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PAGE 1 PROGRESSIVE COLLAPSE OF MOMENT RESISTING STEEL FR AMED BUILDINGS QUANTITATIVE ANALYSIS BASED ENERGY APPROACH By STEFAN TADEUSZ SZYNISZEWSKI A DISSERTAION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1 PAGE 2 2009 Stefan Tadeusz Szyniszewski 2 PAGE 3 Dla mojego Taty Zdzis awa i Mamy Aleksandry oraz Kasi, Oli i mojej kochanej ony Ani, ktrej wsparcie i mi o by y dla mnie bezcenne w trakcie pisania tej pracy To my Father Zdzis aw and Mother Aleksandra, sisters Kasia and Ola and my beloved wife Anna, whose support and love made possibl e the completion of this dissertation 3 PAGE 4 ACKNOWLEDGMENTS This study was conducted at the Center for Infra structure Protection and Physical Security at the University of Florida under contract with the US Army Corps of Engineers, Engineer Research and Development Center (ERDC), Vick sburg, 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 teacher s at the University of Florida from whom I learned so much: Mike McVay, Trey Hamilt on, Kurt Gurley, Youping Chen, John Lybas and others. The author is also i ndebted to his teachers at the Wa rsaw University of Technology: Wojciech Radomski, Tomasz Lewi ski, Wojciech Gilewski, Stanislaw Jemio o, Aleksander Szwed and Wawrzyniec Sadkowski, Andr zej Reterski and Kazimierz Cegie ka. The manuscript was carefully reviewed in its various versions, several times, by Michael Davidson, Nick Henriquez, Hyun Chang Yim and th e University of Florida Editorial Office, resulting in important consistenc y and grammatical improvements. Finally, I want to express my special thanks to my family in Poland, to Aleksandra and Zdzis aw, my parents, for their to tal 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 wonderf ul people, my friends and teachers who have generously given me their advice and hel p, I am pleased to express my gratitude. 4 PAGE 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .............................................................................................................. 4LIST OF TABLES ................................................................................................................ .......... 7LIST OF FIGURES ............................................................................................................... ......... 9ABSTRACT ...................................................................................................................... ............ 19CHAPTER1 INTRODUCTION ............................................................................................................... 20Problem Statement ............................................................................................................. 20Importance .................................................................................................................... ....... 20Research Objectives ........................................................................................................... 20Scope ......................................................................................................................... .......... 212 LITERATURE REVIEW .................................................................................................... 23Overview ...................................................................................................................... ....... 23Event Control ................................................................................................................. ..... 24Indirect Design ............................................................................................................... ..... 24Historical Cases 25Building Code Measures to Pr event Progressive Collapse 30Direct Design: Finite Element Method ............................................................................... 36Static Procedures for Progr essive Collapse Modeling ........................................................ 38Energy Concepts ................................................................................................................. 43Deformation Work (Internal Energy) 45Kinetic Energy 48Energy Based Procedures for Progr essive Collapse Modeling ........................................... 49Energy Flow between Members .......................................................................................... 55Simplified Methods for High Rise Buildings ...................................................................... 59Summary ....................................................................................................................... ...... 673 RESEARCH APPROACH .................................................................................................. 70Proposed Theory ................................................................................................................. 70Virtual Experiments ........................................................................................................... 79Overview 79Material Modeling 79Structural Modeling 84Selected Structures ........................................................................................................... ... 91Summary ....................................................................................................................... ...... 97 5 PAGE 6 4 KINEMATIC RESULTS .................................................................................................... 98Overview ...................................................................................................................... ....... 98Virtual Experiments on the Simplified Steel Framed Buildings ......................................... 98Kinematic Observations .................................................................................................... 100Validation of Results ......................................................................................................... 118Steel Framed Building with Shorter Beam Spans ............................................................. 120Summary ....................................................................................................................... .... 1295 ENERGY APPROACH TO THE ANALYS IS OF PROGRESSIVE COLLAPSE ......... 131Overview ...................................................................................................................... ..... 131Energy Definitions in LSDYNA ...................................................................................... 132Energetic Characteristics of Individual Columns .............................................................. 133Displacement controlled Buckling 133Force controlled Buckling 138Column Buckling Energy 148Energy Flow and Redistribution ....................................................................................... 149Energy Propagation through the Building 167Energy based Column Buckling Crit erion (Full Building Analysis) 171Usefulness of the Energy Buckling Limits 194Energy based Building Failure Limit 197Analytical Solution of ElastoPlastic Column Buckling 201Verification of Energy Approach on Realistic Steel Building .......................................... 207Two Columns Removed. CASE 2 209Three Columns Removed CASE 3 219Summary ....................................................................................................................... .... 2326 CONCLUSIONS AND RECOMMENDATIONS ............................................................ 236Summary ....................................................................................................................... .... 236Progressive Collapse Conclusions .................................................................................... 236Energy Conclusions ........................................................................................................... 238Recommendations ............................................................................................................. 2 40APPENDIXA VERIFICATION OF ENERGY EXTRACTION PROCEDURE .................................... 241B VERIFICATION OF ENERGY APPROACH TO PROGRESSIVE COLLAPSE .......... 249LIST OF REFERENCES ............................................................................................................ 285BIOGRAPHICAL SKETCH ...................................................................................................... 291 6 PAGE 7 LIST OF TABLES Table page 21 Steel building: selected results from nonlinear dynamic analys is (Powell, 2005) ........... 42 22 Calculation of Force Ratios, R1,i in Phase 1 (Dusenberry and Hamburger, 2006) .......... 51 31 Lightly reinforced slab modeling ...................................................................................... 87 32 Moment resistant beams (designated with A) ............................................................... 95 33 Column schedules of the typical SAC building ................................................................ 96 51 W12x58, 156 [in] colu mn buckling results..................................................................... 145 52 W14x74, 156 [in] colu mn buckling results..................................................................... 149 53 Demand Capacity comparison ........................................................................................ 195 54 W12x58, 156 [in] colu mn buckling results..................................................................... 199 55 W14x99, 156 [in] colu mn buckling results..................................................................... 199 56 W14x74, 156 [in] colu mn buckling results..................................................................... 199 57 W12x65, 156 [in] colu mn buckling results..................................................................... 200 58 W12x72, 156 [in] colu mn buckling results..................................................................... 200 59 Steel profiles of columns (designations according to AISC, 2006) ................................ 207 510 Moment resistant beams (designated with A) ............................................................. 208 511 Demand capacity (D/C) ratios ........................................................................................ 233 B1 Removed columns in CASE 1 ........................................................................................ 250 B2 Removed columns in CASE 4 ........................................................................................ 256 B3 W14x74, 156 [in] colu mn buckling results..................................................................... 258 B4 Removed columns in CASE 5 ........................................................................................ 262 B5 W14x74, 156 [in] colu mn buckling results..................................................................... 265 B6 W12x58, 156 [in] colu mn buckling results..................................................................... 266 B7 Removed columns in CASE 6 ........................................................................................ 270 7 PAGE 8 B8 Failure energy limits for the selected columns ............................................................... 272 B9 Removed columns in CASE 7 ........................................................................................ 277 B10 Failure energy limits for the selected columns ............................................................... 280 8 PAGE 9 LIST OF FIGURES Figure page 21 Single degree of freedom system ...................................................................................... 38 22 Loading applied to the SDOF system (re lative time to the natural period T) ................... 39 23 Displacement response to dynamic loads shown in Figure 22 ........................................ 40 24 Typical inelastic forcedeformation relationship .............................................................. 41 25 Simple bar in tension ..................................................................................................... ... 45 26 Stress vs. strain relationship of the rod ............................................................................. 46 27 Strain and energy decomposition into plastic and elas tic components ............................. 48 28 Released mass falling on the spring .................................................................................. 49 29 Deflected shape under static gravity forces ...................................................................... 50 210 Loaddisplacement during pushdown static analysis ...................................................... 52 211 Spring coupled, longitudinally vibrating rods excited by a harmonic point force ............ 58 212 Comparison of exact and simplified solutions for the coupled rod system ...................... 58 213 Progressive collapse of the World Trade Center towers ................................................... 59 214 Typical loaddisplacement diagra m of columns of one story ........................................... 60 215 Continuum model for propa gation of crushing front ........................................................ 63 216 Free body diagram in the crushdown and crushup phase ............................................... 65 31 Energy state in a hypothetic al structural system ............................................................... 70 32 Energy transformation in a generic system ....................................................................... 71 33 Energy transformation between potential, internal and kinetic energies .......................... 72 34 Energy based approach to progressive collapse ................................................................ 73 35 High rise building coll apse, initiating story ...................................................................... 74 36 Energy flowdisplacement of single column (H = height of story) ................................. 75 37 Collapse propagates into consecutive story ...................................................................... 76 9 PAGE 10 38 Energy release with energy absorption by the crushing story itself ................................. 77 39 Collapse propagation due to insufficient energy absorption ............................................. 77 310 Typical stressstrain re lationship for A36 steel ................................................................ 81 311 Threedimensional solid cube buckling modeling ............................................................ 84 312 Integration scheme for W section in the HughesLiu beam element ................................ 86 313 Displacementforce history comparis on of brick and HughesLiu modeling .................. 88 314 Buckling stressslenderness parameter c ....................................................................... 89 315 Selected threestory steel framed building ........................................................................ 91 316 Twodimensional steel frame selected for the analysis .................................................... 92 317 Threedimensional steel frame selected for the analysis .................................................. 93 318 Simplified steel framed building ....................................................................................... 9 3 319 Model of the SAC Modified Boston Building .................................................................. 94 320 Framing plan used for SAC three story building .............................................................. 95 321 Orientation of columns ................................................................................................... .. 96 41 Selected three story steel fr amed building (W14x74 columns) ........................................ 99 42 Two dimensional frame in CASE 1 ................................................................................ 101 43 Three dimensional frame in CASE 1 .............................................................................. 101 44 Displaced shape of the simplified steel framed building in CASE 1 .............................. 102 45 Displacement of the building corner (CASE1) .............................................................. 102 46 Beam moment time history in CASE 1: A) static, B) dynamic phase ........................... 103 47 Collapse sequence of twodimensional frame in CASE 3 .............................................. 106 48 Collapse sequence of the threedimensional frame in CASE 3 ...................................... 107 49 Building with hardened slabs Collapse sequence in CASE 3 ...................................... 108 410 Building with typical slabs Collapse sequence in CASE 3 .......................................... 110 411 Displacement time history of point A1 in CASE 3 ....................................................... 112 10 PAGE 11 412 Moment time history of beam AB at B2 in CASE 3 .................................................... 113 413 Normalized internal forces in A2.1 column : A) hardened slabs, B) typical slabs .......... 114 414 Dynamic phase of normalized internal for ces in A2.1 column: A) hardened slabs, B) typical slabs ................................................................................................................. .... 115 415 Normalized internal forces in B2.1 column : A) hardened slabs, B) typical slabs .......... 116 416 Dynamic phase of normalized internal for ces in B2.1 column: A) hardened slabs, B) typical slabs ................................................................................................................. .... 117 417 Free fall requirement in CASE 1 .................................................................................... 119 418 Adherence to free fall in CASE 3 ................................................................................... 119 419 Selected three story steel framed building for buckling onset analysis .......................... 120 420 Arrested collapse of threedimensiona l building w/ typical slabs in CASE A ............... 121 421 Arrested collapse of threedimensiona l model w/ typical slabs in CASE B ................... 121 422 Collapse sequence of threedimensional model w/ typical slabs in CASE C ................. 122 423 Displacements of A2.1 and B2.1 columns: 1) Complete history, 2) Dynamic, collapse phase. ........................................................................................................................ ...... 124 424 Normalized internal forces in A 2.1 column: 1) CASE A, 2) CASE C ........................... 125 425 Dynamic phase of normalized internal for ces in A2.1 column: 1) CASE A, 2) CASE C ......................................................................................................................................... 126 426 Normalized internal forces in B 2.1 column: 1) CASE A, 2) CASE C ........................... 127 427 Dynamic phase of normalized internal for ces in B2.1 column: 1) CASE A, 2) CASE C ......................................................................................................................................... 128 51 Simulated buckling mode of W12x58, 156 [in] clamped column (beam elements) ...... 134 52 Prescribed column top displacement time history .......................................................... 134 53 Reaction force time history ............................................................................................. 1 35 54 Resistancetop disp lacement function ............................................................................. 136 55 Internal energy (deforma tion work) time history ............................................................ 137 56 Internal energy disp lacement history .............................................................................. 137 11 PAGE 12 57 Acceleration time histor y at the column top ................................................................... 139 58 Acceleration force histor y at the column top .................................................................. 140 59 Velocity force history at the column top ......................................................................... 140 510 Vertical displacement at the column topforce history .................................................. 141 511 Kinetic energy time history ............................................................................................. 141 512 Internal energy time history ............................................................................................ 142 513 Internal energy time histor y in prebuckling phase ........................................................ 143 514 Internal energy rate ..................................................................................................... .... 144 515 Internal energy rate in the prebuckling phase ................................................................ 144 516 W12x58: buckling force for the selecte d, force controlled, loading rates ...................... 146 517 W12x58: Internal buckling energy rate for the selected loading rates ........................... 147 518 W12x58: Internal buckling energy fo r the selected loading rates .................................. 147 519 Buckling force for the selected loading rates in W14x74 column .................................. 150 520 Internal buckling energy for the sele cted loading rates in W14x74 column .................. 150 521 Final shape of the select ed structure after sudden column removal in CASE 1 ............. 151 522 Global energy histories (from GLSTAT) ....................................................................... 152 523 Dynamic phase of the global energies ............................................................................ 152 524 Internal energy time hist ories for all columns ................................................................ 153 525 Kinetic energies for all columns ..................................................................................... 154 526 Final deflection of the sel ected structure in CASE 2 ...................................................... 155 527 Global energies (GLSTAT) ............................................................................................ 155 528 Global energies in dynamic phase (GLSTAT) ............................................................... 156 529 Internal energi es in columns ........................................................................................... 157 530 Instable columns as inferred fr om the internal energy results ........................................ 158 531 Kinetic energies of all columns ....................................................................................... 15 8 12 PAGE 13 532 Internal energy in selected co lumns (initial collapse phase) .......................................... 159 533 Identification of column buckli ng from the kinetic energies .......................................... 159 534 Global energies (from GLSTAT) .................................................................................... 161 535 Internal energi es in columns ........................................................................................... 162 536 Internal energies in selected co lumns with buckling energy threshold .......................... 163 537 Verification of energetic results Collapse sequence assessment ................................. 164 538 Energy propagation zones ............................................................................................... 1 67 539 Energy propagation through th e inclusive building zones .............................................. 168 540 Normalized energy allocation in the building (inclusive zones) .................................... 168 541 Energy rates in the columns ............................................................................................ 169 542 Energy distribution amo ng members in zone 1 .............................................................. 170 543 Global energy redistribution in CASE A ........................................................................ 171 544 Internal energies in CASE A........................................................................................... 172 545 Energy flow in building with typical slabs in CASE A (arrested collapse): A) Axial forcedisplacement, B) Internal energydisplacement .............................................................. 173 546 Energy rates in the columns in CASE A ......................................................................... 174 547 Energy propagation through the inclusive zones in CASE A ......................................... 175 548 Energy propagation through the exclusive zones in CASE A ........................................ 175 549 Normalized energy allocation in the inclusive zones in CASE A .................................. 176 550 Energy split between members in zone 1 (CASE A) ...................................................... 177 551 Arrested collapse of threedimensiona l model w/ typical slabs in CASE B ................... 177 552 Global energies in CASE B ............................................................................................ 178 553 Internal column en ergies in CASE B .............................................................................. 179 554 Energy flow in building with typical slabs in CASE B (arrested collapse): A) Axial forcedisplacement, B) Internal energydisplacement. ............................................................. 180 555 Energy rates in columns in CASE B ............................................................................... 181 13 PAGE 14 556 Energy propagation through the inclusive zones in CASE B ......................................... 182 557 Energy propagation through the exclusive zones in CASE B ........................................ 182 558 Normalized energy allocation in the building in CASE B .............................................. 183 559 Energy split between members in zone 1 (CASE B) ...................................................... 183 560 Collapse sequence of threedimensional model w/ typical slabs in CASE C ................. 184 561 Global energies in CASE C ............................................................................................ 186 562 Internal column en ergies in CASE C .............................................................................. 187 563 Energy flow in building w/ typical slabs: CASE C (tot al failure): A) Axial forcedisplacement, B) Internal energydisplacement. ............................................................. 188 564 Energy rates in columns in CASE C ............................................................................... 189 565 Energy propagation through the exclusive zones in CASE C ........................................ 190 566 Normalized energy allocation in the building in CASE C .............................................. 190 567 Energy split between members in zone 1 (CASE C) ...................................................... 191 568 Elastic component of the absorbed energy in A2.1 column ........................................... 192 569 Normalized energy decomposition into elastic and plastic component .......................... 192 570 Elastic component of absorbed energy in A1.1 beam yy .............................................. 193 571 Normalized energy absorption in A1.1 beam yy ........................................................... 194 572 Buckling force demand/capacity ratios ........................................................................... 195 573 Buckling energy demand/capacity ratios ........................................................................ 195 574 Parallel of energy capacity with axia l capacity of W12x58, 156 [in] column ................ 198 575 Kinematics of column buckling ...................................................................................... 201 576 Bending stretches in the hinge ........................................................................................ 20 2 577 Moment equilibrium ....................................................................................................... 203 578 Elastoplastic material model .......................................................................................... 2 05 579 Force displacement of W12x58 column of 156 [in] height ............................................ 206 580 Energy displacement of W12x58 co lumn of 156 [in] height .......................................... 206 14 PAGE 15 581 Three story, moment resi sting framed building for verification analysis ....................... 207 582 Framing plan used for SAC three story building ............................................................ 208 583 Deflected, final configuration of the building in CASE 2 .............................................. 209 584 Global Energies in CASE 2 ............................................................................................ 209 585 Internal column en ergies in CASE 2 .............................................................................. 210 586 Internal forces in CASE 2: A) column A2.1, B) column B2.1. ...................................... 211 587 Internal forces in the dynamic phase CASE 2: A) A2.1 column, B) B2.1 column. ..... 212 588 Energy absorption in the selected columns CASE 2: A) Axial forcedisplacement, B) Internal energydi splacement. ......................................................................................... 213 589 Energy rates in columns in CASE 2 ............................................................................... 214 590 Building zones used to trace the energy propagation ..................................................... 215 591 Energy propagation through the exclusive zones in CASE 2 ......................................... 215 592 Energy propagation through the inclusive zones in CASE 2 .......................................... 216 593 Normalized energy allocation in the building in CASE 2 .............................................. 216 594 Decomposition of the absorbed energy (deformation work) in A1.1 beam yy ............. 217 595 Decomposition of the absorbed energy (deformation work) in B2.1 column ................ 218 596 Energy split between members in zone 2 (CASE 2)....................................................... 218 597 Global energies in the building ....................................................................................... 21 9 598 Collapse sequence of the steel building in CASE 3 ........................................................ 220 599 Building zones used to trace the energy propagation ..................................................... 221 5100 Energy propagation through the exclusive zones in CASE 3 ......................................... 222 5101 Normalized energy allocation in the building in CASE 3 .............................................. 222 5102 Energy split between members in zone 2 (CASE 3)....................................................... 223 5103 Decomposition of the absorbed energy (deformation work) in A2.1 beam yy ............. 224 5104 Decomposition of the absorbed energy (deformation work) in B1.1 beam xx ............. 224 5105 Closeup view of the energy (deformati on work) decomposition in B2.1 column ........ 225 15 PAGE 16 5106 Energy decomposition of the absorbed en ergy (deformation work) in B2.1 column ..... 225 5107 Decomposition of the absorbed energy (deformation work) in C2.1 column ................ 226 5108 Internal column en ergies in CASE 3 .............................................................................. 227 5109 Normalized internal forces in: A) B2.1 column, B) C2.1 column .................................. 228 5110 Normalized internal forces in dynamic phase: A) B2.1 column, B) C2.1 column ......... 229 5111 Energy flow in the SAC building: CASE 3 (t otal failure): A) Axia l forcedisplacement, B) Internal energydisplacement ..................................................................................... 230 5112 Energy rates in columns in CASE 3 ............................................................................... 231 5113 Buckling force demand/capacity ratios ........................................................................... 233 5114 Buckling energy demand/capacity ratios ........................................................................ 234 5115 Failure energy demand/capacity ratios ........................................................................... 234 A1 Steel frame used for the energy benchmark test ............................................................. 241 A2 Final, displaced shape of the 2D frame used in the energy benchmark test .................. 242 A3 Global energies reported by LSDYNA in GLSTAT file ............................................... 242 A4 Analytical and numerical external work results .............................................................. 243 A5 Analytical and numerical ex ternal work results during the static preloading phase ....... 244 A6 Global (GLSTAT) and sum of local energi es (MATSUM) during th e static preloading245 A7 Comparison of global (GLSTAT) and sum of local energies (MATSUM) .................... 245 A8 Location of the selected node used for energydisplacement histories .......................... 246 A9 Internal, kinetic and total ener gy during the static preloading ........................................ 247 A10 Internal, kinetic and total energy .................................................................................... 247 B1 Removed columns in CASE 1 ........................................................................................ 249 B2 Final configuration of th e steel building (CASE 1) ........................................................ 250 B3 Global energies in CASE 1 ............................................................................................. 250 B4 Internal column en ergies in CASE 1 .............................................................................. 251 B5 Energy rates in columns in CASE 1 ............................................................................... 252 16 PAGE 17 B6 Building zones used to trace the energy propagation ..................................................... 253 B7 Energy propagation through the building in CASE 1 ..................................................... 254 B8 Normalized energy allocation in the building in CASE 1 .............................................. 254 B9 Energy split between members in zone 2 (CASE 1)....................................................... 255 B10 Removed columns in CASE 4 ........................................................................................ 256 B11 Final configuration of the building in CASE 4 ............................................................... 257 B12 Global energies in CASE 4 ............................................................................................. 257 B13 Internal column en ergies in CASE 4 .............................................................................. 258 B14 Energy rates in columns in CASE 4 ............................................................................... 259 B15 Building zones used to trace the energy propagation in CASE 4 ................................... 260 B16 Energy propagation through the building in CASE 4 ..................................................... 260 B17 Normalized energy allocation in the building in CASE 4 .............................................. 261 B18 Energy split between members in zone 2 (CASE 4)....................................................... 261 B19 Removed columns in CASE 5 ........................................................................................ 262 B20 Collapse sequence of the st eel building in CASE 5 ........................................................ 263 B21 Global energies in CASE 5 ............................................................................................. 265 B22 Internal column en ergies in CASE 5 .............................................................................. 266 B23 Energy rates in columns in CASE 5 ............................................................................... 267 B24 Building zones used to trace the energy propagation in CASE 5 ................................... 268 B25 Energy propagation through the building in CASE 5 .................................................. 268 B26 Normalized energy allocation in the building in CASE 5 .............................................. 269 B27 Energy split between members in zone 5 (CASE 5)....................................................... 269 B28 Removed columns in CASE 6 ........................................................................................ 270 B29 Final configuration of the building in CASE 6 ............................................................... 271 B30 Global energies in CASE 6 ............................................................................................. 271 17 PAGE 18 B31 Internal column en ergies in CASE 6 .............................................................................. 272 B32 Internal column en ergies in CASE 6 .............................................................................. 273 B33 Energy rates in columns in CASE 6 ............................................................................... 274 B34 Building zones used to trace the energy propagation in CASE 6 ................................... 275 B35 Energy propagation through the building in CASE 6 ..................................................... 275 B36 Normalized energy allocation in the building in CASE 6 .............................................. 276 B37 Energy split between members in zone 5 (CASE 6)....................................................... 276 B38 Removed columns in CASE 7 ........................................................................................ 277 B39 Collapse sequence of the st eel building in CASE 7 ........................................................ 278 B40 Global energies in CASE 7 ............................................................................................. 279 B41 Internal column en ergies in CASE 7 .............................................................................. 280 B42 Internal column energies in CASE 7. Closeup view ..................................................... 281 B43 Energy rates in columns in CASE 7 ............................................................................... 282 B44 Building zones used to trace the energy propagation in CASE 7 ................................... 283 B45 Energy propagation through the building in CASE 7 ..................................................... 283 B46 Normalized energy allocation in the building in CASE 7 .............................................. 284 B47 Energy split between members in zone 5 (CASE 7)....................................................... 284 18 PAGE 19 Abstract of Dissertation Pres ented 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 FR AMED BUILDINGS QUANTITATIVE ANALYSIS BASED ENERGY APPROACH By Stefan Tadeusz Szyniszewski May 2009 Chair: Theodor Krauthammer Major: Civil Engineering An energy based, quantitative analysis of colla pse 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. Th is study demonstrated that energy based analysis and energetic failure criteria fa cilitate understanding of colla pse propagation and reveal the underlying mechanisms during the co llapse arrest or propagation. Simulations of structural response to the sudde n 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 stru ctural information, probabilistic geometric inaccuracies, and material models) are presented. Kinematic and energetic results with limited experimental validation are ela borated. An easy to use energy based analysis of collapse propagation has been developed and cl early explained in this study. 19 PAGE 20 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 lo cal 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 characteristic s 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) co llapse (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 failureinitiating events The destruction of the WTC on September 11, 2001 and other cases to be discussed confirm that potential th reats cannot be totally eradicated. Research Objectives The main objective of this study was to enable the development of a rational energybased analysis of progressive collaps e of moment resisting steel fr amed buildings by studying simple frame systems focusing on the role of energy flow. 20 PAGE 21 Objective 1 : Examine energy flow and redistribution during progressive collapse, Objective 2 : Provide expressions for column buckli ng energy (deformation work) for typical Wshape columns to facilitate the energy based analysis approaches, Objective 3 : Provide an energybased approach for co llapse assessment of moment resisting steel buildings by analyzing the internal energy (deformation work) of structural members in order to evaluate dynamic alternate paths reforma tions, competing failures, and structural safety, Column buckling energy (deformation work) was proposed as a rate independent buckling criterion in progressive collapse consideratio ns. Moreover, the column failure energy (deformation work) was proposed and verified as the limit of the global building safety. This study demonstrated that energybase d analysis and energy buckling a nd failure criteria facilitate a better understanding of collapse propagation, a nd reveal the underlying mechanisms during the collapse arrest or propagation. Scope This research was limited to understanding the ch aracteristics of progressive collapse in moment resisting steel framed buildings. The in itial 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 s hould be clarified first. The analysis was conducted using finite element simulations. The level of modeling complexity was gradually 21 PAGE 22 enhanced: frames, frames with slabs, and fi nally frames with slab s including semirigid connections. Simulations of structural response to the s udden removal of a key column(s) have been carried out for a number of moment resisting st eel frames and steel framed buildings. The modeling aspects (e.g. retrieving reliable stru ctural information, probabilistic geometric inaccuracies, and material models) are presented in detail. The objectives were accomplishe d within the following scope: 1. The analysis was limited to high fidelity finite element simulations by means of a commercially available software package (LSDYNA), 2. Simplified steel framed buildings with seismic detailing were analyzed to assess the feasibility of energybased approaches to studying progressive collapse, 3. Calculation procedures for buckling energy were proposed for typical Wshape steel columns, 4. Collapse of a three story seismic steel building was simulated after the initiation of numerous localized failures to verify the proposed energybased analysis procedure. 22 PAGE 23 CHAPTER 2 LITERATURE REVIEW Overview Numerous incidents of progressi ve collapse attest to its danger. To name just a few: Ronan Point apartments in the United Kingdo m 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 Mu rrah Federal Building in Oklahoma City, in 1995, where the air blast pressure sufficed to de stroy only a few columns and slabs at lower floors, whereas the upper floors failed by progressi ve collapse; attack on the Pentagon (Mlakar, Dusenberry, and Harris, 2002); WTC collapse in 2001; residential buildin g 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 collaps e and thus prevent the aforementioned disasters. A structure should remain stable to allow fo r 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 pr ogressive collapse ha ve 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, 23 PAGE 24 2. Indirect design Preventing progressive collapse by specifying minimum requirements with respect to strength, stru ctural continuity, ductility, et c. Accidental load and the ensuing structural behavior are not explicitly evaluated, 3. Direct design. Considering resistance against progre ssive collapse and the ability to absorb damage directly by means of a nonlinear fin ite 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 threat s by limiting access to structure, tightening security checks, etc. 2. Specific Local Resistance. Hard spots are desi gned 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, redu ndant members, which should survive the accidental load. Indirect Design Indirect design aims at implementing good desi gn and construction practices, which boost structural safety. Indire ct design is more qualitative than quantitative in its nature. After each catastrophic collapse, reasons of a disproportionate collapse ar e identified and investigators recommend remedies and fixes for recognized probl ems. Examination of historical catastrophic failures and their effects on the design co mmunity exposes how the indirect design recommendations were acquired. 24 PAGE 25 Historical Cases Collapse of the 22story 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, sm ashing 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 (Griff iths, et al., 1968). Building joints between walls and slabs were constructed by filling voids with mortar and tight ening 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 wi th 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 bu ilding elements can better sustain deformations from the failure of a portion of the buildings stru cture. Transverse ties create cantilever action form adjacent walls. Vertical ties provide susp ension from panels abov e, peripheral ties hold floors together, and longitudinal ti es string floor planks large prestressed panels together (Pearson and Delatte, 2005). 25 PAGE 26 The Murrah building bombing resulted in recomm endations of designing for the loss of a column at the building perimeter without progr essive collapse and provisions of sufficient reinforcement for reverse bending in beams and slabs. On April 19, 1995, a truck lo aded with an ammonium nitrate and fuel oil (ANFO) bomb caused the collapse of fully half of the total floor area of the ninestory, reinforced concrete Mu rrah Federal Building in Oklahoma city. The extent of the collapse, resulting from the initial loss of up to three column s due to the original explosion, extended well beyond th e 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 buildings facade. Given its proximity to the blas t, it is generally believed that the concrete in the column was shattered by the blast, leaving on ly the columns bundled reinforcing steel. The short wall segment located only 7 [ft] (2.1 [m]) fu rther into the building was largely intact, as was the thirdfloor transfer girder Absent any alternative load pa th, 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 upw ard force on the floor slabs. As the thick floor slabs were not reinforced to resist upward pressures, differen tial 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 transver se 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 PAGE 27 gravity took over, beam/column connections fail ed in punching shear, and the floor structure draped as a catenary. Of a total of 20 bays in the footprint of the build ing, 10 collapsed over the full height of the building. Destruction of the Murrah Build ing 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 threedimensional space frame th at 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 abso rb overloads with large deformations while maintaining continuity; Beams and slabs must be sufficiently reinfo rced 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 belie f that strong structur al ties unconditionally improve the building safety against progressive collapse. Much attention has been paid to adopting seismic detailing for important buildings in nonseismic zones. Protecting the frame with mechanical fuses is in some ways in conf lict with the good seismic design practice of tying all components together as well as possible. Thus, the catastrophic failure of the Murrah 27 PAGE 28 Building revealed that provisions of sufficient ductility and continuity should not be used as a cureall against progressive collap se and that only direct colla pse 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 an d fire extinguishing. As generally accepted by the community of specialists in structural m echanics, the failure scenario was as follows (National Institute of Standards and T echnology, 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 insula tion was stripped, many structural steel members heated up to 600C, as confirmed by annealing studies of steel debris. The structural steel lost about 85% of its yield streng th at 600C; 3. Differential thermal expansion, combined w ith heatinduced 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 outofplane buckling of framed tube wall. The lateral deflections of some columns due to aircraft impact, the differential thermal expansion, and overstress due to load redist ribution also diminished buckling strength, 28 PAGE 29 4. The combination of the following seven effect s 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 inplane 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 onestory buckling), g. local plastic buckling of heated column webs. 5. The upper part of the tower fell, with little resistance, throug h 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 ex ceeded, by an order of magnitude, the energy that could be absorbed by limited plastic defo rmations and fracturing in the lower part of the tower (Bazant and Zhou, 2002). Although the structural damage inflicted by th e aircraft was severe, it was only local. Without the stripping of a significant porti on 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 cl aims that the structural system adopted for the TwinTowers 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 recomm endations learned from catastrophic collapses. It advocates design and construction practices, which eliminate roots and causes of 29 PAGE 30 historical failures and thus enhance structural safety. Co llapse of Ronan Point in London resulted in provisions of sufficient ductilit y 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 resulte d in provisions of sufficient fireproofing protection of steel members so that structures would remain stable for th e time needed to put out the fire. The main flaw of indirect design lies in its tendency to attribute fi ndings 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 fata l 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 pr edisposition 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 Unite d States or elsewhere. Building codes and standards resort to indirect desi gn and invariably treat general stru ctural integrity and progressive collapse in qualitative rather than quantitative term s. This is due to the current lack of insight into the nature of this phenomenon. The ASCE Standard 705, Minimum Design Loads for Bu ildings 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 PAGE 31 Point collapse. ANSI Standard A58.1 publishe d in 1982, section 1.3: General Structural Integrity, contained a more co mprehensive performance statement, and contained a greatly expanded commentary with references. Recomme ndations were made to provide a good plan layout, returns on walls, at leas t minimal twoway action for fl oors, loadbearing 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 co urse of time the commentary was shortened by eliminating the figures and other specific guidan ce but retaining the discussion of general design approaches to general structural in tegrity (Elingwood and Dusenberry, 2005). This gradual removal of detailed guidance has been caused by fading confidence that the specifics acquired from particular, historical co llapses can be generalized to all types of structures In 2005, the provisions in section 1.4 of ASCE Standard 705 were revised to reflect more recent information and to include more specific suggestions for the e nhancement of general structural integrity. A new s ection 2.5 (and commentary) was added to the load combinations section of ASCE Standard 795, which required a check of strength and st ability of structural systems under lowprobability 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 st ructural 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 Soci ety of Civil Engineers (ASCE), 2005). 31 PAGE 32 Section C2.5 of the commentary recommends th at after an element is notionally removed; the capacity of the remaining structure should be checked using the follo wing load combination: W 0.2+S)0.2orL(0.5+D1.2) or (0.9 (21) 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 buildi ng stability. The load combination in Equation (21 ) has an annual probability of being exceeded equal to 0.05. Structural Concrete (ACI 31802) and Commentary (A CI 318R02) address general structural integrity by specify ing prescriptive detailing requirements (American Concrete Institute (ACI), 2004). The commentary to sect ion 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 itsel f addresses detailing of reinforcement and connections, to effectively tie together the stru ctural members to improve integrity of the overall structure. The require ments address continuation of re inforcement through supports, the location and nature of splicing, and provisions fo r hooks at terminations. Special requirements are cited for precast constructi on. 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 stru ctural integrity. The Interagency Security Committee (Interagency Security Committee, 2001) issued specifications on blast resistance or other specia lized security measures. Progressive collapse analysis is handled indirectly, by reference to ASCE Standard 795 for specific details on 32 PAGE 33 prevention of progressive collapse. It allows pr ogressive collapse analyses for an effective live load that is as low as 25% of the code prescrib ed live load, and for enhanc ed ultimate strengths, to recognize that material strengths often are high er at strain rates associated with impact than they are at strain rates common for app lication 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 evalua te 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 loadcarrying element, is analyzed for the applied load: LDLPC25.02 (22) where LPC = load applied for collapse analysis; D = dead load; and L = designbase live load. The load factor of two on the combined dead and live loads can be t hought 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 dete rmined by the calculation of a demandcapacity ratio (DCR) for each primary and secondary structural element as: CEUDQQDCR / (23) 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 variet y of potential failure modes for steel elements. 33 PAGE 34 Limiting values of the DCR represent correla tion factors that might loosely account for the ductility (energy absorption) inherent in convent ional 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 798. The alternate load path proc edure requires notional removal of one primary vertical and one primary horizonta l element at key locations in th e structure. Connections must be able to develop the capacity of the weaker connected elemen t, and detailing must conform to requirements for seismic design. A capability to withstand load reversal s must be provided. Damage must be contained to the stories abov e 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 eff ect 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 even t or design the structure to with stand 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 i i i i fHPHDPDHFPP1)()()( (24) 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 colla pse, given that hazard Hi and local damage D both occur; The separation of P(FHi) = P(FDHi) P(DHi) and P(Hi) allows one to focus on strategies for reducing risk. P(Hi) depends on siting, controlling the us e of hazardous substances, limiting 34 PAGE 35 access, and other actions that are essentially i ndependent of structural design. In contrast, P(FHi) depends on structural design measures rangi ng from minimum provisions for continuity to a complete postdamage structural evaluation. The probability P(Hi) depends on the specific hazard. Limited data for severe fires, gas explosions, bomb explosions, and vehicular collis ions indicate that the event probability depends on building size, which is measured in dwelling units or square footag e, and ranges from about 0.23 x l06 [dwelling unit/year] to about 7.8 x l06 [dwelling unit/year]. Thus, the probability that a building structure is affected may de pend 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 appropriat e building regulation. If one were to set the conditional limit stat e probability, P(FHi) = 0.1~0.2 [1/year], the annual probability of structural failure from Equation (24 ) would be on the order of 107 up to 106, placing the risk in the lowmagnitude background. Design requirements corresponding to P(FHi) = 0.1~0.2 can be developed using firstorder reliabi lity analysis if the limit stat e function describing structural behavior is available. While current building standards ensure that building failures occur only rarely, no one knows exactly what a socially accepta ble building failure rate might be. However, there is evidence (Eli ngwood and Dusenberry, 2005) that th e risk (measured in terms of probability) below which society normally does not impose any regulatory guidance, is on the order of 107/year. Accepting this target value requires a sociopolitical decisi on that is outside the scope of this work. 35 PAGE 36 Direct Design: Finite Element Method Direct design considers resistance against progressive collapse and the ability to absorb damage directly by means of nonlinear finite el ement methods or any other available methods. The most straightforward approach is to us e nonlinear commercial finite element codes to simulate dynamic behavior after removal of one or more key members. Finite element codes such as LSDYNA, ABAQUS and so on, provide more sophisticated simulation capabilities every year. Moment resisting steel frames have been an alyzed in recent year s using commercially available finite element package ABAQUS (Kra uthammer et al., 2004; Lim and Krauthammer, 2006). Among other structures, a realistic 10story moment re sisting frame was modeled in Abaqus. One, two or three columns at the gr ound level were instantaneously removed at the prescribed time (following the quasistatic a pplication of gravity, dead and live loads). A numerical estimation of internal forces dur ing 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 initiati on as well as the eff ect of column removal time on displacements can be estimated from the computations. Connections play an important role in the overall structural behavior, and their prop erties significantly affect the structural behavior of individual members (Liu, et al., 2005). Whereas most of the connections can be treate d as moment resisting connections (assumed to transfer moments between members) and shea r 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 fa il 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 forcedisp lacement and moment36 PAGE 37 rotation relations using high resolu tion FEM models (solid elements). These relations were later used for springs properties at c onnections in simplified models. High resolution models of perimeter resisti ng frames were also reported by Khandelwal and ElTawil (2005). Columns and girders of th e perimeter resisting frame were modeled with shell elements. Existing general purpose fin ite 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 ad ditional parameters than just extra geometrical information in comparison to simplified representations. To na me just a few: the fracture criterion (depends on element size, geometric details such as b eam access hole and welded stiffeners, etc.), geometric imperfections (to produce more physical local behavior) and others. 37 PAGE 38 Static Procedures for Progressive Collapse Modeling In view of the aforementioned complicati ons and excessive la bor involved in high resolution modeling of progressive collapse; there is a widespread call for simplified procedures to assess structural resistance to progressive co llapse. Complex structural systems do not have an analytical, closed form solution, which would explicitly reveal the un derlying dynamics. On the contrary, computational mechanics result in vast numerical outputs, which are more difficult to generalize. Analytical solutions exist for sing le degree of freedom systems. Explicit solutions of such problems, in terms of symbolic para meters, provide insight into the investigated phenomena. Such results cannot, however, be uncritically extrapolat ed 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 respons e of a single degree of freedom (SDOF Figure 21 ) system to step loading ( Figure 22 ). Figure 21. Single degree of freedom system Response of a SDOF system without damping to a step load ( Figure 22 ) can be calculated by solving the following differential equation (representing equilibrium of forces and/or conservation of momentum): .0constgmFkxxm (25) m mass, k springs stiffness, x(t) spri ngs displacement, g gravity acceleration 38 PAGE 39 Figure 22. Loading applied to the SDOF syst em (relative time to the natural period T) Assuming the absence of an initial veloci ty and initial displacement, solution to Equation (25 ): Tm t m k tx, cos1)(0 k k F2 (26) Maximum response (without damping) is reached every T/2 and its magnitude is: ...,3,2,1,2)cos(1) 2 (0 0 max n k F n k F T ntxx (27) The system reaches similar displacements under step loading and under the application of a doubled quasistatic load, but their tim e histories differ significantly ( Figure 23 ). Displacement shown in Figure 23 is relative to static displacement = u dynamic / u static (assuming 5% of critical damping). Force in the spring under quasistatic doubled load is equal to the maximum dynamic 39 PAGE 40 force because the forcedisplacement spring relation ( ) does not depend on the loading (static vs. dynamic). xkFspring Figure 23. Displacement response to dynamic loads shown in Figure 22 The guidelines for equivalent static approach are essentia lly 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 overstrength ( ) factors are specifie d, typically 1.25 for reinforced concrete and 1.3 for struct ural steel. St rength reduction ( ) factors are the ACI and AISC values (typically 0.9 or 0.85), 40 PAGE 41 4. Deformation Capacities. Only flexural behavior. For sh ear effects, in both concrete and steel, zero ductility is assume d, 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 24 Therefore, when comparing the simulated maximum forces to static capacities of desi gned members, caution and engineering experience must be exercised. First yield Strain hardening Ultimate strength F o r ce Ductile limit Residual strength With cycling D e f o rm at i o n Initial stress Figure 24. Typical inelastic forcedeformation 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 struct ures (Powel, 2005). Kaewkulchai and Williamson (2004) reported that static, material nonlinear analysis produced unconserva tive results for different two dimensional frames as compared to their dynamic, timehistory analysis. Their analysis showed that static loading applied to the damaged stru cture (e.g. with one column removed) resulted in 41 PAGE 42 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 repres entative comparison of demand/capacity (D/C) results for one of the models is presented in Table 21 The ratio of dynamic to static results is the actual dynamic am plification factor. Howe ver these factors vary considerably between internal forces, displacemen ts and rotations (even in a single analysis for the same structure). Additional variation has be en observed between various structures (Ruth, et al., 2006). Table 21. Steel building: sel ected results from nonlinear dy namic 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 almo st impossible. Therefore, the dynamic amplification factor approa ch proves unreliable for a realistic assessment of structural resistance to progressive collapse. Unfort unately there is ungrounde d belief among practicing engineers that static analysis with a dynamic am plification factor of 2 is overly conservative (Marjanishvili, 2004) and its value could ev en be reduced to 1.5 (Ruth, et al., 2006). The poor agreement between timehistory dynamic analysis and static analysis with amplified static dead loads to account for inertia arises from the multidegree of freedom 42 PAGE 43 (MDOF) nature of real systems. Whereas a dyna mic amplification factor (DAF) of two is an excellent approximation for a single degree of fr eedom (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 21 ). Therefore, static methods with a DAF shall not be used fo r the design of progressive collaps e 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 (energeia), which appeared for the first time in the work Nicomachean Ethics (Aristot le, translated by Jonath an Barnes, 1984) of Aristotle in the 4th century BC. Several differe nt forms of energy exis t to explain all known natural phenomena. These forms include (but ar e not limited to) kinetic, potential, thermal, gravitational, sound, light, elastic, and electroma gnetic energy. Although certain energies can be transformed to another, the total energy remains the same. This principl e, the conservation of energy, was first postulated in th e early 19th century, and applie s to any isolated system. The concept of energy emerged out of the id ea 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. GustaveGaspard Coriolis de scribed "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 calor ic) or merely a physical quantity, such as momentum. William Thomson (Lord Kelvin) amalga mated all of these laws into the laws of thermodynamics, which aided in the rapid developm ent of explanations of chemical processes 43 PAGE 44 using the concept of energy by Rudolf Clausius, Josiah Willard Gibbs, and Walther Nernst. It also led to a mathematical formulation of th e concept of entropy by Clausius and to the introduction of laws of radiant ener gy by Joef Stefan. (Smith, 1998). During a 1961 lecture (Feynman, 1964) for undergraduate students at the California Institute of Technology, Richard Feynman, a celeb rated 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 ex ception to this law; it is exact, so far we know. The law is called conservation of energy; it st ates that there is a ce rtain quantity, which we call energy, that does not change in manifold ch anges which nature undergoes. That is a most abstract idea, because it is a math ematical principle; it says that there is a numerical quantity, which does not change when some thing happens. It is not a desc ription of a mechanism, or anything concrete; it is just a st range fact that we can calculate some number, and when we finish watching nature go through her tricks and cal culate the number again, it is the same. The total energy of a system can be subdivi ded and classified in various ways. For example, it is sometimes convenient to distingui sh potential energy (which is a function of coordinates only) from kinetic en ergy (which is a function of c oordinate time derivatives only) and internal energy (which is a function of body deformations). The transfer of energy can ta ke 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 "sys tem" and adjacent regions is work. A familiar example is mechanical work. In simple cases this is written as: WE (28) E = the amount of energy transferred W = work done on the system 44 PAGE 45 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 ri gid 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 th e work done upon it, although that energy appears only in the form of heat. Deformation Work (Internal Energy) A simple bar in tension ( Figure 25 ) is discussed herein to intr oduce the concept of internal energy (deformation work). This definition is used by LSDYNA to report energy results. The bar has crosssectional area A, length L, volume V and is loaded axially with tensile force P. L P Figure 25. Simple bar in tension The internal energy (deformati on work) of the bar is defined as its strain energy: 45 PAGE 46 dVdE int (29) For the given bar under uniform tension, th e internal energy (deformation work) is: dALEint (210) An elastoplastic material is characterized by stressstrain rela tionship depicted in Figure 26 The internal energy (deformation work) of the bar under uniform tension is purely elastic in phase 01 and exhibits elastopla stic behavior in phase 12. 01 2y E tEstrainE Figure 26. Stress vs. strain relationship of the rod The incremental stressstrain relationship in elastic phase 01: E d d (211) = strain increment, d = stress increment and E = elastic modulus. d Hence, the internal energy (deforma tion work) in elastic phase 01: E ALd E ALdALE 2 2 0 10 int (212) The incremental stressstrain relationship in elastoplastic phase 12: 46 PAGE 47 tE d d (213) Thus, the internal energy (deformation work) in elastoplastic phase 12: t y y tEE ALd E d E ALdALEy y2 22 2 2 0 21 int (214) The total strain can be decomposed into elas tic (recoverable) and pl astic (irrecoverable): peddd (215) The irreversible, plastic strain component in phase 12 is: d EEE d E d dddt t e p 11 (216) The plastic portion of internal energy (def ormation work) can be calculated as follows: 2 112 2 int y t t p pEE ALd EE ALdALEy (217) Elastic energy in elastoplastic phase 12: 2 11 2 22 2 2 2 2 intint int y t t y y p eEE AL EE ALEEE (218) E AL EE ALEy y e2 2 22 2 2 2 int (219) Only the elastic portion of internal energy (d eformation work) can be considered as stored energy because it can be potentially retrieved. If a bar is unloaded, only the elastic portion of bars internal energy will be recovered. The plastic portion of deformation is irreversibly 47 PAGE 48 dissipated as heat. Figure 27 depicts decomposition into elastic and plastic components. Plastic energy corresponds to the irreversible strains. On ly elastic energy is pote ntially recoverable. y E tE elastic p lastic E ep Figure 27. Strain and energy decompositi on 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 ri gid 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 ac ting 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. Ki netic energy of a simple bar is defined as: dVvEkin22 1 (220) = mass density, v= particle velocity. 48 PAGE 49 Energy Based Procedures for Progressive Collapse Modeling Energy based procedures to asse s resistance to progressive colla pse directly originate from the analysis of energy flow in a sing le degree of freedom system (SDOF, Figure 28 ) during downward motion caused by the gravity. Figure 28. Released mass falling on the spring If a mass resting on a spring (with stiffness k) is instantaneou sly released, gravity forces will produce a step load (m = mass [kg], g = gravita tional 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( 22maxxxforkx xmg 12 2xm (221) However, when the mass reaches its ma ximum deflection, its velocity is 0. 0)( )(max max max txxtx (222) Thus at this instance released gravitational energy is equal to absorbed internal spring energy and kinetic energy vanishes because the velocity is 0. 2 max max2 1 kx mgx (223) 49 PAGE 50 Therefore, in order to find maximum displacement, it is sufficient to find the displacement satisfying Equation (223 ). The maximum displacement calculated by solving Equation (223 ) and its corresponding force experienced by the spring are: mgkxF k mg x 2 2max max max (224) Dusenberry and Hamburger (2006) attempted to generalize the above SDOF solution to MDOF. If the structure is able to arrest collaps e, kinetic energy at its fi nal deflection vanishes. In other words, the released gravitational energy equals the absorbed stra in energy, to satisfy the conservation of energy. However, the maximum de flected shape corresponding to failure is not known in advance, unless dynamic, timehistory analysis is performed. To illustrate this static pushdown procedure, a grillage structure depicted in Figure 29 was analyzed after removal of a centr al support by Dusenberry (2006). 40 [kip] Girder Beam 1.29 [in] Figure 29. Deflected shape under static gravity forces In order to estimate the final deflected shape, the structure was stat ically pusheddown in bays adjacent to the removed column. Pushdown was stopped when released gravitational energy equaled absorbed strain ener gy. If internal forces at th is deformed state were supported by the structure, it was conclude d that the structure was able to survive the considered sudden column removal. If such a deformed state produced internal forces greater than carrying 50 PAGE 51 capacities or released gr avitational 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 22 Table 22. Calculation of Force Ratios, R1,i in Phase 1 (Dusenberry and Hamburger, 2006) Member Location Plastic capacity FPi [kipft] 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 j, where j = stage of loading) occurred simultaneously at the center and end suppor ts of the girder and has a value of: 64.1)min(,1 1 iR (225) 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 initia lly fixed connections were changed to hinges, when of static gravity load was a pplied. The corresponding deflection at the center of the grillage was (64.1)min(,1 1iRFigure 210 ): ][4.5][12.2,11,1cminiE ie (226) ie ,1 = displacement at ith node, corresponding j=1st set of hinge formations, iE ,1 = displacement at ith node, corresponding to static gravity load; The effective forces causing this deformation were: ][8.291][6.65][4064.11,1kN kip kip gmFi ie (227) 51 PAGE 52 Figure 210. Loaddisplacement during pushdown static analysis It should be noted that the above force has no physical meaning in re lation 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 quasistatic application, no kinetic energy wa s produced. Therefore, the increment of the strain energy was calculated using the area under the forcedeformation curve: i iEi iE i igm gm SE Energy Strain1 ,1 1 ,11 1 1 102 )( 2 n n 21 1 (228) Since the considered example ha d 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][5.69 21,111 10mkN inkip gm SE Energy StrainE 12 (229) 52 PAGE 53 Simultaneous with the energy absorption, potential, gravitational energy was released: ][57.9][7.84 )(1,111 1 ,11 10mkN inkip gm gm PE Energy PotentialE n i iE i (230) Since more energy was released by gravity than absorbed by strain energy, the surplus resulted in kinetic energy: n i iE igm KE Energy Kinetic Energy Kinetic Energy Strain Energy Potential1 1 ,11 10) 2 1( (231) ][72.1][2.15) 2 1(1 1,11110mkN inkip gmKEE (232) Kinetic energy is present in the system after the formation of the 1st hinge, and collapse further progressed. Hinges were placed at lo cations 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 re sisting 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 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 82.1 min min,1 ,01 2 ,0 iE i Pi iEj ijPi jF FF F FF (233) The new effective deformations were calculated as follows: ijjEjjjej ,0, (234) ][14.7][81.21,2cm ine (235) 53 PAGE 54 In phases after the formation of hinges, the energies were: n i ijeieji jjgm PE Energy Potential1 ),1(, )1() ( (236) n i ijeiejijj jjgm SE Energy Strain1 ),1(, )1( )1(2 1 (237) n i jj ijeieji jjgm KE Energy Kinetic1 )1( ),1(, )1(2 1 1) ( (238) With the deflected shape at the formation of the second yield hinge defined, potential energy, strain energy and kinetic energy were estimated: 1,2 e][57.029.272.1 ][29.2 ][27.1342.585.7 ][42.5 ][70.1213.357.9 ][13.320 21 20 21 20 21mkN KE mkN KE mkN SE mkN SE mkN PE mkN PE (239) The calculated strain energy accumulated by th e system at the end of the second phase exceeded the change in potential energy, and th e calculated kinetic energy was negative. Of course, kinetic energy cannot be less than zero: th is negative value meant that the structure has arrested collapse prior to forma tion of the second set of hinges. The aforementioned approach can be generalized to a multidegree of freedom system (MDOF). Mass is assumed to be lumped in each columnslab connection. Such an MDOF approach would produce analogous formul as with summation throughout time (index j) and throughout the structure (index i). In this ap proach, the determination of collapse potential is resolved by calculating the sum of the kinetic en ergy changes in each phase, and comparing it to zero. k j n i j j ijeieji kgm KE11 )1( ),1(, 02 1 1) ( (240) 54 PAGE 55 The accuracy of the aforementioned method de pends on the correctness of the assumed deformed shape. It can be either conservati ve 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 resu lt 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. Th e formulation of the power flow analysis is based on a differential, controlvolume 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 cont rol volume V of a physical system. One considers the timeaveraged vibrational energy pe r 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 S ung, 1987) to result in the power balance: dissPq t w (241) tw Which relates the quasisteady 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 PAGE 56 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 exam ined 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 th e 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: tj cexxFtxu t Stxu x SE )(),( ),(0 2 2 2 2 (242) EcS is the rod stiffness per unit leng th (S = crosssectional area), S is the density per unit length ( = density per unit volume), u(x,t) is the longit udinal displacement, F (xx0)ej t is the harmonic point force applied at point x0. The general solution of Equation (242) is: tjjkx jkxeBeAetxu) (),( (243) Where k is a complex wave number, and c is the phase speed in the rod (where c2 = E/ ). 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 writ ten in terms of displacement as: ),(),( txu t txu x ESq (244) The total energy density at a point is the sum of the strain and kinetic energies: TVe (245) 56 PAGE 57 The displacement solutions can be used to develop expressions for the energy density in the rod. The strain (V) and kine tic (T) energy densities are: 2 2),( 2 1 ),( 2 1 txu t Stxu x ESe (246) In lightly damped structures, th e hysteretic damping coefficient << 1, and thus the imaginary part (describing damping contributio n) 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: xk xkeBeA c ESq2 22 2 2 22 1 (247) and xk xkeBeASe2 22 2 2 2 22 1 (248) Where the operator indicates a time averaged quantity. (247 ) and (248) can be used to develop the simple relationship between local values of time averaged power and energy density in a rod: e dx dc q (249) The time averaged power is proportional to th e gradient of the time averaged density. Equation (249) 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 (249 ). The only approximation made wa s that hysteretic damping is small, <<1. To develop the governing equations which model the energy density in a rod, it is necessary to perform an energy ba lance on a differential rod element. The time rate of change of 57 PAGE 58 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 ca n be written as: dissq x e t (250) Where diss is the power dissipated within the differential control volume. The time derivative of energy density is zer o since the power is being cons idered here for a steady state condition. To solve the particular problem at hand: 02 22 disse dx dc (251) To demonstrate the applicabil ity of the proposed mathematical model, the energy flow relation between two coupled rods ( Figure 211 ) was analyzed by Wohlever (1992). Energy flow results for these coupled rods are shown in Figure 212 F ( xx0 ) e j t x1=0 x2=5 x3=10 1 2 Figure 211. Spring coupled, longitudinally vibr ating rods excited by a harmonic point force 0.0 0.5 1.0 5.0 10.0 0.0 0.0 5.0 10 .0 2.5 1.25 0.0 Length Length Powe r Energy density ( x103 ) exact simplified Figure 212. Comparison of exact and simplif ied solutions for the coupled rod system 58 PAGE 59 Simplified Methods for High Rise Buildings Baant and Verdure (2007) proposed an energy based analysis of progressive collapse of highrise structures. Their concept was illustrate d using the collapse of the World Trade Center (WTC) towers in 2001. Collapse was divide d into crushdown and crushup phases ( Figure 213). When columns at the damaged floor buckle, they yield to the gravity of the upper portion of the structure and a fracture wave propagate s. 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 213 ). CrushDown Phase CrushUp Phase Figure 213. 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 co llapse of high rise structures. The mass of columns was assumed to be lumped, half and half, into th e mass of the upper and lower floors. To analyze progressive collapse, the complete loaddisplacement diagram F(u) must be known ( Figure 214 ). It begins by elastic shorte ning, 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 PAGE 60 F0 mg u0 ucFloor displacement, u ufh h Deceleration p hase Acceleration p haseCrushing force, F(u) Fc Maxwell line Figure 214. Typical loaddisplacemen t diagram of columns of one story When the difference )()(zgmuF )()(zgmuF causes deceleration of mass m(z) if positive and acceleration if negative. The equatio n 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: )( )( zm uF gu (252) The energy loss of the columns, (u) up to displacement u is: )()( )()())()(()(0 0uWuduFwherezgmuWudzgmuFuu u u u (253) z = constant = column top coordinate; W(u) = energy dissipated by the column s = area under the loaddisplacement Figure 214 ; 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 thermodyna mic viewpoint, as the in ternal energy, or free energy, and thus (u) represents the potential energy loss. If F(u) < gm(z) for all u, (u) continuously decreases. If not, then (u) first increases and then decreases during the collapse of each story. 60 PAGE 61 The Criterion of Arrested Collapse Collapse will be arrested if and only if the im parted kinetic energy from collapse through a previous story(ies) does not suffice for reachi ng the interval of accelerated motion, i.e., the interval of decreasing (u). So, the crushing of columns within one story will get arrested before completion if and only if )()()( zgmuWuEc c K (254) EK = kinetic energy of the impacting mass m(z) (uc) = net energy absorption up to uc during the crushing of one story This is the criterion for preventing progressi ve collapse from starti ng. Graphically, this criterion means that EK must be smaller than the area unde r the loaddeflection diagram lying above the horizontal line F = gm(z) ( Figure 214 ). 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: pgWW (255) f guzgmW )( = loss of gravitational energy when the upper part of the tower is moved down by distance uf = final displacement at full compaction; fu f pduuFuWW0)()(= area under the complete loaddisplacement 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 th e next was ensured. The aforementioned representation was homogenized to the global continuum model using energetically equivalent mean crushing force. A nonsoftening energetically equivalent characterization of snapthrough in discrete elements was pursued. It corresponds to 61 PAGE 62 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 loaddisplacement diagram F(z) can be replaced by a simp le diagram that is storywise energetically equivalent, and is represente d by the horizontal line F = Fc. Here Fc is the mean crushing force (or resistance) at level z, such that the dissipa ted energy per story repres ented by the rectangular area under the horizontal line F = Fc, is equal to the total area Wp under the actual loaddisplacement curve: fu ff p cduuF uu W F0)( 1 (256) The energyequivalent replacement avoids unsta ble snap through and is analogous to what, in physics of phase transitions, is called the Ma xwell 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 th e collapsing story from the beginning to the end of column crushing as long as Fc is not large enough to ar rest 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. OneDimensional continuum model for cr ushing front propagation is based on the following simplifying hypotheses: (1) the only disp lacements 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 im plies that the blocks in Figure 215 may be treated as rigid, i.e., the deformations of the blocks away from the crushi ng front may be neglected). (3) The relation of 62 PAGE 63 resisting normal force F (transmitted by all the columns of each floor) to the relative displacement u between two adjacent floors obeys a known loaddisplacement diagram ( Figure 214), terminating with a specified compaction ratio (which must be adjusted to take into account lateral shedding of a cer tain known fraction of rubble outsi de 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 onedimensionally idealized progress of collapse of a tall building (of initial height H) is shown in Figure 215 where: = z(t) = coordinate of the cr ushing front measured from the initial tower top; = y(t) = coordinate of the crushing front me asured from the current tower top; H z0 s0 z A C B A C z0s = s0 y0=z0 (Hz0) y r0 B B B B B C C y r= r0 z0 H ( a ) ( b ) ( c ) ( d ) (e) CrushDown Phase CrushUp Phase dz/dt Figure 215. Continuum model fo r 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 th e story is reduced to h, with denoting the compaction ratio (in finitestrain theory, is called the stretch). After that, the load 63 PAGE 64 can increase without bounds. In a onedimensional model pursued here, one may use the following estimate 0 11 V Vout (257) V0 = initial volume of the tower; V1 volume of the rubble on the ground into which the whole tower mass has been compacted; out = correction representing mainly the fraction of the rubble that ha s been ejected during collapse outside the perimeter of the towe r and thus does not resist compaction. The rubble that has not been ejected duri ng collapse but was pushed outside the tower perimeter only after landi ng on the heap on the ground should not be counted in out. 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 th e collapse, is an upper bound on V1, but probably much too high a bound for serving as an esti mate (Bazant and Verdure, 2007). Let = ( ) = initial mass density at coordinate = continuously smeared mass of undisturbed tower per unit height During crushdown, the ejecte d mass alters the inertia and weight of the moving compacted Part B, whic h requires a correction to m(z), whereas during crushup no correction is needed because Part B is not moving: d zmwherezzford d zmz z z out z 0 0 00 0 0 0)()( )( 1)()( (258) The initial location of the first floor crashing into the one below is at = z(t0) = z0 = y0. The resisting force F and compaction ratio are known functions of z. A and C label the lower and upper undisturbed parts of the tower, respectively. B denote s the zone of crushed stories compacted from initial thickness s0 to the current thickness, s(t): 0 )()( )()(0ztzd tstz z (259) 64 PAGE 65 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)()()()()(0tz ztztz t tstz t tvC (260) The differential equation for z(t) crushdown can be obtained from a dynamic free body diagram (Figure 216 ). In the crushdown phase, the compacted Zone B and the upper Part A of the tower move together as one rigid body accreting mass, with combined momentum: zzmtvzmC)1()()()( (261) CrushDown CrushUp tz cvm m g cFcF cvm mg cFcF ty Figure 216. Free body diagram in the crushdown and crushup 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 dAlembert principle yields the following differential equation for compaction front propagation in the crushdown Phase I of progressive collapse: ) ()()()1()( down crushzFgzm dt zm dtc dz d (262) The initial conditions for the crushdown Phase I are z = z0 and v = 0. Downward propagation will start if and only if: 65 PAGE 66 )()(0 0zFgzmc (263) The differential equation for y(t) crushup phase can be obtained from a dynamic free body diagram ( Figure 216 ). In the crushup phase, the crushing front at = y is moving up (debris is stacking up) with velocity yzvstacking )( (264) Thus the downward velocity of Part C is: yzyzyvC )(1)( (265) It should be noted that part C is decelerating upon c ontact with stationary rubble stack B. Thus, using the dAlembert principle, a force equa l to inertia is applied downward to bring the system into equilibrium. The differential equati on of progressive collaps e in the crushup phase: )()( )(1)(upcrushyFg dt dy y dt d ymc (266) 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 res ponse 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 dur ation 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 th e crushing of one story (representing the energy dissipated by the complete crushing a nd 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 PAGE 67 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 en ergy, not the strength, nor stiffness. (Bazant and Verdure, 2007). It is also instructive to cons ider progressive collapse also from the point of view of an implosion contractor who regularly demo lishes buildings through explosivesinduced 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 wa iting 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, crea ting the maximum amount of fragmentation, and minimizing the potential fly of debris. Conversel y, the structural designer wants to contain the potential energy. However, it should be noted that demolition diffe rs 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 highrise slender structures are at the highe st risk of abnormal loading resu lting in progressive collapse. Limited data indicates that th e probability of abnormal load ing depends on building size, measured in dwelling units or square foot age, and ranges from approximately 0.23 x l06 to 67 PAGE 68 7.8 x l06 [dwelling unit/year] (Elingwood and Dusenberry, 2005). Thus, high rise buildings with numerous dwelling units are the most likely st ructures to face catastr ophic/abnormal loads. On the other hand, dynamic modeling of structural response to such loadings and progressive collapse resistance assessment are highly expens ive for such structures. Difficulty arises specifically from the overwhelming number of stru ctural members in a high rise building, which easily can reach thousands of elements, connectio ns, 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, detai ling, etc) and direct design (e xplicit 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 simp licity, 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 desi gner (Powell, 2005). On the other hand, energy based methods proposed by Dusenberry a nd Hamburger (2006) as well as Baant and Verdure (2007) have significant potential to provide simple yet realistic assessment tools for 68 PAGE 69 practicing engineers because en ergy based approaches capture the physics of progressive collapse. 69 PAGE 70 CHAPTER 3 RESEARCH APPROACH Proposed Theory The following energy based theory is proposed to provide insight in to the phenomenon of progressive collapse. A large amount of gravitational energy, hereinafter called potential energy, is present in a building. Duri ng construction, cranes lift struct ural 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 a nd absorbed as internal (strain) energy. A schematic structure, shown in Figure 31 is used to illustrate the stable, steady energy state, which remains throughout th e typical life of a structure. Figure 31. Energy state in a hypothetical structural system The energy in the stable steady stat e for the hypothetical structure from Figure 31 is: ]internal[]potential[ 2 12 1 total c n i totalE hkhhMgE (31) 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 potentia l energy (gravity acting on the mass of the 70 PAGE 71 structure through deflections). Simultaneousl y, 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 di fference between the released energy and the absorbed deformation energy is tr ansferred into kinetic energy. In other words, the potential energy will be released upon failure (change of the potential energy) a nd will flow (external work) to redistribute energy until either total failure or a new stable state is achieved. In generic energy transformations ( Figure 32 ), 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 recove red under favorab le conditions. Energy Source Energy Storage Energy Loss as Heat Recoverable Output Energy Transfer Figure 32. Energy transforma tion in a generic system Potential energy is an energy source in the proposed approach ( Figure 33 ). External work is a measure of the energy transfer from the potential energy rese rvoir to the building. Energy absorbed through deformations is hereafte r called internal energy (deformation work). Internal energy can be decomposed into recovera ble (elastic) and dissipated energy (plastic). Energy resulting from moti on (velocities) is hereaf ter called kinetic energy. 71 PAGE 72 Potential Energy Energy Loss (Plastic Energy) External Work Energy Transfer Kinetic Energy Internal Energy (Deformation Work) External Work Figure 33. Energy transformation between pot ential, internal and kinetic energies Localized failure triggers the release of poten tial energy. The structur e absorbs the energy. Part of the absorbed energy is lost through plas tic 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 measur e of energy transfer fr om 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 bui lding, there will be kineti c 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 abso rbed 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 34 72 PAGE 73 Figure 34. Energy based appro ach to progressive collapse The hypothetical collapse of a high rise building ( Figure 35 ) 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 ener gy. The energy propagation z one, which enlarges as time progresses, is depicted with a shaded region in Figure 35 Therefore, the first goal is to understand how the energy propagates after the in itiating failure. Such propagation will depend on structural configuration, memb er 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 PAGE 74 simulating a number of typical st eel framed structures, using finite element analysis software (LSDYNA), and examining the energy flow. Figure 35. High rise buildi ng 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 st eady state if released potential energy and energy transferred from othe r members is equal to the energy absorbed in this member. In other words, if there is no ki netic 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 ende d 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 PAGE 75 produced, and collapse propagates through motion. T hus generation of kinetic energy indicates an unstable member state An imaginary storywise collapse is utilized to better illustrate the aforementioned steady state criterion. The storywise collapse can be thought of as be ing initiated by a severe fire uniformly distributed through the floor. In th is 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 columns capac ity enough to initiate the collapse through one story, then the mass supporte d 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 ener gy absorption mechanism will be activated. A schematic of energy distribution for collap se through the initiati ng story is shown in Figure 36 The slope of the released gravitational energy per unit displacement equals the dead weight distributed over the tributary area of the collapsing floor. Figure 36. Energy flowdisplacement of single column (H = height of story) 75 PAGE 76 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 de notes first collapsing story). It is expected that the force resistance of columns, which is the slope of the absorbed energy, will decrease with incr eased displacements due to postbuckling 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 r each a stable configuration and its collapse propagates. If crushing of the initiating story is followed by the failure of the underlying story ( Figure 37 ), 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 37. Collapse propagates into consecutive story 76 PAGE 77 A schematic example of energy distribution in arrested collapse vs. displacement through the consecutive story is shown in Figure 38 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 diss ipate the released gravitational energy, then the surplus of kinetic energy will increase, and there is no state in which kinetic energy vanishes ( Figure 39 ). Figure 38. Energy release with energy absorption by the crushing story itself Figure 39. Collapse propagation du e to insufficient energy absorption 77 PAGE 78 The proposed energy based steady state criterion ca n be written as the following postulate. A structural member will attain a stable stat e if and only if there exists a displacement uarrest(t) such that energy transferred from adjacent memb ers 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 kineticEstrainE membertheto transfered E potential releasedE (32) 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 indi cates that the structural system has the ability to absorb the released gravitational energy. Tracing energy absorption requires time integr ation 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 depe ndent. Thus, the increment of kinetic energy attained over time : dt 2 ),(2uM dt d uuE dt d uMgE dt d E dt dabsor out in (33) Generally speaking, Equation (33 ) 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 observati ons and compatibility conditions. Eabsorb represents material relation, which describes pertinent energy absorption of a particular structural member. 78 PAGE 79 Virtual Experiments Overview Physics based simulation techniques were used to research the energy flow and behavior of major structural members during progressive collaps e propagation or arrest. Representative steel structures were selected for analysis. All possible efforts were made to represent the physical behavior of real structures as ac curately 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 preloading 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 ch aracterized by large deflections, pronounced material nonlinearities and c ontact between members. LSDYNA finite element code (Hallquist, 2006) offers sophisticated algorithms, which enable numerical modeling of collapse propagati on or arrest. Therefore, LS DYNA was chosen to perform simulation studies in this work. Material Modeling The finite element code, LSDYNA (Hallquist 2006) was employed to model structural response to abnormal loading. Progressive co llapse is a dynamic phenomenon with strong material and geometric nonlinearities. Thus, a su fficiently sophisticated analysis method has to be applied in order to reprodu ce the physical behavior of a bu ilding. A typical stressstrain relation of A36 steel, as repor ted by Salmon and Johnson (1990), wa s utilized in the models. A large strain, piecewise linear, material mode l 24 from the LSDYNA (Hallquist, 2006) library has been applied to represent A36 steel materi al behavior. Strength enhancement under high 79 PAGE 80 speed strains was not included in the material modeling. Model 24 operates on logarithmic stress and strain measures, thus it accounts for larg e strain effects. The piecewise linear model is limited only to the monotonically increasing logarithmic stresslogarithmic strain behavior due to the restrictions of stress based plasticity. The loadingunloading criterion must be specified, such as: unloading ford loadingford 0 0 (34) Loadingunloading criterion in the stress space is unable to differentiate between elastic unloading and plastic softening (Khan and Hu ang, p.278, 1995). Thus, only the monotonic stressstrain portion of the materi al relationship is allowed as input because the yielding is evaluated in the stress space. The initial yield point for A36 steel is For stress values lower than the current yield stress deformation is linear a nd the correspondence between ] [250][360MPa ksiYn y and is one to one. For stress values higher than the deformation is nonlinear and history dependent. A36 steel properties, generalized to the threedimensional case, were represented with MaxwellHubervon Mises yield criterion with piecewise orthotropic hardening. In other words, deviatoric stresses are determin ed that satisfy the yield function: n Y 0 3 2 12 p effy ijijSS (35) ijS= deviatoric stress; = effective plastic strain; p effy = uniaxial yield stress The growth of subsequent yield surface is ch aracterized 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 PAGE 81 Logarithmic, large deformation stress measures were used in the yield criterion. LSDYNA offers elements with large strain cap abilities, which are compatible with the large strain plasticity model. Since stresses and st rains are calculated with respect to the current configuration and not with respec t to the initial configuration, all input stressstrain material information needs to be converted from engin eering into logarithmic measures. Uniaxial engineering stressstrain data shall be converted to logarithmic measures as follows (Khan and Huang, p.4, 1995): eng eng l lo eng ol l A P AllA A P l l l dl 1 1lnln00 00 log log (36) Engineering and logarithmic stre ss strain curves are shown in Figure 310 0 10 20 30 40 50 60 70 80 00.050.10.150.20.25Yied stress y[ksi]Effective plastic strain eng log LSDYNA Figure 310. Typical stressstra in relationship for A36 steel Strains are updated at each time increment (rela tively small in explicit analysis as opposed to the implicit larger steps). Although there is no consensus on the best objective corotational 81 PAGE 82 stress rate (Khan and Huang, p.242, 1995), the Jauma nn rate of the deviatoric stress is utilized by LSDYNA: pijppjipijijSSSS (37) ijS= corotational increment of deviatoric stress; = increment of deviatoric stress ijS pj= spin tensor (rigid rotation) It is an exact transformation (Gurtin and Spear, 1983) of th e 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 configuratio ns (differential deformation re latively small with small time increments), it provides suffici ent accuracy for engineering a pplications (Hallquist, 2006). First, the trial deviatoric stress value is calc ulated 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: pi n jppj n ipij n ij n ij n ij n ijSSGSSSS 21* (38) 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 ij = increment of deviatoric strain The effective trial stress is defined by 2 1 1*1* *2 3 n ij n ijSSs (39) 82 PAGE 83 If exceeds yield stress the Hubervon Mises flow rule s*n y(35) is violated. The trial stress shall be scaled back to the yield surface by means of the following radial return algorithm: 1* 1 1 n ij n y n ijS s S (310) In order to compute the yield stre ss 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 LevyMises 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 n y p n y p effEG s 3* (311) G = shear modulus; p E = current tangent modulus (in the effective strain space) Total plastic strain is updated: p eff np eff np eff,1, (312) The yield stress evolution is calculated usi ng the current tangent modulus of the user supplied piecewise linear yield stresseffective plastic strain relationship ( Figure 310 ): p eff npn y n yE 1 (313) Once the uniaxial yield stress is known at step n+1, radial retu rn of the deviatoric stress to the yield surface is carried out (310 ). Thus, for the given displacements and element velocities, stresses and internal forces are calculated. Material failure is characterized by the prescr ibed effective plastic failure strain. For the A36 steel, the failure strain value was utilized (2.0,failurep effFigure 310 ). 83 PAGE 84 Structural Modeling A feasibility study was initially conducted in order to fi nd 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 (Hall quist, 2006) were used for the 3D modeling of single column buckling to provide a benchmark solution for more computationally efficient models. The element is defined by eight nodes ha ving 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 311 Figure 311. Threedimensional solid cube buckling modeling Although 3D cubes provide high fidelity of simu lation, computational expense to simulate a full structure is prohibitive. Therefore, HughesLiu beam elements were investigated as a more 84 PAGE 85 computationally efficient alternative. However, it had to be ensured that the simplified approach utilizing fiber type Hughes Liu beams is sufficien tly accurate and captures real column behavior. The HughesLiu is incrementally objective (rigid body rotations do not ge nerate strains), allowing for the treatment of large strains. It al so includes finite transver se shear strains. The HughesLiu beam element is based on a degenerati on of the isoparametric 8node solid element. To degenerate the 8node brick geometry into 2node beam geometry, the four nodes at 1 and at 1 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 i ijy u G (314) where are the incremental displacements, and are the deformed coordinates. The incremental strain and spin tensors are define d as the symmetric and skewsymmetric parts, respectively, of : iu Gjyij jiij ij jiij ijGG GG 2 1 2 1 (315) The incremental spin tensor ij 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 integra tion along the axis and multiple points in the cross section. For the Wsection (wide flange secti on), trapezoidal integration thr ough the section by means of nine integrations points is carried out ( Figure 312 ). 85 PAGE 86 wt ftd t sw123456 8 9 7 Figure 312. Integration scheme for W section in the HughesLiu beam element Columns were modeled as elements capable of exhibiting a variation of strains and their corresponding stresses through the section. Th us, the HughesLiu 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 cente rline were directly modeled. Material failure was controlled by the prescribed va lue 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 a pproximately as a 6 [in] (150 [mm]) thick shell with uniform material properties inferred from sm earing mechanical properties of the composite deck. A bilinear plastic mode l 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 cutoff 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 31 In conjunction with the custom inte gration, laminated glass (material model 32) was employed. Steel material model was used for the bottom la yer; whereas other laye rs were modeled using 86 PAGE 87 concrete material properties ( Table 31 ). 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 cutoff strain of 0.01. Table 31. Lightly reinforced slab modeling Layer coordinate Area fraction Young modulus [ksi] Poisson ratio Yield stress [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 HughesLiu beam formulation. The simulation was force controlled with monotonic, quasista tic load application. Force imperfection at midheight was introduced to initiate the buckling. Imperfection force was selected to produce initial camber of 400/1500 [cm] at the midheight. Although the brick elements provide higher re sult resolution, th e displacementforce histories comparison in Figure 313 shows that overall column resistance can be modeled very accurately with HughesLiu elements up to the onse t of the global buckling. Fiber elements are not capable of capturing the local buckling (such as that shown in Figure 311 ). 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 PAGE 88 0 2 4 6 8 10 0 1000 2000Displacement [mm]Force [kN] HughsLiu elements Brick elements Figure 313. Displacementforce history comparison of brick and HughesLiu modeling Such good agreement between the brick elements and HughesLiu formulation can be attributed to the column crosssection properties. A W14x74 is a compact section, which means that the ratio of web depth to its thickness as we ll 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 3D mode l with brick elements only in the postbuckling 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 HughesLiu formulations is welcomed because it enables the modeling of a full stru cture with computationally efficient elements, yet ensures that obtained results represent the physic al reality of building collapse Hence, HughesLiu elements were used to represent the building frame members in the simulation Whereas force induced imperfection is acceptabl e in single column buckling, it is not a feasible approach in ensuring proper buckling ini tiation in the full structur al model. It should also be noted that geometric imperfections play an important role in buckling behavior (Jung, 88 PAGE 89 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 outofplumbness such that buckling could be initiated. At the same time, no global buckling behavior was introduced in the model. A normal distribut ion of crookedness (with 95% of values within 1/1500) and outofplumbness (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 r eallife manufacturing imperfections. In order to verify the ability of the proposed modeling techni que to simulate buckling and postbuckling behavior, a number of buckling simulation were ca rried out for different boundary conditions and lengths. Simula tions produced responses both in terms of buckling loads and postbuckling behavior. A comparison of simula ted buckling loads with a code curve (AISC, 2006) is shown in Figure 314 Simulated results match closely w ith 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 0.01.02.03.04.05.0Fcr / FySlenderness ( c) LRFD Buckling Load lsDyna Cantiliver lsDyna Top pinned lsDyna Clamped Figure 314. Buckling stressslenderness parameter c 89 PAGE 90 The wideflange sections have residual stresse s (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 part s, cool more slowly than the web regions. Furthermore, the flange tips having greater exposure to the air cool more rapidly than the re gion at the junction of the flange to web. Consequently, compressive residual stress exists at flange tips and at middepth of the web (the region that cools fastest), while tensile residual stress exists in the flange and the web at the regions wher e they join. Since accounting fo r the geometric imperfections was sufficient to obtain a satisfactor y agreement with the LRFD curve ( Figure 314 ), 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 ag ree well with Hoffs experimental observations (1955) that not a single elas tic specimen buckled suddenly and no inelastic specimen buckled slowly. The difference between elastic and inel astic 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 grad ually. Both buckling forces and postbuckling behavior have been shown to be in good agreement with experimental data. The simulated results captured experimentally de rived 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 PAGE 91 Selected Structures Simplified Steel Framed Structure A simplified 3story steel framed building was se lected 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 anal ysis 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 315 33 [ f t ] w18x35 w18x35 w21x68 w21x68 33 [ ft ] 1 2 3 4 A B C D E w18x35 Figure 315. Selected threestory steel framed building Deck selfweight 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 PAGE 92 ASCE (2005), was applied to account for the pr esence 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 us ed 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 ANSYSLSDYNA. Preprocesse d models were exported to the LSDYNA input files and solved by the LSDYNA 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 preloadi ng was purely quasistatic. Subsequently, the analysis was restarted in th e 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 switche d at the prescribed time of 20 [s]. ANSYS models of the analyzed structures are shown in Figure 316 Figure 317 and Figure 318 Figure 316. Twodimensional steel frame selected for the analysis 92 PAGE 93 Figure 317. Threedimensional stee l frame selected for the analysis Figure 318. Simplified steel framed building 93 PAGE 94 Realistic 3story SAC verification building In order to apply and test th e findings derived from the pr eviously analyzed 3D steel frames, a realistic moment resisting steel framed building was subsequently analyzed. A 3story 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 th e January 17, 1994 Northr idge, California (Los Angeles) Earthquake. The SAC commissioned thr ee consulting firms to perform code designs, following the local code requirements for the ci ty 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 infille r beams and composited decks. The floortofloor heights are taken from centerline of beam/girder to cen terline of beam/girder. The ANSYSLSDYNA model of the SAC Boston modified st eel framed building is depicted in Figure 319 Figure 319. Model of the SA C Modified Boston Building 94 PAGE 95 The buildings was designed for a typical office o ccupancy 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 /f looring /fireproofing, mechanical /electrical/ plumbing systems and partitions (20 [psf], 1 [kPa]). The framing plan of the 3story verification building is shown in Figure 320 and in Table 32 Column schedules are depicted in Figure 321 and in Table 33 An assortment of ten Wshapes was implemented in the building: five column secti ons 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 A A A w 1 8 x 35 A A A w 18x35 w18x35 ( infill beams ) Figure 320. Framing plan used for SAC three story building Table 32. Moment resistant beams (designated with A) Floor Beam A 2 w18x35 3 w21x57 roof w21x62 95 PAGE 96 y 30 [ ft ] 2 3 4 5 A B C D E 1 F G Moment connection Penthouse p erimeter x 30 [ft] Figure 321. Orientation of columns Table 33. 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 PAGE 97 Summary Progressive collapse is a comp lex 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 cap abilities to simulate and solve structural propagation, physical interpretation and insight is ye t 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 se ismic 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 buckl ing data and basic laws of physics. 97 PAGE 98 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 simu lations 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 fullscale simulation, a subset of cases was selected for energetic analysis. Virtual Experiments on the Simplified Steel Framed Buildings Once characteristic properties of an individua l column were established, global simulations of structural response to localized abnormal lo adings 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 th at 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., 2D frame, 3D frame and full 3D model with slabs. The objective was to select proper modeli ng resolution to ensure that investigated energy flow results are applicable to real steel framed structures. For the readers convenience, description of the simplified three story steel framed building selected for the discovery analysis is br iefly repeated herein. The framing plan of the analyzed structure is shown in Figure 41 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 PAGE 99 without splices and varia tion in shape selection. Each stor y 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 w ith light reinforcement. 33 [ f t ] w18x35 w1 8x35 w18x35 w21x68 w21x68 33 [ ft ] 1 2 3 4 A B C D E Figure 41. Selected three story steel framed building (W14x74 columns) To understand the structural response, the re moval 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 assu med as moment resisting to investigate if maximum continuity always provides enhanced levels of protection. 99 PAGE 100 Kinematic Observations A traditional approach to struct ural analysis was exercised to establish the baseline for the energy considerations to follow. Kinematic obser vations consisted of the analysis of internal forces and displacements. For checking the capacity of a structure to with stand 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: WADororSorLADk k2.0)2.19.0()2.05.0(2.1 (41) 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 d ead 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][136505.0865.00.1kPa psf LD Load (42) The following modeling resolutions were impl emented for comparison: 2D frame (faade along column line 1), 3D frame with joists a nd 3D building with joists and slabs. The structures were subjected to the sudden removal of column A1 ( Figure 41 ) at a prescribed time after application of static preloading. Tributary areas were used to appl y loads to the 2D and 3D frames. The 2D frame was loaded with point loads transferred from the W18x35 beams. 3D frames were loaded with uniformly distributed line loads applied to the W18x35 beams. Initial and final displaced configurations obtained fr om dynamic time history analyses from 2D and 100 PAGE 101 3D frames are shown in Figure 42 and Figure 43 In each case, colla pse was arrested with significant beam deformations in the bay s ubjected to the sudden column removal. Figure 42. Two dimensional frame in CASE 1 Figure 43. Three dimensional frame in CASE 1 In the case of the building with floor slabs, lo ad was applied directly to the slabs. Inertia properties of the loads were preserved by lumpin g appropriate mass into decks such that their weight per unit area equaled the sum of dead and liv e loads. Steel framed buildings with heavily reinforced and typical slabs arrested the collapse propagation ( Figure 44 ). 101 PAGE 102 Figure 44. Displaced shape of the simp lified steel framed building in CASE 1 Vertical displacements of the building corner (above the removed column A1) are shown in Figure 45 Due to robust connections, the collapse wa s arrested. The lesse r extent of plastic deformations can be seen in the building with str onger slabs. 35 30 25 20 15 10 5 0 2020.52121.522Displacement [in]Time [s] 3D w/ hardened slabs 3D w/ typical slabs 3D Frame 2D Frame Figure 45. Displacement of the building corner (CASE1) 102 PAGE 103 On the basis of displacement results only, it could be concluded that 2D and 3D frames produced more conservative results and highe r demands on the members. However, the inspection of moment time historie s in beam AB in column line 1, at the supported end (point B) leads to different conclusions ( Figure 46 ). 7000 6000 5000 4000 3000 2000 1000 0 1000 0 5101520Moment [kipin]Time [s] 3D w/ hardened slabs 3D w/ typical slabs 3D Frame 2D FrameA 7000 6000 5000 4000 3000 2000 1000 0 1000 2020.220.420.620.821Moment [kipin]Time [s] 3D w/ hardened slabs 3D w/ typical slabs 3D Frame 2D FrameB Figure 46. Beam moment time history in CASE1: A) static, B) dynamic phase 103 PAGE 104 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 preloading phase. Nevertheless, it became evident ( Figure 46 ) that it lead to erroneous results in the dynamic phase of progressive collapse. Models with loads redistributed according to the tributary area appro ach 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 a nd inertial effects. Lateral stiffening provided by slabs also contributed to the differences between the models in the dynamic phase. Thus full scaled, detailed si mulations, with properly modeled slabs were necessary to analyze the collapse arrest or propagation. Simplif ied models lead to erroneous results in the dynamic phase of calculations. Results from reduced models of 2D and 3D frame with manually redistributed loads from slabs and infill beams were in good agreem ent with results from the full scale building model in the static phase only. However, comp arison of results in the dynamic phase revealed that a traditional tributary area approach, used to transfer the loads from slabs and infill beams to the reduced model, was not effective. Among the three analyses, dynamic displacement results exhibited different oscilla tion periodicity and momenttime histories were distinctive. Thus, using the tributary areas to redistribute loads is inadequate, in spite of the employment of high fidelity nonlinear dynamic analysis. Inadequacy of the tributary area approach to redistribute the loads to the fram e 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 adjacen t exterior column on the ground level (A1 and 104 PAGE 105 B1) were removed after applic ation of static preloading. The following General Services Administration (2003) load combination was applied: ][25.5][136505.0865.00.1kPa psf LD Load (43) 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 le vel rendered ineffective: 1.2 D + 1.0 L CASE 3 was intended to produce the total co llapse; therefore the most structurally detrimental General Services Administrati on (2003) load combination was applied: ][35.7[psf]153501.0862.1L1.0+D1.2 LoadkPa (44) Columns A1 and B1 were removed after static preloading to ini tiate the collapse. Dynamic analyses were carried out to simulate structural behavi or. Collapse sequences for each of the models are shown in Figure 47 through Figure 410 Considerable differences can be noticed between collapse modes of the 2D and 3D frames as well as 3D models incorporating slabs. The collapse sequence of the frame models were brought about larg ely due to the absence of lateral restraint from the slabs. Composite or reinforced concrete decks provided membrane action, which restrained beams from lateral to rsional buckling. Thus 2D and 3D 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 heav ily reinforced slabs. On the other hand, column A2.1 was the first overloaded column in the build ing with typical slabs. Thus, deck stiffness affects structural response to the localized abnormal loading. 105 PAGE 106 Figure 47. Collapse sequence of twodimensional frame in CASE 3 106 PAGE 107 Figure 48. Collapse sequence of the threedimensional frame in CASE 3 107 PAGE 108 Figure 48 Continued B2.1 Figure 49. Building with hardened slabs Collapse sequence in CASE 3 108 PAGE 109 A2.1 C1.1 Figure 49 Continued 109 PAGE 110 A2.1 B2.1 Figure 410. Building with typical sl abs Collapse sequence in CASE 3 110 PAGE 111 C1.1 Figure 410 Continued 111 PAGE 112 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 gl obal building failure. In the bu ilding with light slabs, column A2 buckled first and collapse spread to column s B2 and C1. Successively, failure propagation at the ground level resulted in total building collapse Comparative displaceme nt results at the top of the removed column A1 are shown in Figure 411 The building with stronger slabs attempted to bring the fall to a stop faster than the buildi ng with light decks (20.4 [s] versus 20.8 [s]). 180 160 140 120 100 80 60 40 20 0 20 2020.52121.522Displacement [in]Time [s] 3D w/ hardened slabs 3D w/ typical slabs 3D Frame 2D Frame Figure 411. Displacement time hi story of point A1 in CASE 3 The moment time history of beam AB at s upport B1 on the first floor is presented in Figure 412 Again, significant discrepancies can be seen between the models. Stripped frame models lead to significantly diffe rent results with different peri odicities and amplitudes in the dynamic, collapse phase. Thus, only models in corporating decks and accounting for inertial effects can lead to meaningful and useful results in the analysis of progres sive collapse potential. 112 PAGE 113 10000 8000 6000 4000 2000 0 2000 4000 6000 8000 10000 19.52020.52121.5Moment [kipin]Time [s] 3D w/ hardened slabs 3D w/ typical slabs 3D Frame 2D Frame Figure 412. Moment time history of beam AB at B2 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 notic eable 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 413 through Figure 416 Figure 416 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 dynami c force redistribution. Assessing column safety solely from oscillatory internal forces is a formidable task. 113 PAGE 114 2 1.5 1 0.5 0 0.5 1 05101520Normalized Moment and ForceTime [s] y Axial Force Moment S Moment TA 2 1.5 1 0.5 0 0.5 1 05101520Normalized Moment and ForceTime [s] S T y Axial Force Moment S Moment TB S T Figure 413. Normalized internal forces in A2.1 column: A) hardened slabs, B) typical slabs 114 PAGE 115 y 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 20 20.5 21 21.5Normalized Moment and ForceTime [s] S T Axial Force Moment S Moment T A 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 20 20.5 21 21.5Normalized Moment and ForceTime [s] Axial Force Moment S Moment TB S T Figure 414. Dynamic phase of normalized internal forces in A2.1 column: A) hardened slabs, B) typical slabs 115 PAGE 116 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 05101520Normalized Moment and Force Time [s] y Axial Force Moment S Moment TA 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 05101520Moment or Force [kipin or kip]Time [s] S T y Axial Force Moment S Moment T B S T Figure 415. Normalized internal forces in B2.1 column: A) hardened slabs, B) typical slabs 116 PAGE 117 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 20 20.5 21 21.5Normalized Moment and ForceTime [s] y S T Axial Force Moment S Moment T A 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 20 20.5 21 21.5Moment or Force [kipin or kip]Time [s] Axial Force Moment S Moment TB S T Figure 416. Dynamic phase of normalized internal forces in B2.1 column: A) hardened slabs, B) typical slabs 117 PAGE 118 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 situat ions, progressive collapse does not have such simplified procedures. It is recommended to first verify individual components of the simulation, e.g., buckling behavi or of individual columns to a ssure 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 wit hout additional loads (explosion, truck impact, etc). Structural elements shoul d not fall vertically faster than at free fall (Loizeaux and Osborn, 2006; Bazant and Verdure, 2007). Actually re sistance and energy absorption of resisting members should slow down such motion. Figure 417 and Figure 418 compare simulated displacement results with the freefall curve for CASE 1 and CASE 3. Only the detailed 3D model with slabs and properly applied loads adhered to the freefall 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 freefall physics. Thus, not only slabs play a cruc ial role in the proper simulation of structural response to the loca lized failure, but also inertial effects affect building behavior. 118 PAGE 119 35 30 25 20 15 10 5 0 20 20.5 21 21.5Displacement [in]Time [s] 3D w/ hardened slabs 3D w/ typical slabs 3D Frame 2D Frame Free Fall Figure 417. Free fall re quirement in CASE 1 180 160 140 120 100 80 60 40 20 0 20 2020.52121.522Displacement [in]Time [s] 3D w/ hardened slabs 3D w/ typical slabs 3D Frame 2D Frame Free Fall Figure 418. Adherence to free fall in CASE 3 119 PAGE 120 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 419 was subjected to two corner column removals. Increasingly higher loadi ng 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 ] w18x35 w1 8x35 w18x35 w21x68 w21x68 30 [ ft ] 1 2 3 4 A B C D E Figure 419. Selected three story steel framed building for buck ling onset analysis Two columns: A1.1 and B2.1 on the ground leve l 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 420 through Figure 422 show structural response to th e applied loading scenarios. Interestingly, a small decrease in the span length significantly enhanced structural resistance to collapse propagation. It was necessary to increa se the applied live load from 1.0 L to 2.0 L in order to initiate the progressive failure. Column displacements ( Figure 423 ) did not show characteristics, which could be helpful in determination of the onset of instability. 120 PAGE 121 Figure 420. Arrested collapse of threedimensional building w/ typical slabs in CASE A Figure 421. Arrested collaps e of threedimensional model w/ typical slabs in CASE B 121 PAGE 122 B2.1 A2.1 Figure 422. Collapse sequence of threedime nsional model w/ typical slabs in CASE C 122 PAGE 123 Figure 422 Continued 123 PAGE 124 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 05101520Displacement [in]Time [s] p y A2.1top B2.1top1 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 2020.52121.52222.523Displacement [in]Time [s] A2.1top B2.1top2 Figure 423. Displacements of A2.1 and B2.1 columns: 1) Complete history, 2) Dynamic, collapse phase. Figure 424 through Figure 427 compare internal forces in co lumns in the case of arrested and progressive collapse. Differences between CASE A and CASE C in the column A2.1 are 124 PAGE 125 very minor. Also, column B2.1 reached axial buc kling 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 in ternal force histories are studied. 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 05101520Normalized Moment and ForceTime [s] y Axial Force Moment S Moment T1 2 1.5 1 0.5 0 0.5 1 05101520Normalized Moment and ForceTime [s] S T y Axial Force Moment S Moment T2 Figure 424. Normalized internal forces in A2.1 column: 1) CASE A, 2) CASE C 125 PAGE 126 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 20 20.5 21 21.5Normalized Moment and ForceTime [s] Axial Force Moment S Moment T1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 20 20.5 21 21.5Normalized Moment and ForceTime [s] S T y Axial Force Moment S Moment T2 Figure 425. Dynamic phase of normalized internal forces in A2.1 column: 1) CASE A, 2) CASE C 126 PAGE 127 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 05101520Moment or Force [kipin or kip]Time [s] y Axial Force Moment S Moment T1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 05101520Moment or Force [kipin or kip]Time [s] S T y Axial Force Moment S Moment T2 Figure 426. Normalized internal forces in B2.1 column: 1) CASE A, 2) CASE C 127 PAGE 128 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 20 20.5 21 21.5Moment or Force [kipin or kip]Time [s] Axial Force Moment S Moment T1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 20 20.5 21 21.5Moment or Force [kipin or kip]Time [s] Axial Force Moment S Moment T2 S T Figure 427. Dynamic phase of normalized internal forces in B2.1 column: 1) CASE A, 2) CASE C 128 PAGE 129 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 nonlinear and dynamic eff ects. Momenttime histories varied significantly between models. Whereas, nu merous 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 asse ss 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 3D model with harder slabs flattened around 0.4 [s] after column removal. It directly corresponds to the time when column B2 buckled, followed by buckling of colu mns B1, C1 and A2. Thus the structure attempted to arrest the collapse around 0.5 [s] af ter its onset but was overloaded and these four columns failed. Collapse propagated further unt il total collapse. On the other hand, the simplified 2D and 3D frame simulations did not provide meaningful results. On the contrary, they showed unrealistic collapse modes. It is commonly held that tyi ng the structure provides ductility and increases resistance to progressive collapse. This prin ciple is valid for relatively sma ll 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 ti es, 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 betw een the collapsing portion of the structure and the 129 PAGE 130 rest of the structure so as to not pull the re maining building down. Such connection failure would isolate collapse propagation and protect the remaining porti on of the building. Mass properties play an important role in structural response to the localized, abnormal loading. Only a sufficiently detailed model w ith 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 investigati on. 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 proce sses and progressive collapse propagation. Progressive collapse analysis is characterized by large displacements, strains and pronounced dynamic effects. Therefore, reso rting to simplified techniques that might be adequate in static design would inevitably lead to erroneous resu lts and misleading conclusions. It has been concluded that only fully nonlinea r dynamic time history analysis of sufficiently detailed models, which account for material and geometric nonlinearitie s, can lead to meaningful results. Only the detailed 3D model with slab s and properly applied loads adhere d to the freefall requirement and captured the collapse seque nce with great realism. 130 PAGE 131 CHAPTER 5 ENERGY APPROACH TO THE ANALYSIS OF PROGRESSIVE COLLAPSE Overview Numerical simulations were car ried out to test a proposed energy based methodology. The objective of this study was to esta blish the general energy based anal ysis 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 ne w concept of buckling internal energy (deformation work) is introdu ced in this study. In this re search, columns were tested as single members under various loading conditions and their individual behavi or was characterized in both force and energy domains. Energetic analyses were carried out on selected structures. Simulation results supported the proposed ener getic approach. Global energy measures were correlated to collapse propagation intensity. It wa s demonstrated that when global kinetic energy vanished, the structures reached the final stable stat e, either arrested collaps e, partial collapse, or total catastrophic failure. A detailed view of energy propagation was undertaken through analysis of energy timehistories in all structural members. Bucklin g energies enabled practi cal application of the proposed energy criteria in a quantitative manner. It is shown that for a simplified building, the entire structural simulation is de scribed by energy timehistories. Internal energy (deformation work) timehist ories 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 cap tures the essence of collapse propagation in an accurate, yet easy to digest manner. 131 PAGE 132 Energy Definitions in LSDYNA External work is the work done by applied forc es. The total flow of energy into a system (external work) must equal the to tal amount of energy in the system (sum of internal and kinetic energy. Internal energy (deformation work) is calculated by LSDYNA using the following definition: dVdE int (51) The total strain can be decomposed into elas tic (recoverable) and pl astic (irrecoverable): peddd (52) ed= elastic strain increment, = plastic strain increment. pd Therefore, LSDYNA internal energy include s elastic strain energy and work done in permanent deformation: dVddVddVdEp e int (53) dVde= elastic strain energy, dVdp = 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 ri gid 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: dVvEkin22 1 (54) = 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 PAGE 133 Energetic Characteristics of Individual Columns During abnormal loading events of framed st ructures, columns play a crucial role in structural stability. Propagation of column buc kling through the building floor usually leads to total collapse (Krauthammer et. al, 2004). Theref ore, column buckling was characterized in the energy domain before physicsbased, fullscale st ructural simulations were carried out. Displacement controlled buckling and force c ontrolled 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) co lumn, made of A36 steel, and 156 [in] ( 4 [m]) in height, was selected for preliminary analysis. An individual column within a structural fram e is loaded, in part, by gravity forces; but, the loading process is not solely force controlled. Adjacent b eams and members affect column displacement, and the column is subject to part ially force controlled and partially displacement controlled behavior. Therefore, both displacem ent and force controlled buckling simulations were carried out for the selected, typical W12x58 column. Displacement controlled Buckling Displacement controlled virtual buckling e xperiments were conducted using the finite element software LSDYNA (Hall quist, 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 51 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 52 LSDYNA output file designati ons are also given to clarify how the information was obtained. 133 PAGE 134 Figure 51. Simulated buckling mode of W12x5 8, 156 [in] clamped column (beam elements) 0 2 4 6 8 10 80 70 60 50 40 30 20 10 0 Time [s]Displacement [in] Displacement nodout Figure 52. Prescribed column top displacement time history 134 PAGE 135 It should be noted that no sudden instability was observed in the displacement controlled simulations. Columns gradually deflected fo llowing the clampedclamped buckling mode as shown in Figure 51 On the other hand, the column force resistance, as inferred from the reaction force at the column bottom, dropped abrup tly after approx. 0.02 [s]. Reaction force time history is shown in Figure 53 Although the column suddenly lost load carrying capacity, kinematic instability was restrained by the pr escribed boundary displacem ent conditions. Thus, a column within a structural frame may effec tively 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 10 0 100 200 300 400 500 600 Time [s]Force [kip] Force spcforc Figure 53. Reaction force time history The resistance force displacement curve closel y replicated the resistance time history ( Figure 54 ). It can be seen that column resistance reduced drastically af ter 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 displacementload time history. 135 PAGE 136 Internal energy (deformation work ) is defined as the work needed to change structural shape without inducing kinetic ener gy. Internal energy time histor y of the analyzed column is shown in Figure 55 Once the column buckled, the energy absorption slope significantly decreased. When the prescribed displacemen t reached its final value, Internal energy (deformation work) becomes constant. In contrast to the forcetime histor y, 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 postbuckling phase may be lower than predicted by b eam 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]Force [kip] Force spcforc Figure 54. Resistancet op displacement function The internal energydisplacement relationship is depicted in Figure 55 It is similar to the internal energy time history. Ho wever, once the final value of th e prescribed displacement time history is reached, all data poi nts in the time domain reduce to one data point corresponding to 136 PAGE 137 the final column displacement in Figure 56 The slope of the inte rnal energydisplacement curve is equal to the force resi stance of the analyzed column. 0 2 4 6 8 10 0 2000 4000 6000 8000 10000 Time [s]Internal energy [kipin] Internal energy matsum Figure 55. Internal energy (d eformation work) time history 0 2 4 6 8 10 0 500 1000 1500 2000 2500 Displacement [in]Internal energy [kipin] Internal energy matsum Figure 56. Internal en ergy displacement history 137 PAGE 138 Force controlled Buckling Column stability is critical for building safety. Force controlled buckling simulations, using the finite element package LSDYNA, 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 th e influence of inertial effects on column resistance and energy absorption, various loading ra tes from 10 to 10000 [kip/s] ( 45 to 45000 [kN/s]) were applied at the column top. Each loading rate produced num erous kinematic and energetic results. Due to the volume of individual results, on ly 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 co lumn top is shown in Figure 57 The acceleration is of negligible magnit ude until the buckling resistance is reached and the column buckles in a dynamic fashion. Th e 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 midheight) was formed and material fa ilure followed shortly afterwards. Since ground surface was not introduced in th e individual column tests, th e failed portion of the column continued to accelerate until the simulation term ination time was reached. Therefore maximum acceleration, velocity and displacem ent did not provide useful information in the analysis of the structure, in which acceleration is limited by collis ion 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 PAGE 139 0 20 40 60 80 100 2.5 2 1.5 1 0.5 0 0.5 x 107 Acceleration Stefan Szyniszewski Time [s]Acceleration [in/s2] acceleration nodout Figure 57. Acceleration time history at the column top Figure 58 shows accelerationforce 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 58 Closer inspection of ASCII NODOUT text output files enabled more accurate buckling ti me and in turn buckling load determination as 598[kip] (2660 [kN]). It was onl y 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 59 depicts velocity applied force history and Figure 510 shows the displacement appl ied force history. All kinematic results enabled clear identification of the buckling onset, the corresponding buckling load, and additional energetic characteristics, as discussed below. 139 PAGE 140 0 200 400 600 800 1000 2.5 2 1.5 1 0.5 0 0.5 x 107 Acceleration Stefan Szyniszewski Force [kip]Acceleration [in/s2] acceleration nodout Figure 58. Acceleration for ce history at the column top 0 200 400 600 800 1000 1000 800 600 400 200 0 yy Force [kip]Velocity [in/s] velocity nodout Figure 59. Velocity force history at the column top 140 PAGE 141 0 0 200 400 600 800 1000 1 0.8 0.6 0.4 0. 2 Force [kip]Displacement [in] displacement nodout Figure 510. Vertical displacement at the column topforce history The prebuckling phase can be char acterized as quasistatic with negligible kinetic energy. Once buckling initiates, the kine tic energy abruptly increased ( Figure 511 ). 0 20 40 60 80 100 0 2 4 6 8 10 12 14 16 18 Time [s]Energy [kipin] Kinetic Energy nodout 1010xFigure 511. Kinetic energy time history 141 PAGE 142 The internal energy (defor mation work), shown in Figure 512 confirmed the kinematic results. The internal energy significantly increased after the on set of buckling. 0 20 40 60 80 100 0 1000 2000 3000 4000 5000 6000 7000 Time [s]Internal energy [kipin] matsum Figure 512. Internal energy time history However, a closer inspection of the intern al energy (deformation work) results clearly revealed that after the buckling energy was reached ( Figure 513 ), energy increased dramatically until material column failure. This limiting ener gy, corresponding to the onset of buckling, was approximately 58.9 [kipin] (6.65 [kJ]) (obtai ned 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 kine tic energies rise sharply. A single column within a steel structural fram e must resist not only axial forces, but also end moments in both the weak and strong direc tions. These end forces will vary with time 142 PAGE 143 because progressive collapse is a dynamic phenome non. 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 80 100 0 10 20 30 40 50 60 70 80 Time [s]Internal energy [kipin] matsum buckling energy Figure 513. Internal energy time history in prebuckling phase Time derivative of the internal energy (defor mation work) is called the energy rate in this study. It is instructive to sc rutinize the internal energy time rate, which indicates how fast applied energy can be absorbed by the column through deformations. Figure 514 shows internal energy rate as obtained from LSDYNA (MATSUM te xt 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 colu mn absorbed external energy, beyond buckling, 143 PAGE 144 up to material failure. In the prebuckling phase, the energy rate monotonically increased up to approximately 3 [kipin/s] (0.34 [kJ/s] ) and then the energy rate spiked ( Figure 515 ). 0 20 40 60 80 100 0 5000 10000 15000 Time [s]Internal Energy Rate [kipin/s] matsum Figure 514. Inte rnal energy rate 0 200 400 600 800 1000 0 2 4 6 8 10 Force [kip]Internal Energy Rate [kipin/s] matsum Figure 515. Internal energy ra te in the prebuckling phase 144 PAGE 145 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 51 Simulated quasi static results compared well with the LRFD AISC (2006) code value. Discrepancies between the si mulated buckling loads and code values were in the order of 3% in the quasistatic range. Once the loading rate reached 5000 [kip/s] (22240 [kN/s]), the increase in buckling resistan ce 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 proporti onally to the loading rate as more work was done on the syst em at an increasingly shorter time interval. Table 51. W12x58, 156 [in] column buckling results Loading Rate LSDYNA Buckling Force Static AISC Buckling Force Difference between Dynamic and AISC Static Force Buckling Energy Buckling Energy Rate [kip/s] [kip] [kip] [kipin] [kipin/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 load ing rates results from inertial resistance, which resists the motion and thus delays the ons et of buckling. Whereas transient loads can exceed the static buckling load for short time dur ation, permanent load th at 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 516 For loading rates greater than 3000 [kip/s] (13340 [kN/s]), force resistance increase d noticeably. Loading rates during typical full structure collapse propa gation are of 1000 to 2000 [kip/s] (5000 to 10000 [kN/s]). 145 PAGE 146 Loadin g rate [ ki p /s ] Buckling Force [kip] 10 20 30 50 70 100 200 300 500 1000 2000 5000 10000 0 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 760 800 LSDYNA Dynamic Buckling Force LRFD AISC Static Buckling Force Figure 516. W12x58: buckling force for the selected, force contro lled, loading rates Simulated LSDYNA internal energy rates ve rsus applied loading rate are shown in Figure 517. Internal energy rates corres ponding to the buckling onset increased with the loading rate. The faster the column was loaded, the faster exte rnal 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 518 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 fu ll scale simulations were on the order of 1000 to 2000 [kip/s]. Thus a buckling energy of 59 [kipin] (6.67 [kJ]) was established as the buckling critical threshold value for the W12x58 column. 146 PAGE 147 Loadin g rate [ ki p /s ] Energy rate [kipin] 10 20 30 50 70 100 200 300 500 1000 2000 5000 10000 0 500 1000 1500 2000 Figure 517. W12x58: Internal buckling ener gy rate for the selected loading rates Loadin g rate [ ki p /s ] Energy [kipin] 10 20 30 50 70 100 200 300 500 1000 2000 5000 10000 20 24 28 32 36 40 44 48 52 56 60 64 Figure 518. W12x58: Internal buckling energy for the selected loading rates 147 PAGE 148 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 correspo nding energetic characteristics. Simulated buckling resistance in both displacem ent and force controlled simulations was in good agreement with experimental LRFD AISC values. Simulated results tended to produce higher buckling resistance under highspeed axia l loading rates (5000 to 10000 [kip/s]), which are not available in the AISC sp ecifications. However, this incr ease 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 s hort duration and transien t nature. In other words, if the axial load exceed s permanently the static bucklin g 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 highspeed loading rates, which are characterized by inertial enhancement. From an alternate perspective, displacement response de creases due to inertial resistance and force increases in proportion, thus the product of these two quantities c onstitutes a stab le measure of the column condition. Once a single column is incorporated into th e structural system, it will be subject to potentially complicated loading ti me histories, which may include not only axial forces but also end moments. These end forces are transient in nature with various peri odicities. Therefore a column buckling criterion in the force domain re quires the definition of weak and strong axis moments axial force dynamic interaction diagra ms. An internal ener gy 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 PAGE 149 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) re moval. Special emphasis was put on column behavior because the correspond ing crucial role in the coll apse 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 corres ponding to the buckling onset are shown in Table 52 Table 52. W14x74, 156 [in] column buckling results Loading Rate LSDYNA Buckling Force Static AISC Buckling Force Difference between Dynamic and Static AISC Force Buckling Energy [kip/s] [kip] [kip] [kipin] 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 estab lish the buckling threshold energy level. It is proposed that buckling ener gy 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. Buck ling force resistance for the selected loading rates is depicted in Figure 519 For loading rates ra nging 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 [kipin] (8.7 [kJ]) for all loading rates ( Figure 520 ). The buckling energy criteri on is more conveni ent than that associated with buckling force because it is loading rate independent. 149 PAGE 150 Loadin g rate [ ki p /s ] Buckling Force [kip] 1 2 3 4 5 7 10 20 30 50 70 100 200 500 1000 2000 5000 10000 0 100 200 300 400 500 600 700 800 900 1000 1100 LSDYNA Dynamic Buckling Force LRFD AISC Static Buckling Force Figure 519. Buckling force for the se lected loading rates in W14x74 column Loadin g rate [ ki p /s ] Buckling Energy [kipin] 1 2 3 4 5 7 10 20 30 50 70 100 200 500 1000 2000 5000 10000 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Figure 520. Internal buckli ng energy for the selected lo ading rates in W14x74 column 150 PAGE 151 CASE 1. Building with Heavily reinforced Slabs (Load = 1.0 D + 0.5 L) As opposed to a traditional internal force invest igation, energy based analysis of results is presented herein. A steel framed building with hard ened slabs was analyzed first. In the first analysis, a single corner column de signated as A1.1 was removed ( Figure 419 for layout of columns). A designates the vertical column line A, indicates 1st horizontal row, and .1 indicates the first story level ( Figure 521 ). Removed column Figure 521. Final shape of the selected stru cture after sudden column removal in CASE 1 Global redistribution of external work between internal and kinetic energies is depicted in Figure 522 The static preloading phase was dominate d by stable internal energy redistribution with negligible levels of kinetic energy. Co llapse was arrested in the dynamic phase. The building bay affected by the column removal underwe nt oscillatory motion after collapse arrest. This vibratory behavior can be attributed to the overly strong slabs, which provided significant capacity for elastic vibrations. Figure 523 depicts the global dynamic energy flow in mo re detail. It can be seen that in the case of arrested collapse, there can be tr ansient kinetic energy associated with elastic vibrations. Such energy will be damped out by th e inherent structural and material damping. 151 PAGE 152 0 5 10 15 20 0 500 1000 1500 2000 Time [s]Energy [kipin] Kintetic Energy Internal Energy Total Energy Figure 522. Global energy histories (from GLSTAT) 20 20.5 21 21.5 22 22.5 23 0 500 1000 1500 2000 Time [s]Energy [kipin] Kintetic Energy Internal Energy Total Energy Figure 523. Dynamic phase of the global energies In order to investigate energy distribution thr oughout the building in more detail, internal and kinetic energies were examined for all columns. Figure 524 depicts the energy level in all 152 PAGE 153 columns undergoing compressive loading. Sixt y legends were omitted to avoid cluttering the plot. The buckling energy lim it is shown for comparison in Figure 524 None of the internal energies exceeded 50% of the buckling energy. Thus structur e was deemed safe after the considered column removal. None of the colu mns 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 th at 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 provid ed clear insight into the building safety. This approach is free from the loading rate infl uence as opposed to the bu ckling force approach. Buckling energy Figure 524. Internal energy ti me histories for all columns Kinetic energies in the columns are shown in Figure 525 In the static preloading phase no significant kinetic energy was ob served in the system. Since the loading was applied in the 153 PAGE 154 quasistatic fashion, kinetic en ergies 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 travel ling throughout the structure as well as slabs and beam vibrations in the bays affected by the column removal. Figure 525. Kinetic energies for all columns CASE 2. Building with Hardened Slabs (Load = 1.0 D + 0.5 L) Consecutively energetic characteristics of th e 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 st ates to unsafe situations (i.e., resulting in global or partial collapse). Since one corner column removal was arrested, two columns shown in Figure 526 were removed after application of the static preloading. 154 PAGE 155 Columns removed Figure 526. Final deflection of th e selected structure in CASE 2 External work was redistributed between inte rnal and kinetic energies as depicted in Figure 527. 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 sy stem due to the elastic slab oscillations in the bays affected by column removal. 0 10 20 30 40 0 1000 2000 3000 4000 5000 6000 Time [s]Energy [kipin] Kintetic Energy Internal Energy Total Energy Figure 527. Global energies (GLSTAT) 155 PAGE 156 Released gravitational energy was not fully absorbed within the first 0.1 [s] after column removal. Kinetic energy of the system incr eased until plastic deformations and geometric nonlinearities enabled redi stribution into internal energies of adjacent members. Arrested collapse was characterized by the ability of the stru ctural system to absorb released gravitational energy by transferring it into defo rmation energy. Even if tem porary deficiencies in energy absorption occurred and kinetic en ergy of the system monotonically increased, there was still the potential in the system to recover and absorb th e kinetic energy surplus. Such recovery resulted from nonlinear resistance capabilities such as membrane action, geometric nonlinear stiffening and plastic material hardening. Figure 528 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 simulatio n but remained at the vi bratory level. Such vibrations would be damped out by inherent structural and material damping. 20 20.5 21 21.5 22 22.5 23 0 1000 2000 3000 4000 5000 6000 Time [s]Energy [kipin] Kintetic Energy Internal Energy Total Energy Figure 528. Global energies in dynamic phase (GLSTAT) 156 PAGE 157 Internal energy (deformation work) of all columns in compression ( Figure 529 ) 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 ki nematic instability (see page 133). Column B2.1 buckled but adjacent b eams and slabs retained sufficient residual capacity to redistribute structural loads and arrest the collapse. B2.1 A2.1 B2.2 energy bucklingFigure 529. Internal energies in columns Columns, which appeared to be in the unstable state, are depicted in Figure 530 These members constituted intuitive alternate paths in the building. Thus their overloading and buckling was a feasible scenario. However, i ndependent verification of energetic results was necessary to validate the above conclusions and prove the proposed energy based methodology. 157 PAGE 158 A2.1 B2.1 Figure 530. Instable columns as infe rred from the internal energy results Kinetic energies of the columns reflected glob al system periodicity (vibratory motion in the bays devoid of supporting columns, see Figure 531 ). However closer inspection of kinetic energies shown in Figure 531 and Figure 532 conform the finding of column B2.1 buckling. Figure 531. Kinetic energies of all columns 158 PAGE 159 20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 0 0.05 0.1 Time [s]nergy [kipin] C1.1 C1.2 C1.3 A2.3 A2.1A2.1 A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2E C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 C3.1 Figure 532. Internal en ergy in selected columns (initial collapse phase) 20 20.05 20.1 20.15 20.2 0 0.01 0.02 0.03 0.04 0.05 Time [s]Energy [kipin] C1.1 C1.2 C1.3 A2.1 A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 C3.1 B2.1 B2.1 B2.2 B2.3 Figure 533. Identification of column buckling from the kinetic energies 159 PAGE 160 Hence, the kinetic energy spike in columns B2.1, B2.2 and B2.3 conform that before periodic oscillations were excited in the stru cture, these columns were unable to absorb the applied external work and kinetic energy surplu s 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 th e system, which resulted in the global stable configuration. Both energy and kinematic result s 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 exte rnal 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 colu mns restrained from ki nematic instability by neighboring members. It is a worse outcome than CASE 1 (only one column removed), in which all individual members reached stable stat es well below the buckling energy level. Tracing energy flow provided insight into the st ability 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 conditi on. The sequence and path of dynamic reloading was determined. The level of structural sa fety was inferred from the number of unstable columns in the system. It is proposed to deem collapse as safely arrest ed only if all columns remain in the prebuckling stage. If collapse pr opagation is prevented at the expense of one or more columns buckling, such case should be cons idered as unsafe and potentially leading to catastrophic collapse. 160 PAGE 161 CASE 3. Collapse of the Building with Heav ily Reinforced Slabs (Load = 1.2 D + 1.0 L) In order to further verify the findings on altern ate load paths and system stability on both a local and global level, the case of removi ng two columns under increased loading was investigated. The load was amplified from standa rd load combination to the following scenario: ][153[kPa]3.7[kPa]2.50.1[kPa]42.1L1.0+D1.2 Loadkip (55) External work was redistributed between in ternal and kinetic en ergies as shown in Figure 534. 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 wo rk) monotonically increased as collapse progressed through consecutive bays. Global kinetic energy started diminishing as consecutive portions of the structures impacted the ground layer. Once th e collapse encompassed the whole building, the system reached its final stable state at approx. 24 [s]. 20 21 22 23 24 25 0 5 10 15 x 105 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure 534. Global energies (from GLSTAT) 161 PAGE 162 Interestingly enough, energy results of columns ( Figure 535 ) 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 21 0 1000 2000 3000 4000 5000 6000 7000 8000 Time [s]Energy [kipin] C1.1 C1.2 C1.3 D1.1 A2.1 A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 C3.1 B3.1 B2.1 C2.1 A2.1 Figure 535. Internal energies in columns Thus the analyzed simplified steel framed bu ilding 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 536 enabling the buckling sequen ce to be identified with better accuracy. Column buckled in the following order: B2.1, A2.1 (A2.2 and A2.3 relieved 162 PAGE 163 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 21 0 50 100 150 200 250 300 350 400 450 500 Time [s]Energy [kipin] C1.1 C1.2 C1.3 D1.1 A2.1 A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 C3.1 C2.1 C1.3 C3.1 D2.1 C1.1 B3.1 B2.1 A2.1 Figure 536. Internal energies in select ed columns with buckling energy threshold The collapse sequence from the internal energy (deformation work) histories matched perfectly visual observations of the collapse sequen ce in the simulation. The visual inspection of the collapse propagation is depicted in Figure 537 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 di fferent angle at t = 20.7 [s] revealed a buckled shape, which was hardly noticeable from the fron t view. Finally, the single energy plot provided insight into safety and collapse propagation as opposed to less accurate visual inspection. 163 PAGE 164 B2.1 A2.1 Figure 537. Verification of energetic results Collapse sequence assessment 164 PAGE 165 C2.1 Figure 537 Continued B3.1 Front view B3.1 Left side view 165 PAGE 166 C1.1 C3.1 D2.1 Figure 537 Continued 166 PAGE 167 Energy Propagation through the Building The steel framed building was divided into four zones, as depicted in Figure 538. During static preloading, selected zone s 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 3 Zone 4 Y Figure 538. Energy propagation zones Energy propagation through the building is shown in Figure 539 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 bui lding attempted to arrest the collapse ( Figure 540 ). 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 PAGE 168 20 20.5 21 21.5 22 22.5 23 0 2 4 6 8 10 Time [s]Energy [kipin] 510x zone 1 zone 2 zone 3 zone 4 Figure 539. Energy propagation th rough the inclusive building zones 20 20.5 21 21.5 22 22.5 23 0 0.2 0.4 0.6 0.8 1 1.2 1.4 No r m a l i z e d en erg y of al l me mb e r s St e f an Sz yni s z e w s ki Time [s]Relative Energy zone 1 zone 2 zone 3 zone 4 Figure 540. Normalized energy allocati on in the building (i nclusive zones) 168 PAGE 169 The structure was unable to absorb the releas ed gravitational energy at approx. 20.3 [s], when excessive energy localization initiated failur e 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 541 ) were proposed as the measure of ener gy localization intensity. 20 20.2 20.4 20.6 20.8 21 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 Time [ s ] Energy Rate [kipin/s] C1.1 C1.2 C1.3 D1.1 A2.1 A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 C3.1 Figure 541. 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 imbalan ce between released and 169 PAGE 170 redistributed energy, the larger the localized ener gy rate became. Thus, a critical energy rate was proposed as a test measure to identi fy the total failure initiation. Obviously the energy rate of the column which buckled first is a characteristic of the whole structur e and the corresponding energy absorption ability. Figure 542 illustrates the energy redistribution among columns; beams in the x and y direction ( Figure 538 ); and slabs. 20 20.2 20.4 20.6 20.8 21 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time [s]Relative Energy columns beamsX beamsYY slabs Figure 542. Energy distribut ion among members in zone 1 After collapse initiation, energy absorption increased in the beams spanning the ydirection. However, as plastic deformation initi ated, the role of the b eams decreased and energy began to overflow to the columns. If the column s and deforming structure are able to arrest the collapse, energy rate ( Figure 541 ) will diminish to zero. Alternat ively, 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, de termined for the structure at hand, localized or total failure will likely occur. If energy rate incr ease is averted, the collapse will be arrested. 170 PAGE 171 Energy based Column Buckling Criterion (Full Building Analysis) Energetic analyses of the building with lightly reinforced slabs (depicted in Figure 419 ) under increasingly severe loading conditions is disc ussed 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 colla pse 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 543 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 22 0 0.5 1 1.5 2 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work 410xFigure 543. Global energy redistribution in CASE A Internal energies in the build ing columns are depicted in Figure 544 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 lo calized building damage. 171 PAGE 172 20 20.5 21 21.5 0 10 20 30 40 50 60 70 80 90 100 Time [s]Energy [kipin] C1.1 C1.2 C1.3 D1.1 A2.1energy buckling A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 C3.1 Figure 544. Internal energies in CASE A Internal energy (deformation work) results id entified columns B2.1 and A2.1 as the members providing the alternate lo ad 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 scru tinized to prove it. Column buckling shall be characterized by significant defl ection 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 545 Oscillatory, slightly nonlinear, column behavior can be noticed in the force domain. However, upon relo ading, axial re sistance capacity 172 PAGE 173 was retained, when necessary. It should be not ed 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 0.40.30.20.10Force [kip]Displacement [in] y A2.1top B2.1top LRFD Buckling ForceA force buckling 0 20 40 60 80 100 120 140 160 180 200 0.40.30.20.10Energy [kipin]Displacement [in] A2.1top B2.1top Buckling Energyenergy buckling B Figure 545. Energy flow in building with t ypical slabs in CASE A (arrested collapse): A) Axial forcedisplacement, B) Internal energydisplacement 173 PAGE 174 Energy rates in the columns are shown in Figure 546 The energy rate spiked noticeably in the B2.1 column. However, its value of approximately 700 [kipin/s] (0.08 [MJ]) was well below the characteristic energy rate of 11 000 [kip in/s] (1.24 [MJ]). T hus it appears that the building frame was not in the proximity of the to tal collapse, when the collapse progression was averted. 20 20.2 20.4 20.6 20.8 21 0 100 200 300 400 500 600 700 800 900 1000 Time [s]Energy Rate [kipin/s] C1.1 C1.2 C1.3 D1.1 A2.1 A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 C3.1 Figure 546. Energy rates in the columns in CASE A Participation of the building zones in ener gy absorption and redistribution is shown in Figure 547 (inclusive zones) and Figure 548 (exclusive zones). It can be seen that the released potential energy was mainly absorbed by the me mbers in zone 1. Whereas energy in the exclusive zones 12, 23 and 34 remained practically constant. 174 PAGE 175 20 20.5 21 21.5 22 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Time [s]Energy [kipin] zone 01 zone 02 zone 03 zone 04 Figure 547. Energy propagation thr ough the inclusive zones in CASE A 20 20.5 21 21.5 22 0 2000 4000 6000 8000 10000 12000 14000 16000 Time [s]Energy [kipin] zone 01 zone 12 zone 23 zone 34 Figure 548. Energy propagation thr ough the exclusive zones in CASE A 175 PAGE 176 Internal energy (deformation work) increased sign ificantly 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 potenti al 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 549 ). Sudden removal of columns resulted in localization of both energy release and absorption. 20 20.5 21 21.5 22 0 0.2 0.4 0.6 0.8 1 Time [s]Relative Energy zone 01 zone 02 zone 03 zone 04 Figure 549. Normalized energy allocati on in the inclusive zones in CASE A The contribution of columns, beams and sl abs in energy absorption is described in Figure 550. Beams in the ydirection played a major ro le in energy absorption. On the other hand, column contribution dropped afte r initial energy inflow, as conf irmed by the spike in the B2.1 column energy rate. The system was able to redi stribute 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 PAGE 177 20 20.5 21 21.5 22 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time [s]Relative Energy columns beamsX beams YY slabs Figure 550. 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 unfa vorable 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 551 ). However, vertical displacements in the bays with removed columns were larger than in CASE A. Figure 551. Arrested collaps e of threedimensional model w/ typical slabs in CASE B 177 PAGE 178 Global energies are depicted in Figure 552 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 leve l, and bays affected by the loss of two supporting columns oscillated slightly. It is exp ected that material and structur al damping would damp out these background vibrations. 20 20.5 21 21.5 22 0 0.5 1 1.5 2 2.5 3 x 104 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure 552. Global energies in CASE B Internal energies in the building columns are shown in Figure 553 Although the global collapse was arrested, columns B2.1, A2.1, A2.2 exceeded the buckling energy threshold. Thus, these columns entered the postbuckling column state. Columns B2.1, A2.1 and A2.2 still retained sufficient resi dual capacities, which enabled them to support the a pplied loads. However, buckled columns are not acceptable as l ong term load carrying me mbers. Thus if the energy levels shown in Figure 553 came from sensors monitoring a catastrophic, abnormal event, columns: B2.1 and A2.1, A2.2 would be iden tified as in need of retrofit or replacement if possible. 178 PAGE 179 20 20.5 21 21.5 22 0 20 40 60 80 100 120 140 160 180 200 Time [s]Energy [kipin] C1.1 C1.2 C1.3 D1.1 A2.1 A2.2 A2.1A2.3 B2.1 B2.2 B2.3 C2.1 B2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 C3.1 Figure 553. Internal column energies in CASE B The energy based findings on the stability of co lumns B2.1 and A2.1 were verified with the internal forces and displacement results shown in Figure 554 Both columns exhibited kinematic instability after exceeding the bucking en ergy 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 expe rienced significant plastic deformations, which were in contrast to the stable behavior in Figure 545 Thus the effectiven ess and robustness of the energy based stability criterion was conf irmed. Internal energy (deformation work) combines both force and displacement informa tion as opposed to the nonunique force or interaction diagram criterions. 179 PAGE 180 800 700 600 500 400 300 200 100 0 0.40.30.20.10Force [kip]Displacement [in] A2.1top B2.1top LRFD Buckling ForceA 0 20 40 60 80 100 120 140 160 180 200 0.40.30.20.10Energy [kipin]Displacement [in] A2.1top B2.1top Buckling EnergyB force buckling energy bucklingFigure 554. Energy flow in building with t ypical slabs in CASE B (arrested collapse): A) Axial forcedisplacement, B) Internal energydisplacement. Energy rates in the columns are shown in Figure 555 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 [kipin/s] (0.1 [MJ]) was well below the characteristic energy 180 PAGE 181 rate of 11000 [kipin/s] (1.24 [MJ]). Thus it was concluded that the bu ilding was not in danger of total collapse, when the failure propagation was averted. Figure 555. Energy rates in columns in CASE B Energy absorption and redistribution in the predefined building zones ( Figure 538 ) is shown in Figure 556 and Figure 557 Internal energy increased significantly but primarily in zone 1. The normalized energy allocation is shown in Figure 558 The energy absorbed in zone 1 rose significantly and the en ergy in the other excl usive zones (12, 23 and 34) remained practically constant. Thus the corresponding normalized allocation in the overall system energy soared to 0.85. Sudden removal of columns re sulted in localization of energy discharge and absorption. Energy was mainly abso rbed by the beams in ydirection ( Figure 559 ). 181 PAGE 182 x 10 20 20.5 21 21.5 22 0 0.5 1 1.5 2 2.5 3 Time [s]Energy [kipin] zone 01 zone 02 zone 03 zone 04 Figure 556. Energy propagation thr ough the inclusive zones in CASE B 20 20.5 21 21.5 22 0 0.5 1 1.5 2 2.5 x 10 Time [s]Energy [kipin] zone 01 zone 12 zone 23 zone 34 410 x410 xFigure 557. Energy propagation thr ough the exclusive zones in CASE B 182 PAGE 183 20 20.5 21 21.5 22 0 0.2 0.4 0.6 0.8 1 Time [s]Relative Energy zone 01 zone 02 zone 03 zone 04 Figure 558. Normalized energy allo cation in the building in CASE B 20 20.1 20.2 20.3 20.4 20.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time [s]Relative Energy columns beamsX beamsYY slabs Figure 559. Energy split between members in zone 1 (CASE B) 183 PAGE 184 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 ne xt scenario, resulting in a 2.0 Live Load factor. Unlike in CASE A and CASE B, loca lized failure resulted in the total catastrophic collapse ( Figure 560 ). Columns A2.1 and B2.1 failed first and the collapse propagated outwards from the bays with removed columns. A2.1 Figure 560. Collapse sequence of threedime nsional model w/ typical slabs in CASE C 184 PAGE 185 Figure 422 Continued B2.1 185 PAGE 186 Redistribution of global energies is shown in Figure 561 Initial localized damage spread through the building and resulted in total ca tastrophic failure. Th e collapse sequence is illustrated in Figure 422 Kinetic energy rose as the coll apse 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 28 0 0.5 1 1.5 2 x 106 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure 561. Global energies in CASE C Internal energies in the building columns are shown in Figure 562 Released gravitational energy was not absorbed entirely by the building deformations and the structure progressed from unstable to unstable energetic states until tota l catastrophic collapse. Conclusions from the previous CASE A and CASE B were confirmed. Column B2.1 buckled first but its kinematic instability was initially restra ined by the adjacent members. Column buckling A2.1 followed 186 PAGE 187 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 21.5 0 50 100 150 200 250 300 350 400 Time [s]Energy [kipin] C1.1 C1.2 C1.3 D1.1 A2.1 A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 C3.1 B2.1 A2.1 B2.1 energy bucklingFigure 562. 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 563 ). The onset of instabilities was correctly identified by the energy buckling criterion. It should be noted th at axial force in the co lumns were accompanied by strong and weak axis moments, thus the durati on of instability development and axial loading and reloading was coupled with these internal forces ( Figure 425 through Figure 427 ). Plotting internal energies on the single plot enab led clear stability assessment of columns. In order to obtain similar insights using internal forces and displacements, multiple normalized 187 PAGE 188 plots with cross referenced data were needed. Moreover, oscillatory in ternal 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 21.510.50Force [kip]Displacement [in] y A2.1top B2.1top LRFD Buckling ForceA 0 200 400 600 800 1000 1200 21.510.50Energy [kipin]Displacement [in]force buckling A2.1top B2.1top Buckling Energy energy bucklingB Figure 563. Energy flow in building w/ typical slabs: CASE C (total failure): A) Axial forcedisplacement, B) Internal energydisplacement. 188 PAGE 189 Energy rates in the columns are shown in Figure 555 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 [kipin/s] (1.8 [M J]) exceeded the characteristic energy rate of 11000 [kipin/s] (1.24 [MJ]) established from the an alysis of the building with heavily reinforced slabs. Thus monitoring of the energy rate in the columns, wh ich 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 564. Energy rates in columns in CASE C Energy absorption and redistribution in the predefined building zones ( Figure 538 ) is depicted in Figure 556 and Figure 566 Internal energy increased initially in zone 1 only. Once the collapse progressed, energy in zone 2 th rough 4 exhibited significant increases as well. 189 PAGE 190 20 20.5 21 21.5 22 22.5 23 0 2 4 6 8 10 Time [s]Energy [kipin] zone 01 zone 12 zone 23 zone 34 Figure 565. Energy propagation thr ough the exclusive zones in CASE C 20 21 22 23 24 25 0 0.5 1 1.5 Time [s]Relative Energy zone 01 zone 02 zone 03 zone 04 Figure 566. Normalized energy allo cation in the building in CASE C 190 PAGE 191 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 th an 0.2 [s]. The building was unable to absorb the released gravitational energy by deformations of beams in ydirections ( Figure 567 ). Once capacity of beams to absorb the energy was exha usted, excess of energy was redirected to the columns. Once the column buckling propagate d 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 buckl ed and deformed members approximately reverted to the original energy redistribution. 20 21 22 23 24 25 0 0.1 0.2 0.3 0.4 0.5 Time [s]Relative Energy columns beamsX beamsY slabs Figure 567. Energy split between members in zone 1 (CASE C) Absorbed energy (deformation work) can be decomposed into elastic energy (potentially recoverable) and plastic energy (d issipated as heat). Energy absorbed in columns is mainly elastic up to the buckling onset ( Figure 568 and Figure 569 ). 191 PAGE 192 20 20.2 20.4 20.6 20.8 21 0 500 1000 1500 2000 2500 3000 Time [s]Energy [kipin] 410 x Internal Energy (Deformatn Work) io Elastic Energy Figure 568. Elastic component of th e absorbed energy in A2.1 column 20 20.2 20.4 20.6 20.8 21 0 0.2 0.4 0.6 0.8 1 y Time [s]Normalized Energy Deformation Work Elastic Energy Elastic Plastic Figure 569. Normalized energy decompos ition into elastic and plastic component 192 PAGE 193 Elastic energy in columns accounted for mo re than 80% of the absorbed energy (deformation work) in the prebuckling phase. Once a column buckled and was failed by the structural loads, its significant plastic deformati ons resulted in energy dissipation. Elastic energy accounted for less than 5% of th e total deformation work at the end of its failure. Energy absorbed by beams was mainly plastic ( Figure 570 and Figure 571 ). After collapse initiation, elastic energy quickly dropped to approximate ly 10% of the total deformation work. 20 20.2 20.4 20.6 20.8 21 0 500 1000 1500 Time [s]Energy [kipin] Deformation Work Elastic Energy Plastic Energy Figure 570. Elastic component of absorbed energy in A1.1 beam yy Energy in columns was in essence elastic up to the buckling onset. However, the energy absorbed beyond the buckling initia tion (approximately at time 20. 5 [s]) was purely plastic and thus irreversible. Although elastic energy, accu mulated during the prebuckling phase, could be potentially retrieved, this was not the case, because the permanent lo ad resting on columns prevented any significant unloading Energy absorbed in beams ( Figure 570 ) was chiefly plastic from the very beginning. 193 PAGE 194 20 20.2 20.4 20.6 20.8 21 0 0.2 0.4 0.6 0.8 1 Time [s]Normalized Energy Deformation Work Elastic Energy Plastic Energy Elastic Energy Figure 571. Normalized energy absorption in A1.1 beam yy Energy decomposition results from the simplified steel framed building were consistent with the observations inferred from the seis mic design steel building, discussed in the verification section of this study (p. 217). Since only columns contai ned a significant amount of elastic energy (more than 10%), and since colu mns 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 hist ories are not very sensitive to the buckling initiation and failure. On the contrary, in ternal energy (deformation work) significantly increased after buckling. Buckled members were easily identified on the basis of energy results only. To illustrate the superiority of th e energy approach, both force and energy based demand/capacity (D/C) ratio s were aggregated in Table 53 as well as in Figure 572 and Figure 573 to facilitate the comparative analysis. 194 PAGE 195 Table 53. 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 0.68 0.83 0.84 0.87 0.95 1.01 0 0.5 1 1.5 2 2.5 3 CASE ACASE BCASE C A2.1 column B2.1 column Figure 572. Buckling force demand/capacity ratios 0.82 2.18 0.98 1.76 0 0.5 1 1.5 2 2.5 3 CASE ACASE BCASE C A2.1 column B2.1 column 106 99 Figure 573. Buckling ener gy demand/capacity ratios 195 PAGE 196 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 initiat e the buckling in column A2.1 in CASE B. Moreover, only a slight difference of 0.01 separa ted a safe postbuckling state (A2.1 column) in CASE B from the failure in CASE C. T hus force based D/C ratios provide unreliable information on the structural safety afte r the occurrence of th e localized damage. Energy based D/C ratios were very sensitive to the buckling initiation. Internal energies significantly rose, and the viola tion of the energy buckling criteri on directly corresponded to the onset of buckling ( Figure 563 ). The decrease of the residual column capacity in the post buckling phase was characterized by significan t increase in the energy absorption (mainly irreversible, plastic energy dissipated as heat). Prebuckling, postbuck ling 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 in sights into the collapse propagation (e.g. Figure 562 ). 196 PAGE 197 Energy based Building Failure Limit An energy based global stability criterion is established in this section. Although this study has already established the energetic crite rion for buckling detection, column buckling does not always lead to column failure and collapse propagation. The internal energy (deformation work) in the postbuckling phase can reach significant values because columns retain residual resistan ce 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, c ondition to trigger the progressive collapse. A sufficient, energetic collaps e trigger criterion is propos ed and verified herein. Figure 574 depicts the parallel between postbuckling, re sidual 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 lo ad, significantly more than buckling work has to be done on the column be fore it irreversibly lose s capacity to carry the unending load. In other words, transient dynami c 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 re sidual capacity to suppor t the permanent load. In the post buckling phase, loss of load carrying ca pacity 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. Krau thammer et al. (2004) have shown that column failures trigger collapse propagation. Table 54 through Table 58 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 PAGE 198 sustained load and will certainly fail. It shoul d 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 increas ed to account for the loss of the columns. 0 1 2 3 4 5 0 200 400 600 800 Displacement [in]Force [kip] 0 1 2 3 4 5 0 500 1000 1500 2000 Energy Stefan Szyniszewski Displacement [in]Energy [kipin] Permanent Load Failure Energy Figure 574. Parallel of en ergy capacity with axial capacity of W12x58, 156 [in] column 198 PAGE 199 Table 54. W12x58, 156 [in] column buckling results P/Pc r Buckling Force Internal Energy [kip] [kipin] 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 Table 55. W14x99, 156 [in] column buckling results P/Pc r Buckling Force Internal Energy [kip] [kipin] 1.00 1042 132 0.95 990 360 0.90 938 625 0.85 886 891 0.80 834 1330 0.75 782 2020 0.70 729 3080 0.65 677 3820 0.60 625 4050 0.55 573 4440 0.50 521 4910 Table 56. W14x74, 156 [in] column buckling results P/Pc r Buckling Force Internal Energy [kip] [kipin] 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 0.55 427 1190 0.50 388 1270 199 PAGE 200 Table 57. W12x65, 156 [in] column buckling results P/Pc r Buckling Force Internal Energy [kip] [kipin] 1.00 676 74 0.95 643 222 0.90 609 244 0.85 575 428 0.80 541 532 0.75 507 686 0.70 473 923 0.65 440 1240 0.60 406 1610 0.55 372 1680 0.50 338 1820 Table 58. W12x72, 156 [in] column buckling results P/Pc r Buckling Force Internal Energy [kip] [kipin] 1.00 749 82 0.95 712 246 0.90 674 271 0.85 637 474 0.80 600 590 0.75 562 760 0.70 525 1030 0.65 487 1370 0.60 450 1780 0.55 412 1860 0.50 374 2010 200 PAGE 201 Analytical Solution of Elas toPlastic Column Buckling Although energy properties of a gi ven 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 esta blishes the foundation for the codi fication of energetic behavior and additionally offers an alternative for th e 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 clampedclamped boundary conditions correspond well to the real building restraints. Kinematics of the w eak axis buckling is depicted in Figure 575 Three plastic hinge regions are identifie d: top, center and bottom of the column. Downward column motion induces bending moments in the flanges, shown as rectangles in Figure 575 The rest of the column is treated as an extensible rod, carrying only axial forces. w 4mx 4mx 4mx 4mxtx 2L cos 2 L 2 d 2Ld 2L 2d AE PLcos 1 2 Initial configuration Rotation Shortening flangerod w Figure 575. Kinematics of column buckling 201 PAGE 202 Column motion can be decoupled into rota tion (producing bending in the hinges) and shortening of both hinges and rods. The key variables for the equi librium analysis are total top displacement () and lateral deflection of the column center ( ). Lateral displacement and the total displacement (including rotation and shortening effects) are: txw sin cos 1 2 EA PL w (56) cos cos cos1 L EA P Lxxmt (57) A = section area; E = Young modulus; P = axial force Kinematic relationships enable the calcula tion of the strains and internal forces. Rotation stretches and shortens some of the fibe rs in the hinges, hence inducing bending resistance in the hinges. Be nding strains are calculated using kinematic relationships between rotation and hinge fiber stretches depicted in Figure 576 bottom flange top flange y 2 d fb Figure 576. Bending st retches in the hinge 202 PAGE 203 Strains in the hinges, in cluding both bending and axial shortening effects, are: d x yxym m2 4 tan),,(0 (58) 0 = 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 buc kling is discussed to illustrate the concepts in full theoretical depth. Elastoplastic solution re quires 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 mtdy d x yEP 2 2 4 tan2 2 0 (59) Since the column is symmetric, only half of the column is sufficient to derive the moment equilibrium equations. A fr eebody diagram is shown in Figure 577 w 2M 1M P P Figure 577. Moment equilibrium Moment equilibrium is given as: 21MMwP (510) 203 PAGE 204 Applying the kinematic a nd strain expressions (56) and (58) results in: f bf bf mtdyy d x yE wP2 2 4 tan 22 2 0 (511) Thus employing equilibrium, one arrives at the following system of two equations with three unknowns: vertical displacement rotation tx and axial force P : ff ttb d L EA P Lx E P 2 2 cos cos cos1 4 (512) f ft d b E EA PL P 2 2 12 tan 2sin cos 1 23 0 (513) Incrementing the rotation corresponding vertical displacement and axial forcetx P can be found from the equilibrium equations. Force P can be easily extracted from (512) and inserted into (513) Thus for the given rotation finding vertical displacement reduces to the root finding problem. A full force displacement relationship can be hence established by repeating the above procedure for th e set of rotations varying from 0 to the arbitrarily chosen cutoff limit. tx Whereas the elastic solution provides intere sting insights into the buckling phenomena, plastic effects play an important role in the beha vior of real columns. Hence an elastoplastic solution is required to describe force displacement relation. Using equation (57) strains (58) can be rewritten as: d L EA P Lx yPxyt t2 cos cos cos1 4 1 tan),,,(0 (514) The elastoplastic, piecewise lin ear relationship depicted in Figure 578 was postulated to describe material behavior. 204 PAGE 205 40 30 20 10 0 10 20 30 40 0.020.0100.010.02Stress [ksi]Strain Figure 578. Elastopla stic material model Hence equilibrium equations can be rewritte n in their more general, integral form: f bf bftdy P22 2 (515) f bf bftdyy EA PL P2 2 cos 1 22 2 (516) bf = flange width; tf = flange thickness; = plastic hardening coefficient = 1.25 Thus, employing equilibrium, again one arri ves at two equations with three unknowns: vertical displacement rotation tx and axial force A system of two nonlinear equations can be solved for the given rotation P using commonly available engineering packages such as MathCAD and so on. A full force displacement rela tionship can be thus obta ined for the array of chosen rotations varying from 0 to an arbitrary chosen cutoff limit. Comparison of the analytically calculated answer for the W12x58 column with numerical result obtained from LSDYNA (see page 138) is shown in Figure 579 and Figure 580 205 PAGE 206 0 100 200 300 400 500 600 700 00.20.40.60.811.21.41.61.82Force [kip]Displacement [in] Analytical LsDyna Figure 579. 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 0.4Energy [kipin]Displacement [in] Analytical LsDyna Figure 580. Energy displacement of W12x58 column of 156 [in] height 206 PAGE 207 Verification of Energy Approach on Realistic Steel Building The real life steel building de sign, employed by Gupta and Krawinkler (2000), was used to verify the proposed energy approach. Framing plan of the selected building is shown in Figure 581 and Figure 560 Steel profiles applied in the building are listed in Table 59 and Table 510. The perimeter of the frame contai ned moment resisting connections. y 30 [ ft ] 30 [ ft ] 2 3 4 5 A B C D E 1 F G Moment connection Penthouse p erimeter x Figure 581. Three story, mo ment resisting framed buildi ng for verification analysis Table 59. Steel profiles of column s (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 PAGE 208 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 A A A w 1 8 x 35 A A A w 18x35 w18x35 ( infill beams ) Figure 582. Framing plan used for SAC three story building Table 510. Moment resistant beams (designated with A) Floor Beam A 2 w18x35 3 w21x57 roof w21x62 Only two representative cases are discussed he rein 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 le vel of structural safety and compared with the traditional, force based methods. 208 PAGE 209 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). A lthough collapse was arrested, significant plastic deformations developed in the bays directly affected by column removal ( Figure 583 ). Initially induced kinetic energy was absorbed by the bu ilding, which reached a stable, energetic configuration ( Figure 584 ). Figure 583. Deflected, final configur ation of the build ing in CASE 2 Removed Columns 20 21 22 23 24 25 0 0.5 1 1.5 2 x 104 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure 584. Global En ergies in CASE 2 209 PAGE 210 Internal energy (deformation work) of th e analyzed columns is depicted in Figure 585 None of the columns exceeded the buckling ener gy threshold. Hence none of the columns reached the postbuckling phase, characterized by th e 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 22.5 0 50 100 150 Time [s]Energy [kipin] C1.1 C1.2 C1.3 D1.1 A2.1 A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 w12x58 w14x74 w14x99 w14x99 w14x74 w12x58 B2.1 ( w12x58 ) Figure 585. Internal column energies in CASE 2 Internal forces in columns B2.1 and A2.1 (for comparison) are depicted in Figure 586 and Figure 587 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 ta sk to estimate level of safety associated with column B2.1 on the basis of intern al forces time histories. It appeared that 210 PAGE 211 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 0.2 0.4 0.6 0.8 1 05101520Normalized Moment and ForceTime [s] y Axial Force Moment S Moment TA S T 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 05101520Normalized Moment or ForceTime [s] Axial Force Moment S Moment T S T B Figure 586. Internal forces in CASE 2: A) column A2.1, B) column B2.1. 211 PAGE 212 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 2020.52121.522Normalized Moment and ForceTime [s] y Axial Force Moment S Moment TA 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 2020.52121.522Normalized Moment or ForceTime [s] Axial Force Moment S Moment TB S T Figure 587. 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 588 Internal energies, corresponding to the external work em ployed to deform the columns, did not exceed 212 PAGE 213 the predefined buckling energy thre sholds. Since column B2.1 cons isted 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 0Force [kip]Displacement [in] A2.1 B2.1 A2.1 Buckling Force B2.1 Buckling ForceA 0 20 40 60 80 100 120 140 0.15 0.1 0.05 0Energy [kipin]Displacement [in] A2.1 B2.1 A2.1 Buckling Energy B2.1 Buckling EnergyB w12x58 w14x99 Figure 588. Energy absorption in th e selected columns. CASE 2: A) Axial forcedisplacement, B) Internal energydisplacement. 213 PAGE 214 Energy rates in the columns ( Figure 589 ) indicated that energy was pushed toward columns B2.1 and B3.1. Figure 589. Energy rates in columns in CASE 2 Energy propagation through the predefined building zones ( Figure 590 ) is depicted in Figure 591 and Figure 592 Energy of the building increased after column removal to absorb the released gravitational energy. Internal ener gy (deformation work) in the exclusive zones 23, 34 and 45 remained practically constant. Thus no energy was transferred to these zones from the zones affected by the column rem ovals. Normalized energy allocation ( Figure 593 ) has shown energy localization in the inclusive zone 02, as the global energy share of this zone rose from 20% to 70%. There was in essence no ener gy slosh (energy flow between zones) from the inclusive zone 02 to zone 25. 214 PAGE 215 2 3 4 5 A B C D E F G 1 Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Figure 590. Building zones used to trace the energy propagation 20 20.5 21 21.5 22 22.5 23 0 500 1000 1500 2000 2500 3000 Time [s]Energy [kipin] zone 01 zone 12 zone 23 zone 34 zone 45 Figure 591. Energy propagation thr ough the exclusive zones in CASE 2 215 PAGE 216 20 20.5 21 21.5 22 22.5 23 0 1000 2000 3000 4000 5000 6000 Time [s]Energy [kipin] zone 01 zone 02 zone 03 zone 04 zone 05 Figure 592. Energy propagation thr ough the inclusive zones in CASE 2 20 20.5 21 21.5 22 22.5 23 0 0.2 0.4 0.6 0.8 1 Time [s]Relative Energy zone 01 zone 02 zone 03 zone 04 zone 05 Figure 593. Normalized energy allo cation in the building in CASE 2 216 PAGE 217 Energy absorbed (deformation work) in beams was mainly plastic ( Figure 594 ) and thus mainly irreversible (dissipated). Since none of the columns buckled, energy absorbed (deformation work) in columns remained chiefly elastic ( Figure 595 ). Although this elastic energy could be potentially recovered from the sy stem, 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 nontransient loads effectively prevent any signifi cant unloading and, thus, any elastic energy release. Beams in the ydirection played a major role in arresting the colla pse and absorbing the initially released potential energy ( Figure 596 ). Beams in the xdirection aided the energy absorption at approximately 20.5 [s]. Relative en ergy absorbed by beams in the ydirection rose significantly in comparison to t hose of beams in the xdirecti on and columns. Most of the energy was absorbed by beams. 20 20.5 21 21.5 22 22.5 23 0 50 100 150 200 250 300 350 400 y Time [s]Energy [kipin] Absorbed Energy Elastic Energy Plastic (Irreversible) Energy Figure 594. Decomposition of the absorbed energy (deformation work) in A1.1 beam yy 217 PAGE 218 20 20.5 21 21.5 22 22.5 23 0 20 40 60 80 100 Time [s]Energy [kipin] Absorbed Energy Elastic Energy Figure 595. Decomposition of the absorbed energy (deformation work) in B2.1 column 20 20.5 21 21.5 22 22.5 23 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time [s]Relative Energy columns beamsX beamsY Figure 596. Energy split between members in zone 2 (CASE 2) 218 PAGE 219 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 597 Significant levels of kinetic energy were obs erved 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 colu mns impacted the ground layer. 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 x 104 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure 597. Global energies in the building Although column B2.1 failed, only partial collapse was observed in the building ( Figure 598). 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 xdirection was halted. On the other hand, strong perimeter columns in the moment resisting A line were sufficiently robust to withsta nd the demands from the falling 219 PAGE 220 bays. Once bay A1 impacted the soil laye r, demand on the adjacent bays dropped and the building achieved a final, stable state. B2.1 Figure 598. Collapse sequence of the steel building in CASE 3 220 PAGE 221 2 3 4 5 A B C D E F G 1 Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Figure 599. Building zones used to trace the energy propagation Energy in the predefined zones ( Figure 5100 ) significantly rose in zone 02. A stable configuration was reached after 21.6 [s]. Syst em energy was reduced af ter 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 increa se of the absorbed energy in zone 25. Therefore, energy absorption localized in inclusive zone 02. In essence there was no horizontal energy sloshing (repetitive energy movement inbet ween zones) in the building. Normalized energy allocation ( Figure 5101 ) confirmed that energy was localized in zone 02. Energy absorbed in zone 02 accounted for 90% of th e internal building energy, as opposed to 18% when three columns were removed. After the partial collapse, the bui lding reached a stable energy configuration. 221 PAGE 222 20 20.5 21 21.5 22 22.5 23 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 gy y Time [s]Energy [kipin] zone 01 zone 12 zone 23 zone 34 zone 45 Figure 5100. Energy propagation thr ough the exclusive zones in CASE 3 20 20.5 21 21.5 22 22.5 23 0 0.2 0.4 0.6 0.8 1 Time [s]Relative Energy zone 01 zone 02 zone 03 zone 04 zone 05 Figure 5101. Normalized energy al location in the building in CASE 3 222 PAGE 223 Allocation of energy among main structural members is illustrated in Figure 5102 Mainly beams in the ydirection 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 23 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time [s]Relative Energy columns beamsX beamsY Figure 5102. Energy split between members in zone 2 (CASE 3) Energy absorbed in beams was mainly plastic ( Figure 5103 and Figure 5104 ) and thus chiefly impossible to recover (dissipated). En ergy absorption in the failed B2.1 column, ( Figure 5106 and Figure 5105 ) 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 5107 ) exhibited significant levels of elastic energy (40%). Sin ce this column did not lose its load carrying capacity, the nontransient loads effectively preven ted unloading and energy sloshing. 223 PAGE 224 20 20.5 21 21.5 22 22.5 23 0 200 400 600 800 1000 1200 Time [s]Energy [kipin] Absorbed Energy Elastic Energy Plastic Energy Figure 5103. Decomposition of the absorbed energy (deformation work) in A2.1 beam yy 20 20.5 21 21.5 22 22.5 23 0 50 100 150 200 250 300 350 400 Time [s]Energy [kipin] Absorbed Energy Elastic Energy Plastic Energy Figure 5104. Decomposition of the absorbed energy (deformation work) in B1.1 beam xx 224 PAGE 225 200 20 20.5 21 21.5 22 22.5 23 0 50 100 150 Time [s]Energy [kipin] Absorbed Energy Elastic Energy Plastic Energy Figure 5105. Closeup view of the energy (d eformation work) decomposition in B2.1 column 20 20.5 21 21.5 22 22.5 23 0 0.2 0.4 0.6 0.8 1 Time [s]Normalized Energy Absorbed Energy Elastic Energy Plastic Energy Figure 5106. Energy decomposition of the absorb ed energy (deformation work) in B2.1 column 225 PAGE 226 200 20 20.5 21 21.5 22 22.5 23 0 50 100 150 Time [s]Energy [kipin] Absorbed Energy Elastic Energy Plastic Energy Elastic Energy Figure 5107. 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 va riation in the columns energy le vel. It should be noted that permanent loads weigh down upon columns at all times. These nontransi ent 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 prebuckling to post buckling and failure behavi or is very apparent in the energy domain. On the other hand, axia l 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 ch allenging to understand the building behavior (after column(s) removal) on the basis of force results only. 226 PAGE 227 Analysis of internal column energies revealed that internal energy (deformation work) in the B2.1 column exceeded the respective buckling energy threshold ( Figure 5108 ). Second and third story B2.2 and B2.3 columns were also ov erloaded. 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 mo re energy than other co lumns, the respective buckling energy capacity was not exceeded. On th e other hand, column C2.1, consisting of the weaker W12x58 section, was affected by the slab s tearing. Also, column C2.1 exceeded the respective buckling energy threshold. 20 20.5 21 21.5 22 22.5 0 50 100 150 200 250 Time [s]Energy [kipin] C1.1 C1.2 C1.3 D1.1 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 w12x58 w14x74 w14x99 B2.1 ( w12x58 ) C2.1 ( w12x58 ) A3.1 ( w14x99 ) w12x58 w14x99 Figure 5108. Internal column energies in CASE 3 Internal forces in columns B2.1 and C2.1 ( both made of W12x58 sections) are compared in Figure 5109 Both columns carried practically the same axial and bending loads. Comparison 227 PAGE 228 of the dynamic phases ( Figure 5110 ) 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 05101520Normalized Moment and ForceTime [s] Axial Force Moment S Moment TB 1.00 0.50 0.00 0.50 1.00 1.50 05101520Normalized Moment or ForceTime [s] Axial Force Moment S Moment TA S T S T Figure 5109. Normalized internal forces in: A) B2.1 column, B) C2.1 column 228 PAGE 229 1.0 0.5 0.0 0.5 1.0 1.5 2020.52121.522Normalized Moment and ForceTime [s] S T Axial Force Moment S Moment T A 1.00 0.50 0.00 0.50 1.00 1.50 2020.52121.522Normalized Moment or ForceTime [s] Axial Force Moment S Moment TB S T Figure 5110. Normalized inte rnal 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 th e level of structural safety. Nontransient, permanent loads resting on columns B2.1 and C2.1 were estimated. 229 PAGE 230 Loading and unloading paths are shown in Figure 5111 Internal energy of column B2.1 exceeded the failure energy threshol d. Thus the load carrying cap acity was reduced so severely that no stable configuration (supporting the permanent load) was possible. 700 600 500 400 300 200 100 0 100 2.01.51.00.50.0Force [kip]Displacement [in] y B2.1 C2.1 w12x58 Buckling Force B2.1 permanent load C2.1 permanent loadA 0 100 200 300 400 500 600 700 800 2.01.51.00.50.0Energy [kipin]Displacement [in] B2.1 C2.1 Buckling Energy B2.1 Failure C2.1 FailureB Figure 5111. Energy flow in the SA C building: CASE 3 (total failure): A) Axial forcedisplacement, B) Internal energydisplacement 230 PAGE 231 Loss of the column load carryi ng capacity in the postbuckling phase is irreversible. Thus the more energy a given column absorbs, the sm aller the residual capacity becomes for resisting permanent loads in the postbuckling phase. Therefore, tracing the internal column energy provides not only information on the column buckli ng but also on the corresponding safety level. Although column C2.1 exceeded the buckling ener gy 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 5112 ) showed the energy flow localization (energy sinkhole) in the column B2.1. Thus energy ra tes provided complementary information on the global level of structural safety. Only column B2.1 violated the failure energy rate threshold. Figure 5112. Energy rate s in columns in CASE 3 231 PAGE 232 Summary Energy plots reveal more information than can be seen through examination of the deformed structure. For example, in the total co llapse in CASE 3, bucking of the column B3.1 was really hard to notice in the simulation un less the proper perspective was chosen. Column B3.1 buckling was effectively restrained by th e 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 en ergy redistribution in such a way that the 2nd or 3rd story columns were overpowered first. Thus, any improvement or crosssection change of the columns located at A2 or at C1 should be applied thorough all three stories. Analysis of internal energies not only conf irmed 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 Buck ling and Energy Failure Limits It has been shown that internal force hist ories are not very sensitive to the buckling initiation and failure. On the contrary, in ternal energy (deformation work) significantly increased after buckling. Buckled members were easily identified on the basis of energy results only. To illustrate the superiority of th e energy approach, both force and energy based demand/capacity (D/C) ratio s were aggregated in Table 511 as well as in Figure 5113 through Figure 5115 to facilitate the co mparative analysis. 232 PAGE 233 Table 511. 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 subs equent scenarios (one column in CASE A through three columns in CASE C), the for ce 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 postbuckling state (C2.1 column) from the failure in CASE C. Thus force based D/C ratios provi de results, which are very difficult to interpret and translate into a level of structural safety afte r the occurrence of the localized damage. Energy based D/C ratios were very sensitive to the buckling initiation. Internal energies significantly rose, and the viola tion of the energy buckling criteri on directly corresponded to the onset of buckling ( Figure 5111 ). 0.61 0.69 0.94 0.59 0.58 0.93 0.00 0.50 1.00 1.50 2.00 2.50 3.00 CAS E 1CAS E 2CAS E 3 B2.1 column C2.1 column Figure 5113. Buckling fo rce demand/capacity ratios 233 PAGE 234 0.34 0.43 0.31 0.30 1.72 0.00 0.50 1.00 1.50 2.00 2.50 3.00 CAS E 1CAS E 2CAS E 3 B2.1 column C2.1 column Figure 5114. Buckling energy demand/capacity ratios 0.034 0.051 0.032 0.037 0.21 0.00 0.50 1.00 1.50 2.00 2.50 3.00 CAS E 1CAS E 2CAS E 3 B2.1 column C2.1 column buckling limit 11 failure limit 87 Figure 5115. Failure en ergy 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 criter ion. Column C2.1 absorbed only 21% of the 234 PAGE 235 energy needed to reduce its capacity below the nontransient 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). Prebuckling, postbuckling 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 transparen t results and direct insights into the collapse propagation. 235 PAGE 236 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS Summary The main objective of this study was to de velop and implement a rational, energybased approach to progressive collapse of steel framed buildings by assessi ng 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 co llapse 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 closedform analytical expressions, 3. An energybased approach for the collaps e of moment resisting steel buildings was developed and verified. It has been proven that column energy histories provide an effective interpretation tool, which give s accurate insight into dynamic energy redistributions, dynamic alternate paths reform ations, competing failures, and structural safety. It has been shown that the energybased approach surpasses traditional analysis methods through an accurate assessment of progressive collapse propagation. Progressive Collapse Conclusions Physicsbased simulation techni ques were used to investig ate the behavior of major structural members during progressi ve collapse propagation or arre st. A simplified steel framed building and a more complex three story seismi c designed steel building were selected for comparative analysis. These comparative simula tions have shown that only fully nonlinear dynamic time history analysis of sufficiently de tailed models, which account for material and geometric nonlinearities, can lead to meaningful results. Progre ssive collapse is characterized by large displacements, strains and pronounced dynamic effects. Therefore, resorting to simplified techniques, which do not include r ealistic structural and material models and inertial effects of 236 PAGE 237 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 la ws of classical physics. In the simulated events, one or more columns were removed wit hout 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 freefall requirement. This demonstrated that neglecting the inertia of dead and live loads resulted in significant errors a nd violation of the freefall 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 structur e provides ductility and increases resistance to progressive collapse. However, if the initiating damage is beyond th e arresting capabilities of the structure, due to superbly strong structural ti es, 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 c onnections 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 case s. In contrast, the th ree story building with seismic detailing (with both moment and shear conn ections), exhibited only partial collapse after the removal of three columns. Shear conn ections prevented the collapse from further propagation into the building interior. Tying the structure is important in providing adequate robustn ess to prevent catastrophic collapse, provided that structural ties are proper ly designed. Reliance on structural ties can be beneficial, if a building has sufficient capability to redistribute loads. However, the addition of 237 PAGE 238 ductility must be accompanied by an analytical ex amination of building resistance to progressive collapse, and must not be treate d as a cureall. Strengthening structural ties, wit hout structural examination, can lead to worse consequences because strong connections might cause total collapse as opposed to partial co llapse limited by conn ection failures. It is strongly encouraged to exhaust all an alysis tools to suffici ently design connections, such that foreseeable localized failures can be ar rested and occupants lives saved. However, one must recognize that not all abnormal loadings can be predicted in ad vance. Therefore, it may be beneficial to partition a building, such that collapse cannot sp read 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 localiz ed in the bays affected by the column(s) removal in the instances of arre sted collapse. Analysis of th e energy flow through the building revealed that beams play a crucial role in the ab sorption 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 column s in the total energy absorption. Thus, more energy was absorbed by the columns during th e collapse propagation phase. The released potential energy was main ly dissipated by plastic deformations (beams, slabs and column postbuckling behavior). Elas tic energy was stored primarily in columns, which accounted for less than 20% of the total abso rbed energy. In principle the stored, elastic energy was not released because the nontransi ent column loads prevented any significant unloading. 238 PAGE 239 The energy absorption of the beams was shown to be an important factor in arresting collapse. This conclusion agrees well with findi ngs on the collapse sensi tivity 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 increas ed 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 nece ssary condition to in itiate the collapse (but it is not a sufficient). The column fa ilure 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 re spective 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 (depende nt both on column propert ies and on a value of permanent load). Both buckling energy and column failure energy can be conveniently computed beforehand, using the numerical (LSDYNA) and/or analytical (closedform) calculation procedures propos ed in this study. A comparison of the force demand to the memb er capacity is traditionally employed to evaluate a members safety. It has been shown in this study that force based demand capacity (D/C) is not very sensit ive to the fundamental changes in structural behavior. Conversely, 239 PAGE 240 buckling energy D/C values correctly identified buckling in all analy zed cases in a very distinctive manner. Moreover, the energy failure D/C criteri on was violated only by failed columns. In the case of arrested collapse, co mparing 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 anal ysis of alternative designs such that the safest structural solution can be chosen, Research the role of beams and structural connections on energy fl ow 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 PAGE 241 APPENDIX A VERIFICATION OF ENERGY EXTRACTION PROCEDURE The energy flow of a simple 2D 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 resu lts. Therefore the two dimensional case was investigated only to verify the ability of th e finite element software LSDYNA to properly extract and report the energy flow results. The front faade of the simplified steel framed structure ( Figure 41 ) was monotonically loaded at frame joints with concentrated forces ( Figure A1 ). Twelve concentrated forces provided easy to track loading forcetime histor ies. Joint displacement time histories were written in the LSDYNA NODOUT displacem ent output file at every 1 [ms]. Figure A1. Steel frame used for the energy benchmark test The loading was applied in two phases: 1) st atic preloading (over 20 [s]); 2) collapse propagation (after the static prel oading, initiated by the instantaneous corner column removal). After the removal of the single, corner column, frame collapsed partially ( Figure A2 ). Plastic hinges were formed in the beam supports but co llapse did not propagate to the adjacent bays. 241 PAGE 242 Figure A2. Final, displaced shape of the 2D frame used in the energy benchmark test Global frame energies were extracted from th e LSDYNA GLSTAT text file. The increase of the internal energy (deformati on work) reached its final level wh en the collapse was arrested ( Figure A3 ). Kinetic and internal energies added up to the extern al work done on the system. Thus conservation of energy was clearly satisfied in the LSDYNA simulation. Figure A3. Global energies repo rted by LSDYNA in GLSTAT file The kinetic energy was rising until the coll apsing beams impacted the ground. Once members decelerated, the kinetic energy dipped to zero. Thus the instable states were 242 PAGE 243 characterized by the significant kinetic energy le vels. A stable state was achieved when the global kinetic energy vanished. Global energy results (GLSTAT) provided info rmation 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 retr ieved from the nodal NODOUT output file. Thus, the external work was analytically computed by the integration of the known forcedisplacement 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 LSDYNA external work results is shown in Figure A5 and Figure A4 Excellent agreement between the an alytical and the numerical curves verified the ability of LSDYNA to correctly provide global energy time histories. 0 5 10 15 20 25 0 2000 4000 6000 8000 10000 12000 14000 Time [s]External Work [kipin] External Work Analytical External Work GLSTAT Figure A4. Analytical and nu merical external work results 243 PAGE 244 0 5 10 15 20 0 5 10 15 20 Time [s]External Work [kipin] External Work Analytical External Work GLSTAT Figure A5. Analytical and nume rical external work results during the static preloading phase Global energy results were useful in assess ing whether the structure reached a stable energy state. Tracing the global kinetic energy en abled determination of the proper termination time. Once the global kinetic ener gy 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 prove d useful, the distribution of th e local internal and kinetic energies was of the primary importance in this research. LSDYNA MATSUM ASCII output file was used to obtain local energies for all st ructural 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 th e summed external work, kinetic and internal energies from MATSUM with glob al GLSTAT results is shown in Figure A6 and in Figure A7 244 PAGE 245 0 5 10 15 20 0 5 10 15 20 25 30 35 40 45 50 Time [s] Energy [kipin] Internal Energy GLSTAT Kinetic Energy GLSTAT External Work GLSTAT Internal Energy MATSUM Kinetic Energy MATSUM External Work MATSUM Figure A6. Global (GLSTAT) and sum of lo cal energies (MATSUM) during the static preloading 0 5 10 15 20 25 0 2000 4000 6000 8000 10000 12000 14000 Time [s] Energy [kipin] Internal Energy GLSTAT Kinetic Energy GLSTAT External Work GLSTAT Internal Energy MATSUM Kinetic Energy MATSUM External Work MATSUM Figure A7. Comparison of global (GLSTAT) and sum of local energies (MATSUM) 245 PAGE 246 The sums of the individual energies obtain ed 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 calcul ated external work. Conservation of energy was hence satisfied and the energy extraction proced ure was deemed accurate and trustworthy. The energydisplacement histories pr ovided the closeup view on the energy redistribution. The energytime histories incr eased abruptly after the prescribed column removal. Due to the short duration of the unstabl e energy transition phase, energytime 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 A8 Vertical displacement Figure A8. Location of the selected node used for energydisplacement histories The static preloading phase was characterized by the monotonic increas e of the internal energy (deformation work) ( Figure A9 ). Vertical displacement of the selected joint was small (0.00013 of the column height). The kinetic ener gy 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 A10 246 PAGE 247 0 0.005 0.01 0.015 0.02 0 2 4 6 8 10 12 Vertical displacement [in]Energy [kipin] Internal Energy GLSTAT Kinetic Energy GLSTAT External Work GLSTAT Internal Energy MATSUM Kinetic Energy MATSUM External Work MATSUM Figure A9. Internal, ki netic and total energy duri ng the static preloading 0 50 100 150 0 2000 4000 6000 8000 10000 Vertical displacement [in]Energy [kipin] Internal Energy GLSTAT Kinetic Energy GLSTAT External Work GLSTAT Internal Energy MATSUM Kinetic Energy MATSUM External Work MATSUM Figure A10. Internal, ki netic and total energy 247 PAGE 248 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 intern al energy (deformation work). The system reached its stable energy state corresponding to the pa rtial frame collapse. The demise of kinetic energy evidently indicated that the system arrived to the stable energetic state. 248 PAGE 249 APPENDIX B VERIFICATION OF ENERGY APPROA CH 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 analo gous (and repetitive), only CASE 1 is described in more detail. The results for th e remaining cases are desc ribed 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 B2 ). The internal energy (deformation work) of the system increas ed by 40%. Insignificant levels of kinetic energy were induced ( Figure B3 ). y 30 [ ft ] 30 [ ft ] 2 3 4 5 A B C D E 1 F G Moment connection Penthouse p erimeter x Figure B1. Removed columns in CASE 1 249 PAGE 250 Table B1. 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 B2. Final configuration of the steel building (CASE 1) 20 21 22 23 24 25 0 500 1000 1500 2000 2500 3000 3500 4000 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure B3. Global energies in CASE 1 250 PAGE 251 Internal energies in the selected columns are depicted in Figure B4 Columns in the potential collapse initiation zone consisted of three different sect ion 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 postbuckling phase. 20 20.5 21 21.5 22 22.5 0 50 100 150 Time [s]Energy [kipin] B1.1 B1.2 B1.3 C1.1 C1.2 C1.3 A2.1 A2.2 A2.3 B2.1 B2.2 B2.3 C2.1 C2.2 C2.3 D2.1 A3.1 A3.2 A3.3 B3.1 B3.2 B3.3 w12x58 w14x74 w14x99 w14x99 w14x74 w12x58 Figure B4. Internal column energies in CASE 1 Energy rates are depicted in Figure B5 There was a spike in the internal energy (deformation work) rate in column B2.1. Howe ver, it was well below the value of 10000 [kip251 PAGE 252 in/s] (1.14 [MJ]), which was observed in the failing columns of the simplified steel framed building. Thus energy rate criteri on confirmed the safety of the building, after the collapse was arrested. Figure B5. Energy rates in columns in CASE 1 The steel building was divided into five zones, as depicted in Figure B6 Energy time histories in each of these zones were extracted from numerical an alyses. The portion of the total energy in each zone provided insight into the energy flow th rough the building. Normalized energy split between the zones gave information on the energy localizatio n. 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 581 ). Such 252 PAGE 253 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 4 Zone 5 Figure B6. Building zones used to trace the energy propagation Energy propagation through the building is shown in Figure B7 Internal energy (deformation work) increased mainly in the zone 1. Rise of the energi es in zones 2 through 5 resulted from the energy increase in zone 1. Normalized energy allocation, shown in Figure B8 highlights very little localization. Energy rel eased by the gravity was mainly absorbed by beams in the y direction as depicted in Figure B9 Participation of columns in the energy storage slightly decreased due to the column removal and its pertinent intern al energy (deformation work). 253 PAGE 254 20 20.5 21 21.5 22 22.5 23 0 500 1000 1500 2000 2500 3000 Time [s]Energy [kipin] zone 1 zone 2 zone 3 zone 4 zone 5 Figure B7. Energy propagation through the building in CASE 1 20 20.5 21 21.5 22 22.5 23 0 0.2 0.4 0.6 0.8 1 Time [s]Relative Energy zone 1 zone 2 zone 3 zone 4 zone 5 Figure B8. Normalized energy allo cation in the building in CASE 1 254 PAGE 255 20 20.5 21 21.5 22 0 0.1 0.2 0.3 0.4 0.5 0.8 0.6 0.7 Time [s]Relative Energy columns beamsX beams YY Figure B9. Energy split between members in zone 2 (CASE 1) 255 PAGE 256 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 af ter application of the static preloading (Load = 1.0 D + 0.5 L). Collapse was arrested. y 30 [ ft ] 30 [ ft ] 2 3 4 5 A B C D E 1 F G Moment connection Penthouse p erimeter x Figure B10. Removed columns in CASE 4 Table B2. Removed columns in CASE 4 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 256 PAGE 257 Figure B11. Final configurati on of the building in CASE 4 20 21 22 23 24 25 0 0.5 1 1.5 2 x 104 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure B12. Global en ergies in CASE 4 257 PAGE 258 Table B3. W14x74, 156 [in] column buckling results P/Pc r Buckling Force Internal Energy Comment [kip] [kipin] 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 23 0 50 100 150 200 250 300 350 400 450 500 Time [s]Energy [kipin] A1.1 A1.2 A1.3 B1.1 B1.2 B1.3 B2.1 B2.2 B2.3 B3.1 B3.2 B3.3 A4.1 A4.2 A4.3 B4.1 B4.2 B4.3 w12x58 w14x74 w14x99 A1.3 ( w14x74 ) A4.3 ( w14x74 ) Figure B13. Internal column energies in CASE 4 258 PAGE 259 20 1000 20.2 20.4 20.6 20.8 21 0 100 200 300 400 500 600 700 800 900 Time [s]Energy Rate [kipin/s] A1.3 ( w14x74 ) A1.1 A1.2 A1.3 B1.1 B1.2 B1.3 B2.1 B2.2 B2.3 B3.1 B3.2 B3.3 A4.1 A4.2 A4.3 B4.1 B4.2 B4.3 Figure B14. Energy rates in columns in CASE 4 259 PAGE 260 2 3 4 5 A B C D E 1 F G Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Figure B15. Building zones used to trace the energy propagation in CASE 4 20 20.5 21 21.5 22 0 1000 2000 3000 4000 5000 6000 7000 8000 Time [s]Energy [kipin] zone 1 zone 2 zone 3 zone 4 zone 5 Figure B16. Energy propagation through the building in CASE 4 260 PAGE 261 20 20.5 21 21.5 22 0 0.2 0.4 0.6 0.8 1 Time [s]Relative Energy zone 1 zone 2 zone 3 zone 4 zone 5 Figure B17. Normalized energy allocation in the building in CASE 4 20 20.5 21 21.5 22 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time [s]Relative Energy columns beamsX beamsYY Figure B18. Energy split between members in zone 2 (CASE 4) 261 PAGE 262 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 remove d after application of the static preloading (Load = 1.0 D + 0.5 L). The an alyzed building collapsed. y 30 [ ft ] 2 3 4 5 A B C D E 1 F G Moment connection Penthouse perimeter x 30 [ft] Figure B19. Removed columns in CASE 5 Table B4. 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 PAGE 263 Figure B20. Collapse sequence of the steel building in CASE 5 263 PAGE 264 Figure B 20. Continued 264 PAGE 265 20 22 24 26 28 30 0 2 4 6 8 10 12 14 x 105 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure B21. Global en ergies in CASE 5 Table B5. W14x74, 156 [in] column buckling results P/Pc r Buckling Force Internal Energy Comment [kip] [kipin] 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 PAGE 266 Table B6. W12x58, 156 [in] column buckling results P/Pc r Buckling Force Internal Energy Comment [kip] [kipin] 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 25 0 500 1000 1500 2000 2500 3000 Time [ s ] Energy [kipin] A1.1 A1.2 A1.3 B1.1 B1.2 B1.3 B2.1 B2.2 B2.3 B3.1 B3.2 B3.3 B4.1 B4.2 B4.3 A5.1 A5.2 A5.3 B5.1 B5.2 B5.3 w12x58 w14x74 w14x99 A1.3 ( w14x74 ) B4.3 ( w12x58 ) A5.1 ( w12x58 ) Figure B22. Internal column energies in CASE 5 266 PAGE 267 A1.3 ( w14x74 ) Figure B23. Energy rates in columns in CASE 5 267 PAGE 268 2 3 4 5 A B C D E F G 1 Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Figure B24. Building zones used to trace the energy propagation in CASE 5 20 22 24 26 28 30 32 34 0 1 2 3 4 5 6 7 8 x 10 Time [s]Energy [kipin] zone 1 zone 2 zone 3 zone 4 zone 5 510 xFigure B25. Energy propagation through the building in CASE 5 268 PAGE 269 20 22 24 26 28 30 32 34 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time [s]Relative Energy zone 1 zone 2 zone 3 zone 4 zone 5 Figure B26. Normalized energy allocation in the building in CASE 5 20 22 24 26 28 30 32 34 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time [s]Relative Energy columns beamsX beamsYY Figure B27. Energy split between members in zone 5 (CASE 5) 269 PAGE 270 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 af ter 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 Moment connection Penthouse perimeter x 30 [ft] x Figure B28. Removed columns in CASE 6 Table B7. Removed columns in CASE 6 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 270 PAGE 271 Figure B29. Final configurati on of the building in CASE 6 20 21 22 23 24 25 0 0.5 1 1.5 2 2.5 3 3.5 4 x 104 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure B30. Global en ergies in CASE 6 271 PAGE 272 Table B8. Failure energy lim its 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 w12x65 element 988 459 419 1068 785 1028 663 permanent load [kip] 363 320 320 331 319 342 337 buckling energy [kipin] 82 66 66 66 66 74 74 failure energy [kipin] 2010 993 993 972 995 1800 1820 20 20.5 21 21.5 22 0 10 20 30 40 50 60 70 80 90 100 Time [s]Energy [kipin] D2.1 D2.2 D2.3 E2.1 E2.2 E2.3 F2.1 F2.2 F2.3 C3.1 C3.2 C3.3 F3.1 F3.2 F3.3 D4.1 D4.2 D4.3 E4.1 E4.2 E4.3 F4.1 F4.2 F4.3 w12x58 w12x65 w12x72 w14x99 D4.1 ( w12x72 ) Figure B31. Internal column energies in CASE 6 272 PAGE 273 20.3 20.4 20.5 20.6 20.7 20.8 0 10 20 30 40 50 60 70 80 90 100 Time [s]Energy [kipin] D2.1 D2.2 D2.3 E2.1 E2.2 E2.3 F2.1 F2.2 F2.3 C3.1 C3.2 C3.3 F3.1 F3.2 F3.3 D4.1 D4.2 D4.3 E4.1 E4.2 E4.3 F4.1 F4.2 F4.3 w12x58 w12x65 w12x72 w14x99 Figure B32. Internal column energies in CASE 6 D2.1 ( w12x58 ) E4.1 ( w12x65 ) D4.1 ( w12x72 ) E2.1 ( w12x58 ) 273 PAGE 274 20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 0 50 100 150 200 250 300 350 400 450 500 Time [s]Energy Rate [kipin/s] D2.1 D2.2 D2.3 E2.1 E2.2 E2.3 F2.1 F2.2 F2.3 C3.1 C3.2 C3.3 F3.1 F3.2 F3.3 D4.1 D4.2 D4.3 E4.1 E4.2 E4.3 F4.1 F4.2 F4.3 Figure B33. Energy rates in columns in CASE 6 274 PAGE 275 2 3 4 5 A B C D E 1 F G Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Figure B34. Building zones used to trace the energy propagation in CASE 6 20 20.5 21 21.5 22 22.5 23 0 2000 4000 6000 8000 10000 Time [s]Energy [kipin] zone 1 zone 2 zone 3 zone 4 zone 5 Figure B35. Energy propagation through the building in CASE 6 275 PAGE 276 20 21 22 23 24 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time [s]Relative Energy zone 1 zone 2 zone 3 zone 4 zone 5 Figure B36. Normalized energy allocation in the building in CASE 6 20 22 24 26 28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time [s]Relative Energy columns beamsX beamsYY Figure B37. Energy split between members in zone 5 (CASE 6) 276 PAGE 277 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 remove d after application of the static preloading (Load = 1.0 D + 0.5 L). The an alyzed building collapsed. y 30 [ ft ] 30 [ ft ] 2 3 4 5 A B C D E 1 F G Moment connection Penthouse p erimeter x Figure B38. Removed columns in CASE 7 Table B9. Removed columns in CASE 7 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 277 PAGE 278 time = 20 [ s ] time = 20.9 [ s ] time = 22 [ s ] time = 23.3 [ s ] Figure B39. Collapse sequence of the steel building in CASE 7 278 PAGE 279 time = 24 [ s ] time = 28 [ s ] Figure 598 B39. Continued 20 22 24 26 28 0 0.5 1 1.5 2 x 106 Time [s]Energy [kipin] Internal Energy Kintetic Energy External Work Figure B40. Global en ergies in CASE 7 279 PAGE 280 Table B10. Failure energy lim its 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 w12x65 element 988 459 419 1068 785 2630 663 permanent load [kip] 363 320 320 331 319 158 337 buckling energy [kipin] 82 66 66 66 66 132 74 failure energy [kipin] 2010 993 993 972 995 >10000 1820 20 20.5 21 21.5 22 22.5 23 23.5 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time [s]Energy [kipin] D2.1 D2.2 D2.3 E2.1 E2.2 E2.3 F2.1 F2.2 F2.3 C3.1 C3.2 C3.3 F3.1 F3.2 F3.3 D4.1 D4.2 D4.3 F4.1 F4.2 F4.3 E5.1 E5.2 E5.3 w12x58 w12x65 w12x72 w14x99 E2.1 ( w12x58 ) D4.1 ( w12x72 ) Figure B41. Internal column energies in CASE 7 280 PAGE 281 20 20.5 21 21.5 22 22.5 0 50 100 150 200 250 300 350 400 450 500 Time [s]Energy [kipin] E2.1 ( w12x58 ) D2.1 D2.2 D2.3 E2.1 E2.2 E2.3 F2.1 D4.1 ( w12x72 ) F2.2 F2.3 C3.1 C3.2 C3.3 F3.1 F3.2 F3.3 D4.1 D4.2 D4.3 F4.1 F4.2 F4.3 E5.1 E5.2 E5.3 w12x58 w12x65 w12x72 w14x99 Figure B42. Internal column ener gies in CASE 7. Closeup view 281 PAGE 282 D4.1 ( w12x72 ) E2.1 ( w12x58 ) Figure B43. Energy rates in columns in CASE 7 282 PAGE 283 2 3 4 5 A B C D E F G Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 1 Figure B44. Building zones used to trace the energy propagation in CASE 7 20 22 24 26 28 0 1 2 3 4 5 6 7 8 Time [s]Energy [kipin] zone 1 zone 2 zone 3 zone 4 zone 5 510 xFigure B45. Energy propagation through the building in CASE 7 283 PAGE 284 1 6 20 22 24 26 28 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time [s]Relative Energy zone 1 zone 2 zone 3 zone 4 zone 5 Figure B46. 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Journal of Engineering Mechanics 130(10), 11771187. 290 PAGE 291 291 BIOGRAPHICAL SKETCH Stefan Szyniszewski was born in Poland. He received his bachelors and masters degree from the Technical University of Warsaw, Poland. He also studied at the RWTHAachen 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 PolishGerman Academic Exchange Commission and DAAD Award from the German Ministry of Edu cation. His research on progressive collapse simulations of steel framed structures was recognized with the Best Paper Award by the International Association of Bridge and St ructural Engineers (IABSE) during an annual conference in Helsinki, Finland in 2008. He was also the recipient of the Fulbright Award. 