmentation, and

minimizing the potential fly of debris. Conversely, the structural designer wants to contain the

potential energy.

However, it should be noted that demolition differs from accidental progressive collapse.

In the case of implosion an existing building is prepared for demolition by cutting specific

elements, by elimination of certain key members, by tying walls with columns to force the

desired collapse mode, by removal of secondary partitions and walls, etc. Therefore, specific

collapse observations from implosions cannot be directly used to assess the resistance to

accidental progressive collapse.

Summary

The World Trade Center (WTC) collapse of 2001 and statistical data confirm that high-rise

slender structures are at the highest risk of abnormal loading resulting in progressive collapse.

Limited data indicates that the probability of abnormal loading depends on building size,

measured in dwelling units or square footage, and ranges from approximately 0.23 x l0-6 to

67

7.8 x l0-6 [dwelling unit/year] (Elingwood and Dusenberry, 2005). Thus, high rise buildings

with numerous dwelling units are the most likely structures to face catastrophic/abnormal loads.

On the other hand, dynamic modeling of structural response to such loadings and progressive

collapse resistance assessment are highly expensive for such structures. Difficulty arises

specifically from the overwhelming number of structural members in a high rise building, which

easily can reach thousands of elements, connections, etc. This poses modeling, validation and

computational challenges.

In order to mitigate the risk of progressive collapse, three strategies can be employed:

event control (protecting building form accidental loads), indirect design (ensuring sufficient

ductility, structural ties, detailing, etc) and direct design (explicit design against accidental

loading using e.g. fem software). Design codes burden engineers with an obligation to prevent

disproportionate collapse but fail to provide direct design procedures, which would produce a

structure resistant to progressive collapse. They instead provide general, qualitative provisions,

which are based on historic failures. The community of designers is left without straightforward

procedures of how to design safely.

Several methods have been proposed to evaluate resistance to progressive collapse. In

general, they fall into two categories: static and energy based procedures. Although static

procedures are very appealing to engineers because of their simplicity, they do not describe the

physics of progressive collapse. Therefore, they can be conservative or unconservative. In fact,

their error is unbounded and unknown for the designer (Powell, 2005). On the other hand,

energy based methods proposed by Dusenberry and Hamburger (2006) as well as Bažant and

Verdure (2007) have significant potential to provide simple yet realistic assessment tools for

68

practicing engineers because energy based approaches capture the physics of progressive

collapse.

69

CHAPTER 3 RESEARCH APPROACH

Proposed Theory

The following energy based theory is proposed to provide insight into the phenomenon of

progressive collapse. A large amount of gravitational energy, hereinafter called potential energy,

is present in a building. During construction, cranes lift structural members from the ground to

their respective elevations. Once a member is placed on supports, only a very small fraction of

that potential energy is transferred through external work and absorbed as internal (strain)

energy. A schematic structure, shown in Figure 3-1, is used to illustrate the stable, steady energy

state, which remains throughout the typical life of a structure.

Figure 3-1. Energy state in a hypothetical structural system

The energy in the stable steady state for the hypothetical structure from Figure 3-1 is:

( ) ( )

]internal[]potential[21 2

1

+=

Δ+Δ−= ∑=

total

c

n

itotal

E

hkhhMgE (3-1)

M = rigid mass; g = gravity acceleration; kc = column axial stiffness; n = number of columns.

Accidental, abnormal loads can, however, cause the removal of key elements and produce

excessive displacements, which release potential energy (gravity acting on the mass of the

70

structure through deflections). Simultaneously, deforming members attempt to absorb the

released energy by transferring it into internal, strain energy (work needed to change structural

shapes) and redistributing it to other members. If the released energy cannot be fully absorbed

by the structure, motion is produced. The difference between the released energy and the

absorbed deformation energy is transferred into kinetic energy. In other words, the potential

energy will be released upon failure (change of the potential energy) and will flow (external

work) to redistribute energy until either total failure or a new stable state is achieved.

In generic energy transformations (Figure 3-2), the energy is transferred from the energy

source to the energy storage. Part of the energy is lost as heat, and part of the energy remains in

the system and can be recovered under favorable conditions.

Energy Source

EnergyStorage

Energy Loss as Heat

Recoverable Output

Energy Transfer

Figure 3-2. Energy transformation in a generic system

“Potential energy” is an energy source in the proposed approach (Figure 3-3). “External

work” is a measure of the “energy transfer” from the potential energy reservoir to the building.

Energy absorbed through deformations is hereafter called internal energy (deformation work).

Internal energy can be decomposed into recoverable (elastic) and dissipated energy (plastic).

Energy resulting from motion (velocities) is hereafter called kinetic energy.

71

Potential Energy

Energy Loss (Plastic Energy)

External Work

Energy Transfer

Kinetic Energy

Internal Energy(Deformation Work)

External Work

Figure 3-3. Energy transformation between potential, internal and kinetic energies

Localized failure triggers the release of potential energy. The structure absorbs the energy.

Part of the absorbed energy is lost through plastic deformations, which result in the energy loss

as heat. The deficit between the absorbed and released energy is the kinetic energy. Thus

external work should be understood as the measure of energy transfer from potential energy into

the building (internal energy or deformation work) and motions (kinetic energy). As long as

there is a deficit in energy absorption by the building, there will be kinetic energy in the system,

which further advances various motions, resulting in collapse propagation. Once all the released

potential energy and the excess of kinetic energy (if any) are absorbed by the structure, it reaches

a stable configuration. It is not important what portion of the absorbed energy is dissipated and

what is potentially recoverable (should unloading occur), as long as the kinetic energy is

eliminated from the system. The proposed formulation is illustrated in Figure 3-4.

72

Figure 3-4. Energy based approach to progressive collapse

The hypothetical collapse of a high rise building (Figure 3-5) will be used to exemplify the

energy based progressive collapse assessment procedure. The removal of a corner column will

trigger the release of potential energy. Individual members will attempt to absorb and

redistribute the released energy. The energy propagation zone, which enlarges as time

progresses, is depicted with a shaded region in Figure 3-5. Therefore, the first goal is to

understand how the energy propagates after the initiating failure. Such propagation will depend

on structural configuration, member properties, etc. This objective will be achieved by

NO YES

Local failure

Gravitational energy is released

Is the structure able to redistribute and absorb the released energy?

Can it reach a stable steady state?

Unabsorbed energy is transferred into kinetic energy

Reaching a new stable state Released energy is fully absorbed by the structure

Collapse propagates because velocities result in further advance of various motions

End of Collapse

73

simulating a number of typical steel framed structures, using finite element analysis software

(LS-DYNA), and examining the energy flow.

Figure 3-5. High rise building collapse, initiating story

If the energy transferred into a given column is known, the following procedure is used to

assess whether the current state of the member is stable. It is in a steady state if released

potential energy and energy transferred from other members is equal to the energy absorbed in

this member. In other words, if there is no kinetic energy (no motion) in the member, then it

reached a stable state.

A given column is in a steady state if released potential energy and energy transferred from

other members is equal to the energy absorbed in this member. In other words, if there is no

kinetic energy (no motion) in the member then it reached a stable state. If all members in the

system reached steady state, then collapse has ended in either a new stable state or reached total

failure. On the other hand, if energy absorption of the member is insufficient, kinetic energy is

74

produced, and collapse propagates through motion. Thus generation of kinetic energy indicates

an unstable member state

An imaginary story-wise collapse is utilized to better illustrate the aforementioned steady

state criterion. The story-wise collapse can be thought of as being initiated by a severe fire

uniformly distributed through the floor. In this simplistic approach, energy is distributed

uniformly through the floor, and the structural performance can be described by tracing the

behavior of an individual column. If fire reduced the column’s capacity enough to initiate the

collapse through one story, then the mass supported by each column, denoted as M, will move

downward by a displacement u. Gravity action will release potential energy. Simultaneously

with the energy release, a strain based energy absorption mechanism will be activated.

A schematic of energy distribution for collapse through the initiating story is shown in

Figure 3-6. The slope of the released gravitational energy per unit displacement equals the dead

weight distributed over the tributary area of the collapsing floor.

Figure 3-6. Energy flow- displacement of single column (H = height of story)

75

Because the resisting force of the analyzed column (slope of the absorbed deformation

energy) is smaller than the applied gravity to start the collapse, at the end of the 1st story

crushing, a surplus of kinetic energy is found, Ek1 (where subscript 1 denotes first collapsing

story). It is expected that the force resistance of columns, which is the slope of the absorbed

energy, will decrease with increased displacements due to post-buckling weakening and material

nonlinear behavior. Thus, for all displacements through the 1st collapsing story, there is a surplus

of kinetic energy, the column is unable to reach a stable configuration and its collapse

propagates. If crushing of the initiating story is followed by the failure of the underlying story

(Figure 3-7), a column can reach a steady state only if the applied kinetic energy, Ek1, increased

by released gravitational energy, can be absorbed as deformation energy of the column (at full

capacity, not hindered by fire).

Figure 3-7. Collapse propagates into consecutive story

76

A schematic example of energy distribution in arrested collapse vs. displacement through

the consecutive story is shown in Figure 3-8. For a displacement ≈ 0.4H, the kinetic energy in

the column vanishes, and, thus, it arrives at the stable state. Energy absorption is sufficient to

arrest the collapse. If the deforming structure is unable to dissipate the released gravitational

energy, then the surplus of kinetic energy will increase, and there is no state in which kinetic

energy vanishes (Figure 3-9).

Figure 3-8. Energy release with energy absorption by the crushing story itself

Figure 3-9. Collapse propagation due to insufficient energy absorption

77

The proposed energy based steady state criterion can be written as the following postulate.

A structural member will attain a stable state if and only if there exists a displacement uarrest(t)

such that energy transferred from adjacent members increased by released gravitational energy

corresponding to uarrest is fully absorbed as strain energy. In other words, steady state is achieved

if the kinetic energy of the member is zero:

0==−+ kineticEstrainEmemberthetotransferedEpotentialreleasedE (3-2)

Otherwise, if for all displacements u(t) of the column to the current position, there is

residual kinetic energy, then the column is in an unstable state, and the collapse propagates due

to motion. Increase of kinetic energy in the system indicates a rise of systemic instability. On

the other hand, if kinetic energy diminished, it indicates that the structural system has the ability

to absorb the released gravitational energy.

Tracing energy absorption requires time integration to capture energy propagation through

the structure. Energy absorption at the given instance of time shall be described as not only

dependent on displacement, but also time dependent. Thus, the increment of kinetic energy

attained over time : dt

⎟⎟⎠

⎞⎜⎜⎝

⎛=−+−

2),(

2uMdtduuE

dtduMgE

dtdE

dtd

absoroutin&

&& (3-3)

Generally speaking, Equation (3-3) needs to be evaluated at each member to calculate the

current state of kinetic energy and to assess if the current energy state is stable or unstable. Ein

and Eout depend on the energy flow in the structure during progressive collapse, and shall be

determined from energy flow observations and compatibility conditions. Eabsorb represents

material relation, which describes pertinent energy absorption of a particular structural member.

78

Virtual Experiments

Overview

Physics based simulation techniques were used to research the energy flow and behavior of

major structural members during progressive collapse propagation or arrest. Representative steel

structures were selected for analysis. All possible efforts were made to represent the physical

behavior of real structures as accurately as possible. Simulated results were verified with basic

principles of physics and validated with available experimental results. Implicit time integration

was used to simulate the static pre-loading phase. Then, selected columns were removed from

the model, and the analysis was restarted. To model the post column removal phase, explicit

time integration was used (Hallquist, 2006). A collapse phase is characterized by large

deflections, pronounced material non-linearities and contact between members. LS-DYNA finite

element code (Hallquist, 2006) offers sophisticated algorithms, which enable numerical

modeling of collapse propagation or arrest. Therefore, LS-DYNA was chosen to perform

simulation studies in this work.

Material Modeling

The finite element code, LS-DYNA (Hallquist, 2006) was employed to model structural

response to abnormal loading. Progressive collapse is a dynamic phenomenon with strong

material and geometric nonlinearities. Thus, a sufficiently sophisticated analysis method has to

be applied in order to reproduce the physical behavior of a building. A typical stress-strain

relation of A-36 steel, as reported by Salmon and Johnson (1990), was utilized in the models. A

large strain, piecewise linear, material model 24 from the LS-DYNA (Hallquist, 2006) library

has been applied to represent A-36 steel material behavior. Strength enhancement under high

79

speed strains was not included in the material modeling. Model 24 operates on logarithmic stress

and strain measures, thus it accounts for large strain effects.

The piecewise linear model is limited only to the monotonically increasing logarithmic

stress- logarithmic strain behavior due to the restrictions of stress based plasticity. The loading-

unloading criterion must be specified, such as:

unloadingfordloadingford

00

<≥

σσσσ

(3-4)

Loading-unloading criterion in the stress space is unable to differentiate between elastic

unloading and plastic softening (Khan and Huang, p.278, 1995). Thus, only the monotonic

stress-strain portion of the material relationship is allowed as input because the yielding is

evaluated in the stress space.

The initial yield point for A-36 steel is . For stress values lower

than the current yield stress , deformation is linear and the correspondence between

][250][360 MPaksiY ==σ

nyσ σ and ε

is one to one. For stress values higher than , the deformation is nonlinear and history

dependent. A-36 steel properties, generalized to the three-dimensional case, were represented

with Maxwell-Huber-von Mises yield criterion with piecewise orthotropic hardening. In other

words, deviatoric stresses are determined that satisfy the yield function:

nYσ

( )( )0

321

2

≤−=p

effyijij SS

εσφ (3-5)

ijS = deviatoric stress; = effective plastic strain; peffε yσ = uniaxial yield stress

The growth of subsequent yield surface is characterized by the uniaxial logarithmic yield

stress as a piecewise linear hardening function of logarithmic, effective plastic strain. This

function is used to track loading history dependent, plastic material behavior in each Gauss point.

80

Logarithmic, large deformation stress measures were used in the yield criterion.

LS-DYNA offers elements with large strain capabilities, which are compatible with the large

strain plasticity model. Since stresses and strains are calculated with respect to the current

configuration and not with respect to the initial configuration, all input stress-strain material

information needs to be converted from engineering into logarithmic measures. Uni-axial

engineering stress-strain data shall be converted to logarithmic measures as follows (Khan and

Huang, p.4, 1995):

( )

( )engeng

l

loeng

o

ll

APAllA

AP

ll

ldl

εσσ

εε

+=====

+=== ∫

1

1lnln

0000log

log

(3-6)

Engineering and logarithmic stress strain curves are shown in Figure 3-10.

0

10

20

30

40

50

60

70

80

0 0.05 0.1 0.15 0.2 0.25

Yie

d st

ress

σy

[ksi

]

Effective plastic strain

englogLS-DYNA

Figure 3-10. Typical stress-strain relationship for A36 steel

Strains are updated at each time increment (relatively small in explicit analysis as opposed

to the implicit larger steps). Although there is no consensus on the best objective co-rotational

81

stress rate (Khan and Huang, p.242, 1995), the Jaumann rate of the deviatoric stress is utilized by

LS-DYNA:

pijppjipijij SSSS Ω−Ω−=∇ & (3-7) ∇ijS = corotational increment of deviatoric stress; = increment of deviatoric stress ijS&

pjΩ = spin tensor (rigid rotation)

It is an exact transformation (Gurtin and Spear, 1983) of the stress tensor in the global

configuration into the stress tensor in the corotational representation (rotating with principal

axes) for small deformations only. However, this formulation converges to exact solutions as the

time increment decreases. Since the transformation is applied in explicit dynamic solution to

incrementally updated configurations (differential deformation relatively small with small time

increments), it provides sufficient accuracy for engineering applications (Hallquist, 2006).

First, the trial deviatoric stress value is calculated at step n+1 on the basis of results from

step n and the elastic material relationship. A dynamic finite difference procedure will provide

velocities and displacements at step n+1. Calculated strain increments from step n to step n+1

are cleaned from rigid rotations (i.e. corotational strains are used). On the other hand, stresses

are evaluated at the step n configuration, thus:

pinjppj

nipij

nij

nij

nij

nij SSGSSSS Ω+Ω+′+=+= ∇+ ε&21* (3-8)

where the left superscript, *, denotes a trial stress value ∇n

ijS = corotational increment of deviatoric stress at from step n to n+1 G = shear modulus

ijε ′& = increment of deviatoric strain

The effective trial stress is defined by

21

1*1**

23

⎟⎠⎞

⎜⎝⎛= ++ n

ijnij SSs (3-9)

82

If exceeds yield stress , the Huber-von Mises flow rule s* nyσ (3-5) is violated. The trial

stress shall be scaled back to the yield surface by means of the following radial return algorithm:

1**

11 +

++ = n

ij

nyn

ij Ss

Sσ

(3-10)

In order to compute the yield stress at the end of the current step , evolution of the

yield surface has to be followed by computing the effective plastic strain increment. Effective

plastic strain serves as the hardening parameter in isotropic hardening evolution. The plasticity

constitutive Levy-Mises flow rule and additive decomposition of strain increments into elastic

and plastic components (Hallquist, 2006) are used to solve for the plastic strain increment:

1+nyσ

( )( )p

nyp

eff EGs+

−=Δ

3

* σε (3-11)

G = shear modulus; pE = current tangent modulus (in the effective strain space)

Total plastic strain is updated:

peff

npeff

npeff εεε Δ+=+ ,1, (3-12)

The yield stress evolution is calculated using the current tangent modulus of the user

supplied piecewise linear yield stress- effective plastic strain relationship (Figure 3-10):

peff

npny

ny E εσσ Δ+=+ ,1 (3-13)

Once the uni-axial yield stress is known at step n+1, radial return of the deviatoric stress to

the yield surface is carried out (3-10). Thus, for the given displacements and element velocities,

stresses and internal forces are calculated.

Material failure is characterized by the prescribed effective plastic failure strain. For the

A-36 steel, the failure strain value was utilized (2.0, =failurepeffε Figure 3-10).

83

Structural Modeling

A feasibility study was initially conducted in order to find modeling resolution, which

captures the physics of progressive collapse and, at the same time, is computationally efficient

for the full structure simulations.

Column buckling is a critical phenomenon in building collapse, and its proper modeling

shall be ensured. SOLID164 elements (Hallquist, 2006) were used for the 3-D modeling of

single column buckling to provide a benchmark solution for more computationally efficient

models. The element is defined by eight nodes having the following degrees of freedom at each

node: translations, velocities, and accelerations in the nodal x, y, and z directions. This high

resolution model captures buckling behavior with great realism, including localized effects as

shown in Figure 3-11.

Figure 3-11. Three-dimensional solid cube buckling modeling

Although 3-D cubes provide high fidelity of simulation, computational expense to simulate

a full structure is prohibitive. Therefore, Hughes-Liu beam elements were investigated as a more

84

computationally efficient alternative. However, it had to be ensured that the simplified approach

utilizing fiber type Hughes Liu beams is sufficiently accurate and captures real column behavior.

The Hughes-Liu is incrementally objective (rigid body rotations do not generate strains),

allowing for the treatment of large strains. It also includes finite transverse shear strains. The

Hughes-Liu beam element is based on a degeneration of the isoparametric 8-node solid element.

To degenerate the 8-node brick geometry into 2-node beam geometry, the four nodes at 1−=ξ

and at 1=ξ are combined into a single node with three translational and three rotational degrees

of freedom.

The strain and spin increments are calculated from the incremental displacement gradient

(updated at each time step):

j

iij y

uG

∂Δ∂

= (3-14)

where are the incremental displacements, and are the deformed coordinates. The

incremental strain and spin tensors are defined as the symmetric and skew-symmetric parts,

respectively, of :

iuΔ

G

jy

ij

( )

( )jiijij

jiijij

GG

GG

−=Δ

+=Δ

2121

ω

ε (3-15)

The incremental spin tensor ijωΔ is used as an approximation of the spin tensor to the

rotational contribution of the Jaumann rate of the stress tensor. The spatial integration is

performed with one point integration along the axis and multiple points in the cross section. For

the W-section (wide flange section), trapezoidal integration through the section by means of nine

integrations points is carried out (Figure 3-12).

85

wt

ft

d

t

s

w

1 2 345

6

8 97

Figure 3-12. Integration scheme for W section in the Hughes-Liu beam element

Columns were modeled as elements capable of exhibiting a variation of strains and their

corresponding stresses through the section. Thus, the Hughes-Liu formulation is capable of

modeling yield propagation through the section. Because stresses were integrated over the

section, yielding of flanges and yield propagation toward the centerline were directly modeled.

Material failure was controlled by the prescribed value of effective plastic failure strain. If the

average effective plastic strain of nine integration points was greater than the critical value, the

element was deleted.

Two types of slabs were considered: heavily reinforced and lightly reinforced. The

strongly reinforced composite slab was modeled approximately as a 6 [in] (150 [mm]) thick shell

with uniform material properties inferred from smearing mechanical properties of the composite

deck. A bilinear plastic model with a Young modulus of 3500 [ksi] (24.13 [GPa]), Poisson ratio

of 0.2, yield stress of 10 [ksi] (72 [MPa]), failure stress of 10.5 [ksi] (75 [MPa]) and cut-off

strain of 0.003 were employed in the analysis. Lightly reinforced slab was modeled as a 4 [in]

(100 [mm]) thick shell with the custom integration scheme, depicted in Table 3-1. In

conjunction with the custom integration, laminated glass (material model 32) was employed.

Steel material model was used for the bottom layer; whereas other layers were modeled using

86

concrete material properties (Table 3-1). A bilinear plastic model with a Young modulus of

3600 [ksi] (24.8 [GPa]), Poisson ratio of 0.19, yield stress of 4 [ksi] (27.6 [MPa]), tangent

modulus of 180 [ksi] (1.2 [GPa]) were employed in the analysis. Concrete failure was modeled

with the cut-off strain of 0.01.

Table 3-1. Lightly reinforced slab modeling Layer coordinate

Area fraction

Young modulus [ksi]

Poisson ratio

Yieldstress[ksi]

Failure strain

Tangent modulus [ksi]

0.95 0.05 3600 0.19 -4 0.01 180 0.80 0.1 3600 0.19 -4 0.01 180 0.50 0.2 3600 0.19 -4 0.01 180 0.00 0.3 3600 0.19 -4 0.01 180 -0.50 0.2 3600 0.19 -4 0.01 180 -0.80 0.1 3600 0.19 -4 0.01 180 -0.945 0.045 3600 0.19 -4 0.01 180 -0.995 0.005 29000 0.30 36 0.20 1450

Buckling of a single W14x74 of 157.5 [in] (400 [cm]) height, clamped at the base and

pinned at the top, was simulated by means of both brick elements and Hughes-Liu beam

formulation. The simulation was force controlled with monotonic, quasi-static load application.

Force imperfection at mid-height was introduced to initiate the buckling. Imperfection force was

selected to produce initial camber of 400/1500 [cm] at the mid-height.

Although the brick elements provide higher result resolution, the displacement-force

histories comparison in Figure 3-13 shows that overall column resistance can be modeled very

accurately with Hughes-Liu elements up to the onset of the global buckling. Fiber elements are

not capable of capturing the local buckling (such as that shown in Figure 3-11). However typical

W hot rolled shapes are designed such that global buckling controls a member failure. W shape

manufacturers provide sufficient web and flange thicknesses such that local buckling occurs only

after the global buckling (sufficiently small width to thickness ratios of the webs and flanges to

prevent the local buckling before the global buckling onset).

87

0

2

4

6

8

10

0 1000 2000

Dis

plac

emen

t [m

m]

Force [kN]

Hughs-Liu elements

Brick elements

Figure 3-13. Displacement- force history comparison of brick and Hughes-Liu modeling

Such good agreement between the brick elements and Hughes-Liu formulation can be

attributed to the column cross-section properties. A W14x74 is a compact section, which means

that the ratio of web depth to its thickness as well as flange width to its thickness are sufficiently

small to prevent local buckling before the global instability occurs. It means that a beam

formulation is less accurate then the 3-D model with brick elements only in the post-buckling

phase. However, since the column resistance drops significantly after buckling onset, such

inexactness appears to be of second order.

Good agreement between the brick and Hughes-Liu formulations is welcomed because it

enables the modeling of a full structure with computationally efficient elements, yet ensures that

obtained results represent the physical reality of building collapse. Hence, Hughes-Liu elements

were used to represent the building frame members in the simulation

Whereas force induced imperfection is acceptable in single column buckling, it is not a

feasible approach in ensuring proper buckling initiation in the full structural model. It should

also be noted that geometric imperfections play an important role in buckling behavior (Jung,

88

1952). Whereas introducing global buckling modes from an eigen analysis may lead to a model

capable of buckling initiation, geometric imperfections were introduced to the building

randomly. Each individual column and beam was randomly crooked and with initial out-of-

plumbness such that buckling could be initiated. At the same time, no global buckling behavior

was introduced in the model. A normal distribution of crookedness (with 95% of values within

1/1500) and out-of-plumbness (with 95% values within 1/500) were introduced as recommended

by the American Institute of Steel Construction (2006) and Ballast (1994). These distributions

correspond to assembly accuracy and real-life manufacturing imperfections.

In order to verify the ability of the proposed modeling technique to simulate buckling and

post-buckling behavior, a number of buckling simulation were carried out for different boundary

conditions and lengths. Simulations produced responses both in terms of buckling loads and

post-buckling behavior. A comparison of simulated buckling loads with a code curve (AISC,

2006) is shown in Figure 3-14. Simulated results match closely with the AISC curve, which is in

essence based on experimental results (Hall, 1981).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0

Fcr /

Fy

Slenderness (λc)

LRFD Buckling Load

ls-Dyna Cantiliver

ls-Dyna Top pinned

ls-Dyna Clamped

Figure 3-14. Buckling stress- slenderness parameter λc

89

The wide-flange sections have residual stresses (Salmon, Johnson, 1996) that remain in a

member after it has been formed into a finished product. Such stresses result from plastic

deformations, which result from uneven cooling that occurs after hot rolling of structural shapes.

The flanges, being the thicker parts, cool more slowly than the web regions. Furthermore, the

flange tips having greater exposure to the air cool more rapidly than the region at the junction of

the flange to web. Consequently, compressive residual stress exists at flange tips and at mid-

depth of the web (the region that cools fastest), while tensile residual stress exists in the flange

and the web at the regions where they join. Since accounting for the geometric imperfections

was sufficient to obtain a satisfactory agreement with the LRFD curve (Figure 3-14), residual

stresses were not included in the model.

Visual inspection of the simulated behavior of columns matched closely with experimental

data reported by Hoff (1955). Simulations of very slender columns exhibit slow buckling with

significant deformations. Conversely, columns with slenderness ratios in the inelastic zone

buckled abruptly. These numerical results agree well with Hoff’s experimental observations

(1955) that “not a single elastic specimen buckled suddenly and no inelastic specimen buckled

slowly. The difference between elastic and inelastic buckling was obvious to the observers even

without measurements. Inelastic columns suddenly jumped from an almost straight

configuration into a highly curved state and the jump was always accompanied by an audible

thud. In contrast, elastic columns buckled gradually.” Both buckling forces and post-buckling

behavior have been shown to be in good agreement with experimental data. The simulated

results captured experimentally derived buckling behavior with great realism. Such realism is

essential in correct modeling of progressive collapse because propagation of column buckling

plays a crucial role in collapse arrest or propagation.

90

Selected Structures

Simplified Steel Framed Structure

A simplified 3-story steel framed building was selected for the “discovery analysis”, which

was used to explore the energy flow during collapse and to verify the proposed hypothesis. A

uniform selection of steel sections was used to produce a realistic, though simple, generic

building for the comparative analysis of energy flows under various initiating conditions (single

or multiple, first story, column removals). The framing plan of the analyzed structure is shown

in Figure 3-15.

33 [f

t]

w18

x35

w18

x35

w21x68w21x68

33 [ft]

1

2

3

4

A B C D E

w18

x35

Figure 3-15. Selected three-story steel framed building

Deck self-weight was estimated, using concrete density and slab dimensions, to be 3 [kPa]

(~65 [psf]). Ceilings/flooring/fireproofing, mechanical/electrical/plumbing systems and

partitions were estimated to impose an approximately 1 [kPa] (~21 [psf]) pressure load on the

composite deck. Thus, dead load D = 4 [kPa] (86 [psf]) was employed in the analysis. A

minimum uniformly distributed live load for office, L = 50 [psf] ≈ 2.5 [kPa], as recommended by

91

ASCE (2005), was applied to account for the presence of building occupants, office equipment,

and so on.

Dead and live loads from slabs were transferred to W18x35 beams placed on W21x68

girders. Girders in turn were attached to W14x74 columns (designation of steel beams according

to AISC, 2005). The W14x74 steel sections were used through three stories without splices and

variation in shape selection. Each story was 156 [in] (4 [m]) high.

Numerical models of the investigated steel frames and the steel framed building were

constructed using ANSYS-LS-DYNA. Preprocessed models were exported to the LS-DYNA

input files and solved by the LS-DYNA numerical solver. Implicit analysis was used in the

preloading phase. Building permanent loads were applied over 20 [s] using the implicit solver.

Kinetic energy was not observed in this phase and thus preloading was purely quasi-static.

Subsequently, the analysis was restarted in the explicit mode with a number of key columns

removed from the model. All internal forces, energy levels, etc. were preserved from the

preloading phase and only the solver was switched at the prescribed time of 20 [s]. ANSYS

models of the analyzed structures are shown in Figure 3-16, Figure 3-17 and Figure 3-18.

Figure 3-16. Two-dimensional steel frame selected for the analysis

92

Figure 3-17. Three-dimensional steel frame selected for the analysis

Figure 3-18. Simplified steel framed building

93

Realistic 3-story SAC verification building

In order to apply and test the findings derived from the previously analyzed 3-D steel

frames, a realistic moment resisting steel framed building was subsequently analyzed. A 3-story

steel framed building used in SAC (Gupta and Krawinkler, 2000) research was selected for the

verification analysis. The SAC Steel Project was funded by FEMA to solve the problems of

welded frame structures that surfaced in the January 17, 1994 Northridge, California (Los

Angeles) Earthquake. The SAC commissioned three consulting firms to perform code designs,

following the local code requirements for the city of Boston (National 1993). All prevailing

requirements for gravity, wind, and seismic design were considered.

Rigid offsets between beam and slab centerlines were considered to account for the

composite action between in-filler beams and composited decks. The floor-to-floor heights are

taken from centerline of beam/girder to centerline of beam/girder. The ANSYS-LS-DYNA

model of the SAC Boston modified steel framed building is depicted in Figure 3-19.

Figure 3-19. Model of the SAC Modified Boston Building

94

The buildings was designed for a typical office occupancy live load of 50 [psf] (2.4 [kPa]).

The floors were assumed to support a superimposed dead load of 83 [psf] (4 [kPa]), which

included a concrete - steel composite slab, steel decking, ceilings /flooring /fireproofing,

mechanical /electrical/ plumbing systems and partitions (20 [psf], 1 [kPa]). The framing plan of

the 3-story verification building is shown in Figure 3-20 and in Table 3-2. Column schedules are

depicted in Figure 3-21 and in Table 3-3. An assortment of ten W-shapes was implemented in

the building: five column sections and five beam sections.

30 [f

t]

30 [ft]

2

3

4

5

A B C D E

1

F G

Moment connection Penthouse

w21x44 w21x44 A A A w21x44

w21x68 w21x68 w21x68 w21x68 w21x68 w21x68

w21x68 w21x68 w21x68 w21x68 w21x68 w21x68

w21x68 w21x68 w21x68 w21x68 w21x68 w21x68

w21x44 w21x44 A A A w21x44

AA

A

w18

x35

AA

Aw

18x3

5 w18x35 (in-fill beams)

Figure 3-20. Framing plan used for SAC three story building

Table 3-2. Moment resistant beams (designated with “A”) Floor Beam “A” 2 w18x35 3 w21x57 roof w21x62

95

y

30 [ft]

2

3

4

5

A B C D E

1

F G

Moment connectionPenthouse perimeter

x

30 [f

t]

Figure 3-21. Orientation of columns

Table 3-3. Column schedules of the typical SAC building A B C D E F G 5 w12x58 w12x58 w14x74 w14x99 w14x99 w14x74 w12x58 4 w14x74 w12x58 w12x65 w12x72 w12x65 w12x58 w14x74 3 w14x99 w12x58 w12x65 w12x72 w12x65 w12x58 w14x99 2 w14x99 w12x58 w12x58 w12x58 w12x58 w12x58 w14x99 1 w14x74 w12x58 w14x74 w14x99 w14x99 w14x74 w14x74

96

Summary

Progressive collapse is a complex phenomenon. There is a widespread desire among civil

engineers to develop an analysis methodology of progressive collapse potential. Whereas

advanced finite element method codes offer capabilities to simulate and solve structural

propagation, physical interpretation and insight is yet to be established. To fill this gap, the

energy flow approach is proposed. The concept is applied to a simplified steel framed building

for testing and further verified on a real life, three story seismic resistant building.

Whereas experimental testing provides the most reliable results, it is very expensive to

perform with real structures. Therefore, simulation tools were used to conduct “virtual

experiments.” Such numerical results were verified against higher resolution models and

validated with available experimental buckling data and basic laws of physics.

97

CHAPTER 4 KINEMATIC RESULTS

Overview

Numerical simulations were carried out to provide more insight into the progressive

collapse phenomena. Simulations of full structural systems were carried out. Kinematic results

are presented primarily to ensure that the simulations observed basic laws of physics in a global

sense, and additionally, to provide a meaningful source of information. General conclusions on

measures that can prevent disproportionate collapse are given. On the basis of full-scale

simulation, a subset of cases was selected for energetic analysis.

Virtual Experiments on the Simplified Steel Framed Buildings

Once characteristic properties of an individual column were established, global simulations

of structural response to localized abnormal loadings were carried out. The objective was to

establish realistic simulation techniques, which capture the physical behavior of steel framed

structures. Soundness of the simulations ensures that findings and results obtained in this work

can be directly applied to physical structural systems. A simplified steel framed building was

analyzed at various levels of idealization, e.g., 2-D frame, 3-D frame and full 3-D model with

slabs. The objective was to select proper modeling resolution to ensure that investigated energy

flow results are applicable to real steel framed structures.

For the reader’s convenience, description of the simplified three story steel framed

building selected for the “discovery analysis” is briefly repeated herein. The framing plan of the

analyzed structure is shown in Figure 4-1. The dead load, D = 86 [psf] ≈ 4 [kPa] and uniformly

distributed live load for office, L = 50 [psf] ≈ 2.5 [kPa], were employed in the analysis. Dead

and live loads from slabs were transferred to W18x35 beams placed on W21x68 girders. Girders

in turn were attached to W14x74 columns. Column W14x74 was used through three stories

98

without splices and variation in shape selection. Each story was 156 [in] (4 [m]) high. Two

types of composite slabs were considered: heavily reinforced slab of 6 [in] (150 [mm]) depth and

4 [in] (100 [mm]) slab with light reinforcement.

33 [f

t]

w18

x35

w18

x35

w18

x35

w21x68w21x68

33 [ft]

1

2

3

4

A B C D E

Figure 4-1. Selected three story steel framed building (W14x74 columns)

To understand the structural response, the removal of one corner column at the ground

level, and the removal of two corner columns at the ground level were considered. The first case

represented arrested collapse, whereas removal of two columns led to collapse propagation and

total collapse of frames and ultimately the entire steel structure. In order to investigate the

collapse behavior of the building with typical, lightly reinforced slabs, additional cases (with

increased loading) were considered to initiate total collapse.

Strong structural connections are generally considered to enhance robustness and

resistance to progressive collapse (Ellingwood and Dusenberry, 2005). Therefore, all

connections in the simplified model were assumed as moment resisting to investigate if

maximum continuity always provides enhanced levels of protection.

99

Kinematic Observations

A traditional approach to structural analysis was exercised to establish the baseline for the

energy considerations to follow. Kinematic observations consisted of the analysis of internal

forces and displacements.

For checking the capacity of a structure to withstand the effects of an extraordinary event,

General Services Administration (2003) recommends LD 5.00.1 + load combination. American

Society of Civil Engineers (2006) advocates the following, very similar load combinations:

WADororSorLAD kk 2.0)2.19.0()2.05.0(2.1 ++++ (4-1)

D = dead load; L = live load; Ak = load effect resulting from extraordinary event; S = snow load; W = wind load; CASE 1. Corner column on the ground level removed: 1.0 D + 0.5 L

It is difficult to assess which portion of the dead load contributes to the collapse and thus

should be amplified by 20%, and which portion of the dead load mitigates the collapse

propagation and thus should be reduced by 10%. Therefore the General Services Administration

(2003) load combination was used in the subsequent analysis:

][25.5][136505.0865.00.1 kPapsfLDLoad ==⋅+=+= (4-2)

The following modeling resolutions were implemented for comparison: 2-D frame (façade

along column line 1), 3-D frame with joists and 3-D building with joists and slabs. The

structures were subjected to the sudden removal of column A-1 (Figure 4-1) at a prescribed time

after application of static preloading. Tributary areas were used to apply loads to the 2-D and 3-

D frames. The 2-D frame was loaded with point loads transferred from the W18x35 beams. 3-D

frames were loaded with uniformly distributed line loads applied to the W18x35 beams. Initial

and final displaced configurations obtained from dynamic time history analyses from 2-D and

100

3-D frames are shown in Figure 4-2 and Figure 4-3. In each case, collapse was arrested with

significant beam deformations in the bay subjected to the sudden column removal.

Figure 4-2. Two dimensional frame in CASE 1

Figure 4-3. Three dimensional frame in CASE 1

In the case of the building with floor slabs, load was applied directly to the slabs. Inertia

properties of the loads were preserved by lumping appropriate mass into decks such that their

weight per unit area equaled the sum of dead and live loads. Steel framed buildings with heavily

reinforced and typical slabs arrested the collapse propagation (Figure 4-4).

101

Figure 4-4. Displaced shape of the simplified steel framed building in CASE 1

Vertical displacements of the building corner (above the removed column A-1) are shown

in Figure 4-5. Due to robust connections, the collapse was arrested. The lesser extent of plastic

deformations can be seen in the building with stronger slabs.

-35

-30

-25

-20

-15

-10

-5

0

20 20.5 21 21.5 22

Dis

plac

emen

t [in

]

Time [s]

3-D w/ hardened slabs3-D w/ typical slabs3-D Frame2-D Frame

Figure 4-5. Displacement of the building corner (CASE-1)

102

On the basis of displacement results only, it could be concluded that 2-D and 3-D frames

produced more conservative results and higher demands on the members. However, the

inspection of moment time histories in beam AB in column line 1, at the supported end (point B)

leads to different conclusions (Figure 4-6).

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

0 5 10 15 20

Mom

ent [

kip-

in]

Time [s]


A

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

20 20.2 20.4 20.6 20.8 21

Mom

ent [

kip-

in]

Time [s]


B

Figure 4-6. Beam moment time history in CASE-1: A) static, B) dynamic phase

103

Although, the static phase was reasonably consistent, the dynamic phase revealed

discrepancies between the three models. Tributary area approach to predict the load flow was

very effective in the static pre-loading phase. Nevertheless, it became evident (Figure 4-6) that it

lead to erroneous results in the dynamic phase of progressive collapse. Models with loads

redistributed according to the tributary area approach had deflections distinctively different from

the models with direct load application. Hence, the load flow in the dynamic phase was not only

affected by tributary area but also depended on accelerations and inertial effects. Lateral

stiffening provided by slabs also contributed to the differences between the models in the

dynamic phase. Thus full scaled, detailed simulations, with properly modeled slabs were

necessary to analyze the collapse arrest or propagation. Simplified models lead to erroneous

results in the dynamic phase of calculations.

Results from reduced models of 2-D and 3-D frame with “manually” redistributed loads

from slabs and in-fill beams were in good agreement with results from the full scale building

model in the static phase only. However, comparison of results in the dynamic phase revealed

that a traditional tributary area approach, used to transfer the loads from slabs and in-fill beams

to the reduced model, was not effective. Among the three analyses, dynamic displacement

results exhibited different oscillation periodicity and moment-time histories were distinctive.

Thus, using the tributary areas to redistribute loads is inadequate, in spite of the employment of

high fidelity non-linear dynamic analysis. Inadequacy of the tributary area approach to

redistribute the loads to the frame can be attributed to the importance of lateral restraints

provided by slabs and to inertial effects.

CASE 2. Two columns on the ground level removed: 1.0 D + 0.5 L

Gradually more severe scenarios were considered to transit from safe events to total

collapse progression. One corner and one adjacent exterior column on the ground level (A-1 and

104

B-1) were removed after application of static preloading. The following General Services

Administration (2003) load combination was applied:

][25.5][136505.0865.00.1 kPapsfLDLoad ==⋅+=+= (4-3)

Collapse was arrested for all models, although larger defections than in CASE 1 were

observed. This intermediate case will be used in the development of energy considerations in the

later in this work.

CASE 3. Two columns on the ground level rendered ineffective: 1.2 D + 1.0 L

CASE 3 was intended to produce the total collapse; therefore the most structurally

detrimental General Services Administration (2003) load combination was applied:

][35.7[psf]153501.0862.1L1.0+D1.2 Load kPa==⋅+⋅== (4-4)

Columns A-1 and B-1 were removed after static pre-loading to initiate the collapse.

Dynamic analyses were carried out to simulate structural behavior. Collapse sequences for each

of the models are shown in Figure 4-7, through Figure 4-10. Considerable differences can be

noticed between collapse modes of the 2-D and 3-D frames as well as 3-D models incorporating

slabs. The collapse sequence of the frame models were brought about largely due to the absence

of lateral restraint from the slabs. Composite or reinforced concrete decks provided membrane

action, which restrained beams from lateral torsional buckling. Thus 2-D and 3-D models

offered acceptable results for the stripped frame behavior but inadequate information to analyze

real building response. More detailed models (accounting for slabs) provided more realistic

insights into possible collapse sequences. Collapse started at the column B2.1 (line B, row 2,

story 1) in the building with heavily reinforced slabs. On the other hand, column A2.1 was the

first overloaded column in the building with typical slabs. Thus, deck stiffness affects structural

response to the localized abnormal loading.

105

Figure 4-7. Collapse sequence of two-dimensional frame in CASE 3

106

Figure 4-8. Collapse sequence of the three-dimensional frame in CASE 3

107

Figure 4-8. Continued

B2.1

Figure 4-9. Building with hardened slabs - Collapse sequence in CASE 3

108

A2.1 C1.1


109

A2.1

B2.1

Figure 4-10. Building with typical slabs - Collapse sequence in CASE 3

110

C1.1


111

In the hardened structure, column B2 buckled due to overloading. Collapse spread around

the perimeter of the collapse zone to columns C1 and A2. Subsequently, the collapse progressed

at the ground level, leading to global building failure. In the building with light slabs, column

A2 buckled first and collapse spread to columns B2 and C1. Successively, failure propagation at

the ground level resulted in total building collapse. Comparative displacement results at the top

of the removed column A1 are shown in Figure 4-11. The building with stronger slabs attempted

to bring the fall to a stop faster than the building with light decks (20.4 [s] versus 20.8 [s]).

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

20 20.5 21 21.5 22

Dis

plac

emen

t [in

]

Time [s]


Figure 4-11. Displacement time history of point A-1 in CASE 3

The moment time history of beam A-B at support B-1 on the first floor is presented in

Figure 4-12. Again, significant discrepancies can be seen between the models. Stripped frame

models lead to significantly different results with different periodicities and amplitudes in the

dynamic, collapse phase. Thus, only models incorporating decks and accounting for inertial

effects can lead to meaningful and useful results in the analysis of progressive collapse potential.

112

-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

19.5 20 20.5 21 21.5

Mom

ent [

kip-

in]

Time [s]


Figure 4-12. Moment time history of beam A-B at B-2 in CASE 3

Internal forces in the columns can be used to evaluate column safety and load

redistribution during collapse propagation or arrest. Traditionally, column stability is assessed

on the basis of moment axial force interaction relationships. In the collapse analysis, internal

forces vary spatially along the column as well as exhibit oscillatory nature in time. Moreover

shear forces and torsional moments are also noticeable in the structural members. Axial force

normalized by the LRFD buckling load as well as strong and weak axis moments normalized by

the LRFD plastic moments are shown in Figure 4-13 through Figure 4-16. Figure 4-16 shows

column B2.1 remaining stable at the almost buckling axial loads in the building with lightly

reinforced decks. In contrast, column B2.1 in the hardened building became instable at the first

approach to the buckling load. It illustrates the complexity of the moment axial forces

interactions present in the dynamic force redistribution. Assessing column safety solely from

oscillatory internal forces is a formidable task.

113

-2

-1.5

-1

-0.5

0

0.5

1

0 5 10 15 20

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

y

Axial ForceMoment SMoment T

A

-2

-1.5

-1

-0.5

0

0.5

1

0 5 10 15 20

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

S

T

y


B

S

T

Figure 4-13. Normalized internal forces in A2.1 column: A) hardened slabs, B) typical slabs

114

y

-1

-0.8

-0.6-0.4

-0.2

00.2

0.4

0.60.8

1

20 20.5 21 21.5

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

S

T


A

-1-0.8

-0.6

-0.4

-0.2

0

0.20.4

0.6

0.8

1

20 20.5 21 21.5

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]


B

S

T

Figure 4-14. Dynamic phase of normalized internal forces in A2.1 column: A) hardened slabs, B) typical slabs

115

-1

-0.8

-0.6-0.4

-0.2

00.2

0.4

0.60.8

1

0 5 10 15 20

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

y


A

-1

-0.8

-0.6

-0.4-0.2

0

0.2

0.40.6

0.8

1

0 5 10 15 20

Mom

ent o

r For

ce [

kip-

in o

r kip

]

Time [s]

S

T

y


B

S

T

Figure 4-15. Normalized internal forces in B2.1 column: A) hardened slabs, B) typical slabs

116

-1

-0.8

-0.6-0.4

-0.2

00.2

0.4

0.60.8

1

20 20.5 21 21.5

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

y

S

T


A

-1-0.8

-0.6

-0.4

-0.2

0

0.20.4

0.6

0.8

1

20 20.5 21 21.5

Mom

ent o

r For

ce [

kip-

in o

r kip

]

Time [s]


B

S

T

Figure 4-16. Dynamic phase of normalized internal forces in B2.1 column: A) hardened slabs, B) typical slabs

117

Validation of Results

Whereas certain stages of the modeling process are readily available (e.g., finite element

solvers), proper validation of numerical simulation are a matter of concern. While simple

calculation models can be employed in many situations, progressive collapse does not have such

simplified procedures. It is recommended to first verify individual components of the

simulation, e.g., buckling behavior of individual columns to assure that simulation provides

realistic results.

Collapse simulation must adhere to the basic laws of classical physics. In simulated

events, one or two key columns are removed without additional loads (explosion, truck impact,

etc). Structural elements should not fall vertically faster than at free fall (Loizeaux and Osborn,

2006; Bazant and Verdure, 2007). Actually resistance and energy absorption of resisting

members should slow down such motion. Figure 4-17 and Figure 4-18 compare simulated

displacement results with the free-fall curve for CASE 1 and CASE 3. Only the detailed 3-D

model with slabs and properly applied loads adhered to the “free-fall” requirement. This

phenomenon resulted from load application as a lumped mass into the modeled decks. The

comparison shows that neglecting the inertia of dead and live loads leads to significant errors and

violation of the free-fall physics. Thus, not only slabs play a crucial role in the proper simulation

of structural response to the localized failure, but also inertial effects affect building behavior.

118

-35

-30

-25

-20

-15

-10

-5

0

20 20.5 21 21.5

Dis

plac

emen

t [in

]

Time [s]

3-D w/ hardened slabs3-D w/ typical slabs3-D Frame2-D FrameFree Fall

Figure 4-17. Free fall requirement in CASE 1

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

20 20.5 21 21.5 22

Dis

plac

emen

t [in

]

Time [s]

3-D w/ hardened slabs3-D w/ typical slabs3-D Frame2-D FrameFree Fall

Figure 4-18. Adherence to free fall in CASE 3

119

Steel Framed Building with Shorter Beam Spans

Additional analysis was carried out for a steel framed building with typical slabs but with

slightly shortened beam spans. Beam spans were decreased from 33 [ft] to 30 [ft] (10 to

9.15 [m]). The structure shown in Figure 4-19 was subjected to two corner column removals.

Increasingly higher loading scenarios were considered to track the system behavior from safe to

unsafe state. Such analysis captured the energetic transition from stable to unstable behavior.

30 [f

t]

w18

x35

w18

x35

w18

x35

w21x68w21x68

30 [ft]

1

2

3

4

A B C D E

Figure 4-19. Selected three story steel framed building for buckling onset analysis

Two columns: A1.1 and B2.1 on the ground level were removed. The gradually increasing

loading scenarios were investigated at: A) 1.2 D + 1.0 L; B) 1.2 D + 1.5 L; C) 1.2 D + 2.0 L.

Figure 4-20 through Figure 4-22 show structural response to the applied loading scenarios.

Interestingly, a small decrease in the span length significantly enhanced structural resistance to

collapse propagation. It was necessary to increase the applied live load from 1.0 L to 2.0 L in

order to initiate the progressive failure. Column displacements (Figure 4-23) did not show

characteristics, which could be helpful in determination of the onset of instability.

120

Figure 4-20. Arrested collapse of three-dimensional building w/ typical slabs in CASE A

Figure 4-21. Arrested collapse of three-dimensional model w/ typical slabs in CASE B

121

B2.1

A2.1

Figure 4-22. Collapse sequence of three-dimensional model w/ typical slabs in CASE C

122


123

-0.2-0.18

-0.16

-0.14

-0.12

-0.1

-0.08-0.06

-0.04

-0.02

0

0 5 10 15 20

Dis

plac

emen

t [in

]

Time [s]

p y

A2.1-topB2.1-top

1

-0.2-0.18-0.16-0.14-0.12

-0.1-0.08-0.06-0.04-0.02

0

20 20.5 21 21.5 22 22.5 23

Dis

plac

emen

t [in

]

Time [s]

A2.1-topB2.1-top

2

Figure 4-23. Displacements of A2.1 and B2.1 columns: 1) Complete history, 2) Dynamic, collapse phase.

Figure 4-24 through Figure 4-27 compare internal forces in columns in the case of arrested

and progressive collapse. Differences between CASE A and CASE C in the column A2.1 are

124

very minor. Also, column B2.1 reached axial buckling in CASE C but its instability was not

noticeable in the visual simulation results. The threshold between column buckling and stable

configuration is very vague when the internal force histories are studied.

-1-0.8

-0.6

-0.4

-0.2

0

0.20.4

0.6

0.8

1

0 5 10 15 20

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

y


1

-2

-1.5

-1

-0.5

0

0.5

1

0 5 10 15 20

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

S

T

y


2

Figure 4-24. Normalized internal forces in A2.1 column: 1) CASE A, 2) CASE C

125

-1

-0.8

-0.6-0.4

-0.2

00.2

0.4

0.60.8

1

20 20.5 21 21.5

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]


1

-1-0.8

-0.6

-0.4

-0.2

0

0.20.4

0.6

0.8

1

20 20.5 21 21.5

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

S

T

yAxial ForceMoment SMoment T

2

Figure 4-25. Dynamic phase of normalized internal forces in A2.1 column: 1) CASE A, 2) CASE C

126

-1-0.8

-0.6

-0.4

-0.2

0

0.20.4

0.6

0.8

1

0 5 10 15 20

Mom

ent o

r For

ce [

kip-

in o

r kip

]

Time [s]

y


1

-1

-0.8

-0.6-0.4

-0.2

00.2

0.4

0.60.8

1

0 5 10 15 20

Mom

ent o

r For

ce [

kip-

in o

r kip

]

Time [s]

S

T

y


2

Figure 4-26. Normalized internal forces in B2.1 column: 1) CASE A, 2) CASE C

127

-1

-0.8

-0.6

-0.4-0.2

0

0.2

0.40.6

0.8

1

20 20.5 21 21.5

Mom

ent o

r For

ce [

kip-

in o

r kip

]

Time [s]


1

-1

-0.8

-0.6

-0.4-0.2

0

0.2

0.40.6

0.8

1

20 20.5 21 21.5

Mom

ent o

r For

ce [

kip-

in o

r kip

]

Time [s]


2

S

T

Figure 4-27. Dynamic phase of normalized internal forces in B2.1 column: 1) CASE A, 2) CASE C

128

Summary

The static phase was characterized by small displacements and moments that were in good

agreement between models. Thus a tributary area approach worked well in the static phase as

expected. The dynamic response was characterized by significant displacements and large

strains, which resulted in pronounced non-linear and dynamic effects. Moment-time histories

varied significantly between models. Whereas, numerous simplifications are acceptable in the

static analysis, they are not warranted in the collapse analysis. Instead, detailed and direct

dynamic analysis needs to be employed to assess the susceptibility to progressive collapse.

Due to the oscillatory nature of internal forces in the analyzed collapse cases, it is difficult

to infer the alternative load paths from the force results only. On the other hand, applying

sufficient load for initiation and propagation of collapse, alternate load paths reveal themselves

through collapse modes. It should be noted that the displacements of the 3-D model with harder

slabs flattened around 0.4 [s] after column removal. It directly corresponds to the time when

column B-2 buckled, followed by buckling of columns B-1, C-1 and A-2. Thus the structure

attempted to arrest the collapse around 0.5 [s] after its onset but was overloaded and these four

columns failed. Collapse propagated further until total collapse. On the other hand, the

simplified 2-D and 3-D frame simulations did not provide meaningful results. On the contrary,

they showed unrealistic collapse modes.

It is commonly held that tying the structure provides ductility and increases resistance to

progressive collapse. This principle is valid for relatively small localized damage, as shown in

CASE 1. However, if the initiating damage is large and beyond the arresting capabilities of the

structure, due to superbly strong structural ties, the collapse will propagate through the whole

structure, and it will lead to total collapse, as shown in CASE 3. Actually, it may be more

beneficial to provide connections that fail between the collapsing portion of the structure and the

129

rest of the structure so as to not pull the remaining building down. Such connection failure

would isolate collapse propagation and protect the remaining portion of the building.

Mass properties play an important role in structural response to the localized, abnormal

loading. Only a sufficiently detailed model with the dead and live loads applied as mass loads

ensured adherence to the fundamental free fall requirement. Whereas mass properties of the

main structural members such as columns, girders, beams, etc. can be rather easily incorporated

in the models, mass properties of dead and live loads resulting from office equipment, building

occupants, mechanical installations, etc. require further investigation. It has been proposed to

represent these effects by lumping additional mass to the slabs.

Progressive collapse simulations of moment resisting steel frame buildings were carried

out to provide insight into modeling processes and progressive collapse propagation.

Progressive collapse analysis is characterized by large displacements, strains and pronounced

dynamic effects. Therefore, resorting to simplified techniques that might be adequate in static

design would inevitably lead to erroneous results and misleading conclusions. It has been

concluded that only fully nonlinear dynamic time history analysis of sufficiently detailed models,

which account for material and geometric nonlinearities, can lead to meaningful results. Only

the detailed 3-D model with slabs and properly applied loads adhered to the free-fall requirement

and captured the collapse sequence with great realism.

130

CHAPTER 5 ENERGY APPROACH TO THE ANALYSIS OF PROGRESSIVE COLLAPSE

Overview

Numerical simulations were carried out to test a proposed energy based methodology. The

objective of this study was to establish the general energy based analysis framework of structural

response to localized abnormal loading.

The response of main structural members has to be characterized. Whereas buckling load

is a known concept (originating from the nineteenth century), a new concept of buckling internal

energy (deformation work) is introduced in this study. In this research, columns were tested as

single members under various loading conditions and their individual behavior was characterized

in both force and energy domains. Energetic analyses were carried out on selected structures.

Simulation results supported the proposed energetic approach. Global energy measures were

correlated to collapse propagation intensity. It was demonstrated that when global kinetic energy

vanished, the structures reached the final stable state, either arrested collapse, partial collapse, or

total catastrophic failure.

A detailed view of energy propagation was undertaken through analysis of energy

time-histories in all structural members. Buckling energies enabled practical application of the

proposed energy criteria in a quantitative manner. It is shown that for a simplified building, the

entire structural simulation is described by energy time- histories.

Internal energy (deformation work) time- histories combined with the proposed buckling

energy and failure energy limit enabled accurate prediction of collapse sequence, competing

failures, and safety levels of individual columns. Accuracy of the energetic analysis is shown to

be superior to crude kinematic assessments. The proposed method captures the essence of

collapse propagation in an accurate, yet easy to digest manner.

131

Energy Definitions in LS-DYNA

External work is the work done by applied forces. The total flow of energy into a system

(external work) must equal the total amount of energy in the system (sum of internal and kinetic

energy. Internal energy (deformation work) is calculated by LS-DYNA using the following

definition:

( )dVdE ∫ ∫= εσint (5-1)

The total strain can be decomposed into elastic (recoverable) and plastic (irrecoverable):

pe ddd εεε += (5-2)

edε = elastic strain increment, = plastic strain increment. pdε

Therefore, LS-DYNA internal energy includes elastic strain energy and work done in

permanent deformation:

( ) ( ) ( )dVddVddVdE pe ∫ ∫∫ ∫∫ ∫ +== εσεσεσint (5-3)

( )dVd e∫ ∫ εσ = elastic strain energy, ( )dVd p∫ ∫ εσ = permanent deformation work.

However, not only deformation work (internal energy) results from external work done on

a system. If there is a beam falling down in a rigid motion with velocity v, external work (done

by gravity) results in kinetic energy but no strains and thus no internal energy (deformation

work) is induced in the system. During a collapse there is both strain related energy and velocity

related energy. Kinetic energy is reported using the following equation:

dVvEkin ∫= 2

21 ρ (5-4)

ρ = mass density, v = particle velocity.

The global energy data is printed in the glstat files. The energy in each material (and thus

each member) is printed in the matsum files.

132

Energetic Characteristics of Individual Columns

During abnormal loading events of framed structures, columns play a crucial role in

structural stability. Propagation of column buckling through the building floor usually leads to

total collapse (Krauthammer et. al, 2004). Therefore, column buckling was characterized in the

energy domain before physics-based, full-scale structural simulations were carried out.

Displacement controlled buckling and force controlled buckling of a W12x58 column were

investigated to establish fundamental knowledge on an energetic behavior of a single column

under axial loading. A W12x58 (AISC, 2006) column, made of A36 steel, and 156 [in] (≈4 [m])

in height, was selected for preliminary analysis.

An individual column within a structural frame is loaded, in part, by gravity forces; but,

the loading process is not solely force controlled. Adjacent beams and members affect column

displacement, and the column is subject to partially force controlled and partially displacement

controlled behavior. Therefore, both displacement and force controlled buckling simulations

were carried out for the selected, typical W12x58 column.

Displacement controlled Buckling

Displacement controlled virtual buckling experiments were conducted using the finite

element software LS-DYNA (Hallquist, 2006). Displacement load at the top of the clamped

column was applied at the rate of 10 [in/s] (≈25.4 [cm/s]). The buckling mode observed in the

simulation is shown in Figure 5-1. Displacement was applied gradually at the constant

displacement rate up to half of the column height (78 [in] ≈ 198 [cm]). The displacement of the

column top is shown in Figure 5-2. LS-DYNA output file designations are also given to clarify

how the information was obtained.

133

Figure 5-1. Simulated buckling mode of W12x58, 156 [in] clamped column (beam elements)

0 2 4 6 8 10-80

-70

-60

-50

-40

-30

-20

-10

0

Time [s]

Dis

plac

emen

t [in

]

Displacement - nodout

Figure 5-2. Prescribed column top displacement time history

134

It should be noted that no sudden instability was observed in the displacement controlled

simulations. Columns gradually deflected following the clamped-clamped buckling mode as

shown in Figure 5-1. On the other hand, the column force resistance, as inferred from the

reaction force at the column bottom, dropped abruptly after approx. 0.02 [s]. Reaction force time

history is shown in Figure 5-3. Although the column suddenly lost load carrying capacity,

kinematic instability was restrained by the prescribed boundary displacement conditions. Thus, a

column within a structural frame may effectively buckle without exhibiting sudden kinematic

instability. This can happen if the column is sufficiently restrained by the adjacent members

(e.g., girders, slabs).

0 2 4 6 8 100

100

200

300

400

500

600

Time [s]

Forc

e [k

ip]

Force - spcforc

Figure 5-3. Reaction force time history

The resistance force displacement curve closely replicated the resistance time history

(Figure 5-4). It can be seen that column resistance reduced drastically after a displacement of

approximately 0.2 [in] (≈0.51 [cm]) is reached by the column. The column remained stable

because the top displacement was prescribed in the displacement-load time history.

135

Internal energy (deformation work) is defined as the work needed to change structural

shape without inducing kinetic energy. Internal energy time history of the analyzed column is

shown in Figure 5-5. Once the column buckled, the energy absorption slope significantly

decreased. When the prescribed displacement reached its final value, Internal energy

(deformation work) becomes constant. In contrast to the force-time history, the slope of energy

absorption remained positive in the post buckling phase. Consequently, the column absorbed

significant energy beyond the onset of its buckling. In reality, the energy absorption in the post-

buckling phase may be lower than predicted by beam elements due to effects of local buckling

(flange, web).

0 2 4 6 8 10

0

100

200

300

400

500

600

700

800

Displacement [in]

Forc

e [k

ip]

Force - spcforc

Figure 5-4. Resistance-top displacement function

The internal energy-displacement relationship is depicted in Figure 5-5. It is similar to the

internal energy time history. However, once the final value of the prescribed displacement - time

history is reached, all data points in the time domain reduce to one data point corresponding to

136

the final column displacement in Figure 5-6. The slope of the internal energy- displacement

curve is equal to the force resistance of the analyzed column.

0 2 4 6 8 100

2000

4000

6000

8000

10000

Time [s]

Inte

rnal

ene

rgy

[kip

-in]

Internal energy - matsum

Figure 5-5. Internal energy (deformation work) time history

0 2 4 6 8 100

500

1000

1500

2000

2500

Displacement [in]

Inte

rnal

ene

rgy

[kip

-in]

Internal energy - matsum

Figure 5-6. Internal energy displacement history

137

Force controlled Buckling

Column stability is critical for building safety. Force controlled buckling simulations,

using the finite element package LS-DYNA, were carried out for a typical W12x58 column

(AISC, 2006). The column was assumed to be made of A36 steel and 156 [in] (≈4 [m]) in

height. In order to explore the influence of inertial effects on column resistance and energy

absorption, various loading rates from 10 to 10000 [kip/s] (≈45 to 45000 [kN/s]) were applied at

the column top. Each loading rate produced numerous kinematic and energetic results. Due to

the volume of individual results, only the most pertinent results for the 10 [kip/s] (45 [kN/s])

loading rate are discussed in detail.

Kinematic Results for 10 [kip/s] (≈ 45 [kN/s])

The onset of buckling was clearly identified by visual inspection of simulated results as

well as examination of kinematic output. Vertical acceleration of the column top is shown in

Figure 5-7. The acceleration is of negligible magnitude until the buckling resistance is reached

and the column buckles in a dynamic fashion. The buckling onset time was identified as the time

at which the first sudden increase occurs in the acceleration. At this time, a plastic hinge (at

column mid-height) was formed and material failure followed shortly afterwards. Since ground

surface was not introduced in the individual column tests, the failed portion of the column

continued to accelerate until the simulation termination time was reached. Therefore maximum

acceleration, velocity and displacement did not provide useful information in the analysis of the

structure, in which acceleration is limited by collision with other members or the ground. On the

other hand, a sudden increase in acceleration magnitude was helpful in determination of the

onset of buckling.

138

0 20 40 60 80 100-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

7 Acceleration - Stefan Szyniszewski

Time [s]

Acc

eler

atio

n [in

/s2 ]

acceleration - nodout

Figure 5-7. Acceleration time history at the column top

Figure 5-8 shows acceleration-force history. Since the load application was force

controlled in this case, the acceleration - force history is essentially the force time history

multiplied by the loading rate of 10 [kip/s] (≈ 45 [kN/s]). A buckling load of approximately

600 [kip] (2669 [kN]) was determined from Figure 5-8. Closer inspection of ASCII NODOUT

text output files enabled more accurate buckling time and in turn buckling load determination as

598[kip] (2660 [kN]). It was only 3% larger than the AISC (2006) buckling load of 581.6 [kip]

(2587 [kN]). Thus simulated buckling resistance was reasonably close to the code specified

value, which is based on experimental curve fitting (Hall, 1981). Figure 5-9 depicts velocity

applied force history and Figure 5-10 shows the displacement applied force history. All

kinematic results enabled clear identification of the buckling onset, the corresponding buckling

load, and additional energetic characteristics, as discussed below.

139

0 200 400 600 800 1000-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

7 Acceleration - Stefan Szyniszewski

Force [kip]

Acc

eler

atio

n [in

/s2 ]

acceleration - nodout

Figure 5-8. Acceleration force history at the column top

0 200 400 600 800 1000-1000

-800

-600

-400

-200

0y y

Force [kip]

Vel

ocity

[in/

s]

velocity - nodout

Figure 5-9. Velocity force history at the column top

140

0

0 200 400 600 800 1000-1

-0.8

-0.6

-0.4

-0.2

Force [kip]

Dis

plac

emen

t [in

]

displacement - nodout

Figure 5-10. Vertical displacement at the column top- force history

The pre-buckling phase can be characterized as quasi-static with negligible kinetic energy.

Once buckling initiates, the kinetic energy abruptly increased (Figure 5-11).

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

Time [s]

Ener

gy [k

ip-in

]

Kinetic Energy - nodout

1010x

Figure 5-11. Kinetic energy time history

141

The internal energy (deformation work), shown in Figure 5-12, confirmed the kinematic

results. The internal energy significantly increased after the onset of buckling.

0 20 40 60 80 1000

1000

2000

3000

4000

5000

6000

7000

Time [s]

Inte

rnal

ene

rgy

[kip

-in]

matsum

Figure 5-12. Internal energy time history

However, a closer inspection of the internal energy (deformation work) results clearly

revealed that after the buckling energy was reached (Figure 5-13), energy increased dramatically

until material column failure. This limiting energy, corresponding to the onset of buckling, was

approximately 58.9 [kip-in] (6.65 [kJ]) (obtained from the MATSUM ASCII output file).

Therefore, it is proposed that the onset of buckling can be identified by not only tracking the

axial force but also by investigating the internal energy (deformation work) stored in a column.

Once the internal buckling energy threshold is exceeded, the column energetic state becomes

unstable and both internal and kinetic energies rise sharply.

A single column within a steel structural frame must resist not only axial forces, but also

end moments in both the weak and strong directions. These end forces will vary with time

142

because progressive collapse is a dynamic phenomenon. Thus a proper buckling identification in

the force domain requires moment force interaction diagrams, which account for the dynamic

nature of applied loading. Moreover, end moments and axial forces may exhibit different

periodicity, with respective moment and axial peak forces occurring at different instances of

time. In contrast, buckling internal energy values hold for complex loading time histories as is

shown in later sections.

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

Time [s]

Inte

rnal

ene

rgy

[kip

-in]

matsum

buckling energy

Figure 5-13. Internal energy time history in pre-buckling phase

Time derivative of the internal energy (deformation work) is called the energy rate in this

study. It is instructive to scrutinize the internal energy time rate, which indicates how fast

applied energy can be absorbed by the column through deformations. Figure 5-14 shows internal

energy rate as obtained from LS-DYNA (MATSUM text output file). The internal energy rate

spiked at the onset of buckling. Once the column hinged (due to material failure), the energy

absorption rate diminished. Therefore the column absorbed external energy, beyond buckling,

143

up to material failure. In the pre-buckling phase, the energy rate monotonically increased up to

approximately 3 [kip-in/s] (0.34 [kJ/s]) and then the energy rate spiked (Figure 5-15).

0 20 40 60 80 1000

5000

10000

15000

Time [s]

Inte

rnal

Ene

rgy

Rate

[kip

-in/s

]

matsum

Figure 5-14. Internal energy rate

0 200 400 600 800 10000

2

4

6

8

10

Force [kip]

Inte

rnal

Ene

rgy

Rate

[kip

-in/s

]

matsum

Figure 5-15. Internal energy rate in the pre-buckling phase

144

Collected Results for Loading Rates: 10 to 10000 [kip/s] (45 to 45000 [kN/s])

The results for analyzed loading rates applied to the W12x58 column (156 [in] ≈ 4 [m]) are

summarized in Table 5-1. Simulated quasi static results compared well with the LRFD AISC

(2006) code value. Discrepancies between the simulated buckling loads and code values were in

the order of 3% in the quasi-static range. Once the loading rate reached 5000 [kip/s]

(22240 [kN/s]), the increase in buckling resistance became noticeable. However, the internal

energy (deformation work) corresponding to the onset of buckling was found to be insensitive to

the loading rate. The internal energy rate at the onset of buckling increased proportionally to the

loading rate as more work was done on the system at an increasingly shorter time interval.

Table 5-1. W12x58, 156 [in] column buckling results

Loading Rate

LS-DYNA Buckling Force

Static AISC Buckling Force

Difference between Dynamic and AISC Static Force

Buckling Energy

Buckling Energy Rate

[kip/s] [kip] [kip] [kip-in] [kip-in/s] 10 598 581.7 3% 57 2 100 598 581.7 3% 59 21 1000 600 N/A 3% 59 200 2000 600 N/A 3% 59 400 3000 600 N/A 3% 59 500 5000 654 N/A 12% 60 850 10000 802 N/A 38% 61 2000

An increase in the buckling force under higher loading rates results from inertial resistance,

which resists the motion and thus delays the onset of buckling. Whereas transient loads can

exceed the static buckling load for short time duration, permanent load that exceeds the static

buckling load will inevitably lead to global buckling. Inertial resistance has only a transient,

short duration nature. Simulated buckling loads for the range of applied loading rates and LRFD

AISC static buckling load are summarized in Figure 5-16. For loading rates greater than

3000 [kip/s] (13340 [kN/s]), force resistance increased noticeably. Loading rates during typical

full structure collapse propagation are of 1000 to 2000 [kip/s] (5000 to 10000 [kN/s]).

145

Loading rate [kip/s]

Buc

klin

g Fo

rce

[kip

]

10 20 30 50 70 100 200 300 500 1000 2000 5000 100000

4080

120160200240280320360400440480520560600640680720760800

LS-DYNA Dynamic Buckling ForceLRFD AISC Static Buckling Force

Figure 5-16. W12x58: buckling force for the selected, force controlled, loading rates

Simulated LS-DYNA internal energy rates versus applied loading rate are shown in Figure

5-17. Internal energy rates corresponding to the buckling onset increased with the loading rate.

The faster the column was loaded, the faster external energy was transferred into the deformation

and internal column energy.

Simulated internal energies corresponding to the buckling onset for the selected column

loading rates are shown in Figure 5-18. Although the energy transfer rate increased with the

rising loading rates, the absolute value of the energy at the buckling onset remained a stable

characteristic of column resistance to buckling.

Loading rates experienced by columns in the full scale simulations were on the order of

1000 to 2000 [kip/s]. Thus a buckling energy of 59 [kip-in] (6.67 [kJ]) was established as the

buckling critical threshold value for the W12x58 column.

146


Ener

gy r

ate

[kip

-in]

10 20 30 50 70 100 200300 500 1000 2000 5000 100000

500

1000

1500

2000

Figure 5-17. W12x58: Internal buckling energy rate for the selected loading rates


Ener

gy [k

ip-in

]

10 20 30 50 70 100 200300 500 1000 2000 5000 1000020

24

28

32

36

40

44

48

52

56

60

64

Figure 5-18. W12x58: Internal buckling energy for the selected loading rates

147

Column Buckling Energy

Displacement controlled and force controlled buckling simulations of a typical w12x58

column were investigated. The objective was to establish fundamental knowledge on column

buckling behavior and the corresponding energetic characteristics.

Simulated buckling resistance in both displacement and force controlled simulations was in

good agreement with experimental LRFD AISC values. Simulated results tended to produce

higher buckling resistance under high-speed axial loading rates (5000 to 10000 [kip/s]), which

are not available in the AISC specifications. However, this increase can be rationally explained

as the effect of inertial resistance, which prevents a column from assuming its buckling shape.

The inertial resistance “enhancement” has only short duration and transient nature. In other

words, if the axial load exceeds permanently the static buckling load, the column will buckle

eventually after a short delay due to the inertial resistance.

Unlike buckling resistance, buckling internal energy exhibits essentially the same value

even for the high-speed loading rates, which are characterized by inertial enhancement. From an

alternate perspective, displacement response decreases due to inertial resistance and force

increases in proportion, thus the product of these two quantities constitutes a stable measure of

the column condition.

Once a single column is incorporated into the structural system, it will be subject to

potentially complicated loading time histories, which may include not only axial forces but also

end moments. These end forces are transient in nature with various periodicities. Therefore a

column buckling criterion in the force domain requires the definition of weak and strong axis

moments – axial force dynamic interaction diagrams. An internal energy buckling criterion is

proposed as an alternative approach in this study. This concept will be further explored in the

full scale collapse propagation or arrest simulation described in later sections.

148

Energy Flow and Redistribution

A quantitative energy based approach was used to analyze the response of the simplified

steel framed building to sudden column(s) removal. Special emphasis was put on column

behavior because the corresponding crucial role in the collapse propagation or arrest

(Krauthammer et. al, 2004). All the columns in the simplified steel framed building were

W14x74 sections of 156 [in] (≈4 [m]) height. Axial buckling of the selected w14x74 column

under loading rates from 1 to 10000 [kip/s] (4.5 to 45000 [kN]) was simulated. Simulated

buckling loads and internal energies corresponding to the buckling onset are shown in Table 5-2.

Table 5-2. W14x74, 156 [in] column buckling results

Loading Rate

LS-DYNA Buckling Force

Static AISC Buckling Force

Difference between Dynamic and Static AISC Force

Buckling Energy

[kip/s] [kip] [kip] [kip-in] 1 770 744 3% 76 10 770 744 3% 76 100 777 N/A 4% 76 1000 810 N/A 9% 77 10000 1100 N/A 48% 77

Single member simulations were used to establish the buckling threshold energy level. It

is proposed that buckling energy obtained from single member axial experiments (or from

validated simulations) can be used as a failure criterion for members incorporated into complex

framed structural systems. Buckling force resistance for the selected loading rates is depicted in

Figure 5-19. For loading rates ranging from 1000 to 10000 [kip/s] (4500 to 45000 [kN])

buckling resistance was enhanced by inertial resistance for the short duration loads only. The

buckling energy threshold was estimated to be approximately 77 [kip-in] (8.7 [kJ]) for all

loading rates (Figure 5-20). The buckling energy criterion is more convenient than that

associated with buckling force because it is loading rate independent.

149


Buc

klin

g Fo

rce

[kip

]

1 2 3 4 5 7 10 20 30 5070100 200 500 1000 2000 5000100000

100

200

300

400

500

600

700

800

900

1000

1100

LS-DYNA Dynamic Buckling ForceLRFD AISC Static Buckling Force

Figure 5-19. Buckling force for the selected loading rates in W14x74 column


Buc

klin

g En

ergy

[kip

-in]

1 2 3 4 5 7 10 20 30 5070100 200 500 1000 2000 50001000005

101520253035404550556065707580

Figure 5-20. Internal buckling energy for the selected loading rates in W14x74 column

150

CASE 1. Building with Heavily reinforced Slabs (Load = 1.0 D + 0.5 L)

As opposed to a traditional internal force investigation, energy based analysis of results is

presented herein. A steel framed building with hardened slabs was analyzed first. In the first

analysis, a single corner column designated as “A1.1” was removed (Figure 4-19 for layout of

columns). “A” designates the vertical column line A, “1” indicates 1st horizontal row, and “.1”

indicates the first story level (Figure 5-21).

Removed column

Figure 5-21. Final shape of the selected structure after sudden column removal in CASE 1

Global redistribution of external work between internal and kinetic energies is depicted in

Figure 5-22. The static preloading phase was dominated by stable internal energy redistribution

with negligible levels of kinetic energy. Collapse was arrested in the dynamic phase. The

building bay affected by the column removal underwent oscillatory motion after collapse arrest.

This vibratory behavior can be attributed to the overly strong slabs, which provided significant

capacity for elastic vibrations.

Figure 5-23 depicts the global dynamic energy flow in more detail. It can be seen that in

the case of arrested collapse, there can be transient kinetic energy associated with elastic

vibrations. Such energy will be damped out by the inherent structural and material damping.

151

0 5 10 15 200

500

1000

1500

2000

Time [s]

Ener

gy [k

ip-in

]

Kintetic EnergyInternal EnergyTotal Energy

Figure 5-22. Global energy histories (from GLSTAT)

20 20.5 21 21.5 22 22.5 230

500

1000

1500

2000

Time [s]

Ener

gy [k

ip-in

]


Figure 5-23. Dynamic phase of the global energies

In order to investigate energy distribution throughout the building in more detail, internal

and kinetic energies were examined for all columns. Figure 5-24 depicts the energy level in all

152

columns undergoing compressive loading. Sixty legends were omitted to avoid cluttering the

plot. The buckling energy limit is shown for comparison in Figure 5-24. None of the internal

energies exceeded 50% of the buckling energy. Thus structure was deemed safe after the

considered column removal. None of the columns were in an unstable equilibrium state. It

should be noted that final energy levels did not significantly increase. Whereas visual inspection

of the simulated results proved that collapse was arrested, it did not provide quantification of the

safety of the building. Scrutiny of internal energy (deformation work) levels and their

comparison with the column buckling energy provided clear insight into the building safety.

This approach is free from the loading rate influence as opposed to the buckling force approach.

Buckling energy

Figure 5-24. Internal energy time histories for all columns

Kinetic energies in the columns are shown in Figure 5-25. In the static preloading phase

no significant kinetic energy was observed in the system. Since the loading was applied in the

153

quasi-static fashion, kinetic energies were negligible. In the dynamic phase, most of the

vibratory energy was excited in beams and slabs. Columns remained stable. Their kinetic

energies were induced by elastic waves travelling throughout the structure as well as slabs and

beam vibrations in the bays affected by the column removal.

Figure 5-25. Kinetic energies for all columns

CASE 2. Building with Hardened Slabs (Load = 1.0 D + 0.5 L)

Consecutively energetic characteristics of the building with hardened slabs in CASE 2

were investigated. The removal of A1.1 and B1.1 columns was intended to constitute a more

severe scenario than CASE 1. The objective of this study was to explore the full range of

structural behaviors from safe states to unsafe situations (i.e., resulting in global or partial

collapse). Since one corner column removal was arrested, two columns shown in Figure 5-26

were removed after application of the static preloading.

154

Columns removed

Figure 5-26. Final deflection of the selected structure in CASE 2

External work was redistributed between internal and kinetic energies as depicted in Figure

5-27. The static preloading phase was dominated by stable internal energy rearrangement with

insignificant levels of kinetic energy. Collapse propagation was prevented in the dynamic phase.

Low levels of kinetic vibrations remained in the system due to the elastic slab oscillations in the

bays affected by column removal.

0 10 20 30 400

1000

2000

3000

4000

5000

6000

Time [s]

Ener

gy [k

ip-in

]


Figure 5-27. Global energies (GLSTAT)

155

Released gravitational energy was not fully absorbed within the first 0.1 [s] after column

removal. Kinetic energy of the system increased until plastic deformations and geometric

nonlinearities enabled redistribution into internal energies of adjacent members. Arrested

collapse was characterized by the ability of the structural system to absorb released gravitational

energy by transferring it into deformation energy. Even if temporary deficiencies in energy

absorption occurred and kinetic energy of the system monotonically increased, there was still the

potential in the system to recover and absorb the kinetic energy surplus. Such recovery resulted

from non-linear resistance capabilities such as membrane action, geometric non-linear stiffening

and plastic material hardening.

Figure 5-28 depicts the global dynamic energy flow in more detail. Kinetic energy begins

to decay to its vibratory level at approximately 0.2 [s] after removal of the columns). Kinetic

energy did not completely vanish in the simulation but remained at the vibratory level. Such

vibrations would be damped out by inherent structural and material damping.

20 20.5 21 21.5 22 22.5 230

1000

2000

3000

4000

5000

6000

Time [s]

Ener

gy [k

ip-in

]


Figure 5-28. Global energies in dynamic phase (GLSTAT)

156

Internal energy (deformation work) of all columns in compression (Figure 5-29) revealed

that two columns (B2.1 and A2.1) have exceeded the buckling energy limit. It has already been

shown in this study that kinematic restraints can prevent noticeable kinematic instability (see

page 133). Column B2.1 buckled but adjacent beams and slabs retained sufficient residual

capacity to redistribute structural loads and arrest the collapse.

B2.1

A2.1 B2.2

energybuckling

Figure 5-29. Internal energies in columns

Columns, which appeared to be in the unstable state, are depicted in Figure 5-30. These

members constituted intuitive alternate paths in the building. Thus their overloading and

buckling was a feasible scenario. However, independent verification of energetic results was

necessary to validate the above conclusions and prove the proposed energy based methodology.

157

A2.1 B2.1

Figure 5-30. Instable columns as inferred from the internal energy results

Kinetic energies of the columns reflected global system periodicity (vibratory motion in

the bays devoid of supporting columns, see Figure 5-31). However closer inspection of kinetic

energies shown in Figure 5-31 and Figure 5-32 conform the finding of column B2.1 buckling.

Figure 5-31. Kinetic energies of all columns

158

20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.80

0.05

0.1

Time [s]

nerg

y [k

ip-in

]

C1.1C1.2C1.3

A2.3

A2.1

A2.1A2.2A2.3B2.1B2.2B2.3C2.1C2.2

E C2.3D2.1A3.1A3.2A3.3B3.1B3.2B3.3C3.1

Figure 5-32. Internal energy in selected columns (initial collapse phase)

20 20.05 20.1 20.15 20.20

0.01

0.02

0.03

0.04

0.05

Time [s]

Ener

gy [k

ip-in

]

C1.1C1.2C1.3A2.1A2.2A2.3B2.1B2.2B2.3C2.1C2.2C2.3D2.1A3.1A3.2A3.3B3.1B3.2B3.3C3.1

B2.1

B2.1

B2.2

B2.3

Figure 5-33. Identification of column buckling from the kinetic energies

159

Hence, the kinetic energy spike in columns B2.1, B2.2 and B2.3 conform that before

periodic oscillations were excited in the structure, these columns were unable to absorb the

applied external work and kinetic energy surplus was produced. However the structure was able

to redistribute the energy to adjacent members and the collapse was arrested. The increase of

global kinetic energy was in good agreement with the energy spike in the B2 columns. However,

an alternate energy path was established in the system, which resulted in the global stable

configuration. Both energy and kinematic results have shown that once column B2.1 failed,

columns A2.1, A2.2 and A2.3 became the secondary load path. The buckling energy threshold

was slightly exceeded in column A2.1. However, due to the presence of adjacent beams and

slabs it was able to withstand the external work transferred to the column.

Although the collapse was arrested after removal of two corner columns, energetic analysis

revealed that the system ended with two columns restrained from kinematic instability by

neighboring members. It is a worse outcome than CASE 1 (only one column removed), in which

all individual members reached stable states well below the buckling energy level.

Tracing energy flow provided insight into the stability of individual members in an easy

and convenient manner. Analysis of a single plot with internal energy - time histories of all

columns sufficed to assess the system condition. The sequence and path of dynamic reloading

was determined. The level of structural safety was inferred from the number of unstable

columns in the system. It is proposed to deem collapse as safely arrested only if all columns

remain in the pre-buckling stage. If collapse propagation is prevented at the expense of one or

more columns buckling, such case should be considered as unsafe and potentially leading to

catastrophic collapse.

160

CASE 3. Collapse of the Building with Heavily Reinforced Slabs (Load = 1.2 D + 1.0 L)

In order to further verify the findings on alternate load paths and system stability on both a

local and global level, the case of removing two columns under increased loading was

investigated. The load was amplified from standard load combination to the following scenario:

][153[kPa]3.7[kPa]2.50.1[kPa]42.1L1.0+D1.2 Load kip==⋅+⋅== (5-5)

External work was redistributed between internal and kinetic energies as shown in Figure

5-34. Collapse propagated throughout the whole structure until total catastrophic failure

occurred. A detailed collapse sequence has been already elaborated on page 105. Global

internal energy (deformation work) monotonically increased as collapse progressed through

consecutive bays. Global kinetic energy started diminishing as consecutive portions of the

structures impacted the ground layer. Once the collapse encompassed the whole building, the

system reached its final stable state at approx. 24 [s].

20 21 22 23 24 250

5

10

15x 10

5

Time [s]

Ener

gy [k

ip-in

]

Internal EnergyKintetic EnergyExternal Work

Figure 5-34. Global energies (from GLSTAT)

161

Interestingly enough, energy results of columns (Figure 5-35) confirmed the soundness of

findings from the previously analyzed, arrested collapse. Column B2.1 buckled first and

buckling propagated to: A2.1, C2.1, B3.1 and so on.

20 20.2 20.4 20.6 20.8 210

1000

2000

3000

4000

5000

6000

7000

8000

Time [s]

Ener

gy [k

ip-in

]

C1.1C1.2C1.3D1.1A2.1A2.2A2.3B2.1B2.2B2.3C2.1C2.2C2.3D2.1A3.1A3.2A3.3B3.1B3.2B3.3C3.1

B3.1

B2.1 C2.1

A2.1

Figure 5-35. Internal energies in columns

Thus the analyzed simplified steel framed building under increased load exhibited the same

dynamic load paths. However, in this case, the building was unable to redistribute the energy

and collapse spread around the perimeter of the localized damage zone. A closer view on energy

time histories is shown in Figure 5-36, enabling the buckling sequence to be identified with

better accuracy. Column buckled in the following order: B2.1, A2.1 (A2.2 and A2.3 relieved

162

after A2.1 buckled), C2.1, B3.1, C1.1 (C1.2 and C1.3 relived after C1.1 buckled), C3.1 and so

on.

20 20.2 20.4 20.6 20.8 210

50

100

150

200

250

300

350

400

450

500

Time [s]

Ener

gy [k

ip-in

]


C2.1

C1.3

C3.1

D2.1 C1.1

B3.1

B2.1 A2.1

Figure 5-36. Internal energies in selected columns with buckling energy threshold

The collapse sequence from the internal energy (deformation work) histories matched

perfectly visual observations of the collapse sequence in the simulation. The visual inspection of

the collapse propagation is depicted in Figure 5-37. Actually, buckling of column B3.1 was

overlooked in the kinematic analysis because it was effectively restrained by the adjacent

members. However, closer inspection from a different angle at t = 20.7 [s] revealed a buckled

shape, which was hardly noticeable from the front view. Finally, the single energy plot provided

insight into safety and collapse propagation as opposed to less accurate visual inspection.

163

B2.1

A2.1

Figure 5-37. Verification of energetic results – Collapse sequence assessment

164

C2.1


B3.1

Front view

B3.1

Left side view

165

C1.1

C3.1

D2.1


166

Energy Propagation through the Building

The steel framed building was divided into four zones, as depicted in Figure 5-38. During

static preloading, selected zones carried energy proportional to the volume fraction of each zone.

However once global energetic instability was triggered by removal of the selected columns,

energy distribution varied in time.

X

1

2

3

4

A B C D E

Zone 1

Zone 2

Zone 3Zone 4

Y

Figure 5-38. Energy propagation zones

Energy propagation through the building is shown in Figure 5-39. Zone 2 includes Zone 1;

Zone 3 includes Zones 1 and Zone 2 and so on. The zones are inclusive. Absorbed energy

initially localized in zone 1. Total failure brought back even more energy redistribution for

zones 1 to 4. A significant increase of internal energies due to progressing failure was observed.

The energy localized in zone 1 when the building attempted to arrest the collapse (Figure 5-40).

As columns B2.1 and A2.1 were failed, potential energy was released across the whole structure

resulting in energy increase in other zones and the total failure.

167

20 20.5 21 21.5 22 22.5 230

2

4

6

8

10

Time [s]

Ener

gy [k

ip-in

]

510x

zone 1zone 2zone 3zone 4

Figure 5-39. Energy propagation through the inclusive building zones

20 20.5 21 21.5 22 22.5 230

0.2

0.4

0.6

0.8

1

1.2

1.4

Normalized energy of all members - Stefan Szyniszewski

Time [s]

Rela

tive

Ener

gy

zone 1zone 2zone 3zone 4

Figure 5-40. Normalized energy allocation in the building (inclusive zones)

168

The structure was unable to absorb the released gravitational energy at approx. 20.3 [s],

when excessive energy localization initiated failure progression. As the energy transfer to the

adjacent members was hindered by the plastic softening, more potential energy at the

increasingly faster rate was released into members located in zone 1. Such intensified energy

transfer to the column B2.1 resulted in its failure. Internal energy rates (Figure 5-41) were

proposed as the measure of energy localization intensity.

20 20.2 20.4 20.6 20.8 210

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Time [s]

Ener

gy R

ate

[kip

-in/s

]


Figure 5-41. Energy rates in the columns

Once the building was unable to redistribute the released gravitational energy, the column

energy rate soared to satisfy the energy conservation. In other words, energy rate was inversely

proportional to the energy redistribution rate. The larger the imbalance between released and

169

redistributed energy, the larger the localized energy rate became. Thus, a critical energy rate was

proposed as a test measure to identify the total failure initiation. Obviously the energy rate of the

column which buckled first is a characteristic of the whole structure and the corresponding

energy absorption ability. Figure 5-42 illustrates the energy redistribution among columns;

beams in the x and y direction (Figure 5-38); and slabs.

20 20.2 20.4 20.6 20.8 210

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsYY

slabs

Figure 5-42. Energy distribution among members in zone 1

After collapse initiation, energy absorption increased in the beams spanning the y-

direction. However, as plastic deformation initiated, the role of the beams decreased and energy

began to overflow to the columns. If the columns and deforming structure are able to arrest the

collapse, energy rate (Figure 5-41) will diminish to zero. Alternatively, in the case of collapse

initiation, significant rates larger than 11000 [kip-in/s] (1.24 [MJ]) were reached. Hence, if

energy rate is approaching the critical value, determined for the structure at hand, localized or

total failure will likely occur. If energy rate increase is averted, the collapse will be arrested.

170

Energy based Column Buckling Criterion (Full Building Analysis)

Energetic analyses of the building with lightly reinforced slabs (depicted in Figure 4-19)

under increasingly severe loading conditions is discussed herein. Loads have been increased in

the consecutive loading scenarios A, B and C (discussed on page 120). The goal was to

gradually transit from safe, arrested collapse to catastrophic failure propagation.

CASE A. Columns A1.1 and B1.1 removed (Load = 1.2 D + 1.0 L)

Global energies are shown in Figure 5-43. Collapse was averted at 1 [s] after the removal

of two columns at the ground floor. Kinetic energy diminished to a vibratory level at the same

time. External work and internal energy (deformation work) oscillated, which indicated the

energy transfer between the potential and internal energies (energy flowed back and forth).

20 20.5 21 21.5 220

0.5

1

1.5

2

Time [s]

Ener

gy [k

ip-in

]


410x

Figure 5-43. Global energy redistribution in CASE A

Internal energies in the building columns are depicted in Figure 5-44. None of the

columns exceeded the buckling energy threshold. Thus it can be concluded that all columns

reached a satisfactory, stable state in response to the localized building damage.

171

20 20.5 21 21.50

10

20

30

40

50

60

70

80

90

100

Time [s]

Ener

gy [k

ip-in

]

C1.1C1.2C1.3D1.1A2.1energybucklingA2.2A2.3B2.1B2.2B2.3C2.1C2.2C2.3D2.1A3.1A3.2A3.3B3.1B3.2B3.3C3.1

Figure 5-44. Internal energies in CASE A

Internal energy (deformation work) results identified columns B2.1 and A2.1 as the

members providing the alternate load paths after the loss of two supporting columns. According

to the proposed energetic buckling criterion, these columns did not buckle and remained stable.

Although some energetic reasoning has already been elaborated in this work to support the

proposed theory, internal forces are herein scrutinized to prove it. Column buckling shall be

characterized by significant deflection increase accompanied by diminishing axial load carrying

capacity. Therefore axial force displacement and internal energy displacement plots for both

columns B2.1 and A2.1 are shown in Figure 5-45. Oscillatory, slightly nonlinear, column

behavior can be noticed in the force domain. However, upon reloading, axial resistance capacity

172

was retained, when necessary. It should be noted that bending moments may prevent the column

from reaching the full axial buckling resistance. Force results verified the energy based finding

that analyzed columns remained stable.

-800

-700

-600

-500

-400

-300

-200

-100

0

-0.4 -0.3 -0.2 -0.1 0

Forc

e [ki

p]

Displacement [in]

y

A2.1-topB2.1-topLRFD Buckling Force

A

forcebuckling

020406080

100120140160180200

-0.4 -0.3 -0.2 -0.1 0

Ener

gy [k

ip-in

]

Displacement [in]

A2.1-topB2.1-topBuckling Energy

energybuckling

B

Figure 5-45. Energy flow in building with typical slabs in CASE A (arrested collapse): A) Axial force-displacement, B) Internal energy-displacement

173

Energy rates in the columns are shown in Figure 5-46. The energy rate spiked noticeably

in the B2.1 column. However, its value of approximately 700 [kip-in/s] (0.08 [MJ]) was well

below the characteristic energy rate of 11 000 [kip-in/s] (1.24 [MJ]). Thus it appears that the

building frame was not in the proximity of the total collapse, when the collapse progression was

averted.

20 20.2 20.4 20.6 20.8 210

100

200

300

400

500

600

700

800

900

1000

Time [s]

Ener

gy R

ate

[kip

-in/s

]


Figure 5-46. Energy rates in the columns in CASE A

Participation of the building zones in energy absorption and redistribution is shown in

Figure 5-47 (inclusive zones) and Figure 5-48 (exclusive zones). It can be seen that the released

potential energy was mainly absorbed by the members in zone 1. Whereas energy in the

exclusive zones 1-2, 2-3 and 3-4 remained practically constant.

174

20 20.5 21 21.5 220

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time [s]

Ener

gy [k

ip-in

]

zone 0-1zone 0-2zone 0-3zone 0-4

Figure 5-47. Energy propagation through the inclusive zones in CASE A

20 20.5 21 21.5 220

2000

4000

6000

8000

10000

12000

14000

16000

Time [s]

Ener

gy [k

ip-in

]


Figure 5-48. Energy propagation through the exclusive zones in CASE A

175

Internal energy (deformation work) increased significantly but mainly in zone 1. It should

be noted that there is no energy sloshing in the horizontal directions (between the exclusive

zones). Since most of the potential energy is released and absorbed in the zone 1, the portion of

overall internal energy (deformation work) of the system soared from 0.2 to 0.85 in the zone 1

(Figure 5-49). Sudden removal of columns resulted in localization of both energy release and

absorption.

20 20.5 21 21.5 220

0.2

0.4

0.6

0.8

1

Time [s]

Rela

tive

Ener

gy


Figure 5-49. Normalized energy allocation in the inclusive zones in CASE A

The contribution of columns, beams and slabs in energy absorption is described in Figure

5-50. Beams in the y-direction played a major role in energy absorption. On the other hand,

column contribution dropped after initial energy inflow, as confirmed by the spike in the B2.1

column energy rate. The system was able to redistribute and absorb the released energy such

that work done on the critical columns did not exceed buckling capacity. Therefore, the global

structure reached steady and stable energetic state.

176

20 20.5 21 21.5 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsYY

slabs

Figure 5-50. Energy split between members in zone 1 (CASE A)

CASE B. Columns A1.1 and B1.1 removed (Load = 1.2 D + 1.5 L)

Applied loading was increased by 0.5 L in the next scenario. The objective was to analyze

the given building at hand under increasingly unfavorable loading conditions. Such investigation

was aimed at capturing the transition from stable collapse arrest to unstable states and total

failure. In spite of the loading increase, collapse was also arrested in CASE B (Figure 5-51).

However, vertical displacements in the bays with removed columns were larger than in CASE A.

Figure 5-51. Arrested collapse of three-dimensional model w/ typical slabs in CASE B

177

Global energies are depicted in Figure 5-52. Collapse propagation was also avoided at this

case at approx. 1 [s] after the two columns at the ground floor were removed. Kinetic energy

shrank to the low vibratory level, and bays affected by the loss of two supporting columns

oscillated slightly. It is expected that material and structural damping would damp out these

background vibrations.

20 20.5 21 21.5 220

0.5

1

1.5

2

2.5

3x 10

4

Time [s]

Ener

gy [k

ip-in

]


Figure 5-52. Global energies in CASE B

Internal energies in the building columns are shown in Figure 5-53. Although the global

collapse was arrested, columns B2.1, A2.1, A2.2 exceeded the buckling energy threshold. Thus,

these columns entered the post-buckling column state. Columns B2.1, A2.1 and A2.2 still

retained sufficient residual capacities, which enabled them to support the applied loads.

However, buckled columns are not acceptable as long term load carrying members. Thus if the

energy levels shown in Figure 5-53 came from sensors monitoring a catastrophic, abnormal

event, columns: B2.1 and A2.1, A2.2 would be identified as in need of retrofit or replacement if

possible.

178

20 20.5 21 21.5 220

20

40

60

80

100

120

140

160

180

200

Time [s]

Ener

gy [k

ip-in

]

C1.1C1.2C1.3D1.1A2.1A2.2

A2.1 A2.3B2.1B2.2B2.3C2.1

B2.1 C2.2C2.3D2.1A3.1A3.2A3.3B3.1B3.2B3.3C3.1

Figure 5-53. Internal column energies in CASE B

The energy based findings on the stability of columns B2.1 and A2.1 were verified with

the internal forces and displacement results shown in Figure 5-54. Both columns exhibited

kinematic instability after exceeding the bucking energy threshold. From a traditional buckling

perspective, the axial capacity of column B2.1 fell gradually and was accompanied by noticeable

irreversible deformations. Column A2.1 experienced significant plastic deformations, which

were in contrast to the stable behavior in Figure 5-45. Thus the effectiveness and robustness of

the energy based stability criterion was confirmed. Internal energy (deformation work)

combines both force and displacement information as opposed to the non-unique force or

interaction diagram criterions.

179

-800

-700

-600

-500

-400

-300

-200

-100

0

-0.4 -0.3 -0.2 -0.1 0

Forc

e [ki

p]

Displacement [in]


A

020406080

100120140160180200

-0.4 -0.3 -0.2 -0.1 0

Ener

gy [k

ip-in

]

Displacement [in]


B

forcebuckling

energybuckling

Figure 5-54. Energy flow in building with typical slabs in CASE B (arrested collapse): A) Axial force-displacement, B) Internal energy-displacement.

Energy rates in the columns are shown in Figure 5-55. Energy rate spike in the B2.1

column was manifested at approximately 0.2 [s] after the removal of two ground floor columns.

However, the value of approx. 850 [kip-in/s] (0.1 [MJ]) was well below the characteristic energy

180

rate of 11000 [kip-in/s] (1.24 [MJ]). Thus it was concluded that the building was not in danger

of total collapse, when the failure propagation was averted.

Figure 5-55. Energy rates in columns in CASE B

Energy absorption and redistribution in the predefined building zones (Figure 5-38) is

shown in Figure 5-56 and Figure 5-57. Internal energy increased significantly but primarily in

zone 1. The normalized energy allocation is shown in Figure 5-58. The energy absorbed in zone

1 rose significantly and the energy in the other exclusive zones (1-2, 2-3 and 3-4) remained

practically constant. Thus the corresponding normalized allocation in the overall system energy

soared to 0.85. Sudden removal of columns resulted in localization of energy discharge and

absorption. Energy was mainly absorbed by the beams in y-direction (Figure 5-59).

181

x 10

20 20.5 21 21.5 220

0.5

1

1.5

2

2.5

3

Time [s]

Ener

gy [k

ip-in

]


Figure 5-56. Energy propagation through the inclusive zones in CASE B

20 20.5 21 21.5 220

0.5

1

1.5

2

2.5x 10

Time [s]

Ener

gy [k

ip-in

]


410x

410x

Figure 5-57. Energy propagation through the exclusive zones in CASE B

182

20 20.5 21 21.5 220

0.2

0.4

0.6

0.8

1

Time [s]

Rela

tive

Ener

gy


Figure 5-58. Normalized energy allocation in the building in CASE B

20 20.1 20.2 20.3 20.4 20.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsYY

slabs

Figure 5-59. Energy split between members in zone 1 (CASE B)

183

CASE C. Columns A1.1 and B1.1 removed (Load = 1.2 D + 2.0 L)

Applied loading was increased by 0.5 L in the next scenario, resulting in a 2.0 Live Load

factor. Unlike in CASE A and CASE B, localized failure resulted in the total catastrophic

collapse (Figure 5-60). Columns A2.1 and B2.1 failed first and the collapse propagated

outwards from the bays with removed columns.

A2.1

Figure 5-60. Collapse sequence of three-dimensional model w/ typical slabs in CASE C

184


B2.1

185

Redistribution of global energies is shown in Figure 5-61. Initial localized damage spread

through the building and resulted in total catastrophic failure. The collapse sequence is

illustrated in Figure 4-22. Kinetic energy rose as the collapse propagation unfolded. Once the

whole building collapsed, kinetic energy finally diminished to the low vibratory level.

Significantly larger external work was done on the system as compared with the arrested

collapses in CASE A and CASE B. Global energy reached a stable state after the whole building

was taken down to the ground.

20 22 24 26 280

0.5

1

1.5

2x 10

6

Time [s]

Ener

gy [k

ip-in

]


Figure 5-61. Global energies in CASE C

Internal energies in the building columns are shown in Figure 5-62. Released gravitational

energy was not absorbed entirely by the building deformations and the structure progressed from

unstable to unstable energetic states until total catastrophic collapse. Conclusions from the

previous CASE A and CASE B were confirmed. Column B2.1 buckled first but its kinematic

instability was initially restrained by the adjacent members. Column buckling A2.1 followed

186

shortly with significant energy localization. Collapse spread around the perimeter of the bays

affected by the removal of two columns on the ground level.

20 20.5 21 21.50

50

100

150

200

250

300

350

400

Time [s]

Ener

gy [k

ip-in

]


B2.1

A2.1

B2.1

energybuckling

Figure 5-62. Internal column energies in CASE C

Energetic findings were verified against the internal forces and displacement results for the

columns B2.1 and A2.1 (Figure 5-63). The onset of instabilities was correctly identified by the

energy buckling criterion. It should be noted that axial force in the columns were accompanied

by strong and weak axis moments, thus the duration of instability development and axial loading

and re-loading was coupled with these internal forces (Figure 4-25 through Figure 4-27).

Plotting internal energies on the single plot enabled clear stability assessment of columns. In

order to obtain similar insights using internal forces and displacements, multiple normalized

187

plots with cross referenced data were needed. Moreover, oscillatory internal forces did not

provide insight into the level of member and structural safety for arresting collapse.

-800

-700

-600

-500

-400

-300

-200

-100

0

-2 -1.5 -1 -0.5 0

Forc

e [ki

p]

Displacement [in]

y


A

0

200

400

600

800

1000

1200

-2 -1.5 -1 -0.5 0

Ener

gy [k

ip-in

]

Displacement [in]

forcebuckling


energybuckling

B

Figure 5-63. Energy flow in building w/ typical slabs: CASE C (total failure): A) Axial force-displacement, B) Internal energy-displacement.

188

Energy rates in the columns are shown in Figure 5-55. An energy rate spike in the A2.1

column was evident at approximately 0.2 [s] after the removal of two ground floor columns. Its

value of approximately 17000 [kip-in/s] (1.8 [MJ]) exceeded the characteristic energy rate of

11000 [kip-in/s] (1.24 [MJ]) established from the analysis of the building with heavily reinforced

slabs. Thus monitoring of the energy rate in the columns, which exceeded the buckling energy

limit in several cases, proved to be effective in detecting the global structural instability.

Energy rate corresponding to the onset of global instability

Figure 5-64. Energy rates in columns in CASE C

Energy absorption and redistribution in the predefined building zones (Figure 5-38) is

depicted in Figure 5-56 and Figure 5-66. Internal energy increased initially in zone 1 only.

Once the collapse progressed, energy in zone 2 through 4 exhibited significant increases as well.

189

20 20.5 21 21.5 22 22.5 230

2

4

6

8

10

Time [s]

Ener

gy [k

ip-in

]


Figure 5-65. Energy propagation through the exclusive zones in CASE C

20 21 22 23 24 250

0.5

1

1.5

Time [s]

Rela

tive

Ener

gy


Figure 5-66. Normalized energy allocation in the building in CASE C

190

Internal energy (deformation work) localized in zone 1 initially. The zone 1 portion of the

global energy increased from 0.2 to 0.82 in less than 0.2 [s]. The building was unable to absorb

the released gravitational energy by deformations of beams in y-directions (Figure 5-67). Once

capacity of beams to absorb the energy was exhausted, excess of energy was redirected to the

columns. Once the column buckling propagated around the bays with removed columns, the

share of energy absorbed by the columns increased from 0.1 to 0.4. Since the columns were

unable to absorb the transferred energy, collapse spread through the building resulting, in total

catastrophic failure. Final energy redistribution among buckled and deformed members

approximately reverted to the original energy redistribution.

20 21 22 23 24 250

0.1

0.2

0.3

0.4

0.5

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsY

slabs

Figure 5-67. Energy split between members in zone 1 (CASE C)

Absorbed energy (deformation work) can be decomposed into elastic energy (potentially

recoverable) and plastic energy (dissipated as heat). Energy absorbed in columns is mainly

elastic up to the buckling onset (Figure 5-68 and Figure 5-69).

191

20 20.2 20.4 20.6 20.8 210

500

1000

1500

2000

2500

3000

Time [s]

Ener

gy [k

ip-in

]

410x

Internal Energy (Deformat n Work)ioElastic Energy

Figure 5-68. Elastic component of the absorbed energy in A2.1 column

20 20.2 20.4 20.6 20.8 210

0.2

0.4

0.6

0.8

1y

Time [s]

Nor

mal

ized

Ene

rgy

Deformation WorkElastic Energy

Elastic

Plastic

Figure 5-69. Normalized energy decomposition into elastic and plastic component

192

Elastic energy in columns accounted for more than 80% of the absorbed energy

(deformation work) in the pre-buckling phase. Once a column buckled and was failed by the

structural loads, its significant plastic deformations resulted in energy dissipation. Elastic energy

accounted for less than 5% of the total deformation work at the end of its failure. Energy

absorbed by beams was mainly plastic (Figure 5-70 and Figure 5-71). After collapse initiation,

elastic energy quickly dropped to approximately 10% of the total deformation work.

20 20.2 20.4 20.6 20.8 210

500

1000

1500

Time [s]

Ener

gy [k

ip-in

]


Plastic Energy

Figure 5-70. Elastic component of absorbed energy in A1.1 beam y-y

Energy in columns was in essence elastic up to the buckling onset. However, the energy

absorbed beyond the buckling initiation (approximately at time 20.5 [s]) was purely plastic and

thus irreversible. Although elastic energy, accumulated during the pre-buckling phase, could be

potentially retrieved, this was not the case, because the permanent load resting on columns

prevented any significant unloading. Energy absorbed in beams (Figure 5-70) was chiefly plastic

from the very beginning.

193

20 20.2 20.4 20.6 20.8 210

0.2

0.4

0.6

0.8

1

Time [s]

Nor

mal

ized

Ene

rgy


Plastic Energy

Elastic Energy

Figure 5-71. Normalized energy absorption in A1.1 beam y-y

Energy decomposition results from the simplified steel framed building were consistent

with the observations inferred from the seismic design steel building, discussed in the

verification section of this study (p. 217). Since only columns contained a significant amount of

elastic energy (more than 10%), and since columns accounted for less than 20% of total energy

absorption, it is safe to say that deformation work (internal energy) was mainly irreversible.

Usefulness of the Energy Buckling Limits

It has been shown that internal force histories are not very sensitive to the buckling

initiation and failure. On the contrary, internal energy (deformation work) significantly

increased after buckling. Buckled members were easily identified on the basis of energy results

only. To illustrate the superiority of the energy approach, both force and energy based

demand/capacity (D/C) ratios were aggregated in Table 5-3 as well as in Figure 5-72 and Figure

5-73 to facilitate the comparative analysis.

194

Table 5-3. Demand Capacity comparison Loading Scenario Buckling Demand/Capacity

Force Energy A2.1 column B2.1 column A2.1 column B2.1 column CASE A 1.2D+1.0L 0.68 0.87 0.82 0.98 CASE B 1.2D+1.5L 0.83 0.95 2.18 1.76 CASE C 1.2D+2.0L 0.84 1.01 106 99

0.680.83 0.840.87 0.95 1.01

0

0.5

1

1.5

2

2.5

3

CASE A CASE B CASE C

A2.1 columnB2.1 column

Figure 5-72. Buckling force demand/capacity ratios

0.82

2.18

0.98

1.76

0

0.5

1

1.5

2

2.5

3

CASE A CASE B CASE C

A2.1 columnB2.1 column

106 99

Figure 5-73. Buckling energy demand/capacity ratios

195

As the loading increased in the subsequent scenarios (CASE A through CASE C), the force

based demand capacity ratios (D/C) rose respectively. Whereas 0.87 was not sufficient to buckle

column B2.1 in CASE A, 0.83 was enough to initiate the buckling in column A2.1 in CASE B.

Moreover, only a slight difference of 0.01 separated a safe post-buckling state (A2.1 column) in

CASE B from the failure in CASE C. Thus force based D/C ratios provide unreliable

information on the structural safety after the occurrence of the localized damage.

Energy based D/C ratios were very sensitive to the buckling initiation. Internal energies

significantly rose, and the violation of the energy buckling criterion directly corresponded to the

onset of buckling (Figure 5-63). The decrease of the residual column capacity in the post

buckling phase was characterized by significant increase in the energy absorption (mainly

irreversible, plastic energy dissipated as heat). Pre-buckling, post-buckling and failure states

resulted in very distinctive energy states. The traditional force based approach lead to a very

minor difference in D/C between the cases and contradictory results (0.87 did not corresponded

to the buckled column in CASE A but 0.83 did in CASE B). On the contrary, the energy results

provided very clear results and direct insights into the collapse propagation (e.g. Figure 5-62).

196

Energy based Building Failure Limit

An energy based global stability criterion is established in this section. Although this

study has already established the energetic criterion for buckling detection, column buckling

does not always lead to column failure and collapse propagation. The internal energy

(deformation work) in the post-buckling phase can reach significant values because columns

retain residual resistance after buckling. Although buckling of a single column reduces internal

load carrying capacity, it does not automatically indicate the onset of building collapse.

Buckling is a necessary, but not sufficient, condition to trigger the progressive collapse.

A sufficient, energetic collapse trigger criterion is proposed and verified herein. Figure

5-74 depicts the parallel between post-buckling, residual column resistance in the displacement

controlled experiment and the internal energy of the column. Since displacements were

controlled, no kinetic energy was induced.

Should the permanent column load be at e.g. 60% of its buckling load, significantly more

than buckling work has to be done on the column before it irreversibly loses capacity to carry the

unending load. In other words, transient dynamic effects can cause temporary overloading and

result in internal energy (deformation work) increase of the column. After certain work (failure

limit) is done on the column, there is not enough residual capacity to support the permanent load.

In the post buckling phase, loss of load carrying capacity is irreversible to the point that no

possibility remains for the column to ever support the permanent load. Such an energetic state

has to trigger localized column failure. Krauthammer et al. (2004) have shown that column

failures trigger collapse propagation. Table 5-4 through Table 5-8 provide critical energy levels

for the given permanent loads as the fraction of the buckling loads. If for the given load, critical

energy is exceeded, the column has irreversibly lost the capacity necessary to support the

197

sustained load and will certainly fail. It should be noted that permanent load can exceed the

tributary area load from the preloading phase, as the excess load from the removed columns has

to be supported by the adjacent columns, if collapse is to be arrested. Thus the column load used

for the critical energy estimate shall be increased to account for the loss of the columns.

0 1 2 3 4 50

200

400

600

800

Displacement [in]

Forc

e [k

ip]

0 1 2 3 4 50

500

1000

1500

2000 Energy - Stefan Szyniszewski

Displacement [in]

Ener

gy [k

ip-in

]

Permanent Load

Failure Energy

Figure 5-74. Parallel of energy capacity with axial capacity of W12x58, 156 [in] column

198

Table 5-4. W12x58, 156 [in] column buckling results P/Pcr Buckling Force Internal Energy [kip] [kip-in] 1.00 603 66 0.95 573 188 0.90 544 207 0.85 513 296 0.80 482 322 0.75 452 471 0.70 422 552 0.65 392 656 0.60 362 832 0.55 332 972 0.50 302 1030



199



200

Analytical Solution of Elasto-Plastic Column Buckling

Although energy properties of a given column can be determined from finite element or

experimental analysis, it is also advantageous to provide a closed form solution when practical.

Such a theoretical solution establishes the foundation for the codification of energetic behavior

and additionally offers an alternative for the community of engineers unwilling or unable to

efficiently use finite element codes.

Weak axis buckling of the clamped column is investigated herein. Building columns are

restrained from rotations by beams and slabs, thus the clamped-clamped boundary conditions

correspond well to the real building restraints. Kinematics of the weak axis buckling is depicted

in Figure 5-75. Three plastic hinge regions are identified: top, center and bottom of the column.

Downward column motion induces bending moments in the flanges, shown as rectangles in

Figure 5-75. The rest of the column is treated as an extensible rod, carrying only axial forces.

θw

θ 4mx

4mx

4mx

4mx

tx

2L

θcos2L

2d

2L

d

2L

2d

⎟⎠⎞

⎜⎝⎛ −

AEPL θcos1

2

Initial configuration Rotation Shortening

flange

rod

w

Figure 5-75. Kinematics of column buckling

201

Column motion can be decoupled into rotation (producing bending in the hinges) and

shortening of both hinges and rods. The key variables for the equilibrium analysis are total top

displacement ( ) and lateral deflection of the column center ( ). Lateral displacement and the

total displacement (including rotation and shortening effects) are:

tx w

θθ sincos12

⋅⎟⎠⎞

⎜⎝⎛

⋅⋅

−⋅=EA

PLw (5-6)

( ) θθθ coscoscos1 ⋅⎟⎠⎞

⎜⎝⎛ ⋅

⋅+−⋅+= L

EAPLxx mt (5-7)

A = section area; E = Young modulus; P = axial force

Kinematic relationships enable the calculation of the strains and internal forces.

Rotation θ stretches and shortens some of the fibers in the hinges, hence inducing bending

resistance in the hinges. Bending strains are calculated using kinematic relationships between

rotation θ and hinge fiber stretches depicted in Figure 5-76.

bottom flange

top flange

y

θ

2d

∂

fb

Figure 5-76. Bending stretches in the hinge

202

Strains in the hinges, including both bending and axial shortening effects, are:

( )d

xyxy m

m2

4tan),,( 0 ⋅⎟

⎠⎞

⎜⎝⎛ −−⋅= θθθε (5-8)

θ0 = initial geometric imperfection (1/1500)

In order to find the deflections of the column for the given axial load P , equilibrium

equations have to be employed. First, elastic buckling is discussed to illustrate the concepts in

full theoretical depth. Elasto-plastic solution requires the use of spreadsheet software such as

Microsoft Excel, MathCAD, etc, and hence not all the conceptual steps can be shown in detail.

Internal axial force from integrations of stresses equals the externally applied load:

( ) f

bf

bf

m tdyd

xyEP 22

4tan

2

2

0 ⋅⋅⎟⎠⎞

⎜⎝⎛ −−⋅⋅−= ∫

−

θθ (5-9)

Since the column is symmetric, only half of the column is sufficient to derive the moment

equilibrium equations. A free-body diagram is shown in Figure 5-77.

θ

w

2M

1M P

P

Figure 5-77. Moment equilibrium

Moment equilibrium is given as:

21 MMwP = +⋅ (5-10)

203

Applying the kinematic and strain expressions (5-6) and (5-8) results in:

( ) f

bf

bf

m tdyyd

xyEwP 22

4tan2

2

2

0 ⋅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎠⎞

⎜⎝⎛ −−⋅⋅⋅=⋅ ∫

−

θθ (5-11)

Thus employing equilibrium, one arrives at the following system of two equations with

three unknowns: vertical displacement , rotation tx θ and axial force P :

( ) fft tbd

LEA

PLxEP 22coscoscos14

⋅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎠⎞

⎜⎝⎛ ⋅

⋅−−⋅−⋅−= θθθ (5-12)

( )f

f td

bE

EAPLP 22

12tan

2sincos12

30 ⋅⋅−

⋅=⋅⎟⎠⎞

⎜⎝⎛

⋅⋅

−⋅⋅θθ

θθ (5-13)

Incrementing the rotationθ , corresponding vertical displacement and axial forcetx P can

be found from the equilibrium equations. Force P can be easily extracted from (5-12), and

inserted into (5-13). Thus for the given rotationθ , finding vertical displacement reduces to

the root finding problem. A full force displacement relationship can be hence established by

repeating the above procedure for the set of rotations varying from 0 to the arbitrarily chosen cut-

off limit.

tx

Whereas the elastic solution provides interesting insights into the buckling phenomena,

plastic effects play an important role in the behavior of real columns. Hence an elasto-plastic

solution is required to describe force – displacement relation. Using equation (5-7), strains (5-8)

can be rewritten as:

( ) ( )d

LEA

PLxyPxy tt2coscoscos1

41tan),,,( 0 ⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎠⎞

⎜⎝⎛ ⋅

⋅−−⋅−⋅−−⋅= θθθθθθε (5-14)

The elasto-plastic, piecewise linear relationship depicted in Figure 5-78 was postulated to

describe material behavior.

204

-40

-30

-20

-10

0

10

20

30

40

-0.02 -0.01 0 0.01 0.02

Stre

ss [k

si]

Strain

Figure 5-78. Elasto-plastic material model

Hence equilibrium equations can be rewritten in their more general, integral form:

( ) f

bf

bf

tdyP 22

2

γεσ ⋅−= ∫−

(5-15)

( ) f

bf

bf

tdyyEA

PLP 22cos12

2

2

γεσθ⋅⋅⋅=⎟

⎠⎞

⎜⎝⎛

⋅⋅

−⋅⋅ ∫−

(5-16)

bf = flange width; tf = flange thickness; γ = plastic hardening coefficient = 1.25

Thus, employing equilibrium, again one arrives at two equations with three unknowns:

vertical displacement , rotation tx θ and axial force . A system of two nonlinear equations can

be solved for the given rotation

P

θ using commonly available engineering packages such as

MathCAD and so on. A full force displacement relationship can be thus obtained for the array of

chosen rotations varying from 0 to an arbitrary chosen cut-off limit.

Comparison of the analytically calculated answer for the W12x58 column with numerical

result obtained from LS-DYNA (see page 138) is shown in Figure 5-79 and Figure 5-80.

205

0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Forc

e [ki

p]

Displacement [in]

Analytical

Ls-Dyna

Figure 5-79. Force displacement of W12x58 column of 156 [in] height

0

20

40

60

80

100

120

140

160

180

200

0 0.1 0.2 0.3 0.4

Ener

gy [k

ip-in

]

Displacement [in]

AnalyticalLs-Dyna

Figure 5-80. Energy displacement of W12x58 column of 156 [in] height

206

Verification of Energy Approach on Realistic Steel Building

The real life steel building design, employed by Gupta and Krawinkler (2000), was used to

verify the proposed energy approach. Framing plan of the selected building is shown in Figure

5-81 and Figure 5-60. Steel profiles applied in the building are listed in Table 5-9 and Table

5-10. The perimeter of the frame contained moment resisting connections.

y

30 [f

t]

30 [ft]

2

3

4

5

A B C D E

1

F G


x

Figure 5-81. Three story, moment resisting framed building for verification analysis

Table 5-9. Steel profiles of columns (designations according to AISC, 2006) A B C D E F G 5 w12x58 w12x58 w14x74 w14x99 w14x99 w14x74 w12x58 4 w14x74 w12x58 w12x65 w12x72 w12x65 w12x58 w14x74 3 w14x99 w12x58 w12x65 w12x72 w12x65 w12x58 w14x99 2 w14x99 w12x58 w12x58 w12x58 w12x58 w12x58 w14x99 1 w14x74 w12x58 w14x74 w14x99 w14x99 w14x74 w14x74

207

30 [f

t]

30 [ft]

2

3

4

5

A B C D E

1

F G

Moment connection Penthouse

w21x44 w21x44 A A A w21x44

w21x68 w21x68 w21x68 w21x68 w21x68 w21x68

w21x68 w21x68 w21x68 w21x68 w21x68 w21x68

w21x68 w21x68 w21x68 w21x68 w21x68 w21x68

w21x44 w21x44 A A A w21x44

AA

A

w18

x35

AA

Aw

18x3

5

w18x35 (in-fill beams)

Figure 5-82. Framing plan used for SAC three story building

Table 5-10. Moment resistant beams (designated with “A”) Floor Beam “A” 2 w18x35 3 w21x57 roof w21x62

Only two representative cases are discussed herein in more detail (CASE 2 and CASE 3).

The remaining column removal scenarios, which further confirm the effectiveness and

robustness of the proposed energy based approach, are elaborated in Appendix B (page 249).

The energy analysis was applied to interpret the level of structural safety and compared with the

traditional, force based methods.

208

Two Columns Removed. CASE 2

Two columns on the ground floor (A1.1 and B1.1) were removed after the application of

static preloading (Load = 1.0 D + 0.5 L). Although collapse was arrested, significant plastic

deformations developed in the bays directly affected by column removal (Figure 5-83). Initially

induced kinetic energy was absorbed by the building, which reached a stable, energetic

configuration (Figure 5-84).

Figure 5-83. Deflected, final configuration of the building in CASE 2

Removed Columns

20 21 22 23 24 250

0.5

1

1.5

2x 10

4

Time [s]

Ener

gy [k

ip-in

]


Figure 5-84. Global Energies in CASE 2

209

Internal energy (deformation work) of the analyzed columns is depicted in Figure 5-85.

None of the columns exceeded the buckling energy threshold. Hence none of the columns

reached the post-buckling phase, characterized by the irreversible loss of load carrying capacity.

However, column B2.1 (W12x58 section) absorbed more energy than other columns. Thus the

dynamic, load redistribution mostly affected column B2.1.

20 20.5 21 21.5 22 22.50

50

100

150

Time [s]

Ener

gy [k

ip-in

]

C1.1C1.2C1.3D1.1A2.1A2.2A2.3B2.1B2.2B2.3C2.1C2.2C2.3D2.1A3.1A3.2A3.3B3.1B3.2B3.3w12x58w14x74w14x99

w14x99

w14x74

w12x58

B2.1 (w12x58)

Figure 5-85. Internal column energies in CASE 2

Internal forces in columns B2.1 and A2.1 (for comparison) are depicted in Figure 5-86 and

Figure 5-87. In addition to moderate axial loading (60% of the buckling load), Column B2.1 was

subjected to significant strong axis moments. It is a formidable task to estimate level of safety

associated with column B2.1 on the basis of internal forces time histories. It appeared that

210

column B2.1 was capable of withstanding the combination of the significant axial load and

strong axis bending moment.

-1-0.8

-0.6

-0.4-0.2

0

0.20.4

0.6

0.81

0 5 10 15 20

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

y


A

S

T

-1-0.8

-0.6

-0.4

-0.2

0

0.20.4

0.6

0.8

1

0 5 10 15 20

Nor

mal

ized

Mom

ent o

r For

ce

Time [s]


S

T

B

Figure 5-86. Internal forces in CASE 2: A) column A2.1, B) column B2.1.

211

-1-0.8

-0.6

-0.4

-0.2

0

0.20.4

0.6

0.8

1

20 20.5 21 21.5 22

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

y


A

-1-0.8

-0.6

-0.4

-0.2

0

0.20.4

0.6

0.8

1

20 20.5 21 21.5 22

Nor

mal

ized

Mom

ent o

r For

ce

Time [s]


B

S

T

Figure 5-87. Internal forces in the dynamic phase. CASE 2: A) A2.1 column, B) B2.1 column.

Loading and unloading of columns A2.1 and B2.1 is shown in Figure 5-88. Internal

energies, corresponding to the external work employed to deform the columns, did not exceed

212

the predefined buckling energy thresholds. Since column B2.1 consisted of a relatively weaker

section, the corresponding safety margin was lower than that of column A2.1.

-1200

-1000

-800

-600

-400

-200

0

-0.15 -0.1 -0.05 0

Forc

e [ki

p]

Displacement [in]

A2.1B2.1A2.1 Buckling ForceB2.1 Buckling Force

A

0

20

40

60

80

100

120

140

-0.15 -0.1 -0.05 0

Ener

gy [k

ip-in

]

Displacement [in]

A2.1B2.1A2.1 Buckling EnergyB2.1 Buckling Energy

B

w12x58

w14x99

Figure 5-88. Energy absorption in the selected columns. CASE 2: A) Axial force-displacement, B) Internal energy-displacement.

213

Energy rates in the columns (Figure 5-89) indicated that energy was pushed toward

columns B2.1 and B3.1.

Figure 5-89. Energy rates in columns in CASE 2

Energy propagation through the predefined building zones (Figure 5-90) is depicted in

Figure 5-91 and Figure 5-92. Energy of the building increased after column removal to absorb

the released gravitational energy. Internal energy (deformation work) in the exclusive zones 2-3,

3-4 and 4-5 remained practically constant. Thus no energy was transferred to these zones from

the zones affected by the column removals. Normalized energy allocation (Figure 5-93) has

shown energy localization in the inclusive zone 0-2, as the global energy share of this zone rose

from 20% to 70%. There was in essence no energy slosh (energy flow between zones) from the

inclusive zone 0-2 to zone 2-5.

214

2

3

4

5

A B C D E F G

1

Zone 1

Zone 2

Zone 3

Zone 4Zone 5

Figure 5-90. Building zones used to trace the energy propagation

20 20.5 21 21.5 22 22.5 230

500

1000

1500

2000

2500

3000

Time [s]

Ener

gy [k

ip-in

]

zone 0-1zone 1-2zone 2-3zone 3-4zone 4-5

Figure 5-91. Energy propagation through the exclusive zones in CASE 2

215

20 20.5 21 21.5 22 22.5 230

1000

2000

3000

4000

5000

6000

Time [s]

Ener

gy [k

ip-in

]


Figure 5-92. Energy propagation through the inclusive zones in CASE 2

20 20.5 21 21.5 22 22.5 230

0.2

0.4

0.6

0.8

1

Time [s]

Rela

tive

Ener

gy


Figure 5-93. Normalized energy allocation in the building in CASE 2

216

Energy absorbed (deformation work) in beams was mainly plastic (Figure 5-94) and thus

mainly irreversible (dissipated). Since none of the columns buckled, energy absorbed

(deformation work) in columns remained chiefly elastic (Figure 5-95). Although this elastic

energy could be potentially recovered from the system, there was relatively low variation in the

column energy levels. It should be noted that permanent loads are present on the columns at all

times. These non-transient loads effectively prevent any significant unloading and, thus, any

elastic energy release.

Beams in the y-direction played a major role in arresting the collapse and absorbing the

initially released potential energy (Figure 5-96). Beams in the x-direction aided the energy

absorption at approximately 20.5 [s]. Relative energy absorbed by beams in the y-direction rose

significantly in comparison to those of beams in the x-direction and columns. Most of the

energy was absorbed by beams.

20 20.5 21 21.5 22 22.5 230

50

100

150

200

250

300

350

400y

Time [s]

Ener

gy [k

ip-in

]

Absorbed EnergyElastic Energy

Plastic (Irreversible) Energy

Figure 5-94. Decomposition of the absorbed energy (deformation work) in A1.1 beam y-y

217

20 20.5 21 21.5 22 22.5 230

20

40

60

80

100

Time [s]

Ener

gy [k

ip-in

]


Figure 5-95. Decomposition of the absorbed energy (deformation work) in B2.1 column

20 20.5 21 21.5 22 22.5 230

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsY

Figure 5-96. Energy split between members in zone 2 (CASE 2)

218

Three Columns Removed. CASE 3

Three columns at the ground floor (A1.1, A2.1 and B1.1) were removed to induce a

catastrophic collapse (under Load = 1.0 D + 0.5 L). Global energies are depicted in Figure 5-97.

Significant levels of kinetic energy were observed in the building, and internal energy

(deformation work) of the system increased by more than 2000%. Kinetic energy began to

diminish as bays with removed columns impacted the ground layer.

20 21 22 23 24 250

1

2

3

4

5

6

7

8

9x 10

4

Time [s]

Ener

gy [k

ip-in

]


Figure 5-97. Global energies in the building

Although column B2.1 failed, only partial collapse was observed in the building (Figure

5-98). Column buckling resulted in the collapse of four bays. Shear connections between the

A3.1, A3.2 and A3.2 columns and the adjacent beams and slabs failed. Therefore, collapse

propagation in the x-direction was halted. On the other hand, strong perimeter columns in the

moment resisting “A” line were sufficiently robust to withstand the demands from the falling

219

bays. Once bay A1 impacted the soil layer, demand on the adjacent bays dropped and the

building achieved a final, stable state.

B2.1

Figure 5-98. Collapse sequence of the steel building in CASE 3

220

2

3

4

5

A B C D E F G

1

Zone 1

Zone 2

Zone 3

Zone 4Zone 5

Figure 5-99. Building zones used to trace the energy propagation

Energy in the predefined zones (Figure 5-100) significantly rose in zone 0-2. A stable

configuration was reached after 21.6 [s]. System energy was reduced after partial collapse was

arrested upon impact with the ground and failed elements were removed from the simulation.

On the other hand, there was no significant increase of the absorbed energy in zone 2-5.

Therefore, energy absorption localized in inclusive zone 0-2. In essence there was no horizontal

energy sloshing (repetitive energy movement in-between zones) in the building. Normalized

energy allocation (Figure 5-101) confirmed that energy was localized in zone 0-2. Energy

absorbed in zone 0-2 accounted for 90% of the internal building energy, as opposed to 18%

when three columns were removed. After the partial collapse, the building reached a stable

energy configuration.

221

20 20.5 21 21.5 22 22.5 230

2000

4000

6000

8000

10000

12000

14000

16000

18000gy y

Time [s]

Ener

gy [k

ip-in

]


Figure 5-100. Energy propagation through the exclusive zones in CASE 3

20 20.5 21 21.5 22 22.5 230

0.2

0.4

0.6

0.8

1

Time [s]

Rela

tive

Ener

gy


Figure 5-101. Normalized energy allocation in the building in CASE 3

222

Allocation of energy among main structural members is illustrated in Figure 5-102.

Mainly beams in the y-direction absorbed released gravitational energy as the respective relative

energy share rose. Upon column B2.1 buckling, column participation in the energy absorption

increased too, providing temporary relief for the beams. Overall, beams absorbed most of the

released energy in all analyzed cases.

20 20.5 21 21.5 22 22.5 230

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsY

Figure 5-102. Energy split between members in zone 2 (CASE 3)

Energy absorbed in beams was mainly plastic ( Figure 5-103 and Figure 5-104) and thus

chiefly impossible to recover (dissipated). Energy absorption in the failed B2.1 column, (Figure

5-106 and Figure 5-105) was principally elastic until the buckling onset. After the buckling

initiation, absorbed energy rose significantly due to significant plastic deformations. Energy

absorption in the buckled (but not failed) C2.1 column (Figure 5-107) exhibited significant levels

of elastic energy (40%). Since this column did not lose its load carrying capacity, the non-

transient loads effectively prevented unloading and energy sloshing.

223

20 20.5 21 21.5 22 22.5 230

200

400

600

800

1000

1200

Time [s]

Ener

gy [k

ip-in

]


Plastic Energy

Figure 5-103. Decomposition of the absorbed energy (deformation work) in A2.1 beam y-y

20 20.5 21 21.5 22 22.5 230

50

100

150

200

250

300

350

400

Time [s]

Ener

gy [k

ip-in

]


Plastic Energy

Figure 5-104. Decomposition of the absorbed energy (deformation work) in B1.1 beam x-x

224

200

20 20.5 21 21.5 22 22.5 230

50

100

150

Time [s]

Ener

gy [k

ip-in

]


Plastic Energy

Figure 5-105. Close-up view of the energy (deformation work) decomposition in B2.1 column

20 20.5 21 21.5 22 22.5 230

0.2

0.4

0.6

0.8

1

Time [s]

Nor

mal

ized

Ene

rgy


Plastic Energy

Figure 5-106. Energy decomposition of the absorbed energy (deformation work) in B2.1 column

225

200

20 20.5 21 21.5 22 22.5 230

50

100

150

Time [s]

Ener

gy [k

ip-in

]


Plastic Energy

Elastic Energy

Figure 5-107. Decomposition of the absorbed energy (deformation work) in C2.1 column

Although the elastic energy in column C2.1 could be potentially recovered from the

system, there was relatively low variation in the column’s energy level. It should be noted that

permanent loads weigh down upon columns at all times. These non-transient loads effectively

prevent any significant unloading and, thus, any elastic energy release. Energy absorbed by a

column significantly rose after the buckling initiation. Therefore the kinematic column

instability resulted in a very distinctive change in the energy domain. Further column failure

resulted in a very significant additional increase of internal energy (deformation work). Hence,

column transition from safe pre-buckling to post buckling and failure behavior is very apparent

in the energy domain. On the other hand, axial force capacity is bounded by the buckling force.

Significant transitions in the column behavior correspond to minor variations in the dynamic

force time histories. Therefore, it is very challenging to understand the building behavior (after

column(s) removal) on the basis of force results only.

226

Analysis of internal column energies revealed that internal energy (deformation work) in

the B2.1 column exceeded the respective buckling energy threshold (Figure 5-108). Second and

third story B2.2 and B2.3 columns were also overloaded. However, when column B2.1 buckled,

columns B2.2 and B2.3 were relieved. Column A3.1 played an important role in preventing the

collapse propagation. Although it absorbed more energy than other columns, the respective

buckling energy capacity was not exceeded. On the other hand, column C2.1, consisting of the

weaker W12x58 section, was affected by the slabs tearing. Also, column C2.1 exceeded the

respective buckling energy threshold.

20 20.5 21 21.5 22 22.50

50

100

150

200

250

Time [s]

Ener

gy [k

ip-in

]

C1.1C1.2C1.3D1.1B2.1B2.2B2.3C2.1C2.2C2.3D2.1A3.1A3.2A3.3B3.1B3.2B3.3w12x58w14x74w14x99

B2.1 (w12x58)

C2.1 (w12x58)

A3.1 (w14x99)

w12x58

w14x99

Figure 5-108. Internal column energies in CASE 3

Internal forces in columns B2.1 and C2.1 (both made of W12x58 sections) are compared in

Figure 5-109. Both columns carried practically the same axial and bending loads. Comparison

227

of the dynamic phases (Figure 5-110) showed that dynamic loads in column C2.1 nearly reached

the axial buckling capacity. This significant axial loading was accompanied by considerable

strong axis bending moments but the column did not fail.

-1.0

-0.5

0.0

0.5

1.0

1.5

0 5 10 15 20

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]


B

-1.00

-0.50

0.00

0.50

1.00

1.50

0 5 10 15 20

Nor

mal

ized

Mom

ent o

r For

ce

Time [s]


A

S

T

S

T

Figure 5-109. Normalized internal forces in: A) B2.1 column, B) C2.1 column

228

-1.0

-0.5

0.0

0.5

1.0

1.5

20 20.5 21 21.5 22

Nor

mal

ized

Mom

ent a

nd F

orce

Time [s]

S

T


A

-1.00

-0.50

0.00

0.50

1.00

1.50

20 20.5 21 21.5 22

Nor

mal

ized

Mom

ent o

r For

ce

Time [s]


B

S

T

Figure 5-110. Normalized internal forces in dynamic phase: A) B2.1 column, B) C2.1 column

Unlike ambiguous internal force time histories, energetic analysis explained why column

C2.1 did not fail. An energy based approach also provided information on the level of structural

safety. Non-transient, permanent loads resting on columns B2.1 and C2.1 were estimated.

229

Loading and unloading paths are shown in Figure 5-111. Internal energy of column B2.1

exceeded the failure energy threshold. Thus the load carrying capacity was reduced so severely

that no stable configuration (supporting the permanent load) was possible.

-700

-600

-500

-400

-300

-200

-100

0

100

-2.0 -1.5 -1.0 -0.5 0.0

Forc

e [ki

p]

Displacement [in]

y

B2.1C2.1w12x58 Buckling ForceB2.1 permanent loadC2.1 permanent load

A

0

100

200

300

400

500

600

700

800

-2.0 -1.5 -1.0 -0.5 0.0

Ener

gy [k

ip-in

]

Displacement [in]

B2.1C2.1Buckling EnergyB2.1 FailureC2.1 Failure

B

Figure 5-111. Energy flow in the SAC building: CASE 3 (total failure): A) Axial force-displacement, B) Internal energy-displacement

230

Loss of the column load carrying capacity in the post-buckling phase is irreversible. Thus

the more energy a given column absorbs, the smaller the residual capacity becomes for resisting

permanent loads in the post-buckling phase. Therefore, tracing the internal column energy

provides not only information on the column buckling but also on the corresponding safety level.

Although column C2.1 exceeded the buckling energy threshold, the work in the post buckling

phase did not bring the column to the verge of failure. On the contrary, there was a significant

energetic reserve of approximately 450 [kip-in] (50 [kJ]) in the C2.1 column.

Energy rates in columns (Figure 5-112) showed the energy flow localization (“energy

sinkhole”) in the column B2.1. Thus energy rates provided complementary information on the

global level of structural safety. Only column B2.1 violated the failure energy rate threshold.

Figure 5-112. Energy rates in columns in CASE 3

231

Summary

Energy plots reveal more information than can be seen through examination of the

deformed structure. For example, in the total collapse in CASE 3, bucking of the column B3.1

was really hard to notice in the simulation unless the proper perspective was chosen. Column

B3.1 buckling was effectively restrained by the neighboring members and the column did not

completely fail. However, a left side view evidently disclosed that B3.1 column buckled.

Moreover, energy plots revealed competing failure modes A2.1, A2.2 and A2.3 as well as

C1.1, C1.2 and C1.3. Each of the 2nd and 3rd story columns could buckle before the 1st story

column if random imperfections diverted the energy redistribution in such a way that the 2nd or

3rd story columns were overpowered first. Thus, any improvement or cross-section change of the

columns located at A2 or at C1 should be applied thorough all three stories.

Analysis of internal energies not only confirmed kinematic findings, but in fact revealed

more information. Buckling sequence and propagation was assessed on a single time history

plot. Furthermore, members competing for failure and energy redistribution were identified. In

simple terms, analysis of internal energies provided a simple yet robust tool to understand and

analyze building response to the abnormal loading.

Usefulness of the Energy Buckling and Energy Failure Limits

It has been shown that internal force histories are not very sensitive to the buckling

initiation and failure. On the contrary, internal energy (deformation work) significantly

increased after buckling. Buckled members were easily identified on the basis of energy results

only. To illustrate the superiority of the energy approach, both force and energy based

demand/capacity (D/C) ratios were aggregated in Table 5-11 as well as in Figure 5-113 through

Figure 5-115 to facilitate the comparative analysis.

232

Table 5-11. Demand capacity (D/C) ratios Loading Scenario Demand/Capacity Ratios

Buckling Force Buckling Energy Failure Energy B2.1 C2.1 B2.1 C2.1 B2.1 C2.1 CASE 1 0.61 0.59 0.34 0.31 0.034 0.032 CASE 2 0.69 0.58 0.43 0.30 0.051 0.037 CASE 3 0.94 0.93 87 1.72 11 0.21

As more columns were removed in the subsequent scenarios (one column in CASE A

through three columns in CASE C), the force based demand capacity ratios (D/C) rose

respectively. Whereas a D/C ratio of 0.93 was not sufficient to fail column C2.1 in CASE 3, a

ratio of 0.94 was enough to initiate the failure of column B2.1 in the same CASE. Only a slight

difference of 0.01 separated safe post-buckling state (C2.1 column) from the failure in CASE C.

Thus force based D/C ratios provide results, which are very difficult to interpret and translate

into a level of structural safety after the occurrence of the localized damage.

Energy based D/C ratios were very sensitive to the buckling initiation. Internal energies

significantly rose, and the violation of the energy buckling criterion directly corresponded to the

onset of buckling (Figure 5-111).

0.61 0.690.94

0.59 0.580.93

0.00

0.50

1.00

1.50

2.00

2.50

3.00

CASE 1 CASE 2 CASE 3

B2.1 columnC2.1 column

Figure 5-113. Buckling force demand/capacity ratios

233

0.34 0.430.31 0.30

1.72

0.00

0.50

1.00

1.50

2.00

2.50

3.00



Figure 5-114. Buckling energy demand/capacity ratios

0.034 0.0510.032 0.0370.21

0.00

0.50

1.00

1.50

2.00

2.50

3.00



buckling limit

11

failure limit

87

Figure 5-115. Failure energy demand/capacity ratios

Both columns C2.1 and B2.1 buckled in CASE 3, and the buckling was easily identified by

a comparison of the absorbed energy with the energy buckling limit. However, only the failed

column B2.1 violated the energy collapse criterion. Column C2.1 absorbed only 21% of the

234

energy needed to reduce its capacity below the non-transient level of loading. Thus column C2.1

was deemed as buckled but safe in the energy domain.

Decrease of the residual column capacity in the post buckling phase was characterized by a

significant increase in energy absorption (mainly irreversible, plastic energy). Pre-buckling,

post-buckling and failure states manifested themselves in very distinctive energy states. The

traditional force based approach lead to very minor difference in D/C between the discussed

cases. The energy results provided very transparent results and direct insights into the collapse

propagation.

235

CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS

Summary

The main objective of this study was to develop and implement a rational, energy-based

approach to progressive collapse of steel framed buildings by assessing individual members and

full structural behavior focusing on the role of energy flow in these phenomena. This was

accomplished as follows:

1. Energy propagation during progressive collapse was described and explained on both global and local levels,

2. Column buckling and failure criteria in terms of internal energy have been derived for single columns. The criteria were verified in full scale simulations and described with closed-form analytical expressions,

3. An energy-based approach for the collapse of moment resisting steel buildings was developed and verified. It has been proven that column energy histories provide an effective interpretation tool, which gives accurate insight into dynamic energy redistributions, dynamic alternate paths reformations, competing failures, and structural safety. It has been shown that the energy-based approach surpasses traditional analysis methods through an accurate assessment of progressive collapse propagation.

Progressive Collapse Conclusions

Physics-based simulation techniques were used to investigate the behavior of major

structural members during progressive collapse propagation or arrest. A simplified steel framed

building and a more complex three story seismic designed steel building were selected for

comparative analysis. These comparative simulations have shown that only fully nonlinear

dynamic time history analysis of sufficiently detailed models, which account for material and

geometric nonlinearities, can lead to meaningful results. Progressive collapse is characterized by

large displacements, strains and pronounced dynamic effects. Therefore, resorting to simplified

techniques, which do not include realistic structural and material models and inertial effects of

236

dead and live loads (mass of slabs, mechanical installations, desks, office equipment, etc), would

certainly lead to erroneous results and misleading conclusions.

Collapse simulation must adhere to the basic laws of classical physics. In the simulated

events, one or more columns were removed without additional loads (explosion, truck impact,

etc). Structural elements must not fall vertically faster than at free fall. Structural resistance and

energy absorption of resisting members slow down such motion. Only the model with load

application as a lumped mass (instead of a pressure load) into the modeled slabs adhered to the

“free-fall” requirement. This demonstrated that neglecting the inertia of dead and live loads

resulted in significant errors and violation of the free-fall physics. Thus, accounting for inertial

effects plays a crucial role in the proper simulation of structural response to the localized failure.

It is widely held that tying the structure provides ductility and increases resistance to

progressive collapse. However, if the initiating damage is beyond the arresting capabilities of the

structure, due to superbly strong structural ties, the collapse will propagate through the whole

structure, and will lead to total collapse. Such cases have been demonstrated in this study for the

simplified steel framed building (with moment connections only). Whereas the removal of one

corner column was arrested by the structures easily, the removal of two columns led to

catastrophic collapse in some of the analyzed cases. In contrast, the three story building with

seismic detailing (with both moment and shear connections), exhibited only partial collapse after

the removal of three columns. Shear connections prevented the collapse from further

propagation into the building interior.

Tying the structure is important in providing adequate robustness to prevent catastrophic

collapse, provided that structural ties are properly designed. Reliance on structural ties can be

beneficial, if a building has sufficient capability to redistribute loads. However, the addition of

237

ductility must be accompanied by an analytical examination of building resistance to progressive

collapse, and must not be treated as a cure-all. Strengthening structural ties, without structural

examination, can lead to worse consequences because strong connections might cause total

collapse as opposed to partial collapse limited by connection failures.

It is strongly encouraged to exhaust all analysis tools to sufficiently design connections,

such that foreseeable localized failures can be arrested and occupants’ lives saved. However,

one must recognize that not all abnormal loadings can be predicted in advance. Therefore, it

may be beneficial to partition a building, such that collapse cannot spread throughout the whole

building. Such an approach ensures that, in the worst case scenario, only a portion of the

building collapses as opposed to the entire building.

Energy Conclusions

Internal energy (deformation work) was localized in the bays affected by the column(s)

removal in the instances of arrested collapse. Analysis of the energy flow through the building

revealed that beams play a crucial role in the absorption of the released gravitational energy. For

the cases of arrested collapse, beams accounted for approximately 70% of the energy absorption,

whereas columns and slabs for only 30%. For the instances of total collapse, the energy

absorbed during column buckling increased the contribution of columns in the total energy

absorption. Thus, more energy was absorbed by the columns during the collapse propagation

phase. The released potential energy was mainly dissipated by plastic deformations (beams,

slabs and column post-buckling behavior). Elastic energy was stored primarily in columns,

which accounted for less than 20% of the total absorbed energy. In principle the stored, elastic

energy was not released because the non-transient column loads prevented any significant

unloading.

238

The energy absorption of the beams was shown to be an important factor in arresting

collapse. This conclusion agrees well with findings on the collapse sensitivity to span lengths

and slab strengths. Whereas, a noteworthy increase in the slab stiffness did not avert the

collapse, a 10% change of the bay spans increased significantly the resistance to progressive

collapse. Beams (sensitive to span variations) accounted for up to 70% of the energy absorption,

whereas, slabs and columns accounted for up to 30%.

The energy buckling limit was proposed as a necessary condition to initiate the collapse

(but it is not a sufficient). The column failure energy was introduced and verified as the

sufficient collapse criterion. A buckled column must be able to carry permanent load (slab

weight, etc.) after the transient effects pass. However, should an extensive amount of work be

done on the column, its load carrying capacity will be irreversibly reduced below the value of the

permanent load. Exceeding this energy threshold for the axial forces caused by permanent loads

means that the column will fail. Therefore, in the case of an arrested collapse, comparison of the

energy absorbed by the buckled column to the respective column failure energy enables one to

evaluate the building safety.

Buckling energies are characteristic values of columns, and failure energy limits are

fundamental properties of a structure (dependent both on column properties and on a value of

permanent load). Both buckling energy and column failure energy can be conveniently

computed beforehand, using the numerical (LS-DYNA) and/or analytical (closed-form)

calculation procedures proposed in this study.

A comparison of the force demand to the member capacity is traditionally employed to

evaluate a member’s safety. It has been shown in this study that force based demand capacity

(D/C) is not very sensitive to the fundamental changes in structural behavior. Conversely,

239

buckling energy D/C values correctly identified buckling in all analyzed cases in a very

distinctive manner. Moreover, the energy failure D/C criterion was violated only by failed

columns. In the case of arrested collapse, comparing the energy stored (deformation work) in a

given column to the failure limit enabled direct evaluation of the column safety.

Recommendations

It is recommended that further research be carried out on the following subjects:

• More cases (i.e. more building types and structural configurations) to confirm the applicability of the proposed approach to a large variety of structural systems,

• Investigation of the impact of secondary structural members such as: walls, partitions, sliding objects, etc. on collapse behavior,

• Use of energy D/C values for a comparative analysis of alternative designs such that the safest structural solution can be chosen,

• Research the role of beams and structural connections on energy flow and redistribution during a collapse,

• Designing for energy flow paths during collapse (e.g. by the use of connection properties or damping devices) to maximize structural resistance to progressive collapse.

240

APPENDIX A

VERIFICATION OF ENERGY EXTRACTION PROCEDURE

The energy flow of a simple 2-D frame was analyzed to verify the energy extraction

procedure. It was shown in this study (see page 129) that only the sufficiently detailed three

dimensional models provided meaningful results. Therefore the two dimensional case was

investigated only to verify the ability of the finite element software LS-DYNA to properly

extract and report the energy flow results.

The front façade of the simplified steel framed structure (Figure 4-1) was monotonically

loaded at frame joints with concentrated forces (Figure A-1). Twelve concentrated forces

provided easy to track loading force-time histories. Joint displacement time histories were

written in the LS-DYNA NODOUT displacement output file at every 1 [ms].

Figure A-1. Steel frame used for the energy benchmark test

The loading was applied in two phases: 1) static preloading (over 20 [s]); 2) collapse

propagation (after the static preloading, initiated by the instantaneous corner column removal).

After the removal of the single, corner column, frame collapsed partially (Figure A-2). Plastic

hinges were formed in the beam supports but collapse did not propagate to the adjacent bays.

241

Figure A-2. Final, displaced shape of the 2-D frame used in the energy benchmark test

Global frame energies were extracted from the LS-DYNA GLSTAT text file. The increase

of the internal energy (deformation work) reached its final level when the collapse was arrested

(Figure A-3). Kinetic and internal energies added up to the external work done on the system.

Thus conservation of energy was clearly satisfied in the LS-DYNA simulation.

Figure A-3. Global energies reported by LS-DYNA in GLSTAT file

The kinetic energy was rising until the collapsing beams impacted the ground. Once

members decelerated, the kinetic energy dipped to zero. Thus the instable states were

242

characterized by the significant kinetic energy levels. A stable state was achieved when the

global kinetic energy vanished.

Global energy results (GLSTAT) provided information on the external work done on the

system, the internal energy (caused by the deformations of the structural members), and the

kinetic energy (caused by the motions). Since the force time histories were prescribed in

simulations and were explicitly known, external work was calculated analytically. Displacement

time histories from the simulation were retrieved from the nodal NODOUT output file. Thus,

the external work was analytically computed by the integration of the known force-displacement

histories in each joint and adding them up to obtain the global external work at each instance of

time. The comparison of the calculated and LS-DYNA external work results is shown in Figure

A-5 and Figure A-4. Excellent agreement between the analytical and the numerical curves

verified the ability of LS-DYNA to correctly provide global energy time histories.

0 5 10 15 20 250

2000

4000

6000

8000

10000

12000

14000

Time [s]

Exte

rnal

Wor

k [k

ip-in

]

External Work - AnalyticalExternal Work - GLSTAT

Figure A-4. Analytical and numerical external work results

243

0 5 10 15 200

5

10

15

20

Time [s]

Exte

rnal

Wor

k [k

ip-in

]

External Work - AnalyticalExternal Work - GLSTAT

Figure A-5. Analytical and numerical external work results during the static preloading phase

Global energy results were useful in assessing whether the structure reached a stable

energy state. Tracing the global kinetic energy enabled determination of the proper termination

time. Once the global kinetic energy vanished, the simulation was safely terminated because the

structure has reached its final energetic equilibrium state. In general collapse can be arrested,

partial or the whole structure can collapse. In all instances the final values of the global energies

stabilize and kinetic energy fades away.

Whereas, the global energies proved useful, the distribution of the local internal and kinetic

energies was of the primary importance in this research. LS-DYNA MATSUM ASCII output

file was used to obtain local energies for all structural members, such as: individual columns,

beams, joists and slabs. The sum of all local energies from MATSUM shall equal the global

energies from GLSTAT. The comparison of the summed external work, kinetic and internal

energies from MATSUM with global GLSTAT results is shown in Figure A-6 and in Figure A-7.

244

0 5 10 15 200

5

10

15

20

25

30

35

40

45

50

Time [s]

Ene

rgy

[kip

-in]

Internal Energy - GLSTATKinetic Energy - GLSTATExternal Work - GLSTATInternal Energy - MATSUMKinetic Energy - MATSUMExternal Work - MATSUM

Figure A-6. Global (GLSTAT) and sum of local energies (MATSUM) during the static preloading

0 5 10 15 20 250

2000

4000

6000

8000

10000

12000

14000

Time [s]

Ene

rgy

[kip

-in]


Figure A-7. Comparison of global (GLSTAT) and sum of local energies (MATSUM)

245

The sums of the individual energies obtained from the MATSUM were in the excellent

agreement with the global energies from the GLSTAT. Thus the individual energies were

extracted properly from the simulation results. Moreover, the sum of the individual internal and

kinetic energies equaled the analytically calculated external work. Conservation of energy was

hence satisfied and the energy extraction procedure was deemed accurate and trustworthy.

The energy- displacement histories provided the close-up view on the energy

redistribution. The energy- time histories increased abruptly after the prescribed column

removal. Due to the short duration of the unstable energy transition phase, energy- time histories

did not provide clear insight into the collapse behavior. The global energies were hence

presented with respect to the vertical displacement at the end beam shown in Figure A-8.

Vertical displacement

Figure A-8. Location of the selected node used for energy- displacement histories

The static preloading phase was characterized by the monotonic increase of the internal

energy (deformation work) (Figure A-9). Vertical displacement of the selected joint was small

(0.00013 of the column height). The kinetic energy was practically negligible in this phase.

Thus transition from one stable energy state to another energy state was accompanied by the

absence of kinetic energies. Global energies with respect to the selected joint vertical

displacement are shown in Figure A-10.

246

0 0.005 0.01 0.015 0.020

2

4

6

8

10

12

Vertical displacement [in]

Ener

gy [k

ip-in

]


Figure A-9. Internal, kinetic and total energy during the static preloading

0 50 100 1500

2000

4000

6000

8000

10000

Vertical displacement [in]

Ener

gy [k

ip-in

]


Figure A-10. Internal, kinetic and total energy

247

After the column removal, the external work was released at approximately constant rate.

It indicated the quick formation of the plastic hinges in the beams, which provided relatively

constant force resistance. Once the falling bay impacted the ground surface, kinetic energy

instantaneously dropped to zero. In the same instance, the sudden deformations resulted in the

significant increase of the internal energy (deformation work). The system reached its stable

energy state corresponding to the partial frame collapse. The demise of kinetic energy evidently

indicated that the system arrived to the stable energetic state.

248

APPENDIX B VERIFICATION OF ENERGY APPROACH TO PROGRESSIVE COLLAPSE

Collapse scenarios presented in this appendix verify the usefulness of the proposed energy

approach. Since interpretation of all cases is in essence analogous (and repetitive), only CASE 1

is described in more detail. The results for the remaining cases are described in figures only.

One Column Removed. CASE 1

The A1.1 corner column was removed (Figure B-1, Table B-1) after application of the

static preloading (Load = 1.0 D + 0.5 L). Collapse was arrested (Figure B-2). The internal

energy (deformation work) of the system increased by 40%. Insignificant levels of kinetic

energy were induced (Figure B-3).

y

30 [f

t]

30 [ft]

2

3

4

5

A B C D E

1

F G


x

Figure B-1. Removed columns in CASE 1

249

Table B-1. Removed columns in CASE 1 A B C D E F G 5 w12x58 w12x58 w14x74 w14x99 w14x99 w14x74 w12x58 4 w14x74 w12x58 w12x65 w12x72 w12x65 w12x58 w14x74 3 w14x99 w12x58 w12x65 w12x72 w12x65 w12x58 w14x99 2 w14x99 w12x58 w12x58 w12x58 w12x58 w12x58 w14x99 1 w14x74 w12x58 w14x74 w14x99 w14x99 w14x74 w14x74

Figure B-2. Final configuration of the steel building (CASE 1)

20 21 22 23 24 250

500

1000

1500

2000

2500

3000

3500

4000

Time [s]

Ener

gy [k

ip-in

]


Figure B-3. Global energies in CASE 1

250

Internal energies in the selected columns are depicted in Figure B-4. Columns in the

potential collapse initiation zone consisted of three different section types: W14x99, W14x74

and W12x58. Thus three different levels of buckling energy were calculated for each section

respectively. Column internal energies shall be compared to their respective limiting buckling

values. In CASE 1, none of the thresholds were exceeded within the columns. For this reason it

was concluded that the collapse was safely arrested and none of the columns entered the post-

buckling phase.

20 20.5 21 21.5 22 22.50

50

100

150

Time [s]

Ener

gy [k

ip-in

]

B1.1B1.2B1.3C1.1C1.2C1.3A2.1A2.2A2.3B2.1B2.2B2.3C2.1C2.2C2.3D2.1A3.1A3.2A3.3B3.1B3.2B3.3w12x58w14x74w14x99

w14x99

w14x74

w12x58

Figure B-4. Internal column energies in CASE 1

Energy rates are depicted in Figure B-5. There was a spike in the internal energy

(deformation work) rate in column B2.1. However, it was well below the value of 10000 [kip-

251

in/s] (1.14 [MJ]), which was observed in the failing columns of the simplified steel framed

building. Thus energy rate criterion confirmed the safety of the building, after the collapse was

arrested.

Figure B-5. Energy rates in columns in CASE 1

The steel building was divided into five zones, as depicted in Figure B-6. Energy time

histories in each of these zones were extracted from numerical analyses. The portion of the total

energy in each zone provided insight into the energy flow through the building. Normalized

energy split between the zones gave information on the energy localization. Distribution of

energy among the main structural members such as beams in the x direction, beams in the y

direction and columns was also analyzed (directions are shown in Figure 5-81). Such

252

comparisons shed light onto the importance and participation of each member group in the

energy redistribution.

2

3

4

5

A B C D E

1

F G

Zone 1

Zone 2

Zone 3

Zone 4Zone 5

Figure B-6. Building zones used to trace the energy propagation

Energy propagation through the building is shown in Figure B-7. Internal energy

(deformation work) increased mainly in the zone 1. Rise of the energies in zones 2 through 5

resulted from the energy increase in zone 1. Normalized energy allocation, shown in Figure B-8,

highlights very little localization. Energy released by the gravity was mainly absorbed by beams

in the y direction as depicted in Figure B-9. Participation of columns in the energy storage

slightly decreased due to the column removal and its pertinent internal energy (deformation

work).

253

20 20.5 21 21.5 22 22.5 230

500

1000

1500

2000

2500

3000

Time [s]

Ener

gy [k

ip-in

]

zone 1zone 2zone 3zone 4zone 5

Figure B-7. Energy propagation through the building in CASE 1

20 20.5 21 21.5 22 22.5 230

0.2

0.4

0.6

0.8

1

Time [s]

Rela

tive

Ener

gy


Figure B-8. Normalized energy allocation in the building in CASE 1

254

20 20.5 21 21.5 220

0.1

0.2

0.3

0.4

0.5

0.8

0.6

0.7

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsYY

Figure B-9. Energy split between members in zone 2 (CASE 1)

255

CASE 4. Columns A2.1 and A3.1 removed (Load = 1.0 D + 0.5 L)

The A2.1 and A3.1 columns were removed after application of the static preloading

(Load = 1.0 D + 0.5 L). Collapse was arrested.

y

30 [f

t]

30 [ft]

2

3

4

5

A B C D E

1

F G


x



256

Figure B-11. Final configuration of the building in CASE 4

20 21 22 23 24 250

0.5

1

1.5

2x 10

4

Time [s]

Ener

gy [k

ip-in

]



257

Table B-3. W14x74, 156 [in] column buckling results P/Pcr Buckling Force Internal Energy Comment [kip] [kip-in] 1.00 776 85 0.95 738 229 0.90 699 254 0.85 660 382 0.80 621 414 0.75 582 617 0.70 543 711 0.65 505 845 0.60 466 1090 0.50 388 1270 0.40 310.55 1500 0.30 232.91 1940 Failure limit of A1.3 and A4.3

20 20.5 21 21.5 22 22.5 230

50

100

150

200

250

300

350

400

450

500

Time [s]

Ener

gy [k

ip-in

]

A1.1A1.2A1.3B1.1B1.2B1.3B2.1B2.2B2.3B3.1B3.2B3.3A4.1A4.2A4.3B4.1B4.2B4.3w12x58w14x74w14x99

A1.3 (w14x74)

A4.3 (w14x74)


258

20

1000

20.2 20.4 20.6 20.8 210

100

200

300

400

500

600

700

800

900

Time [s]

Ener

gy R

ate

[kip

-in/s

]

A1.3 (w14x74)A1.1A1.2A1.3B1.1B1.2B1.3B2.1B2.2B2.3B3.1B3.2B3.3A4.1A4.2A4.3B4.1B4.2B4.3


259

2

3

4

5

A B C D E

1

F G

Zone 1

Zone 2

Zone 3

Zone 4Zone 5

Figure B-15. Building zones used to trace the energy propagation in CASE 4

20 20.5 21 21.5 220

1000

2000

3000

4000

5000

6000

7000

8000

Time [s]

Ener

gy [k

ip-in

]



260

20 20.5 21 21.5 220

0.2

0.4

0.6

0.8

1

Time [s]

Rela

tive

Ener

gy



20 20.5 21 21.5 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsYY


261

CASE 5. Columns A2.1, A3.1 and A4.1 removed (Load = 1.0 D + 0.5 L)

The A2.1, A3.1 and A4.1 columns were removed after application of the static preloading

(Load = 1.0 D + 0.5 L). The analyzed building collapsed.

y

30 [ft]

2

3

4

5

A B C D E

1

F G


x

30 [ft]

Figure B-19. Removed columns in CASE 5 Table B-4. Removed columns in CASE 5 A B C D E F G 5 w12x58 w12x58 w14x74 w14x99 w14x99 w14x74 w12x58 4 w14x74 w12x58 w12x65 w12x72 w12x65 w12x58 w14x74 3 w14x99 w12x58 w12x65 w12x72 w12x65 w12x58 w14x99 2 w14x99 w12x58 w12x58 w12x58 w12x58 w12x58 w14x99 1 w14x74 w12x58 w14x74 w14x99 w14x99 w14x74 w14x74

262

Figure B-20. Collapse sequence of the steel building in CASE 5

263

Figure B-20. Continued

264

20 22 24 26 28 300

2

4

6

8

10

12

14x 10

5

Time [s]

Ener

gy [k

ip-in

]



Table B-5. W14x74, 156 [in] column buckling results P/Pcr Buckling Force Internal Energy Comment [kip] [kip-in] 1.00 776 85 0.95 738 229 0.90 699 254 0.85 660 382 0.80 621 414 0.75 582 617 0.70 543 711 0.65 505 845 0.60 466 1090 0.50 388 1270 0.40 310.55 1510 0.30 232.91 1940 Failure limit of A1.3

265

Table B-6. W12x58, 156 [in] column buckling results P/Pcr Buckling Force Internal Energy Comment [kip] [kip-in] 1.00 603 66 0.95 573 188 0.90 544 207 0.85 513 296 0.80 482 322 0.75 452 471 0.70 422 552 0.65 392 656 0.60 362 832 0.50 302 1030 0.40 241.21 1220 0.30 180.91 1590 Failure limit of B4.3, A5.1

20 21 22 23 24 250

500

1000

1500

2000

2500

3000

Time [s]

Ener

gy [k

ip-in

]

A1.1A1.2A1.3B1.1B1.2B1.3B2.1B2.2B2.3B3.1B3.2B3.3B4.1B4.2B4.3A5.1A5.2A5.3B5.1B5.2B5.3w12x58w14x74w14x99

A1.3 (w14x74)

B4.3 (w12x58)

A5.1 (w12x58)


266

A1.3 (w14x74)


267

2

3

4

5

A B C D E F G

1

Zone 1

Zone 2

Zone 3

Zone 4 Zone 5


20 22 24 26 28 30 32 340

1

2

3

4

5

6

7

8x 10

Time [s]

Ener

gy [k

ip-in

]


510x


268

20 22 24 26 28 30 32 340

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time [s]

Rela

tive

Ener

gy



20 22 24 26 28 30 32 340

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsYY


269

CASE 6. Columns D3.1 and E3.1 removed (Load = 1.0 D + 0.5 L)

The D3.1 and E3.1 columns were removed after application of the static preloading

(Load = 1.0 D + 0.5 L). Collapse was arrested.

30 [ft]

2

3

4

5

A B C D E

1

F G


x

30 [ft]

x



270

Figure B-29. Final configuration of the building in CASE 6

20 21 22 23 24 250

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

Time [s]

Ener

gy [k

ip-in

]



271

Table B-8. Failure energy limits for the selected columns location D4.1 E2.1 D2.1 F4.1 F3.1 E4.1 C3.1 section w12x72 w12x58 w12x58 w12x58 12x58 w12x65 w12x65element 988 459 419 1068 785 1028 663 permanent load [kip]

363

320

320 331 319 342

337

buckling energy [kip-in]

82

66

66

66

66

74

74

failure energy [kip-in]

2010

993 993 972 995 1800

1820

20 20.5 21 21.5 220

10

20

30

40

50

60

70

80

90

100

Time [s]

Ener

gy [k

ip-in

]

D2.1D2.2D2.3E2.1E2.2E2.3F2.1F2.2F2.3C3.1C3.2C3.3F3.1F3.2F3.3D4.1D4.2D4.3E4.1E4.2E4.3F4.1F4.2F4.3w12x58w12x65w12x72w14x99

D4.1 (w12x72)


272

20.3 20.4 20.5 20.6 20.7 20.80

10

20

30

40

50

60

70

80

90

100

Time [s]

Ener

gy [k

ip-in

]

D2.1D2.2D2.3E2.1E2.2E2.3F2.1F2.2F2.3C3.1C3.2C3.3F3.1F3.2F3.3D4.1D4.2D4.3E4.1E4.2E4.3F4.1F4.2F4.3w12x58w12x65w12x72w14x99


D2.1 (w12x58)

E4.1 (w12x65)

D4.1 (w12x72)

E2.1 (w12x58)

273

20 20.1 20.2 20.3 20.4 20.5 20.6 20.70

50

100

150

200

250

300

350

400

450

500

Time [s]

Ener

gy R

ate

[kip

-in/s

]

D2.1D2.2D2.3E2.1E2.2E2.3F2.1F2.2F2.3C3.1C3.2C3.3F3.1F3.2F3.3D4.1D4.2D4.3E4.1E4.2E4.3F4.1F4.2F4.3


274

2

3

4

5

A B C D E

1

F G

Zone 1

Zone 2

Zone 3

Zone 4

Zone 5


20 20.5 21 21.5 22 22.5 230

2000

4000

6000

8000

10000

Time [s]

Ener

gy [k

ip-in

]



275

20 21 22 23 240

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time [s]

Rela

tive

Ener

gy



20 22 24 26 280

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsYY


276

CASE 7. Columns D3.1, E3.1 and E4.1 removed (Load = 1.0 D + 0.5 L)

The D3.1, E3.1 and E4.1 columns were removed after application of the static preloading

(Load = 1.0 D + 0.5 L). The analyzed building collapsed.

y

30 [f

t]

30 [ft]

2

3

4

5

A B C D E

1

F G


x



277

time = 20 [s]

time = 20.9 [s]

time = 22 [s]

time = 23.3 [s]

Figure B-39. Collapse sequence of the steel building in CASE 7

278

time = 24 [s]

time = 28 [s]

Figure 5-98B-39. Continued

20 22 24 26 280

0.5

1

1.5

2x 10

6

Time [s]

Ener

gy [k

ip-in

]



279

Table B-10. Failure energy limits for the selected columns location D4.1 E2.1 D2.1 F4.1 F3.1 E5.1 C3.1 section w12x72 w12x58 w12x58 w12x58 12x58 w14x99 w12x65element 988 459 419 1068 785 2630 663 permanent load [kip]

363

320

320 331 319 158

337

buckling energy [kip-in]

82

66

66

66

66

132

74

failure energy [kip-in]

2010

993 993 972 995 >10000

1820

20 20.5 21 21.5 22 22.5 23 23.50

200

400

600

800

1000

1200

1400

1600

1800

2000

Time [s]

Ener

gy [k

ip-in

]

D2.1D2.2D2.3E2.1E2.2E2.3F2.1F2.2F2.3C3.1C3.2C3.3F3.1F3.2F3.3D4.1D4.2D4.3F4.1F4.2F4.3E5.1E5.2E5.3w12x58w12x65w12x72w14x99

E2.1 (w12x58)

D4.1 (w12x72)


280

20 20.5 21 21.5 22 22.50

50

100

150

200

250

300

350

400

450

500

Time [s]

Ener

gy [k

ip-in

]

E2.1 (w12x58) D2.1D2.2D2.3E2.1E2.2E2.3F2.1

D4.1 (w12x72) F2.2F2.3C3.1C3.2C3.3F3.1F3.2F3.3D4.1D4.2D4.3F4.1F4.2F4.3E5.1E5.2E5.3w12x58w12x65w12x72w14x99

Figure B-42. Internal column energies in CASE 7. Close-up view

281

D4.1 (w12x72)

E2.1 (w12x58)


282

2

3

4

5

A B C D E F G

Zone 1

Zone 2

Zone 3

Zone 4

Zone 5 1


20 22 24 26 280

1

2

3

4

5

6

7

8

Time [s]

Ener

gy [k

ip-in

]


510x


283

1.6

20 22 24 26 280

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [s]

Rela

tive

Ener

gy



20 22 24 26 280

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [s]

Rela

tive

Ener

gy

columnsbeamsX

beamsYY


284

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Structures Used as an Energy Absorber.” Journal of Applied Mechanics, Transactions ASME, 74(4), 628-635.

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BIOGRAPHICAL SKETCH

Stefan Szyniszewski was born in Poland. He received his bachelor’s and master’s degree

from the Technical University of Warsaw, Poland. He also studied at the RWTH-Aachen in

Germany and at the Kanazawa University in Japan. He completed his PhD studies at the

University of Florida, USA. He received Monbusho Award from the Japanese Ministry of

Education, GFPS Fellowship from the Polish-German Academic Exchange Commission and

DAAD Award from the German Ministry of Education. His research on progressive collapse

simulations of steel framed structures was recognized with the Best Paper Award by the

International Association of Bridge and Structural Engineers (IABSE) during an annual

conference in Helsinki, Finland in 2008. He was also the recipient of the Fulbright Award.

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