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Collapse assessment of concrete buildings : an application to non-ductile reinforced concrete moment… Baradaran Shoraka, Majid 2013

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COLLAPSE ASSESSMENT OF CONCRETE BUILDINGS: AN APPLICATION TO NON-DUCTILE REINFORCED CONCRETE MOMENT FRAMES  by  Majid Baradaran Shoraka  B.Sc., Sharif University of Technology, 2003 M.Sc., Sharif University of Technology, 2006  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)  August 2013  ? Majid Baradaran Shoraka, 2013Abstract    iiABSTRACT Existing reinforced concrete buildings lacking details for ductile response during earthquake shaking represent prevalent construction type in high seismic zones around the world. Seismic rehabilitation of these existing buildings plays an important role in reducing urban seismic risk; however, with the massive inventory of existing concrete buildings and high costs of seismic rehabilitation, it is necessary to start by identifying and retrofitting those buildings which are most vulnerable to collapse.  The collapse of most non-ductile concrete buildings will be controlled by the loss of support for gravity loads prior to the development of a side-sway collapse mechanism. ?Gravity load collapse? may be precipitated by axial-load failure of columns, punching-shear failure of slab-column connections, or axial-load failure beam-column joints.  In this dissertation, system-level collapse criteria are developed and implemented in a structural analysis platform to allow for a more accurate detection of collapse in these existing moment frames. Detailed models for primary components, which may precipitate gravity-load collapse of the concrete moment frame, are first required to achieve this objective and develop the collapse assessment framework. An analytical model based on mechanics is developed to reliably capture the lateral load?deformation response of a broad range of reinforced concrete columns with limited ductility due to degradation of shear resistance, either before or after flexural yielding.  The robust collapse performance assessment could be used for many structural applications. In this dissertation, it is used to identify collapse indicators, design and response parameters that are correlated with ?elevated? collapse probability. The collapse assessment framework is also used to identify the relative collapse risk of different rehabilitation techniques.  Finally, the framework is used to estimate the impact of collapse criteria on the expected financial losses for existing concrete frame buildings in high seismic zones.   This dissertation includes important contributions to (1) modeling techniques for components in existing concrete frames through the development of a mechanical model for existing concrete columns, (2) development of system-level collapse criteria, and (3) application of collapse fragilities in defining collapse indicators, improving loss estimation of existing concrete frames, and differentiating the collapse performances of existing and retrofitted concrete frames. Preface    iiiPREFACE Chapters 3, 4, 7, and 8 of this thesis are based on versions of four manuscripts that have been accepted or planned to be submitted for publication in peer-reviewed journals: ? Chapter 3: Baradaran Shoraka, M., and Elwood, K. J. (2013). ?Mechanical model for non-ductile reinforced concrete columns.? Journal of Earthquake Engng. DOI: 10.1080/13632469.2013.794718. ? Chapter 4: Baradaran Shoraka, M., Yavari, S., Elwood, K. J. and Yang, T. Y., ?System-Level Collapse Assessment of Non-Ductile Concrete Frames.?  Plan to be submitted. ? Chapter 7: Baradaran Shoraka, M., Yang, T. Y. and Elwood, K. J. (2013). ?Seismic loss estimation of non-ductile reinforced concrete buildings.? Earthquake Engng. Struct. Dyn., 42: 297?310. doi: 10.1002/eqe.2213. ? Chapter 8: Baradaran Shoraka, M., Elwood, K. J. and Yang, T. Y. ?Collapse Assessment of Non-Ductile, Retrofitted and Ductile Reinforced Concrete Moment Frames.?  Plan to be submitted. The author of this thesis is responsible for reviewing the literature, developing models, conducting analysis, data processing, and interpreting the results. The manuscripts were drafted by the author of this thesis and finalized in an iterative process with the thesis advisors, Dr. Kenneth J. Elwood and Dr. Tony Yang. The author of this thesis is responsible for preparing the tables and figures. Table of Contents    ivTABLE OF CONTENTS Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iii Table of Contents ......................................................................................................................... iv List of Tables .............................................................................................................................. viii List of Figures ................................................................................................................................ x List of Symbols ........................................................................................................................... xvi Acknowledgments .................................................................................................................... xviii Dedication .................................................................................................................................... xx Chapter 1. Introduction ................................................................................................................ 1 1.1. Overview .............................................................................................................................. 1 1.2. Objectives and Scope ........................................................................................................... 4 1.3. Organization and Outline ..................................................................................................... 7 Chapter 2. Literature Review .................................................................................................... 10 2.1. Overview ............................................................................................................................ 10 2.2. Seismic Vulnerabilities of Existing Non-Ductile Reinforced Concrete Buildings ............ 10 2.3. Component Models in Reinforced Concrete Buildings ..................................................... 13 2.3.1. Nonlinear Beam?Column Elements ............................................................................ 13 2.3.2. Beam?Column Joints .................................................................................................. 19 2.3.3. Slab?Column Connections .......................................................................................... 21 2.4. Collapse Assessment .......................................................................................................... 26 2.4.1. Progressive Collapse .................................................................................................. 26 2.4.2. Non-Simulated Collapse Detection ............................................................................. 28 Chapter 3. Mechanical Model for Non-Ductile Reinforced Concrete Columns ................... 32 3.1. Introduction ........................................................................................................................ 32 3.2. Macro Model for Non-Ductile Columns ........................................................................... 33 3.3. Flexure Response ............................................................................................................... 35 3.4. Reinforcement Slip Response ............................................................................................ 36 Table of Contents    v3.5. Shear Response .................................................................................................................. 38 3.5.1. Pre- Peak Behaviour ................................................................................................... 38 3.5.2. Point of Shear Failure ................................................................................................. 39 3.5.3. Post ? Peak Behaviour ................................................................................................ 44 3.5.4. Combined Response .................................................................................................... 47 3.6. Model Verification ............................................................................................................. 49 3.7. Conclusions ........................................................................................................................ 55 Chapter 4. System-Level Collapse Assessment of Non-Ductile Concrete Frames ................ 56 4.1. Introduction ........................................................................................................................ 56 4.2. Existing Numerical Models for Nonlinear Analysis of Shear-Critical Frames ................. 57 4.2.1. Nonlinear Beam Column Elements ............................................................................. 57 4.2.2. Beam ? Column Joints ................................................................................................ 60 4.3. System-Level Collapse Definition ..................................................................................... 65 4.3.1. Gravity Load Collapse ................................................................................................ 66 4.3.2. Side-Sway Collapse ..................................................................................................... 70 4.4. Comparison of Collapse Definition to Experimental Results ............................................ 72 4.4.1.Details of Shaking Table Test Specimen ...................................................................... 72 4.4.2. Response to Recorded Table Input Motion ................................................................. 73 4.5. Collapse Fragilities ............................................................................................................ 77 4.6. Conclusions ........................................................................................................................ 80 Chapter 5. Collapse Indicators - Methodology......................................................................... 82 5.1. Overview ............................................................................................................................ 82 5.2. Collapse Indicators ............................................................................................................ 83 5.3. Methodology to Identify Suitable Collapse Indicators ...................................................... 86 5.3.1. Step 1 ? Identify a Suite of Potential Collapse Indicators .......................................... 86 5.3.2. Step 2 ? Numerical Model ........................................................................................... 86 5.3.3. Step 3 - Seismic Hazard Calculations and Ground Motion Records Selection .......... 90 5.3.4. Step 4 ? Record-to-Record Variability ........................................................................ 93 5.3.5. Step 5 - Probabilistic Analysis (Ground Motion and Model Uncertainty) ................. 97 5.3.6. Step 6 - Assessment Procedure and Post ? Processing Results ................................ 101 5.4. Conclusions ...................................................................................................................... 117 Chapter 6. Collapse Indicators - Implementation .................................................................. 118 Table of Contents    vi6.1. Overview .......................................................................................................................... 118 6.2. Example Case Studies ...................................................................................................... 119 6.2.1. Numerical Model....................................................................................................... 119 6.2.2. Seismic Hazard Calculations and Ground Motion Records Selection ..................... 121 6.2.3. Record-to-Record (RTR) Variability ......................................................................... 121 6.2.4. Probabilistic Analysis (Ground Motion and Model Uncertainty) ............................ 121 6.3. Simplified Model of 2-D Frames ? Design Parameters ................................................... 124 6.3.1.Collapse Indicators .................................................................................................... 124 6.3.2. Assessment Procedure and Post ? Processing .......................................................... 124 6.3.3. Observations ............................................................................................................. 127 6.4. Simplified Model of 2-D Frames - Response Parameters ............................................... 134 6.4.1. Collapse indicators ................................................................................................... 134 6.4.2. Assessment Procedure and Post ? Processing the Results ....................................... 134 6.4.3. Observations ............................................................................................................. 136 6.5. Conclusions ...................................................................................................................... 141 Chapter 7. Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings ........ 143 7.1. Introduction ...................................................................................................................... 143 7.2. Seismic Loss Assessment Procedure ............................................................................... 144 7.3. Modeling the Collapse of Non-Ductile Reinforced Concrete Frames ............................. 148 7.4. Case Study for the Loss Assessment of Non-Ductile Concrete Buildings ...................... 149 7.4.1. Model, Ground Motions, and Fragility Curves ........................................................ 150 7.4.2. Response of the Building ........................................................................................... 154 7.4.3. Loss Estimation ......................................................................................................... 158 7.5. Summary and Conclusions .............................................................................................. 162 Chapter 8. Collapse Assessment of Non-Ductile, Retrofitted and Ductile Reinforced Concrete Moment Frames ........................................................................................................ 163 8.1. Introduction ...................................................................................................................... 163 8.2. Case Study Structure ........................................................................................................ 164 8.2.1. Non-ductile Perimeter Concrete Moment Resisting Frame Building ....................... 165 8.2.2. Retrofitted Buildings ................................................................................................. 168 8.3. Performance Assessment Based on ASCE 41 ................................................................. 172 8.3.1. Pushover Results ....................................................................................................... 172 Table of Contents    vii8.4. Ground Motion Selection ................................................................................................. 177 8.5. Dynamic Results, Fragility Analysis and System Performance ...................................... 179 8.6. Conclusions ...................................................................................................................... 185 Chapter 9. Summary, Conclusions, and Future Work .......................................................... 186 9.1. Summary .......................................................................................................................... 186 9.2. Findings ........................................................................................................................... 186 9.2.1. Mechanical Model for Non-Ductile Reinforced Concrete Columns (Chapter 3) ..... 186 9.2.2. System-Level Collapse Assessment of Non-Ductile Concrete Frames (Chapter 4) .. 187 9.2.3. Collapse Indicators ? Methodology (Chapter 5) ...................................................... 188 9.2.4. Collapse Indicators ? Implementation (Chapter 6) .................................................. 188 9.2.5. Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings (Chapter 7) ............................................................................................................................................. 189 9.2.6. Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted, and Ductile Moment Frames (Chapter 8) ........................................................ 190 9.3. Future Research ............................................................................................................... 191 Bibliography .............................................................................................................................. 193 Appendix A. Ground Motion Selection and Scaling .............................................................. 204 A.1. Introduction ..................................................................................................................... 204 A.2. Site and Structure Conditions ......................................................................................... 204 A.3. Uniform Hazard Spectra ................................................................................................. 204 A.4. Deaggregation of the Hazard .......................................................................................... 206 A.5. Conditional Mean Spectrum ........................................................................................... 211 A.6. Process of Selecting Ground Motion Recordings ........................................................... 211 A.7. Scaling of the Ground Motion Recordings ..................................................................... 212 Appendix B. Example Buildings .............................................................................................. 216 B.1. 4- and 7- Story Buildings ................................................................................................ 216 B.2. 12- Story Building ........................................................................................................... 218 Appendix C. High Volume Parallel Analysis Using NEEShub ............................................. 226 C.1 Introduction ...................................................................................................................... 226 C.2. NEEShub ......................................................................................................................... 226 C.3. OpenSeesMP ................................................................................................................... 227 C.4. Instructions to Work With NEEShub .............................................................................. 228List of Tables    viiiLIST OF TABLES Table 3-1 Material and geometry properties of square specimens ............................................... 51 Table 4-1 Comparison of three of the leading models to simulate shear failure in existing columns ................................................................................................................................. 59 Table 4-2 Shear stress - strain backbone key points (adapted from Hassan, 2011) ...................... 63 Table 4-3 Pinching4 material model parameter (adapted from Hassan, 2011) ............................ 64 Table 4-4 Parameters and column material properties of tested specimen ................................... 73 Table 5-1 Collapse indicators ....................................................................................................... 85 Table 5-2 Uncertainty modeling of the random variables .......................................................... 101 Table 5-3 Comparison of the different approaches used to indicate limits on the collapse indicators ............................................................................................................................. 108 Table 6-1 Archetype non-ductile RC frame structures ............................................................... 120 Table 6-2 Limits on the collapse indicators ? Design Parameters applying both approaches .... 129 Table 6-3 Limits on the collapse indicators - Response Parameters applying both approaches 138 Table 7-1 Ground motions selected for case study ..................................................................... 152 Table 7-2 Summary of Performance Group assignment ............................................................. 153 Table 8-1 Target drift according to ASCE 41 ............................................................................. 172 Table 8-2 Results of static pushover analysis ............................................................................. 172 Table 8-3 Modeling parameters of numerical acceptance criteria for nonlinear procedures?reinforced concrete columns (for condition ii) from ASCE 41, Supplement 1 (2007 ? ASCE by permission) ..................................................................................................................... 173 Table 8-4 Far Field Ground Motion Set (adapted from Haselton and Deierlein, 2007) ............. 178 Table 8-5 Spectral shape factor ................................................................................................... 180 Table 8-6 Collapse performance metrics .................................................................................... 183 Table 8-7 System performance of the three set of buildings ...................................................... 184 Table A-1 Input parameters required to derive a UHS curve ..................................................... 206 Table A-2 Deaggregation of Uniform Hazard Spectra for different hazard levels ..................... 210 Table A-3 GMs representing six hazard levels suitable for the Van Nuys Building site ........... 213 Table A-4 Scale Factor of GMs for six different hazard levels .................................................. 214 Table B-1 Live load for office occupancy .................................................................................. 219 Table B-2 1965 NBCC lateral load parameters .......................................................................... 220 Table B-3 Lateral load distribution along the height of the building ......................................... 220 List of Tables    ix Table C-1 Available venues on NEEShub (Rodgers et al, 2013) ............................................... 226 List of Figures    xLIST OF FIGURES Figure 1-1 Totally collapsed concrete building in the M 6.3 Christchurch earthquake (Photo: K.J. Elwood) ................................................................................................................................... 2 Figure 1-2 Distribution of damage from four earthquakes falling into three categories from Otani (? 1999 EERI, by permission) ............................................................................................... 2 Figure 1-3 Two samples of buildings with severe damage after an earthquake that were able to maintain their overall stability ................................................................................................ 3 Figure 1-4 Primary objectives and auxiliary targets of the study ................................................... 4 Figure 2-1 Component and system-level seismic deficiencies found in pre-1980 concrete buildings from NIST (? 2010 NIST, by permission) ........................................................... 11 Figure 2-2 Collapse associated with component failures ............................................................. 12 Figure 2-3 Idealized models of beam-column elements from NIST GCR 10-917-5 (? 2010 NIST, by permission) ............................................................................................................ 14 Figure 2-4 Shear failure definition (adapted from Elwood 2004) ................................................ 16 Figure 2-5 Rotation-based shear model from LeBorgne (? 2012 Ph.D. Thesis, by permission). 17 Figure 2-6 Illustration of spring model with degradation from Haselton et al. (? 2008 PEER, by permission) ............................................................................................................................ 18 Figure 2-7 Existing beam-column joints models from Celik and Ellingwood, (? 2008 Taylor & Francis, by permission); (a) Alath and Kunnath (1995); (b) Biddah and Ghobarah (1999); (c) Yousseff and Ghobarah (2001), (d) Lowes and Altoontash (2003); (e) Altoontash (2004), and (f) Shin and LaFave (2004). .............................................................................. 21 Figure 2-8 Model of slab?column connection from Elwood et al. (? 2007 EERI, by permission)............................................................................................................................................... 23 Figure 2-9 Gravity shear ratio versus interstory ratio at punching from Kang and Wallace (? 2006 EERI, by permission) ................................................................................................... 23 Figure 2-10 Details associated with shear transfer on the critical section from Kang (2004 ? Ph.D. thesis by permission) .................................................................................................. 25 Figure 2-11 Modeling of failure of beams from Kim et al.  (? 2009 Elsevier, by permission) ... 28 Figure 3-1 Specimen elevation and cross section details from Sezen and Moehle (? 2006 ACI, by permission) ....................................................................................................................... 36 Figure 3-2 Concrete and steel material models (a) Concrete03 (nonlinear tension softening, model presented in OpenSees), (b) Steel02 (Giuffr?-Menegotto-Pinto model with isotropic strain hardening) (? 2009 PEER, by permission) ................................................................ 37 Figure 3-3 Reinforcement slip response, specimen Sezen No. 1 (Sezen and Moehle,  2006) ..... 37 Figure 3-4 Shear response of example column, proposed model compared with test data, specimen Sezen No. 1 (Sezen and Moehle, 2006) ................................................................ 40 List of Figures    xiFigure 3-5 Plastic hinge region, the numerical integration points, the section used to detect shear failure point and ASFI background (adapted from  Mostafaei and Vecchio, 2008) ............. 42 Figure 3-6 Shear failure detection algorithm ................................................................................ 43 Figure 3-7 Detection of shear failure for (a) diagonal tension failure and (b) diagonal compression failure ............................................................................................................... 44 Figure 3-8 Free-body diagram of column after shear failure from Elwood and Moehle (? 2005 EERI, by permission) ............................................................................................................ 45 Figure 3-9 Combined shear response for specimen Sezen No. 1 (Sezen and Moehle,  2006) ..... 46 Figure 3-10 Proposed shear model compared with test data, specimen Sezen No. 1 (Sezen and Moehle, 2006) ....................................................................................................................... 47 Figure 3-11 Shear, axial and rotational spring in series model with the nonlinear beam-column element .................................................................................................................................. 48 Figure 3-12 Combined response for specimen Sezen No. 1 (Sezen and Moehle, 2006) .............. 48 Figure 3-13 Hysteretic behaviour from Elwood (2002 ? Ph.D. thesis by permission) ................ 49 Figure 3-14 Comparisons between the proposed model and experimental behaviour ................. 54 Figure 3-15 Detection of shear failure for specimens (a) Sezen No.1 and (b) Sezen No.2 .......... 54 Figure 4-1 Examples of existing concrete buildings in the M 6.3 Christchurch earthquake with different structural performances .......................................................................................... 57 Figure 4-2 Rotational spring joint model (adapted from Alath and Kunnath, 1995).................... 60 Figure 4-3 Pinching4 OpenSees model (? 2009 PEER, by permission)...................................... 61 Figure 4-4 Gravity load collapse captured explicitly in the numerical model .............................. 67 Figure 4-5 Gravity load collapse captured explicitly in the numerical model .............................. 68 Figure 4-6 Slab?column connection modeling ............................................................................. 69 Figure 4-7 Side-sway collapse captured in an incremental dynamic analysis based on maximum inter-story drift ratio .............................................................................................................. 71 Figure 4-8 Side-sway collapse captured in an incremental dynamic analysis based on lateral capacity and demand ............................................................................................................. 71 Figure 4-9 Shaking table test specimens (units in mm) from Yavari (2011 ? Ph.D. thesis by permission) ............................................................................................................................ 72 Figure 4-10 Analytical model for the shaking table specimen from Yavari (2011 ? Ph.D. thesis by permission) ....................................................................................................................... 73 Figure 4-11 Response histories for collapse test ........................................................................... 74 Figure 4-12 Shear hysteretic response .......................................................................................... 75 Figure 4-13 Axial load hysteretic response of first-story columns ............................................... 75 Figure 4-14 Gravity load collapse captured explicitly in the numerical model compared with the test data ................................................................................................................................. 76 List of Figures    xiiFigure 4-15 Side-sway and gravity load collapse captured explicitly in the numerical model .... 76 Figure 4-16 Structural model uncertainty in (a) shear and (b) axial failure models ..................... 79 Figure 4-17 Collapse fragility obtained for the shaking table specimen ...................................... 80 Figure 5-1 Methodology for quantitatively selecting and establishing collapse indicators .......... 87 Figure 5-2 Steps to compute the Condition Mean Spectrum (Baker, 2011) ................................. 92 Figure 5-3 Condition mean values of spectral acceleration at all periods, given Sa (T = 1 sec) .. 92 Figure 5-4 CMS vs. UHS for a specific site conditioned for Sa (T = 1 sec) ................................ 93 Figure 5-5 MSA for record-to-record (RTR) uncertainty ............................................................. 94 Figure 5-6 IDA for record-to-record (RTR) uncertainty .............................................................. 96 Figure 5-7 Fragility curve for MSA and IDA ............................................................................... 97 Figure 5-8 Collapse fragility curve for record-to-record (RTR) uncertainty and record-to-record including model (RTR + Model) uncertainty ..................................................................... 100 Figure 5-9 Approach 1 for establishing collapse indicator limits ............................................... 105 Figure 5-10 Approach 2 for establishing collapse indicator limits (the ?good? existing building dictates the acceptable risk and the range for the collapse indicator is determined based on this risk) .............................................................................................................................. 106 Figure 5-11 Approach 3 for establishing collapse indicator limits ............................................. 107 Figure 5-12 Approach 4 for establishing collapse indicator limits (conjectured collapse fragilities for different collapse indicator values) ............................................................................... 107 Figure 5-13 Fragility curve of the maximum interstory drift ratio obtained by three different methods ............................................................................................................................... 111 Figure 5-14 Approach 1 for establishing collapse indicator limits ............................................. 115 Figure 5-15 Approach 2 for establishing collapse indicator limits ............................................. 116 Figure 6-1 Target spectra in addition to the UHSs for the three example buildings (RT = 2475 yrs) ...................................................................................................................................... 122 Figure 6-2 Collapse fragility curve for three non-ductile RC buildings, illustrating key metrics for collapse performance..................................................................................................... 123 Figure 6-3 Mean annual frequency of collapse for the average minimum column transverse reinforcement ratio .............................................................................................................. 131 Figure 6-4 Probability of collapse for the average minimum column transverse reinforcement ratio for a return period of 2475 years ................................................................................ 131 Figure 6-5 Mean annual frequency of collapse for the maximum ratio of story stiffness for two adjacent stories .................................................................................................................... 131 Figure 6-6 Probability of collapse for the maximum ratio of story stiffness for two adjacent stories (CI) for a return period of 2475 years ..................................................................... 131 List of Figures    xiiiFigure 6-7 Mean annual frequency of collapse for the maximum ratio of story strength for two adjacent stories .................................................................................................................... 132 Figure 6-8 Probability of collapse for the maximum ratio of story strength for two adjacent stories (CI) for a return period of 2475 years ..................................................................... 132 Figure 6-9 Mean annual frequency of collapse for the portion of story gravity loads supported by columns with ratio of plastic shear demand to shear capacity > 0.7................................... 132 Figure 6-10 Probability of collapse for the portion of story gravity loads supported by columns with ratio of plastic shear demand to shear capacity > 0.7 ................................................. 132 Figure 6-11 Mean annual frequency of collapse for the maximum ratio of plastic shear capacity (2Mp/L) to column shear strength (Vp/Vn) ........................................................................ 133 Figure 6-12 Probability of collapse for the maximum ratio of plastic shear capacity (2Mp/L) to column shear strength (Vp/Vn) ........................................................................................... 133 Figure 6-13 Mean annual frequency of collapse for the maximum ratio of axial load to strength of transverse reinforcement (45 deg truss model)............................................................... 133 Figure 6-14 Probability of collapse for the maximum ratio of axial load to strength of transverse reinforcement (45 deg truss model) .................................................................................... 133 Figure 6-15 Definition of story shear degradation (4 story building) ......................................... 135 Figure 6-16 Mean annual frequency of collapse for story shear degradation ............................. 139 Figure 6-17 Probability of collapse for story shear degradation for return period of 2475 years............................................................................................................................................. 139 Figure 6-18 Mean annual frequency of collapse for shear failure in columns in one story ....... 139 Figure 6-19 Probability of collapse for percentage of columns with shear failures for return period of 2475 years............................................................................................................ 139 Figure 6-20 Mean annual frequency of collapse for axial failure in columns in one story ........ 140 Figure 6-21 Probability of collapse for percentage of columns with axial failures for return period of 2475 years............................................................................................................ 140 Figure 6-22 Mean annual frequency of collapse for interstory drift ........................................... 140 Figure 6-23 Probability of collapse for interstory drift for return period of 2475 years ............ 140 Figure 6-24 Collapse indicator limit trends with Building characteristics ................................. 142 Figure 7-1 Performance-assessment framework from Cornell and Krawinkler  (2000 ? PEER by permission) .......................................................................................................................... 144 Figure 7-2 An example of the unit repair cost function from Yang et al., (2009 ? ASCE by permission) .......................................................................................................................... 147 Figure 7-3 Shear and axial failure definitions (adapted from Elwood, 2004) ............................ 149 Figure 7-4 (a) South frame elevation (b) column, beam, and joint models (c) sample column cross section (Krawinkler, 2005) ........................................................................................ 151 Figure 7-5 Summary of the fragility curves used for the study .................................................. 153 List of Figures    xivFigure 7-6 Side-sway and gravity load collapse example for case study building (Record CAP000 in Table 1) ............................................................................................................ 155 Figure 7-7 Example of Side - sway collapse captured in an IDA analysis for the case study structure............................................................................................................................... 155 Figure 7-8 Gravity load collapse captured explicitly in the numerical model for the case study structure............................................................................................................................... 156 Figure 7-9 Results of the IDA for different collapse criteria ...................................................... 157 Figure 7-10 Collapse fragilities for seven-story non-ductile RC building, illustrating the effect of different collapse criteria .................................................................................................... 157 Figure 7-11 Repair cost (a) probability distribution (CDF) for 5 hazard levels and PGs for hazard level (b) RT = 72 [yrs], (c) RT = 224 [yrs], (d) RT = 475 [yrs], (e) RT = 975 [yrs], and (f) RT = 2475 [yrs] ................................................................................................................... 160 Figure 7-12 Cumulative Distribution Functions for normalized cost at 5 different hazard levels considering different collapse criteria  [normalized cost = repair cost / (replacement cost + demolition costs)]................................................................................................................ 161 Figure 7-13 Annualized total repair cost .................................................................................... 161 Figure 8-1 Collapse fragility functions for 4-story space frames Liel (2008 ? Ph.D. thesis by permission) .......................................................................................................................... 164 Figure 8-2 Design documentation for 8-story non-ductile perimeter frame structure from Liel (2008 ? Ph.D. thesis by permission) .................................................................................. 166 Figure 8-3 Design documentation for 8-story retrofitted perimeter frame structure (members highlighted in this picture are modified for the different retrofitting measures) ................ 170 Figure 8-4 Force-deformation response of the non-ductile frame .............................................. 174 Figure 8-5 Pushover results for the retrofitted buildings ............................................................ 176 Figure 8-6 Probability of collapse vs. Sa(T1) for non-ductile building ...................................... 180 Figure 8-7 IDA results for the retrofitted buildings .................................................................... 181 Figure 8-8 Collapse fragility curves for different buildings ....................................................... 182 Figure 8-9 Collapse fragility curves normalized by Sa 2% in 50 yrs ......................................... 183 Figure A-1 USGS probabilistic ground motion map (http://eqint.cr.usgs.gov/deaggint/2008/) . 205 Figure A-2 Probabilistic Uniform Hazard Spectra for the site in Van Nuys, California for six levels of annual exceedence probability modified for local site conditions (NEHRP class D soil) ..................................................................................................................................... 205 Figure A-3 Hazard curve for the Sa (T1 = 1 sec) for the Van Nuys Site, based on the USGS website results and modified for local site conditions (NEHRP class D soil) .................... 206 Figure A-4 PSHA deaggregation for Van Nuys, given different return periods for Sa (T1 = 1 sec) (Figure from USGS Custom Mapping and Analysis Tools, http://eqint.cr.usgs.gov/deaggint/2008/.) ............................................................................ 209 List of Figures    xvFigure A-5 Conditional mean values of spectral acceleration at all periods, given Sa(1s) and six hazard levels........................................................................................................................ 212 Figure A-6 GMs scaled to CMS for Sa (T1 = 1 sec) and 4975 yrs return period hazard level ... 215 Figure B-1 Design documentation for 4-story space frame from Liel (2008 ? Ph.D. thesis by permission) .......................................................................................................................... 216 Figure B-2 South frame elevation (adapted from Krawinkler, 2005) ......................................... 217 Figure B-3 Design documentation for 12-story .......................................................................... 222 Figure B-4 Design documentation for 12-story (beam and column layout) ............................... 223 Figure B-5 Cross section of columns (all units are in inches) .................................................... 224 Figure B-6 Cross section of beams (all units are in inches) ....................................................... 225     List of Symbols    xviLIST OF SYMBOLS a shear span Ag  nominal column section area (bh) Ast column transverse reinforcement area b  width of the column section c compression depth of the column section d  effective depth of the column section dc  depth of the column core from center line to the center line of the ties df  flexure depth of the column section    concrete compressive strength fc1  concrete principle tensile stress (determined using MCFT) fc2  concrete principle compressive stress (determined using MCFT) fcx  concrete stress in x-direction (determined using MCFT) fsyl  yield stress of longitudinal reinforcement  fsyt  yield stress of transverse reinforcement h  height of the column section L length of column M moment at the flexure section N compression force acting normal to the crack on the shear failure plane P  axial load applied on the column s  hoop spacing of column V maximum shear demand applied on the column Vsf shear due to shear friction acting on the shear failure plane ? column deformation (horizontal displacement) ? column deformation (horizontal displacement) at shear failure ? column deformation (horizontal displacement) at axial failure    concrete longitudinal strain    axial strain due to the axial and flexural mechanisms at the center of section i  concrete transverse strain List of Symbols    xvii concrete principle tensile strain  concrete principle compressive strain   concrete compression strain at the centroid of the rectangular stress block at section i  shear strain demands in plastic hinge region of the column 	 shear strain in plastic hinge region representative of diagonal compression shear failure 	 shear strain in plastic hinge region representative of diagonal tension shear failure  curvature at the flexure section  effective coefficient of friction   crack angle of the shear failure plane    crack angle in concrete section   transverse reinforcement ratio (Ast/bs) ?g  longitudinal reinforcement ratio 	  resultant shear stress applied on the column 	  shear stress transferred by aggregate interlock across a crack surface v  nominal shear stress (V/bd) Acknowledgments    xviiiACKNOWLEDGMENTS I would like to express my deepest gratitude to my academic father, Dr. Kenneth Elwood, for his outstanding guidance in research and generous advice in teaching and many other aspects offered by him. I feel truly honoured to have the opportunity of being his student. The completion of this thesis would have not been possible without his insightful comments and valuable feedback. He also involved me in many collaborative projects like the Applied Technology Council (ATC) Project 63/2, which have helped me to learn how to work within a group of people with various perspectives. My many thanks to Dr. Tony Yang, my co-supervisor, for his amazing support/advising/ guidance/friendship through all the years of my PhD studies. We spent many hours in his office discussing and developing solutions to various issues in this research. I am indebted to my dissertation oral defence and supervisory committee: Dr. Perry Adebar, Dr. Ricardo Foschi, and Dr. Frank Lam for taking the time to read this thesis and for providing me with very helpful comments. This research has been funded by National Science and Engineering Research Council of Canada (NSERC) and Canadian Seismic Research Network (CSRN). Their generous support is graciously acknowledged. I would like to gratefully recognize my friends and colleagues at the Department of Civil Engineering for their support, helpful discussions and joyful company. First I would like to thank Ehsan, a great friend which you can always rely on. Special thanks to Soheil, Yasmin, Mojtaba, Alireza Ahmadnia, Armin, Ali, Amir, Alireza Forghani, Abbas, Kaveh, Seku, Miguel, Michael, Bishnu, Manuel, and Jose, among many others.  Special thanks to my dear and very close friends Nima, Golbarg, Alireza, and Pooya for their generous friendship during the years I have lived in Vancouver and giving me the energy I needed to complete my degree.  I am utterly thankful to my love, Golnaz, for her continual support, inspiration, and being there for me at hard times and sharing the happy moments throughout the years of studying and living in Vancouver. Words cannot express my appreciation to her.  I am eternally grateful to my parents who always listened to me and shared a piece of advice on the different challenges I faced. Their love and comfort was present every moment and Acknowledgments    xixthey taught me the path of success under God?s will, being patient and working hard to succeed while serving others. This degree and all my achievements would not be possible without their encouragements and continuous support. Last but not least, I would like to thank my brothers, Mohammad and Massoud, for being unbelievably supportive. Dedication    xxDEDICATION To my parents for their unconditional love and support Chapter 1: Introduction 1 Chapter 1. INTRODUCTION 1.1. OVERVIEW Non-ductile reinforced concrete (RC) buildings represent a large portion of the existing building inventory in Canada and the United States. The non-ductile term is used for existing (older) buildings which lack the detailing enforced in newer buildings to ensure a ductile response during earthquake shaking. Recent earthquakes have demonstrated that these structures, constructed prior to the introduction of seismic provisions in modern building codes, are susceptible to irrecoverable damage and potentially collapse during severe earthquake shaking. In 2004, the State of California Multi-Hazard Mitigation Plan estimated there are 40,000 of these buildings in California, with 14,000 in Los Angeles County (Anagnos et al., 2010). Because of the huge inventory of these non-ductile buildings, the need for a detailed procedure to identify the high-risk buildings has been recognized as a high priority (NIST GCR 10-917-7, 2010) and is one of the main motivations to undertake this study. As summarized by Villaverde (2007), buildings have partially or totally collapsed during many earthquakes, including in Valparaiso, Chile in 1985; Mexico City in 1985; Armenia in 1988; Luzon, Philippines in 1990; Northridge, California in 1994; Kobe, Japan in 1995; Kocaeli, Turkey in 1999; Chi-Chi, Taiwan in 1999, and Bhuj, India in 2001. Existing concrete buildings have been severely damaged in recent earthquakes as well, e.g. Concepcion, Chile in 2010 (Rojas et al., 2011) and Christchurch, New Zealand in 2011 (Elwood, 2013). Figure 1-1 shows an example from the February 2011 Christchurch Earthquake. Otani (1999) reported damage statistics of reinforced concrete buildings, with damage defined in three categories:  1. Operational damage (light to minor damage): columns or structural walls were slightly damaged in bending, and some shear cracks might be observed in non-structural walls; 2. Heavy damage (medium to major damage): spalling and crushing of concrete, buckling of reinforcement, or shear failure in columns were observed, and lateral resistance of shear walls might be reduced by heavy shear cracking;  3. Collapse (partial and total collapse), which also included those buildings demolished at the time of investigation. Chapter 1: Introduction 2  Figure 1-1 Totally collapsed concrete building in the M 6.3 Christchurch earthquake (Photo: K.J. Elwood) Figure 1-2 presents the distribution of damage for four earthquakes reported by Otani (1999). As seen in this figure, only a small percentage of the reinforced concrete buildings have collapsed during these earthquakes. The results imply that not all existing concrete buildings will collapse and there is a need to identify the collapse hazardous buildings using seismic rehabilitation standards. Current rehabilitation standards, including the latest version of ASCE/SEI 41 (ASCE, 2006), assist engineers with the seismic assessment of existing buildings based on component acceptance criteria. In this standard, component demands are compared with component acceptance criteria for different performance levels, namely immediate occupancy (IO), life   Figure 1-2 Distribution of damage from four earthquakes falling into three categories from Otani (? 1999 EERI, by permission) Chapter 1: Introduction 3 safety (LS), and collapse prevention (CP). The component with the worst performance level will define the state of the entire structure. In other words, component-based criteria do not take into account the global behaviour and the capability of structures to redistribute gravity and lateral forces after failure of one component. Therefore, the current standards, based on component-based criteria, lack the ability to differentiate system-level limit states, as a result, tend to err on the conservative side. Field observations of structures after severe earthquakes (e.g., Aschheim et al. 2000) have revealed that intense damage to primary components has not necessarily resulted in structural collapse  (e.g., Figure 1-3) as would have been concluded by current seismic assessment procedures.  The need for improvement in collapse assessment for existing concrete buildings has been recognized as a high priority by NIST and FEMA, leading to the initiation of the following projects: NIST GCR 10-917-7, titled ?Program Plan for the Development of Collapse Assessment and Mitigation Strategies for Existing Reinforced Concrete Buildings? (NIST, 2010), ATC-78, titled ?Identification and Mitigation of Non-Ductile Concrete Buildings? (ATC,2012), and ATC-95, titled ?Development of a Collapse Indicator Methodology for Existing Reinforced Concrete Buildings?. NIST GCR 10-917-7 highlights the following reasons why there is a special need for the collapse assessment of existing concrete buildings: ? these buildings represent a significantly greater portion of the  high-risk building stock; ? such buildings are vulnerable to earthquake-induced collapse, posing a threat to public safety and economic loss in future earthquakes; ? at present, there is a huge uncertainty involved in predicting collapse of the different    Figure 1-3 Two samples of buildings with severe damage after an earthquake that were able to maintain their overall stability Chapter 1: Introduction 4 types of older reinforced concrete buildings; and ? full advantage has not yet been taken of past research efforts In this thesis, numerical studies are proposed to address the needs mentioned above. To undertake such numerical studies, there is a need to develop and validate numerical models that are able to capture different modes of collapse. The collapse of most non-ductile concrete buildings will be controlled by the loss of support for gravity loads prior to the development of a side-sway collapse mechanism. Gravity-load collapse may be precipitated by axial-load failure of columns, punching-shear failure of slab?column connections, or axial-load failure beam?column joints.  The robust collapse performance assessment could be used for many structural applications.  In this thesis, it is first used to identify collapse indicators, design and response parameters that are correlated with ?elevated? collapse probability. The collapse assessment framework is also used to identify the relative collapse risk of different rehabilitation techniques.  Finally, the framework is used to estimate the impact of collapse criteria on the expected financial losses for existing concrete frame buildings in high seismic zones. All three applications are considered in this study. 1.2. OBJECTIVES AND SCOPE Figure 1-4 illustrates the general framework of the research described in this thesis. As shown and explained, the main objective will be to develop collapse probabilities for existing reinforced   Figure 1-4 Primary objectives and auxiliary targets of the study Collapse Probability of Non-Ductile Existing RC BuildingsRobust Collapse ModelDetailed Models for Primary ComponentsIdentify AppropriateCollapse Indicatorsand LimitsAssess the Collapse Risk of Retrofitted StructuresEstimate the Seismic Loss of Existing StructuresMain Objective DownstreamUpstreamColumn ModelJoint ModelSlab-column ModelChapter 1: Introduction 5 concrete frame buildings. Detailed models for the primary components of concrete frame buildings will be required to achieve this objective. Development and validation of such models could be entitled ?Upstream? because they feed into the main objective. The new component models will also be used to develop reliable and robust collapse criteria for existing reinforced concrete buildings. The applications extracted from the main objective could be called ?Downstream?, as they use the robust collapse assessments in their development. In the process of developing the proposed objectives, several auxiliary targets are identified, each constituting a contribution to the body of knowledge related to the seismic performance of existing concrete buildings. The auxiliary targets in this research are demonstrated in Figure 1-4 and summarised as follows: ? Macro-models for failure of concrete columns and slab-column connections: Column failure is the primary cause of collapse during earthquakes in many existing reinforced concrete frames. Current modeling approaches of reinforced concrete behaviour provide a reasonably accurate prediction of flexural and longitudinal bar slip response whereas, the determination of shear behaviour needs further development. An objective of this study is to develop a reliable macro model to reproduce the lateral load?deformation response of reinforced concrete columns with limited ductility due to degradation of shear resistance. In addition to beam?column frames, it is essential to consider punching failure as the main source of collapse in slab?column frames. Both models will be used to perform robust performance assessment of collapse risk for existing reinforced concrete frame structures and hence must be computationally efficient (presented in Figure 1-4 as ?Upstream?).   ? Collapse criteria definition: The estimation of collapse probability necessitates a detailed numerical model capable of capturing the collapse of reinforced concrete buildings during earthquakes. Moreover, it is important to understand the mechanisms causing collapse in such structures when subjected to both gravity and seismic loads. As a result, an objective of this research is to overcome the shortcomings of existing collapse criteria for non-ductile concrete frames and to present a reliable and robust collapse detection procedure using research oriented software (presented in Figure 1-4 as ?Upstream?). Chapter 1: Introduction 6 ? Assessment of collapse probabilities: The nonlinear models, used for comprehensive collapse simulations, will be used to estimate the probability of collapse (i.e., collapse fragilities) when ground motion and failure model uncertainty are considered. The process of obtaining such a fragility curve is an objective of this research, and the emphasis is on the use of these fragility curves to assess the collapse safety of existing structures (presented in Figure 1-4 as ?Main Objective?). ? Collapse Indicators: This study will introduce the concept of collapse indicators, design and response parameters that are correlated with elevated collapse probability. The methodology for identifying collapse indicators is based on results of comprehensive collapse simulations. Eventually these collapse indicators could be integrated into an ASCE/SEI 31- (ASCE, 2003) and 41-type assessment procedure, enabling practicing engineers to consider the overall system response when evaluating the collapse prevention performance level (presented in Figure 1-4 as ?Downstream?).  ? Collapse probabilities for different rehabilitation measures: Rehabilitation plays an important role in reducing seismic risk from older concrete buildings. Probability of collapse is currently used to evaluate and set targets for system performance and response measures of new buildings (FEMA P695, 2009). Therefore, it should also be considered for rehabilitation. In order to decide on the most appropriate and economical rehabilitation strategy for an existing structure and to design the rehabilitation system, it is necessary to assess the risk of collapse of each rehabilitation measure. Hence, an objective of this research is to assess and compare the probability of collapse of structures retrofitted based on different rehabilitation measures (presented in Figure 1-4 as ?Downstream?). ? Consideration of collapse in seismic loss assessments: Owing to the seismic vulnerability of existing concrete buildings in areas of high seismic activities, non-ductile reinforced concrete buildings pose a significant threat to the life safety of occupants and damage to such structures can result in large economic losses. The state-of-the-art loss simulation procedure developed for new buildings (ATC 58, 2008; Yang et al., 2009) is extended in this study to estimate the expected losses for existing non-ductile concrete buildings when their vulnerability to collapse is considered. A practical methodology is presented to assist structural engineers to assess the seismic loss of these structural Chapter 1: Introduction 7 systems efficiently using the robust collapse performance assessment procedures. Therefore, an objective of this research is to identify the impact of collapse on seismic loss estimation (presented in Figure 1-4 as ?Downstream?). Such broad objectives require some definition of scope. They are summarized as follows: ? The basic methodology presented in this dissertation could be applied to any seismic system, however, the research described herein is limited to the study of concrete frame building structures. Although other systems are prevalent in the inventory of older concrete buildings, this class of structures has been shown to be particularly vulnerable to collapse in past earthquakes (Sezen et al., 2003).  ? The collapse of most non-ductile concrete buildings will be controlled by the loss of support for gravity loads prior to the development of a side-sway collapse mechanism. This study will focus on axial-load failures of columns in beam?column frames and punching failures in slab?column frames, perhaps the most commonly observed component failures contributing to the collapse of concrete building in past earthquakes.  ? Collapse performance assessments are probabilistic; uncertainties in ground motions and structural modeling are considered in this study. It should be noted that some simplifications are necessary to keep the number of analyses manageable. ? Only two-dimensional numerical models are considered in this thesis. Collapse is actually a three-dimensional response as loads redistribute, but three-dimensional models and torsion are considered beyond the scope of this study. 1.3. ORGANIZATION AND OUTLINE This dissertation will discuss the following: previous related research studies, development of a mechanical column model and a slab?column punching failure model, the development of system-level collapse criteria and validation through comparison with shake table test results, introduction of design and response parameters that are correlated with elevated collapse probability, application of collapse performance assessment to identify the collapse risk of retrofitted structures, and application of collapse performance assessment to quantify the seismic loss of existing structures. The dissertation has been organized as described below. Chapter 2, Literature Review, summarizes previous studies that address the seismic performance of reinforced concrete frames, with special focus on non-ductile frames and the Chapter 1: Introduction 8 collapse vulnerability of these structures. Seismic vulnerabilities of existing concrete buildings and current collapse performance assessment methods and tools are briefly discussed. Chapter 3, Mechanical Model for Non-ductile Concrete Columns, presents and describes a mechanical model which has been developed to detect shear failure based on the axial?shear?flexure interaction model (Mostafaei et al., 2009) and modified compression field theory (Collins and Mitchell 1991). Furthermore, the new model degrades the column shear resistance and initiates axial?load failure based on shear-friction concepts (Elwood 2002). The accuracy of the model is evaluated based on the observed response for several quasi-static column tests. Chapter 4, Assessment of Collapse Probability of Non-ductile Concrete Buildings, describes the development of methods and tools required for performing rigorous collapse performance assessment. System-level collapse criteria, namely gravity-load and side-sway collapse, are numerically defined. These system-level collapse criteria have been validated through comparisons of nonlinear analyses with a shake table test of a two-story two-bay concrete frame tested to collapse. At the end of the chapter, collapse probabilities considering uncertainty in failure models and ground motions are evaluated; these collapse fragilities will be used as an essential tool in the following chapters. Chapter 5, Collapse Indicators ? Methodology, introduces the concept of design and response parameter collapse indicators, design characteristics and response quantities that are correlated with an elevated collapse probability. A step by step procedure is proposed to identify and evaluate design and response parameters. The methodology is formulated in this chapter.  Chapter 6, Collapse Indicators ? Implementation, applies the proposed methodology introduced in Chapter 5 to example existing concrete frame buildings. Both design and response parameter collapse indicators are studied in this chapter, and if proven suitable, limits are suggested for each building case and collapse indicator. The suitable collapse indicators may contribute to improvements in ASCE/SEI 31 and ASCE/SEI 41. Chapter 7, Seismic Loss Estimation for Non-ductile Reinforced Concrete Buildings, extends the ATC-58 (Applied Technology Council 58, 2008) methodology developed for new buildings to existing concrete buildings, considering the impact of collapse criteria, using the methods and tools introduced in Chapter 4. Chapter 8, Collapse Assessment of Concrete Buildings: Comparison of Non-ductile, Retrofitted and Ductile Moment Frames, explores the collapse risk of existing, retrofitted, and Chapter 1: Introduction 9 modern concrete frames using the system-level collapse criteria developed in Chapter 4. This assessment is used to investigate the collapse performance of buildings retrofitted using ASCE/SEI 41. Finally Chapter 9, Conclusions and Future Work, will summarize the important results from the dissertation and recommend topics in need of further investigation to achieve the overall objective of this research. Chapter 2: Literature Review 10 Chapter 2. LITERATURE REVIEW 2.1. OVERVIEW This chapter summarizes previous studies that address the seismic performance of reinforced concrete (RC) frames, with special focus on non-ductile frames and the collapse vulnerability of these structures. The chapter will first focus on the vulnerabilities of the existing concrete frames and will then address the component models that are currently used to simulate the seismic response of these structures. The last section will provide a summary of the current methods of detecting gravity load collapse in the RC moment frames. 2.2. SEISMIC VULNERABILITIES OF EXISTING NON-DUCTILE REINFORCED CONCRETE BUILDINGS A list of critical deficiencies contributing to the collapse vulnerability of concrete buildings was developed by structural engineers in California and summarized in NIST GCR 10-917-7 (2010). Each of the component and system deficiencies shown in Figure 2-1 has been found to lead to collapse or partial collapse of RC buildings in past earthquakes. The critical component deficiencies, A through D, and the system-level deficiencies, E through J, alone or in combination with other deficiencies, can elevate the potential for collapse of a structure during strong ground shaking. Many of the existing RC buildings contain one or more of the deficiencies shown in Figure 2-1. While these conditions can lead to collapse, there are many examples of existing buildings that include some of these deficiencies and have survived strong shaking without collapse. NIST GCR 10-917-7 emphasizes the main challenge as to identify when these deficiencies will lead to building collapse and when they will not. The common deficiencies that can lead to a collapse of a reinforced concrete frame are shear-critical columns (Deficiency A), unconfined beam?column joints (Deficiency B), and slab?column connections (Deficiency C). NIST GCR 10-917-7 (2010) indicates that column failure appears to be the most common cause of older concrete building collapse during major seismic events. Figure 2-2.a shows an example from the 1994 Northridge earthquake. In this example, the failure appears to have been triggered by shear failure of a relatively short segment between two adjacent floors, leading subsequently to shattering of the column and loss of Chapter 2: Literature Review 11 vertical-load-carrying capacity. There have been many studies (Elwood, 2004; Elwood and Moehle, 2005; Ghannoum et al., 2008; Sezen and Moehle, 2006; Wu et al., 2009; Yavari et al., 2009b) of these types of component failure in recent years to identify the failure mechanism and to develop numerical models to capture it in analytical studies. These models are presented later in this chapter.   Figure 2-1 Component and system-level seismic deficiencies found in pre-1980 concrete buildings from NIST (? 2010 NIST, by permission) (a) Collapse associated with column failure, 1994 Northridge earthquakeFigure NIST GCR 10-917-7 (2010)examples where distress to intecollapse, whereas some examples of exterior beamprimary causes of building collapse. Recentassumption that collapse of such frames is less likely to occur due to joint failure than column failure (Yavari et al., 2009a)earthquake. In this example, the failure appears to have been triggered by exterior joint failure, leading subsequently to shattering of the column and loss of verticalThese components will be explained later in this chapter.Hueste et al. (2007) highlight that slabforce-resisting system (LRFSfailures in the slab?column connection rfailures in a 15-story building with waffle flatearthquake. Hall (1995) also points out punching failures in column drop panels in a department store during the 1994 Northridge earthquake; these were due to discontinuous flexural reinforcement at slab?column connections. A significant number of experiments have been conducted to evaluate the performance of slabloading. This information has been used in this thesis to form to represent the behaviour of slabchapter, and a global collapse model for these types of frames is introduced in Chapter 4.Chapter 2: Literature Review  (b) Collapse associated with joint failure, Turkey earthquake (Photo: H. Sezen) from  NISEE (? 1999 EERC, by permission2-2 Collapse associated with component failures  points out that earthquake reconnaissance literature contains no rior beam?column connections was the direct cause of building ?column joint failure have been cited as  tests on non-ductile RC frames have also proven the . Figure 2-2.b shows an example from the 1999 Izmit, Turkey -load ?column frames are not suitable as a main lateral) in regions of high seismic risk because of the potential shear egion. Rodriguez and Diaz (1989) -plate construction following the 1985 Mexico City ?column connections subject to gravity and lateral the basis of numerical models used ?column frames. These models are pre 12  1999 )  -carrying capacity. -note punching shear sented later in this  Chapter 2: Literature Review 13 2.3. COMPONENT MODELS IN REINFORCED CONCRETE BUILDINGS In addition to the overall framework of estimating collapse in structures and the vastly different definitions of this state, it is crucial that the numerical model contains reliable and robust component models that simulate the behaviour of a structure from the elastic region to the post-peak collapse state. The collapse-governing components that should be modeled in concrete frames are nonlinear beam?column elements, beam?column joints, and slab?column connections. These models are presented in the following sections. 2.3.1. Nonlinear Beam?Column Elements Loss of gravity load support due to column failure is one of the primary causes of collapse in existing reinforced concrete frames. Accurate numerical models that are able to reproduce the non-ductile behaviour of these columns will be required to perform assessment of collapse risk for existing RC frame structures. Current modelling approaches for these elements independently estimate lateral displacements due to flexure, bar slip, and shear.  The inelastic flexural response of beam?column elements can be modeled using one of the five idealized model types shown in Figure 2-3. These inelastic models fall into two main categories: 1) lumped plasticity at the ends of the element or 2) distributed plasticity along its length (NIST GCR 10-917-5, 2010). In the concentrated plasticity models, the inelastic deformations are lumped at the ends of the element (Figure 2-3.a, b). On the other hand, in the distributed plasticity models, the inelastic response is simulated either in a finite length hinge model (Figure 2-3.c), or a fibre formulation (Figure 2-3.d) that distributes plasticity by numerical integrations through the member cross sections and along the member length, or finally through the use of the finite element model (Figure 2-3.e), which is the most complex model and breaks down the continuum along the member length and cross sections into finite elements. Currently, the fibre formulation (Figure 2-3.d) is the most commonly used approach because of its computational efficiency.  This approach uses the uniaxial stress?strain relationships for both concrete and steel reinforcement. As a result, this approach allows various concrete regions and steel reinforcement to be modeled separately.   Chapter 2: Literature Review 14  Figure 2-3 Idealized models of beam-column elements from NIST GCR 10-917-5 (? 2010 NIST, by permission) While most current modelling approaches of reinforced concrete behaviour allow for a reasonably accurate prediction of flexural and longitudinal bar slip response, the determination of shear behaviour needs further development. Pincheira et al. (1999) presented a two-dimensional cyclic model that incorporates deformations due to flexure, anchorage slip, and shear, the model includes degradation of lateral stiffness and strength. Lee and Elnashai (2001) used lumped hysteretic representations to evaluate the inelastic flexure and shear response of bridge columns. Lee and Elnashai (2002) further developed the hysteretic shear model for axial force variation to simulate flexure?shear?axial interaction in shear-dominated reinforced concrete columns. These existing models are appropriate for flexure-controlled columns or pure shear failures, but they do not account for the degradation of shear strength with inelastic flexural deformations and, hence, may not accurately predict the shear capacity for columns experiencing flexural yielding before shear failure. In addition to this, these models provide no guidance for simulating response once shear failure is detected. A workshop held by ATC-95 (2013) concluded that models with the following features should be used to simulate the nonlinear response of existing columns leading to shear and subsequent axial failure: ? Computational efficient  ? Calibrated to a wide range of column failure modes ? Ability to transit between shear and flexure failures ? Capable of simulating the degrading lateral-force response including in-cycle (post-peak negative stiffness) and cyclic degradation (reduction of strength due to large number of cycles that result in inelastic response) ? Compatible with joint and bar slip response Chapter 2: Literature Review 15 ? Ability to adjust to different boundary conditions Three models developed by Elwood (2004), LeBorgne and Ghannoum (2009), and Haselton et al. (2008) have overcome the previous issues and contain the main features listed above. These three models are summarized below. Review of Elwood (2004) Elwood (2004) proposed an analytical model to capture the response of columns experiencing flexural yielding before shear failure. Degradation in the shear resistance was initiated when the column drift ratio exceeded the drift at shear failure (?s L? , where ?s is column deformation at shear failure and L is the column height) estimated by Eqn. (2-1) developed based on calibration with laboratory test results (Elwood 2004). ? = + 4? ?	?  ?   (2-1) where ? is the transverse reinforcement ratio, ? is the nominal shear stress (in MPa),  is the concrete compressive strength (in MPa), P is the axial load on the column, and Ag is the gross cross-sectional area. The drift ratio in Eqn. (2-1) was selected as the drift at which the shear resistance dropped below 80% of the maximum shear recorded. The model has been implemented in OpenSees (PEER 2009) using a ?Limit State Material? model (Elwood 2004).  Although it provides a practical model to capture the response of non-ductile columns in building analyses, this model is limited in its application to columns representative of those from the database used to develop Eqn. (2-1) and, in particular, columns expected to experience flexural yielding prior to shear failure. Furthermore, detection of shear failure in this model is based on the column drift demand, although drift ratios may include rigid body rotation (e.g., rotation of joints due to beam flexibility) which does not contribute to column damage. Further details can be found in Elwood (2004). In order to evaluate the limit-state model for non-ductile concrete frames, results from a shaking table test performed on a one-third scale model with three bays and two shear-critical columns and two ductile columns were compared with the data from analysis (Yavari et al., 2009). In particular, shear and axial failure of the columns were closely examined using the empirical capacity models (Elwood, 2004). The finite element program OpenSees was employed to conduct the analyses. Complete discussion of the blind prediction and refined model can be  Chapter 2: Literature Review 16  Figure 2-4 Shear failure definition (adapted from Elwood 2004) found in the study by Yavari et al. (2009), where limitations, weaknesses, and strengths of the analytical model were examined. Review of LeBorgne and Ghannoum (2009) LeBorgne and Ghannoum (2009) also employed the limit-state material in OpenSees (Elwood 2004) but changed the shear failure detection to be based on the plastic rotation at the two ends of the column instead of drift ratio. The model is calibrated to 32 rectangular flexure?shear critical column tests. The shear spring they have used has the ability to monitor both deformations in the plastic hinge region and shear force in the adjacent columns (Figure 2-5.a). As demonstrated by LeBorgne and Ghannoum (2009), the equation used for the total rotation over the plastic hinge region to detect shear failure is shown below:  = 0.044 ? 0.017 ? 0.021   ? 0.002   ? 0.009       (2-2) where Total is the total rotation across the plastic hinge region, s is the transverse reinforcement spacing, d is the depth from the extreme compression fiber to the centroid of tension reinforcement, P represents the axial load, Ag is the gross section area, v is the shear stress (v = V/bd, V = shear force, b = column width), and c is the concrete compressive strength. LeBorgne and Ghannoum's model simulates full degrading behaviour, including in-cycle and cyclic degradation after shear failure is detected, once any of the deformation or force criteria has been reached. The material model used in their approach has several damage functions that include strength and stiffness degradation, all of which have been defined using regression-based equations (Figure 2-5.b). As seen in this figure, before shear failure is detected, the material model remains elastic (Kelasetic). After shear failure is triggered, the model degrades with a value (Kdeg) calibrated to the 32-column database. The pinching unloading (pinchUPN) column lateralload demandLateralLoad?horizshear capacity curveshear failureChapter 2: Literature Review 17 and reloading (pinchRNP) parameters, in addition to the cyclic reloading stiffness damage (dmgRCys) and cyclic strength damage (dmgSCyc) parameters, are represented by regression-based equations for the extracted shear response of the 32 columns in the database.  (a) Model representation  (b) Shear spring constitutive model Figure 2-5 Rotation-based shear model from LeBorgne (? 2012 Ph.D. Thesis, by permission) Chapter 2: Literature Review 18 While detecting shear failure based on plastic hinge rotation instead of drift ratio and the ability to adapt to varying column boundary conditions were both considered steps forward compared with the model presented by Elwood (2004), this model is also limited in its application to columns representative of those from the database used to develop Eqn. (2-2), i.e., the parameters used to define the response are regression based and could not cover the wide spectrum of column properties observed in existing concrete frames. Further details can be found in LeBorgne (2012). Review of Haselton et al., (2008) Haselton et al. (2008) use a concentrated plasticity approach (i.e., rotational springs placed at the ends of elastic line-elements) to simulate the column response. The model is calibrated to 255 rectangular columns tests with only approximately 12% of the database representing columns with a flexure?shear failure mode. The detailed hysteretic nonlinear model representing the rotational springs is based on regression-based equations to estimate the linear and nonlinear parameters based on column properties (Figure 2-6). In-cycle and cyclic degradation are included in the model and are defined by the regression-based equations. Although this model is calibrated to a larger dataset compared with the previous two models, it has the following limitations (ATC 95, 2013): ? Lumped-plasticity models do not take into account the impact of varying axial load in their response, and this will make a difference when a frame experiences column failure   Figure 2-6 Illustration of spring model with degradation from Haselton et al. (? 2008 PEER, by permission) Chapter 2: Literature Review 19 and load redistribution occurs; ? Model parameters are calibrated on a dataset consisting of mainly columns with flexure-dominated failures; ? Model parameters are pre-defined, and therefore, it is not capable of adapting to varying boundary conditions during the simulation; and ? Model parameters are regression based and could not cover the wide spectrum of column properties observed in existing concrete frames. Summary All three models discussed provide regression-based equations to detect shear failure of an existing column. The main drawback to all three is that they do not cover the wide spectrum of existing columns in non-ductile moment frames. Therefore, there is a need to develop a new column model that will be based on physical mechanics rather than empirical relations and will be capable of representing the flexural, shear, and bar slip responses in columns with different failure modes with acceptable accuracy. Chapter 3 addresses this need and introduces a mechanical-based column model.  2.3.2. Beam?Column Joints Little or no transverse shear reinforcement in beam?column joints and/or termination of the beam bottom reinforcement (i.e., splices in longitudinal reinforcement) are the two main problematic reinforcing details in the pre-1967 concrete frames. Therefore, the beam?column joint behaviour is governed by shear and bond?slip responses in existing frames. The typical practice of providing little or no joint shear reinforcement leads to shear deformations, which may be substantial, in the panel zone. This practice also leads to joint shear failure that can restrict the utilization of the flexural capacities of the joining beams and columns. This deficiency (providing little or no joint shear reinforcement) is usually seen in interior beam?column joints. Moreover, the common practice of terminating the beam bottom reinforcement within the joints makes the bottom reinforcement prone to pullout under a seismic excitation. This deficiency is common in the exterior joints. Most of the experimental research and numerical modeling efforts have focused on the interior and exterior joints in two-dimensional frames. There are two approaches for simulating interior and exterior joint response in these frames.  Chapter 2: Literature Review 20 The first approach is using a rotational spring to capture the joint moment versus joint rotation response. Lumped-plasticity rotational hinge models have been proposed in several studies to directly model joint deformations (Alath and Kunnath, 1995; Altoontash, 2004; Ghobarah and Biddah, 1999; Lowes and Altoontash, 2003; Shin and Lafave, 2004; Walker, 2001; Youssef and Ghobarah, 2001). These models account for joint deformation by means of rotational springs placed at the ends of beam and column elements (Figure 2-7). Such models allow for separation of joint response from those of columns and beams and easier interpretation of the results. The second approach uses finite element to simulate the response in beam?column joints. Continuum finite element models that can be linked to beam?column elements through transition elements have also been proposed (Fleury et al., 2000; Elmorsi et al., 1998). Each model has its advantage and limitations; these are summarized in the following: ? The joint model presented by Alath and Kunnath (1995) has the simplest configuration and models the kinematics of the joint region by including rigid zones at the end of the columns and beams in addition to a rotational spring in the joint panel region. Owing to its simplicity, the model is computationally efficient; however, it does not account for variation of axial load, and this could affect the collapse simulation. Furthermore, this model accounts mainly for the shear deformation in the joint panel. ? The model proposed by Biddah and Ghobarah (1999) has added two rotational springs in the ends of the beams to capture the bond-slip response in addition to the shear response in the joint panel. This model involves additional calibration efforts for the bond-slip springs and does not account for the kinematics of the joints by removing the rigid zones. ? The models presented by Youssef and Ghobarah (2001), Lowes and Altoontash (2003), Altoontash (2004), and Shin and LaFave (2004) capture the shear response in the joint panel in addition to individual axial and/or rotational springs to simulate the joint interface reactions. These models are too complicated and, therefore, may not be computationally efficient.  The continuum finite models introduced by Elmorsi et al., 1998, Fleury et al., 2000, and Ziyaeifar and Noguchi (2000) simulate the complex joint response with planar elements, and they have proposed transition zones to ensure the compatibility between the planar element and the line elements representing the beams and columns. Such models were found to be  Chapter 2: Literature Review 21  Figure 2-7 Existing beam-column joints models from Celik and Ellingwood, (? 2008 Taylor & Francis, by permission); (a) Alath and Kunnath (1995); (b) Biddah and Ghobarah (1999); (c) Yousseff and Ghobarah (2001), (d) Lowes and Altoontash (2003); (e) Altoontash (2004), and (f) Shin and LaFave (2004). computationally intensive and have not been tested for modeling of large frame systems, especially at high deformation demands. Of the aforementioned joint models, the Alath and Kunnath (1995) model has been adopted for this research because of its simplicity and computational efficiency, which is a key element in collapse analysis. Additional aspects of the model are presented in section 4.2.2. 2.3.3. Slab?Column Connections The use of two-way slabs without beams to support gravity loads in high-seismic regions has become very popular because of their relatively simple formwork and due to their cost and functional advantages. Research studies and experimental data have shown that slab?column frames provide lateral stiffness and strength contributions to the overall lateral-force-resisting system and should be able to resist the deformation demands during a seismic event, although designed for gravity forces only. In general, slab?column connections could have three modes of failure. Theses failure modes could be punching shear, flexure, or a combination of flexure and Chapter 2: Literature Review 22 punching shear where a punching shear failure occurs at a higher drift level following yielding of the slab reinforcement (Hueste et al., 2007). ACI 369 (2011) states that the analytical model for a slab?column frame element should represent strength, stiffness, and deformation capacity of slabs, columns, slab?column connections, and other components of the frame. This guideline also defines the analytical model of the slab?column frame based on any of the following approaches: ? Effective beam width model: Columns and slabs are represented by line elements rigidly interconnected at the slab?column connection, where the slab width included in the model is adjusted to account for flexibility of the slab?column connection (Allen and Darvall, 1977); ? Equivalent frame model: Columns and slabs are represented by line elements, and stiffness of column or slab elements is adjusted to account for flexibility of the slab?column connection (Park and Gamble 1980); and ? Finite element model: Columns are represented by line elements, and the slab is modeled using plate-bending elements. In a beam?column frame the beams and columns frame directly into one another, but in a slab?column frame the connection occurs around the column. The slab?column connection is modeled using a zero-length rigid-plastic (torsional) spring that is used to monitor the moment transfer at this connection (Elwood et al., 2007; Kang et al., 2009). The slab?column numerical model is shown in Figure 2-8.  Slab?column connections in structures subjected to earthquake loading must transfer forces due to both gravity and lateral loads. This combination can create large shear and unbalanced moment demands at the connection. Hueste et al. (2007) state that without proper detailing, the slab?column connection can be susceptible to punching shear failure during response to lateral loads. Punching failures of slab?column connections have been shown to be primarily a function of the gravity shear ratio on the slab?column critical section and the interstory drift ratio imposed on the connection, as reported by Pan and Moehle (1989). The relationship between gravity shear ratio (Vg/V0) and maximum interstory drift for test data and several models is summarized in Figure 2-9. Chapter 2: Literature Review 23  Figure 2-8 Model of slab?column connection from Elwood et al. (? 2007 EERI, by permission)   Figure 2-9 Gravity shear ratio versus interstory ratio at punching from Kang and Wallace (? 2006 EERI, by permission)   Chapter 2: Literature Review 24 Many researchers have used linear regression analysis on the experimental data for slab?column connections without shear reinforcement to provide expressions for the maximum story drift ratio (DR in percentage) as a function of gravity shear ratio (Vg/V0). For example:                     = 5 ? 7    (	  < 0.6) 0.5               (	  < 0.6)           (Hueste et al., 2007)         (2-3a,b)                      = 3.5 ? 5    (	  < 0.6) 0.5               (	  < 0.6)           (ACI 318-05)  where Vg and vu are given by the following two equations:  =            (2-4)   =  ?          (2-5) where vu is the shear stress, Vu is the shear force acting at the centroid of the critical section (Figure 2-10), Mu is the factored unbalanced bending moment acting about the centroid of the critical section (Figure 2-10), d is the distance from the extreme compression fibre to the centroid of the longitudinal tension reinforcement, b0 is the length of the perimeter of the critical section, c is the distance from the centroid axis of the critical section to the point where shear stress is being computed, J is a property of the critical section analogous to the polar moment of inertia, and  is the fraction of the unbalanced moment considered to be transformed by eccentricity of shear. , J, and b0 are determined by the following equations:  = 1 ? 			        (2-6)  = 2 +          (2-7)  = ! + ! + 		!        (2-8) where b1 and b2 are the widths of the critical section measured in the direction of the span for which Mu is determined and in the perpendicular direction. In the absence of shear reinforcement, the shear strength (V0) is defined as  =          (2-9)  Chapter 2: Literature Review 25   (a) Transfer at interior column (b) Transfer at exterior column Figure 2-10 Details associated with shear transfer on the critical section from Kang (2004 ? Ph.D. thesis by permission)  =  4	2 + "	2 + # 	                  (ACI 318-05)  (2-10) where s equals 40, 30, and 20 for interior, edge, and corner columns, respectively, and c is the ratio of long side to short side of the column. Kang et al. (2009) developed a modeling approach that involves extending the limit-state model developed by Elwood (2002) for column shear failures to model punching failures at slab?column connections. According to this model, strength degradation occurs after a defined limit state is reached. For slab?column connections, this limit state is determined using either the rotation at a given slab?column connection or the interstory drift, in combination with the gravity shear ratio for that connection. Kang et al. (2009) have shown that the numerical model implemented in OpenSees (PEER, 2009) has the ability to effectively and accurately predict the response of shake table tests conducted by Kang and Wallace (2005). Experimental research indicates that slab?column connections have relatively low lateral resistance following a punching shear failure. However, the gravity load resistance depends on the reinforcement layout of the connection in the vicinity of the slab?column joint. Hwang and Moehle (1990) reported that vertical reactions dropped noticeably when punching occurred for their three-bay specimen; however, the interior columns continued to carry most of the vertical loads after punching. Uniform continuous bottom steel was adopted for their specimen and was Chapter 2: Literature Review 26 the main component for vertical load resistance after punching failure. In many old concrete slab?column frames the continuous bottom steel is not used, and punching failure in the slab?column connection will result in approximately negligible vertical resistance. Further details of the slab?column connections and gravity load collapse from punching failure of these components are presented in section 4.3.1. 2.4. COLLAPSE ASSESSMENT A framework for assessing the collapse vulnerability of these buildings has been the center point of attention of many recent research projects (Talaat and Mosalam, 2007; FEMA P695A, 2009; Liel et al., 2011; ATC-78, 2012). These researchers follow two alternative options to model gravity load collapse. The first approach is explicit modeling of gravity load failure, which is usually termed as progressive collapse and involves element removal until the structure reaches a state that cannot resist the gravity load demand and the building is considered to reach the state of collapse. The second alternative addresses gravity failure with post-processing of the simulation results. This is usually referred to as non-simulated collapse modes. The recent advances in both approaches of collapse assessment are summarized in the following sections. 2.4.1. Progressive Collapse The commentary in the American Society of Civil Engineers (ASCE) Standard 7-02 ?Minimum Design Loads for Buildings and Other Structures? describes progressive collapse as ?the spread of an initial local failure from element to element, eventually resulting in the collapse of an entire structure or a disproportionately large part of it?. The local damage or failure initiates a chain reaction of failures that propagates vertically or horizontally through the structural system, leading to an extensive partial or total collapse. Design for collapse resistance is typically accomplished by providing alternative load paths to resist gravity loads, in the event that one or more primary gravity load bearing elements are compromised (Hamburger, 2003). Increasing structural integrity and/or redundancy are essential to accomplish the load redistribution required to redistribute gravity loads under such conditions. Procedures to incorporate progressive collapse considerations into the design process are available in guideline documents published by the U.S. General Services Administration (2003) and the Department of Defence (2005). Bao and Kunnath (2010) state that these documents do not provide sufficient information on Chapter 2: Literature Review 27 procedures, particularly numerical modeling guidelines, to carry out progressive collapse studies of buildings. Numerical simulations investigating progressive collapse have been carried out by several researchers (Bao and Kunnath, 2010; Grierson et al., 2005; Kaewkulchai and Williamson, 2004; Marjanishvili and Agnew, 2006). Most of the published progressive collapse analyses for entire buildings or their components are based on the alternate load path method with column removal. The linear-elastic static, nonlinear static, linear-elastic dynamic, and nonlinear dynamic analyses are four successively more sophisticated analysis procedures used to estimate the progressive collapse hazard. In the linear static method, a step-by-step procedure is used in which members that exceed the Demand-Capacity Ratios (DCR) are removed until all the members left have DCR values smaller than the limit magnitudes, and then the extent of the structural damage is evaluated (Bae et al., 2008). Kaewkulchai and Williamson (2004) apply the beam element formulation for progressive collapse dynamic analysis. Through use of damage-dependent constitutive relationships, the developed beam?column element accounts for the interaction of the axial force and bending moment, including strength and stiffness degradation.  Kim et al. (2009) applied OpenSees for development of an integrated system of progressive collapse analysis. The numerical examples presented are limited to the two- and three-story framed structures represented by two-dimensional planar frames, and the analysis results show that the collapse mechanism depends greatly on the modeling technique for failed members. Two modeling approaches were used in Kim et al. (2009) to model failed components. If the nonlinear hinge model shown in Figure 2-11.a is used for the progressive collapse analysis, the ends of beam members will be modeled as hinges once the damage indices become 1.0, and the moment-resisting capacity of the failed members can be eliminated. However, the axial or shear force-resisting capacities still remain, and the behaviour of the failed member cannot be modeled accurately. In the second approach, a new node is generated at the end of the failed members to separate the member end from the node, as shown in Figure 2-11.b. As Kwasniewski (2010) summarizes in a review of recently published numerical studies of progressive collapse behaviour, for the most part beam element models are used and the  Chapter 2: Literature Review 28   (a) Formation of hinge at failure  (b) Separation of nodes at failure  Figure 2-11 Modeling of failure of beams from Kim et al.  (? 2009 Elsevier, by permission) multilevel strategy is applied, where the structure is analyzed first on the subsystem or component level before global analysis is performed on a simplified global model. Talaat and Mosalam (2007) have also implemented the direct element removal of the structural model upon failure of the element in OpenSees. Their element removal is based on dynamic equilibrium and the resulting transient change in system kinematics. The algorithm applies mainly to axial-load carrying elements and is an automatic procedure during ongoing simulations. The element removal algorithm involves updating nodal masses, removing floating nodes, and removing all associated element and nodal forces after each element reaches the failure state. The main drawbacks of this method are ? Requirement of a bookkeeping operation to essentially update the structural model after each element is removed. This will be computationally intensive. ? Convergence problems will be extreme in this method because of the sudden changes in the structural model after each element removal. 2.4.2. Non-Simulated Collapse Detection Review of FEMA P695 FEMA P695 report (2009), titled ?Quantification of building system performance and response parameters?, introduces state-of-the-art research on building behaviour at the collapse limit state and quantification of this behaviour for new design (FEMA 2009). FEMA P695 has recommended a methodology to reliably quantify building system performance. This report Chapter 2: Literature Review 29 consists of a framework for establishing seismic performance factors (SPFs), such as force modification (R) and over-strength (?0) factors. The approach involves the development of detailed system design information and probabilistic assessment of collapse risk. Application of the incremental dynamic analysis (IDA) approach (Vamvatsikos and Cornell, 2002) and probabilistic assessment of collapse risk establish the SPFs for a proposed system. The methodology only applies to the seismic-force-resisting system of new buildings. It utilizes nonlinear analysis techniques, and discretely considers uncertainties in ground motion, modeling, design, and test data (FEMA 2009). FEMA 695 is a state-of-the-art method for system assessment and has highlighted many difficult technical issues in assessing the collapse performance of building structures. The current study will attempt to address some of these technical issues, specifically for existing reinforced concrete frames. These include (1) incorporation of additional uncertainty by adjusting the collapse capacity due to the effects of spectral shape; (2) evaluation of non-simulated collapse modes by limit state checks without explicit consideration of uncertainty in the ability of models to capture the limit states; (3) focus on side-sway collapse as the only collapse mode for reinforced concrete frames; and (4) discretely considering uncertainty in ground motion and modeling.  In the IDA approach, response spectra of selected ground motion records are scaled in order to reach the collapse state. For rare ground motions the spectral shape is different from the less rare ground motions available in ground motion databases. In order to compensate for this difference, an adjustment factor is applied to the collapse fragilities. This modification factor depends on the period and the ductility capacity of the structure. The simplified spectral shape factor will adjust the collapse margin ratio and, as a result, may lead to an increase in the (aleatory) uncertainty associated with the collapse capacity (Haselton et al., 2011). In the approach recommended in FEMA P695, collapse modes are assessed through either explicit simulation in the nonlinear analyses or evaluation of non-simulated collapse modes using alternative limit-state checks on demand quantities from the nonlinear analyses. If applied to non-ductile reinforced concrete frames, the FEMA P695 methodology would treat shear failure and subsequent axial failure of concrete columns as non-simulated collapse modes that are dealt with by limit-state checks. These limit-state checks will generally result in a low estimate of the median collapse point because non-simulated collapse modes are usually Chapter 2: Literature Review 30 associated with a component failure mode. The inherent assumption is that the first occurrence of this non-simulated failure mode will lead to collapse of the structure, which may not always be the case. Furthermore, the impact of the non-simulated collapse modes on the rest of the structure is not directly accounted for in the analysis. For this reason, local failure modes should also be directly simulated in order to more accurately reflect the behaviour of the structure. In incremental dynamic analysis, side-sway collapse is the governing mechanism, and collapse prediction is based on dynamic instability or excessive lateral displacements, (i.e., the primary expected collapse mode is flexural hinging leading to sidesway collapse). However, in non-ductile reinforced concrete frames, it is expected that columns will frequently lose the capacity to support gravity loads due to shear and axial load failures prior to the development of the flexural mechanism necessary for a side-sway collapse, and as a result, they will likely form other collapse mechanisms (e.g., loss in vertical load carrying capacity). This dominant non-ductile collapse mode has not been directly implemented in FEMA P695. Liel et al., (2011)  The methodology presented in FEMA P695 was further improved in Liel et al. (2011). The study was used to assess the collapse risk for pre-1970s reinforced concrete structures. Collapse modes related to shear and subsequent axial failure in non-ductile columns were included in the non-simulated collapse modes. These collapse modes were detected by post-processing the simulation results using component limit state checks. The loss of gravity load carrying capacity of a column was considered the damage state relevant to structural collapse.  Incorporating these non-simulated collapse modes in the process of collapse detection has the following disadvantages: ? The process is highly conservative because it ignores load redistribution and will dictate a state of collapse as soon as one of the elements reaches the failure limit state ? The results will be inaccurate owing to the lack of realistic representation of post-failure response Summary The need for improvement in collapse assessment for existing concrete buildings has been recognized as a high priority. Both methods mentioned in this section, progressive collapse (by Chapter 2: Literature Review 31 the means of element removal) and collapse detection using non-simulated collapse modes, require further improvement especially for existing RC frames and applying a system-level definition to detect collapse. In a recent study performed by ATC-78 (2012), collapse is defined as when more than half of the columns in a particular story have exceeded the axial failure criterion (defined as the deformation in the axial spring exceeding 2%). This system-level collapse definition has not been verified using shake table tests and requires further improvement. A system-level collapse definition of gravity load collapse is introduced in Chapter 4. Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 32 Chapter 3. MECHANICAL MODEL FOR NON-DUCTILE REINFORCED CONCRETE COLUMNS1 3.1. INTRODUCTION The estimation of collapse probability necessitates a numerical model capable of capturing the collapse of reinforced concrete buildings during earthquakes. The collapse of most non-ductile concrete buildings will be controlled by the loss of support for gravity loads prior to the development of a side-sway collapse mechanism. Loss of gravity load support due to column failure is one of the primary causes of collapse in existing reinforced concrete frames. Accurate structural analysis models which are able to reproduce the non-ductile behaviour of these columns will be required to perform probabilistic assessment of collapse risk for existing reinforced concrete frame structures. The models must reasonably capture the critical non-ductile behaviour expected in such columns (i.e., shear- and axial-load failure) while remaining computationally efficient such that Monte Carlo collapse simulations of multi-story building structures can be realized. While current modelling approaches of reinforced concrete behaviour allow for a reasonably accurate prediction of flexural and longitudinal bar slip response, the determination of shear behaviour needs further development. Elwood (2004) proposed an analytical model to capture the response of columns experiencing flexural yielding before shear failure. Degradation in the shear resistance was initiated when the column drift ratio exceeded the drift at shear failure (? ) estimated by Eqn. (3-1) developed based on calibration with laboratory test results (Elwood, 2004). ? = + 4? ?	?  ?  2     (3-1) The drift ratio in Eqn. (3-1) was selected as the drift at which the shear resistance dropped below 80% of the maximum shear recorded. Although providing a practical model to capture the response of non-ductile columns in building analyses, this model is limited in its application to                                                  1 A version of chapter 3 has been published. Baradaran Shoraka, M., and Elwood, K. J. (2013). ?Mechanical model for non-ductile reinforced concrete columns.? Journal of Earthquake Engng. DOI: 10.1080/13632469.2013.794718.? Journal of Earthquake Engng. DOI: 10.1080/13632469.2013.794718 2 All parameters in this chapter are defined in the Notation List section Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 33 columns representative of those from the database used to develop Eqn. (3-1), and in particular columns expected to experience flexural yielding prior to shear failure. Furthermore, detection of shear failure in this model is based on the column drift demand although drift ratios may include rigid body rotation (e.g., rotation of joints due to beam flexibility) which does not contribute to column damage.   To predict the point of shear failure, Mostafaei et al. (2009) proposed a displacement-based analysis method, which approximately accounts for axial-shear-flexure interaction (ASFI) in the column response. This method is a macro-model-based approach which considers the effect of shear deformations in sectional analysis. A limitation of this method is its complex state determination making it relatively computational inefficient. Currently the ASFI model is not available in commonly used nonlinear analysis software, such as OpenSees (PEER, 2009), for collapse analysis of concrete frame systems.  The column model introduced in this chapter (1) provides an improvement on Elwood (2004) using the concepts introduced in the ASFI method (Mostafaei et al. 2009); (2) is based on physical mechanics rather than empirical relations (e.g., Eqn. (3-1)); and (3) is capable of representing the flexural, shear and bar slip response in columns with different failure modes with acceptable accuracy.  An important aspect of modeling non-ductile columns is to predict the point of ?shear failure? reliably. Different definitions for shear failure in concrete columns are possible. In this dissertation the onset of shear failure is defined as the state when large diagonal shear cracks develop and the column response reaches peak shear resistance immediately followed by a state where increasing drifts are associated with strength degradation. Note that this definition is slightly different from the drift at shear failure (? ) used in Eqn. (3-1). The new column model uses maximum shear deformation demands () at both ends of the column to detect the point of shear failure. 3.2. MACRO MODEL FOR NON-DUCTILE COLUMNS The main objective of this chapter is to develop a macro model that simulates the lateral load-deformation response of reinforced concrete columns; particularly those whose deformation capacity is limited by shear failure and may be vulnerable to axial load failure. The macro model is an extension of the current modelling approaches for reinforced concrete behaviour which Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 34 estimate lateral displacements due to flexure, bar slip, and shear independently (Sezen and Chowdhury 2009). The model developed in this study simulates these three responses by individual springs in series which are later combined to obtain the total lateral response of the column. The main focus is on improving the estimation of shear response using a mechanical-based model.  In this study, the shear response is based on mechanical models to simulate behaviour of reinforced concrete columns in the range from pre-peak to post peak response, including point of shear failure. A simplified piece-wise linear model representing the nonlinear response from the Modified Compression Field Theory (Vecchio, 1990) is used to simulate the pre-peak shear behaviour. To predict the point of shear failure, a recently proposed displacement-based analysis method, ASFI (Mostafaei et al., 2009), which considers the effects of axial-shear-flexure interaction is used in the analysis. The ASFI method is modified so as to monitor local shear deformations across the plastic hinge region of the column as an indicator of the initiation of shear failure. Finally, a shear-friction model is used to represent the degrading slope of the lateral force-deformation response after shear failure.  The models used in this chapter to simulate the shear response are simple enough that they can be implemented in current finite element frameworks such as OpenSees. In order to validate the procedure introduced in this chapter, total lateral load-deformation predictions are compared with measured results from shear-dominated reinforced concrete columns tested previously. It should be noted that the proposed model could also be applied to ?flexural-controlled? columns, although such a detailed model is not necessary for such elements. The following sections describe how the three deformation components (flexure, shear and slip) are captured and combined in the macro model, followed by a comparison with experimental data. Specimen No. 1 from Sezen and Moehle (2006), shown in Figure 3-1, will be used as an example to demonstrate the various aspects of the proposed macro model. The specimen is intended to be representative of interior columns in a gravity load-carrying frame system. The specimen was subjected to constant compressive axial loads and multiple lateral deformation cycles. This specimen is shear dominated (Sezen and Moehle, 2006). Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 35 3.3. FLEXURE RESPONSE The flexural behaviour of a reinforced concrete column cross section can be estimated through a moment?curvature analysis. The primary M- relationship is derived using standard flexural analysis of the fibre cross-sectional model with appropriate constitutive laws for concrete and steel. This approach uses uniaxial stress-strain relationships for both concrete and steel reinforcement, allowing various concrete regions and steel reinforcement to be modeled separately. All concrete fibres were modeled using the ?Concrete01? uniaxial material model in OpenSees, which is based on the modified Kent and Park model (Kent and Park, 1971). The modified Kent and Park model offers a good balance between simplicity and accuracy and its availability in OpenSees makes it convenient for nonlinear analysis. Strength loss at large compressive strains was also taken into account for the concrete material using the Kent and Park model. The cyclic response of the concrete material models used to define the behaviour of the concrete fibres for the outside and center of columns is shown in Figure 3-2.a. Reinforcing steel was modeled using ?Steel02?, a Giuffr?-Menegotto-Pinto Model with isotropic strain hardening (Figure 3-2.b). The flexural sub-element is modelled by means of nonlinear fibre beam-column elements with five sections defining the moment-axial load interaction (Spacone et al., 1996). Based on the flexibility method, the beam-column elements determine the section forces (moment and axial load) from interpolation of the element end forces. The section deformations (curvatures and axial strains) obtained from those forces are then integrated over the length of the element to determine the element end deformations (rotations and axial elongation). Each nonlinear fibre beam-column element includes five sections located at Gauss-Lobatto integration points along the length of the element for optimum integration of the section deformations, which also provides the critical section forces and deformations at the ends of the element.  Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 36  Figure 3-1 Specimen elevation and cross section details from Sezen and Moehle (? 2006 ACI, by permission) 3.4. REINFORCEMENT SLIP RESPONSE Slip deformations result from the extension of the reinforcing bars relative to the anchoring concrete. These deformations contribute to additional lateral displacements that are not included in flexural analysis. The bar slip model used in this chapter was developed by Sezen and Setzler (2008) which considers two uniform bond stress models depending on whether or not the bar is yielding.   Using the Sezen and Setzler model, the moment vs. bar slip rotation relationship (Figure 3-3) can be determined based on column geometry, amount and locations of longitudinal and transverse reinforcement, axial external forces and material properties. It should be noted that Figure 3-3 represents the column rotational deformation resulting from bar slip. As shown in the figure, the moment- bar slip rotation can be approximated by a tri-linear curve determined based on equal area of the calculated and approximated responses. However, the plateau of slip response occurs at the same force level as the plateau of flexural response; hence, given limited strain hardening, it is possible to assume that yielding only occurs in the flexural response and the bar slip response can be represented by a bi-linear curve (the first two section of the tri-linear curve) instead (Figure 3-3). fsyl = 438 MPa fsyt = 476 MPa f?c = 21 MPa P/Ag f?c = 0.15 Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 37    (a) (b)  Figure 3-2 Concrete and steel material models (a) Concrete03 (nonlinear tension softening, model presented in OpenSees), (b) Steel02 (Giuffr?-Menegotto-Pinto model with isotropic strain hardening) (? 2009 PEER, by permission)    Figure 3-3 Reinforcement slip response (column rotation due to bar slip), specimen Sezen No. 1   01002003004005006007000 0.001 0.002 0.003 0.004 0.005 0.006Moment [kN.m]Bar slip rotationSezen's ModelMulti-linearBi-LinearChapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 38 3.5. SHEAR RESPONSE In this chapter, the shear response is regarded to be based on three mechanical models that correspond to particular performance states of reinforced concrete columns: pre-peak response, point of shear failure, and post-peak response. The primary new contributions from this chapter are the combination of the three models to define the shear response, the use of shear deformations to detect shear failure, and the estimation of post-peak response based on a shear-friction model. Each of the three mechanical models is described below. 3.5.1. Pre- Peak Behaviour The Modified Compression Field Theory (MCFT) is used to represent the shear response of RC columns until the peak resistance is reached. The MCFT relates average stresses to average strains in a cracked reinforced concrete element satisfying conditions of compatibility and equilibrium (Vecchio, 1990). In the present study, the MCFT is used to develop a simple tri-linear idealization of column shear response prior to shear failure. It should be noted that the concrete model used in the MCFT equations is based on the modified Kent and Park model in order to account for confinement and to be consistent with the model used for flexural response. An elastic-perfectly-plastic model is assumed for the steel constitutive relationship in the MCFT equations. Based on MCFT, the shear deformation can be related to the axial strains developed in an element using the following equation:  = 2 ?  ?  ?             (3-2) Shear strain estimation at a specific load level requires accurate evaluation of the longitudinal, transverse and compressive principle strains. For a given column shear, V= M/L where L is the distance from the section under examination to the point of zero moment (point of inflection), the shear strain is evaluated as follows: Perform sectional analysis for given material properties, geometrical dimensions, moment and axial external forces. At each load increment, curvature (?), compression depth (c), and longitudinal strain (?x) are determined. The moment-curvature response of the section is determined along with the compression depth (c) and longitudinal strain (?x). Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 39 ? Estimate ?2 based on the compressive strain at the centroid of an equivalent stress block and determine the transverse strain (?y) using the following equations: 2 = $ + ? ? (h ? 0.85c)/2            (3-3) 1 = $ ? 2 ? %1?%%1+%?& + $      (3-4) ' = 1 + 2 ? $             (3-5) o Eqn. (3-3) is based on sectional analysis, Eqn. (3-4) is from MCFT, and Eqn. (3-5) is from strain compatibility.  ? Calculate the shear strain using Eqn. (3-2). The final result of the above procedure will be a prediction of the shear force vs. shear deformation response for the column (Figure 3-4). As observed in Figure 3-4, the lateral force-shear deformation determined from the above procedure could be approximated by a tri-linear curve. The tri-linear approximation is carried out by the equal area method. The first linear section will cover the elastic range followed by second linear segment representing reduced stiffness after shear cracking. Shear cracking is assumed to occur at approximately the same load resulting in flexural cracking. The tri-linear curve obtained from this procedure is used to model the shear response prior to shear failure in the proposed column model. 3.5.2. Point of Shear Failure Shear dominant reinforced concrete columns can potentially have two types of failure: diagonal tension failure and diagonal compression failure. Diagonal tension failure occurs as diagonal cracks form and shear reinforcement is insufficient to maintain small crack widths. Diagonal compression failure, on the other hand, typically occurs in the condition where adequate shear reinforcement is provided and the column is able to resist higher shear forces until the compressive stress in the diagonal compression struts exceeds the compression capacity of concrete. Diagonal compression failures have also been known to occur in columns with high axial loads leading to the softening of concrete in the flexural compression zone and weakening of the diagonal compression strut (Kuo et al. 2006). These two shear failure conditions defined for a reinforced concrete column can be captured using the MCFT (Collins and Mitchell, 1991). Diagonal tension failure, typically the  Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 40  Figure 3-4 Shear response of example column, proposed model compared with test data, specimen Sezen No. 1 (data from Sezen and Moehle,  2006)   governing case for columns with low transverse reinforcement ratios and moderate axial loads, is detected when the applied shear stress exceeds the limit given in Eqn. (3-6) (Collins and Mitchell 1991): 	 =  ? 	 +          (3-6) where 	 is given by Walraven?s Eqn. (Walraven, 1981).  Columns with high axial loads or higher transverse reinforcement ratios may experience diagonal compression failure, which is detected using Eqn. (3-7) (Collins and Mitchell, 1991): 	 =  ?  !  " #                      (3-7) where df = h for short columns with span-depth ratios less than 1.0 and df = d for columns with span-depth ratios more than 1.5 and df  can be determined by interpolation for ratios between 1.0 and 1.5 (Mostafaei et al., 2009). It is desirable to represent the two different shear failure modes discussed above using deformation-based failure surfaces. Previous column models (Elwood, 2004) detected shear failure based on the total drift ratio of the column without distinguishing the contribution of flexure, shear and bar slip from the total drift ratio. LeBorgne and Ghannoum (2009) proposed a method to use a rotation-based shear failure model. They also employed the Limit State material 0501001502002503003500 1 2 3 4Shear Force (kN)Shear Displacement (mm)Experimentaltri-linear curveProposed ModelChapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 41 in OpenSees but changed the shear failure detection to be based on the plastic rotation at the two ends of the column instead of drift ratio. The same concept, local deformation-based shear failure, is applied in the proposed model. More specifically, shear deformation across the plastic hinge region of the column will be used to detect the onset of shear failure.  Concepts from the Axial-Shear-Flexural Interaction (ASFI) method (Mostafaei and Vecchio, 2008; Mostafaei et al., 2009), developed to include the effects of shear deformations in sectional analysis, are used here to estimate strains in the plastic hinge region and enable conversion of Eqns. (3-6) and (3-7) to deformation-controlled relationships. The main hypothesis of the ASFI method is that the concrete principle compression strain () and the longitudinal strain () for an element between two sections could be determined based on average values of the concrete uniaxial compression and longitudinal strains corresponding to the resultant forces of the concrete stress blocks (Figure 3-5).  = $	!$	                (3-8)  = $!$                    (3-9) The    and  can be determined from the results of section analysis. Based on Mohr?s circle for strains, the cracking angle is given by:  = %$	$            (3-10) where ?x and ?2 are determined based on the average strain approach of ASFI (Eqns. (3-8) and (3-9)).  Using Eqn. (3-10) and combining it with the two failure criteria (Eqns. (3-6) and (3-7)), diagonal tension shear failure can be defined based on the following inequality:  ?  =  " &' ? 2 ?                (3-11) and diagonal compression shear failure is defined based on the following inequality:  ?  =  ?  ? ( +(  ? 4         (3-12) It should be noted that converting Eqn. (3-7) to achieve deformation-controlled relationships will have two solutions. The solution presented in Eqn. (3-12) is the appropriate result whenever the column is in compression. The overall step by step process to detect shear failure is illustrated in the flow chart shown in Figure 3-6. The two shear failure modes, diagonal tension and diagonal compression Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 42 failure, are distinguished in the flow chart. The algorithm has been implemented in the OpenSees (PEER, 2009) source code and used with the Limit State material model to detect shear failure.  Specimen 12 and 16 from Mostafaei et al. (2009) will be used as examples to demonstrate the two shear failure modes. The specimens are intended to be representative of columns located in the first floor of a building. The specimens were loaded under constant axial load and static cyclic unidirectional lateral load and they were considered as double-curvature elements with in the inflection point at the center of the columns (Mostafaei, 2006). Table 3-1 (presented in Section 3-6) summarizes the main properties of these specimens. The two shear failure modes are depicted in Figure 3-7.a and 3-7.b for the two column specimens. It should be noted that the horizontal axis refers to the shear strain in the plastic hinge region expressed as a percentage of the length of the plastic hinge, not the total height of the column. As indicated in both figures, each failure mode is represented by a limit state surface. The onset of shear failure is defined whenever the shear response first reaches any of the limit state surfaces.     Figure 3-5 Plastic hinge region, the numerical integration points, the section used to detect shear failure point and ASFI background (adapted from  Mostafaei and Vecchio, 2008) Plastic hinge regionElement subjected to axial, shear and flexurePVM1VPM2PVGauss-Lobattointegration pointPlastic hinge regionCentroidStrainsConcreteStress Block at section iConcreteStress Block at section i+1section isection i+1Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 43     Figure 3-6 Shear failure detection algorithm   Input Material Properties and Geometrical DimensionsBased on Shear Force, Moment and Axial Load Perform Sectional AnalysisObtain Curvature, Compression Depth and Centroid StrainCompute Longitudinal Strain and Compression Strain based on ASFIBased on MCFT obtain Principle Strains, Transverse Strain, Crack Angle, Compression Stress and Aggregate Interlock Shear StressCompute Shear Strain in Plastic Hinge Region for Diagonal Tension FailureCompute Shear Strain in Plastic Hinge Region for Diagonal Compression FailureIf SHEAR FAILUREOutput ShearStrain in Plastic Hinge Region =  ?  ?  ? V +	? V  ?   ?   ?    =  ? ? 	 ?  Proceed to next converged load stepChapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 44 (a) Specimen 12 (Mostafaei et al., 2009) (b) Specimen 16 (Mostafaei et al., 2009)     Figure 3-7 Detection of shear failure for (a) diagonal tension failure and (b) diagonal compression failure   3.5.3. Post ? Peak Behaviour The degrading slope of the shear-drift backbone after shear failure is a key parameter influencing the response of shear-critical columns before axial failure. Elwood and Moehle (2005) developed the following equation based on shear-friction concepts to estimate the drift at axial load failure for columns experiencing shear failure: ? () =  *+,-	+,. /       (3-13) The shear-friction model can be extended to provide an estimate of the degrading slope. Considering the column illustrated in Figure 3-8 just before the total loss of shear capacity, and ignoring the dowel action and axial capacity of the longitudinal reinforcement, the equilibrium equations can be written as follows: ?( ?  sin +  =  cos +  tan         (3-14) ?0 ? P =  cos + sin         (3-15) 0501001502002503000 0.005 0.01 0.015 0.02Shear Force [kN]Shear Strain @ plastic hinge [rad]Diagonal Tension Failure Limit SurfaceDiagonal Compression Failure Limit SurfaceShear Failure0501001502002503003504000 0.005 0.01 0.015 0.02 0.025 0.03Shear Force [kN]Shear Strain @ plastic hinge [rad]Diagonal Tension Failure Limit SurfaceDiagonal Compression Failure Limit SurfaceShear FailureChapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 45 By using the classical shear-friction model (Eqn. (3-16)) and an approximate relationship between drift and the effective coefficient of friction proposed by Elwood and Moehle (2005) (Eqn. (3-17)), the equilibrium equations (Eqns. (3-14) and (3-15)) can be combined to give the following expression for the shear force (Eqn. (3-18)):   =  !           (3-16) ! = 2.1 ?  ? () ? 0       (3-17)  =  ? 2.1 ? " #  1?1.!21.1??$        (3-18) To find the degrading slope of the shear-drift backbone, Eqn. (3-18) is differentiated with respect to the drift ratio: 343?#= ?4.5P ? 11?9.6??#2           (3-19) Finally, using Eqn. (3-13) to express the drift ratio as a function of the axial load and the transverse reinforcement, the following expression (Eqn. (3-20)) provides an estimate of the degrading slope of the shear-drift backbone: 343?#= ?4.5P ? 5%367 ? 4.6 + 12           (3-20) Consistent with the axial-load failure model developed by Elwood and Moehle (2005), Eqn. (3-20) has been developed assuming a crack angle () of 65 degrees. All parameters are illustrated in Figure 3-8.   Figure 3-8 Free-body diagram of column after shear failure from Elwood and Moehle (? 2005 EERI, by permission) Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 46 The total shear lateral response of a reinforced concrete column can be modeled by incorporating the pre-peak and post-peak response explained in the preceding sections. The point of shear failure is also detected based on the proposed modified ASFI model. The total shear response is figuratively illustrated in Figure 3-9. The cyclic shear response model used in this chapter is based on the Hysteretic uniaxial material presented in OpenSees. This cyclic model integrates strength and stiffness degradation with pinching of hysteresis loops based on a damage model. The cyclic shear response for Sezen?s specimen No. 1 (Sezen and Moehle, 2006) is illustrated in Figure 3-10. Comparison of proposed and experimental results shows that the proposed model adequately (for the purpose of simulating collapse in non-ductile moment frames) captures the pre-peak response, point of shear failure and post-peak response. An important note for this mechanical model is that the backbone is estimated based on the dead load on the columns, however, the failure surfaces are defined during the analysis and capture the shear failure for any level of axial load on the columns. Another important note is that this model has only been verified for test data where the columns have experienced shear force in combination with axial compression (explained in section 3.6) but conceptually this mechanical model could also be used for columns which undergo shear deformations when they are in tension.  Figure 3-9 Combined shear response for specimen Sezen No. 1 00.20.40.60.811.20 0.002 0.004 0.006 0.008 0.01Shear Force/VmaxShear Displacement/Column HeightSimplified Piecewise Linear based on MCFTShear Failure based on Modified ASFI MethodDegrading Slope based onShear - Friction ModelChapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 47   Figure 3-10 Proposed shear model compared with test data, specimen Sezen No. 1 (data from Sezen and Moehle,  2006) 3.5.4. Combined Response The total lateral response of a reinforced concrete column can be modeled by coupling flexure, reinforcement slip and shear responses as springs in series, where the force in each spring is the same and the total deformation is the sum of individual spring deformations. Flexural deformations are captured by the nonlinear beam-column element.  Zero-length sections located at the top and bottom of the column attach to the nonlinear beam-column element. The zero-length sections are defined by three uncoupled material models describing 1) the moment-rotation relationship representing reinforcement slip response, 2) the shear-horizontal displacement relationship representing the shear force-displacement response, and 3) the axial load-vertical displacement relationship (Figure 3-11). The axial Limit State model introduced by Elwood and Moehle (2005) is employed to capture any possible axial failures in these non-ductile columns. Because the beam-column element includes the axial flexibility of the column, the pre-failure backbone for the axial spring is defined with a high stiffness to ensure that the spring does not add any axial flexibility to the model until after failure has occurred. For a description of the concept of the vertical spring, refer to Yavari et al. (2009). Each material model can be considered as a spring in series with the nonlinear beam-column element.  -400-300-200-1000100200300400-0.03 -0.015 0 0.015 0.03Shear Force (kN)Shear Displacement/Column HeightExperimentalProposed ModelChapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 48   Figure 3-11 Shear, axial and rotational spring in series model with the nonlinear beam-column element The combined response for the example column is illustrated in Figure 3-12. As shown in this figure, after the shear failure, the global drift is entirely influenced by the shear response and the flexural deformations decrease.    An existing hysteretic model available in OpenSees, based on Takeda et al. (1970) and    Figure 3-12 Combined response for specimen Sezen No. 1  PV Reinforcement slip springNonlinear beam-column elementShear response springAxial response springReinforcement slip springPVZero length elementZero length element00.20.40.60.811.20 0.5 1 1.5 2 2.5 3Column Shear Force/VmaxColumn Drift [%]Total ResponseFlexure ResponseShear ResponseBar Slip ResponseChapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 49  (a) definition of pinching parameters (b) definition of unloading parameters Figure 3-13 Hysteretic behaviour from Elwood (2002 ? Ph.D. thesis by permission) Filippou et al. (1992), was used to capture the hysteretic behaviour of the zero length springs. The hysteretic model includes the following parameters: px = pinching factor for strain (or deformation) during reloading; py = pinching factor for stress (or force) during reloading; and ?), where ? = displacement ductility and ? = parameter to determine the degraded unloading stiffness based on ductility (Figure 3-13). Theoretically, parameters px, py, and ? can be varied between 0 and 1; however, in this study, the following values were selected for all columns: px  = 0.5, py  = 0.2, and ? = 0.5.  These values were suggested by Kang et al. (2009) to achieve the best results (convergence). 3.6. MODEL VERIFICATION The accuracy of the proposed model is evaluated by comparing the calculated and measured total shear-drift response of columns subjected to reverse cyclic loading. Since the proposed model is intended to be used for structural analysis of a building structure, model verification based on a visual comparison of the full hysteretic response is preferred to the selection of any one point during the cyclic response (e.g., point of shear failure). The proposed model was evaluated using a data set of 20 column tests; eight double curvature specimens are selected here to demonstrate the capabilities and limitations of the model. Properties of the selected specimens are listed in Table 3-1 and comparisons of experimental and analytical results are shown in Figure 3-14. These specimens have been selected for presentation here as they provide a range of column dimensions and design details typical in older reinforced concrete buildings, they include columns experiencing shear failure both before and after flexural yielding, and demonstrate the range of accuracy achieved by the proposed model. In addition, both types of shear failure, Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 50 diagonal tension and diagonal compression failure are represented in this sample. It must be noted that (with the exception of Eqn. (3-13) for drift at axial failure) the proposed column model, based on the mechanics of shear, flexure and bar slip, was not calibrated to any column tests prior to this comparison. Not relying on empirical limits for drift capacity means that the proposed model can be used for the analysis of a broad range of column designs and represents an important advancement over previous empirical models for non-ductile columns (e.g., Elwood, 2004). Considering the mechanics-based approach and the inherent variability expected for non-ductile columns experiencing significant shear-related damage, the agreement shown in Figure 3-14 can be considered very good.  Failure modes are determined in the analysis and provided in Table 3-1 for all eight reinforced concrete columns specimens. It is observed from the table that specimens with high axial loads or transverse reinforcement ratios exceeding 0.15% are predicted to experience a diagonal compression shear failure. The only exception to these criteria is specimen Sezen No.1, where the shear failure mode is predicted to be diagonal compression although the axial load and transverse reinforcement ratios are lower than the other cases with this type of failure mode. To understand this discrepancy, the detection of shear failure based on shear strain demands in the plastic hinge is compared for specimens Sezen No.1 and No.2 in Figure 3-15. a indicates that the predicted shear strain response in the plastic hinge region for specimen Sezen No. 1 very nearly intersected the diagonal tension failure limit surface before intersection with the diagonal compression failure limit surface. Considering the uncertainty in estimating the failure surfaces, it is plausible that Sezen No. 1 in fact experienced diagonal tension failure and has been misclassified as diagonal compression failure in this assessment. Inspection of the results in Figure 3-14 indicate that the proposed model generally does a good job of representing the response of the column up to and including the point of shear failure. After this point, the column model experiences degradation of the lateral load resistance governed by Eqn. (3-20). For columns experiencing diagonal compression failures (Specimen No. 16 and Sezen No. 2), Eqn. (3-20) appears to do a reasonable job of representing the rate of shear degradation. For columns experiencing diagonal tension failure, Eqn. (3-20) generally underestimates the rate of shear degradation. Columns experiencing diagonal tension failures are expected to exhibit more cracking in the shear-damaged portion of the column, compared with columns experiencing diagonal compression failures. However, the post-peak response model Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 51 (Eqn. (3-20)) has been developed assuming sliding occurs along one principle crack (see Figure 3-8).  For diagonal tension failures where multiple cracks are anticipated, this post-peak response model may be expected to be less accurate. In addition to this, the results clearly signify the limitation of this model in reproducing the amount of cyclic pinching, especially for the specimens which experience a diagonal tension failure.   Furthermore, the underestimation of the rate shear degradation for diagonal tension failures may in part be attributed to the lack of cyclic degradation (degradation in resistance between cycles to the same drift demand) in the current model. Cyclic degradation may be more significant for columns experiencing diagonal tension failures due to the increased number of cracks expected for such columns and movement that would occur on these cracks with increasing number of cycles. The hysteretic model (Figure 3-13) used for the zero-length shear spring does not account for cyclic degradation. Development of a mechanics-based cyclic degradation model is extremely challenging given the sensitivity to the slip and load transfer on multiple cracks with degrading aggregate interlock.  This was considered beyond the scope of the current study. Future development of the proposed column model should strive to improve the accuracy of the post-peak response in comparison with columns experiencing diagonal tension failures and incorporate cyclic degradation.    Table 3-1 Material and geometry properties of square specimens Specimen b = h [mm] 2a [mm] a/h s [mm] ?g [%] ?sy [%] fsyl(7) [MPa] fsyt [MPa] fc?(8) [MPa] P/Agfc? Predicted Shear Failure Mode No.12(1) 300 900 1.50 150 2.26 0.14 415 410 28 0.21 DT(5) No.16(1) 300 600 1.0 50 1.8 0.43 450 410 27 0.21 DC(6) Purdue 2(2) 457 1473 1.61 203 1.5 0.07 441 490 19 0.37 DT Purdue 4(2) 457 1473 1.61 457 2.5 0.07 441 490 24 0.44 DT Kansas 1(3) 457 2946 3.22 457 2.5 0.07 445 372 33 0.32 DT Kansas 2(3) 457 2946 3.22 457 2.5 0.07 445 372 34 0.21 DT Sezen No. 1(4) 457 2946 3.22 305 2.5 0.17 438 476 21.1 0.15 DC Sezen No. 2(4) 457 2946 3.22 305 2.5 0.17 434 476 21.1 0.61 DC Note: (1) (Mostafaei et al., 2009); (2) (Henkhaus, 2010);  (3) (Matamoros and Woods, 2010); (4) (Sezen and Moehle, 2006) (5) Diagonal Tension; (6) Diagonal Compression; (7) yield strength of longitudinal and transverse reinforcement based on test data; (8) concrete compressive strength based on test data.   Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 52    (a) Specimen No. 12  ? analysis (b) Specimen No. 12  ? test data   (c) Specimen No. 16 ? analysis (d) Specimen No. 16 ? test data   (e) Specimen Purdue 2 ? analysis (f) Specimen Purdue 2 ? test data -300-1500150300-3 -1.5 0 1.5 3Lateral Load (kN)Drift Ratio (%)-300-1500150300-3 -1.5 0 1.5 3Lateral Load (kN)Drift Ratio (%)-400-2000200400-5 -2.5 0 2.5 5Lateral Load (kN)Drift Ratio (%)-400-2000200400-5 -2.5 0 2.5 5Lateral Load (kN)Drift Ratio (%)-600-3000300600-3 -1.5 0 1.5 3Lateral Load (kN)Drift Ratio (%)-600-3000300600-3 -1.5 0 1.5 3Lateral Load (kN)Drift Ratio (%)Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 53   (g) Specimen Purdue 4 ? analysis (h) Specimen Purdue 4 ? test data   (i) Specimen Kansas 1 ? analysis (j) Specimen Kansas 1 ? test data   (k) Specimen Kansas 2 ? analysis (l) Specimen Kansas 2 ? test data -800-4000400800-3 -1.5 0 1.5 3Lateral Load (kN)Drift Ratio (%)-800-4000400800-3 -1.5 0 1.5 3Lateral Load (kN)Drift Ratio (%)-600-3000300600-1.5 -0.75 0 0.75 1.5Lateral Load (kN)Drift Ratio (%)-600-3000300600-1.5 -0.75 0 0.75 1.5Lateral Load (kN)Drift Ratio (%)-400-2000200400-3 -1.5 0 1.5 3Lateral Load (kN)Drift Ratio (%)-400-2000200400-3 -1.5 0 1.5 3Lateral Load (kN)Drift Ratio (%)Chapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 54   (m) Specimen Sezen No. 1 ? analysis (n) Specimen Sezen No. 1 ? test data   (o) Specimen Sezen No. 2  ? analysis (p) Specimen Sezen No. 2 ? test data Figure 3-14 Comparisons between the proposed model and experimental behaviour   (a) Specimen Sezen No. 1 (b) Specimen Sezen No. 2 Figure 3-15 Detection of shear failure for specimens (a) Sezen No.1 and (b) Sezen No.2 -400-2000200400-6 -3 0 3 6Lateral Load (kN)Drift Ratio (%)-400-2000200400-6 -3 0 3 6Lateral Load (kN)Drift Ratio (%)-400-2000200400-2.5 -1.25 0 1.25 2.5Lateral Load (kN)Drift Ratio (%)-400-2000200400-2.5 -1.25 0 1.25 2.5Lateral Load (kN)Drift Ratio (%)0501001502002503003500 0.01 0.02 0.03 0.04 0.05Shear Force [kN]Shear Strain @ plastic hinge [rad]Diagonal tensionfailure limit surfaceDiagonal compressionfailure limit surfaceShearFailure0501001502002503003500 0.01 0.02 0.03 0.04 0.05Shear Force [kN]Shear Strain @ plastic hinge [rad]Diagonal tensionfailure limit surfaceDiagonal compressionfailure limit surfaceShearFailureChapter 3: Mechanical Model for Non-Ductile Reinforced Concrete Columns 55 3.7. CONCLUSIONS With the efforts to develop performance-based seismic design methodologies, and in order to identify existing concrete buildings vulnerable to collapse, it is important to understand the mechanisms causing collapse in concrete frame buildings when subjected to earthquakes. Loss of gravity load support after column shear failure is one of the primary causes of collapse in existing reinforced concrete frames. An analytical model is developed to simulate the nonlinear cyclic response of non-ductile concrete columns and improve the collapse assessment of non-ductile frame buildings.  Models for flexural, bar slip and shear deformations are combined to obtain the total lateral drift response of a column. Most notably the pre-peak shear behaviour, point of shear failure, and post-peak shear behaviour are determined based on mechanics principles, making the proposed column model applicable to a broader range of columns compared with similar empirical models (e.g., Elwood, 2004). Shear failure is determined based on the shear deformations in the plastic hinge zones exceeding deformation limits based on the MCFT, while the post peak response is determined based on shear-friction concepts.  The two types of shear failure, diagonal tension and compression failure, are numerically represented in the mechanical model and are used to detect shear failure. The mechanical model is implemented in nonlinear analysis software (OpenSees) to demonstrate its capability of representing the hysteretic response in columns with different failure modes with acceptable accuracy.  The proposed model is verified using experimental data of eight columns experiencing shear failure either before or after flexural yielding. The comparison of the analytical model with experimental test data indicated that the numerical model adequately captures the pre-peak response and point of shear failure. The column model is found to provide a more reasonable estimation of the test results for the post-peak response of columns sustaining diagonal compression failures compared to columns experiencing diagonal tension failures. Additional research is required to improve the post-peak behaviour for columns which experience diagonal tension failure.  Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 56 Chapter 4. SYSTEM-LEVEL COLLAPSE ASSESSMENT OF NON-DUCTILE CONCRETE FRAMES3  4.1. INTRODUCTION Earthquakes have repeatedly demonstrated the need for robust collapse criteria to detect structures most susceptible to collapse. The 22 February 2011 M 6.3 Christchurch earthquake has many examples of existing non-ductile concrete frames with different structural performances. As shown in Figure 4-1, the Pyne Gould Corporation (PGC) building (built pre-1970 and shown in Figure 4-1.a) has totally collapsed, while the Television New Zealand (TVNZ) building (built pre-1970 and shown in Figure 4-1.b) was able to survive without collapse, even with shear failure in several columns. To evaluate adequately the collapse prevention performance level for existing concrete buildings it is important to understand the mechanisms causing collapse in such structures under both gravity and seismic loads. Furthermore, refined modeling of structural collapse is necessary for the development of new risk-based seismic design codes in the United States targeting a collapse probability of 1% in 50 years (ASCE Standard 7-10, 2010). Because of the cost and time required for experimental tests, it is essential to develop analytical models that can reasonably predict the behaviour of structural elements up to the stage of collapse. Current modelling typically focuses on side-sway collapse mechanisms, despite the fact that shaking table tests and post-earthquake observations indicate that components of the gravity system (e.g., columns) may lose ability to support gravity loads prior to developing a side-sway collapse mechanism. The main objective of this chapter is to overcome the shortcomings of existing collapse criteria for non-ductile concrete frames. The chapter is organized to first display the accuracy the component models required to simulate gravity load collapse (e.g., non-ductile columns or slab?column connections). The components models are then used to define enhanced system-level collapse criteria. This study presents the application of these collapse criteria to previously tested shaking table specimens to examine their accuracy and practicality. Finally, the collapse criteria are used to develop estimates of collapse probability of the previously tested                                                  3 A version of chapter 4 plan to be submitted. Baradaran Shoraka, M., Yavari, S., Elwood, K. J. and Yang, T. Y., ?System-Level Collapse Assessment of Non-Ductile Concrete Frames.?   Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 57    (a) PGC Building (Photo: K.J. Elwood) (b) TVNZ (Photo: K.J. Elwood) Figure 4-1 Examples of existing concrete buildings in the M 6.3 Christchurch earthquake with different structural performances shaking table specimen, when all significant sources of uncertainty are considered. 4.2. EXISTING NUMERICAL MODELS FOR NONLINEAR ANALYSIS OF SHEAR-CRITICAL FRAMES  In order to better understand (and determine) the collapse potential for existing concrete buildings, it is proposed to develop model-building prototypes for analysis using research-oriented software programs (e.g., OpenSees). A critical requirement for a realistic performance assessment of non-ductile concrete buildings under seismic loads is to accurately model shear-related effects in both the gravity- and lateral-force-resisting systems. The estimation of collapse probability necessitates a detailed numerical model capable of capturing these types of failures. The selection of the collapse criteria can significantly alter the final results of the numerical analysis. The following will summarize the component models used to capture gravity load collapse. 4.2.1. Nonlinear Beam Column Elements As reviewed in Chapter 2 and extended in Chapter 3, non-ductile columns are mainly susceptible to shear failure followed by axial failure, and predictive analytical models must have the ability to capture the onsets of such failures and post-failure behaviour. Therefore, nonlinear models should incorporate elements with the capability to estimate the loss of vertical load capacity for such columns while accounting for the P-delta effects as well. Chapter 2 summarized three of the Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 58 recently developed models for capturing shear failure in existing columns. In Table 4-1, two of the models reviewed in Chapter 2, Haselton et al. (2008) and LeBorgne and Ghannoum (2009), are compared with the new mechanical model introduced in Chapter 3. Table 4-1 summarizes the main features of each model.  In addition to the shear failure model, an axial failure model should also be included in the numerical model. Elwood and Moehle (2005) developed Eqn. (3-13) based on shear-friction concepts to estimate the drift at axial load failure for columns experiencing shear failure. Although the mechanical model presented in Chapter 3 has been calibrated to a smaller dataset compared with the other two modeling approaches, this mechanical model provides a better interaction with the axial-load failure model and therefore is used in simulating the existing concrete frames in this study.   Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 59 Table 4-1 Comparison of three of the leading models to simulate shear failure in existing columns             Model 1: Haselton et al. (2008) Model 2: Leborgne and Ghannoum (2009) Model 3: Mechanical Model (Chapter 3) Failure types  Flexure, flexure?shear  Shear, flexure?shear  Shear, flexure?shear  Elements  Elastic beam?column element + zero-length flexural spring  3 nonlinear elements + zero-length shear springs  Nonlinear element + zero-length springs  Calibration database  255 columns  32 columns  20 columns  Model description  The model provides regression-based equations that are used to estimate linear and nonlinear parameters of flexural springs based on column properties and loading conditions.  The shear spring model has the ability during analyses to monitor the deformations between two nodes bracketing the plastic hinge region and forces in the adjacent column element. The model compares the shear force in the column with a limiting shear force and the rotation of the plastic hinge region with a limiting rotation.  This model detects shear or flexure?shear failure based on shear strains in the plastic hinge zone of the columns element. The model can detect when shear capacity is sufficient and flexural deformations govern response; however, it does not currently capture flexural failures (i.e., degradation due to rebar buckling/fracture). Cyclic modeling  Calibrated for the full cyclic behaviour, including in-cycle and cyclic degradation  The model can simulate the full degrading behaviour, including in-cycle and cyclic degradation.  The model can simulate the full degrading behaviour, including in-cycle and cyclic degradation; however, cyclic parameters are not calibrated.  Input by user vs. adaptive model  All model parameters are fixed by user input at the model building phase. Thus, the model does not adjust behaviour to varying boundary conditions during analysis.  The user can either input fixed values for rotation and shear-force limits or use the calibrated version of the model that automatically evaluates limits during analysis; this model uses the ASCE 41 shear strength Eqn. and a regression-based plastic rotation Eqn.. During analysis the model monitors column forces and deformation demands between integration points and adjusts the limit state that triggers strength degradation  OpenSees material  Pinching4 using hysteretic model by Ibarra et al. (2005)  PinchingLimitState Material described in Leborgne (2012)  LimitState Material (Elwood, 2004) with modifications     PV Flexural springElastic beam-column elementPVZero length elementFlexural springZero length elementPVNonlinear beam-column elementShear response springAxial response springReinforcement slip springPVZero length elementZero length elementReinforcement slip springPV Reinforcement slip springNonlinear beam-column elementShear response springAxial response springReinforcement slip springPVZero length elementZero length elementChapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 60 4.2.2. Beam ? Column Joints Although the model suggested by Alath and Kunnath (1995) may not capture the behaviour of the joints thoroughly (the drawback of the scissors model is the inability to model the true kinematics of the joint), it was selected for modeling the unconfined joints in the current study due to its simplicity and practicality. The finite size of the joint panel is taken into account by the introduction of rigid links (see Figure 4-2). Joint shear deformation is simulated by a rotational spring model with degrading hysteresis. The beam?column joint element is implemented in the numerical model through defining duplicate nodes with the same coordinates at the center of the joint. After the two nodes are defined, the element connectivity is set such that one of the nodes is connected to the column rigid link and the other node is connected to the beam rigid link. Next, a zero length rotational spring is used to connect the two nodes so that the column rigid link is connected to one end of the spring and the beam rigid link is connected to the other. The degrees of freedom at the two central nodes are defined to permit only relative rotation between the two nodes through the constitutive model of the rotational spring. The rotational spring transforms the shear deformation into an equivalent rotation as described below. The moment?rotation relationship for the joint panel zone is obtained using the joint shear stress?strain. The envelope to the shear stress?strain relationship is determined empirically, whereas the cyclic response is captured with a hysteretic model that is calibrated to experimental cyclic response (Lowes and Altoontash, 2003). To implement the constitutive relationship for the panel zone in the analysis, a backbone curve and cyclic response based on previous experimental test data (e.g., Walker, 2001) is used. The values proposed by Lowes and Altoontash (2003) are used to define the envelope and the pinched hysteresis model (Pinching4 in OpenSees, Figure 4-3). The model parameters for the Pinching4 model are defined in OpenSees online website (http://opensees.berkeley.edu/wiki/index.php/Pinching4_Material). The Pinching4 material   Figure 4-2 Rotational spring joint model (adapted from Alath and Kunnath, 1995) Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 61  Figure 4-3 Pinching4 OpenSees model (? 2009 PEER, by permission) model has eight positive and negative envelope parameters which are calculated using the proposed backbone curves found in the following paragraphs. The model has different parameters to define pinching behaviour (rDispP to uForceN), parameters to define unloading stiffness degradation (K1 to KLim), parameters to define reloading stiffness degradation (D1 to DLim), parameters to define strength degradation (F1 to FLim), and finally a parameter gE to define energy dissipation rule. The following two equations are used to switch between the joint shear stress?strain constitutive model and moment?rotation constitutive models. As demonstrated by Celik (2007), the rotational spring moment, Mj, can be related to the joint shear stress, ?jh, through  =               (4-1) where Lb is the length from beam inflection point to the column centerline, which can be approximated as half of the beam centerline span. The parameter j is the effective beam lever arm ratio, which can be approximated as 0.9. The column height Lc is measured between column inflection points, which can be approximated by story height. The width and height of the joint panel are bj and hj, respectively. The rotation of the rotational spring, ?j, is taken as equal to the joint shear strain, ?jh:  =               (4-2) Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 62 Eqn. (4-2), proposed by Hassan (2011), assumes that the joint rotation resulting from beam bar slip is explicitly defined by a separate zero length rotational slip spring element. The proposed backbone curve characteristic points for point 1 (ePd1, ePf1) to point 4 (ePd4, ePf4) are described and quantified in Table 4-2 and are converted to moment (ePfi) and deformation (ePdi) using Eqns. (4-2) and (4-3). The equations presented in Table 4-2 by Hassan (2011) have been suggested based on the test results Hassan has conducted. The values of the 22 parameters used to define cyclic response are presented in Table 4-3. These empirical equations with the values proposed by Hassan (2011) are used in modeling all the existing non-ductile beam?column joints in this study.  Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 63  Table 4-2 Shear stress - strain backbone key points (adapted from Hassan, 2011)     Characteristic 1 = 3.5 1 + 0.002 	  =0.5  = 0.5 Cracking Strength 2 	 = 0.0002 	 =0.002 	 = 0.1 Pre-Peak ?Yielding? Strength 3  = 11.25.  =  = 0.14 ? 380   	 ? 0.30.175 ? 340  	 < 0.3 Peak Shear Strength 4  = 0.7 =  + 0.02  	 ? 0.3 + 0.025  	 < 0.3  =  Post-Peak (Residual /Axial Failure) Strength where  to  represent the shear stress in the joints;  to  are the shear strains in the joints; P is the axial load; Ajh represents the joint area (bj*hj); G is the joint shear modulus of rigidity;  represents concrete Poisson?s ratio;  is the joint aspect ratio; and  is an axial load factor (more detail in Hassan, 2011).   =.	  ;        = 0.2;        = ;          (4-3a, b, c) and  = 1 + 0.86 ? 0.3   ? 0.15        (4-4)Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 64 Table 4-3 Pinching4 material model parameter (adapted from Hassan, 2011) Parameter Type Parameter ID Hassan Pinching Parameters rDispP 0.15 rForceP 0.35 uForceP -0.1 rDispN 0.15 rForceN 0.15 uForceN -0.4 Unloading Stiffness Degradation Parameters gK1 0.5 gK2 0.2 gK3 0.1 gK4 -0.4 gKLim 0.99 Reloading Stiffness Degradation Parameters gD1 0.1 gD2 0.4 gD3 1 gD4 0.5 gDLim 0.99 Strength Degradation Parameters gF1 0.05 gF2 0.02 gF3 1 gF4 0.05 gFLim 0.99 Energy Dissipation gE 10 dmgType energy  It should be noted that axial failure of joints is not considered in this study. A recent study by Hassan (2011) found that for joints with beam-to-column depth ratio less than 2.5, joint axial load failure was unlikely to precede column axial-load failure.   Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 65 4.3.  SYSTEM-LEVEL COLLAPSE DEFINITION  As explained in Chapter 2, the limit-state checks used in FEMA P695 (2009) to capture non-simulated failure modes will generally result in a low estimate of the median collapse point because non-simulated failure modes are usually associated with a single component failure mode. The inherent assumption is that the first occurrence of this non-simulated failure mode will lead to collapse of the structure, ignoring load redistribution after failure. Furthermore, the impact of the non-simulated collapse modes on the rest of the structure is not directly accounted for in the analysis. Therefore, it is preferred to have local failure modes directly simulated in order to more accurately reflect impact on the structural performance at the near-collapse limit state. In recent FEMA guidelines (e.g., FEMA 440A, (2007) and FEMA P695, (2009)), side-sway collapse (where collapse is defined based on unrestrained lateral deformations with an increase in ground motion intensity) is typically assumed to be the governing collapse mechanism. However, in non-ductile reinforced concrete frames, it is expected that components without ductile detailing may lose the capacity to support gravity loads prior to the development of a flexural mechanism necessary for a side-sway collapse mode.  Gravity load collapse may be precipitated by axial load failure of columns, punching shear failure of slab?column connections, failure of slab?diaphragm connections, or axial-load failure of beam?column joints. This study will demonstrate approaches for capturing gravity load collapse in beam?column and slab?column frames, herein defined as the point at which the total vertical load capacity at a single story drops below the gravity load demand at the same story. An important step in collapse assessment is to perform a detailed and robust nonlinear analysis on the non-ductile reinforced concrete frame to predict the inelastic behaviour of beam and column elements and capture the onset of collapse. Non-ductile behaviour originating from column shear and subsequent axial failure plays an important role in these structures, and the analytical model must have the ability to capture such behaviour. In Chapter 3, accuracy and reliability of the column model was verified and the beam?column joint model is verified by Hassan (2011). The next step is to specify a system-level collapse definition. Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 66 4.3.1. Gravity Load Collapse When a non-ductile reinforced concrete frame is laterally deformed, local failure is likely to occur in vertical load-carrying components (e.g., non-ductile columns or slab?column connections) as a result of shear/torsion and subsequent axial failure. As the gravity loads carried by these elements are transferred to the neighbouring elements, the localized failure can develop until the structure loses its ability to resist the gravity load.  The progression of damage can be tracked throughout the numerical analysis by comparing floor-level gravity load demands and capacities. The gravity load demand in beam?column frames will be the axial load and the capacity will be adjusted at each time step to account for shear and subsequent axial failures. For slab?column frames, there will be gravity shear demands and the capacity will adjusted at each time step to account for punching shear failure.  The demand at each floor will be the total (aggregate) gravity load at that floor. The capacity will be altered at each load step based on the lateral drift and the column or slab?column connection characteristics (Figure 4-4 and Figure 4-5). For beam?column frames, as seen in Figure 4-4, as the inter-story drifts increase the exterior and interior column capacities decrease, and this will affect the total floor vertical load capacity, which will be the sum of axial load capacity for all columns in each floor. The axial capacity for each column is based on Eqn. (3-13) and as seen in this equation, as the column starts to drift, the axial capacity decreases and when the column drifts back to it?s original state, the axial capacity will increase to it?s initial state. In this study, a subroutine is integrated in the numerical analysis (OpenSees) to detect the first point when floor-level vertical load demands exceed the total vertical load capacity at that floor. This process is shown figuratively in Figure 4-4. As can be seen in the bottom graph of Figure 4-4, the floor capacity decreases, but it will not reach the critical floor demand unless at least one of the columns fails (right before the collapse point).  For slab?column frames, the demand at each floor will essentially be the aggregate shear force on the slab critical section for the slab?column connections in that floor. The capacity, the accumulative shear strength in the slabs, will be altered at each load step based on the lateral drift (shown in Figure 4-5). As seen in this figure, as the interstory drifts increase the slab shear strength decreases, and this will affect the total floor capacity, which will essentially be the sum Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 67 of all slab?column connections in each floor. In this study, a subroutine is integrated in the numerical analysis to detect the first point when floor-level gravity shear demands exceed the total shear strength at that floor. This process is shown figuratively in Figure 4-5. In the bottom graph of Figure 4-5, before reaching the collapse state, the floor shear strength first experiences a drop in the capacity which represents a slab?column connection failure (punching failure), but this will not trigger the gravity load collapse until the next connection failure.   Figure 4-4 Gravity load collapse captured explicitly in the numerical model Interstory DriftTimeVertical LoadTimeExterior Column CapacityInterior Column CapacityVertrical LoadTimeCritical Floor DemandCritical Floor Capacity?Gravity Load? collapseChapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 68  Figure 4-5 Gravity load collapse captured explicitly in the numerical model Kang et al. (2009) have used the limit state model, which was originally developed by Elwood (2004) to detect column shear and axial failures, to capture the punching shear failure at the slab?column connection region. The limit state approach simplifies modeling of punching failures, as all factors that impact story drift are monitored throughout the analysis. Once the limit state surface is reached, a punching failure is detected, and the ability of the slab?column connection to transfer moment (or unbalanced moment) degrades according to a specified relationship.  In addition to the moment, the axial interaction between the slab and the column should also be included in the numerical model, and after punching failure, the axial response should also degrade. This feature has been included in the numerical models used for this study to model the slab?column moment frames. The total response of the slab?column connection can be modeled by coupling moment, axial, and shear responses as springs in series. Zero-length Interstory DriftShearLoadTimeCritical Floor Shear StrengthCritical Floor Gravity Shear Demands?Gravity Load? collapseChapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 69 sections located at the joint of the slab?column are attached to the two adjacent columns and the two adjoining slab?beams. The zero-length sections are defined by three uncoupled material models describing (1) the moment?rotation relationship representing torsional response, (2) the shear ? horizontal displacement relationship representing the shear force ? displacement response (which is rigid in this case), and (3) the axial load ? vertical displacement relationship (Figure 4-6). It should be noted that the zero-length element and the slab?beams are at the same height.   Figure 4-6 Slab?column connection modeling   Nonlinear beam-column elementShear response spring (rigid spring)Axial response springTorsional connection springZero length element (interior joint region)Rigid end zone(slab thickness)Rigid end zone(column width)Nonlinear beam-column elementChapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 70 4.3.2. Side-Sway Collapse Global collapse for most ductile structures and very limited existing frames (e.g., frames with a tall first story) are governed by side-sway collapse. In this type of failure, side-sway instability occurs in one or more storeys (Figure 4-7). In many studies, including the process in collapse analysis used in FEMA P695 (2009) and Haselton (2006), the criteria for lateral collapse is based on the maximum inter-story drift ratio or when the story drift increases without relevant increase in the ground motion intensity. As shown in Figure 4-7, whenever the inter-story drift ratio (inter-story drift ratio for the first story in this example) increases rapidly (without gravity load failure) owing to increasing of the spectral acceleration, the structure is considered to sustain a side-sway collapse. For such cases of global failure, in contrast to a gravity load collapse, the state of (incipient) collapse is subjective, and the collapse probabilities could vary slightly based on the selection of this state.    In order to overcome such issues and to have an objective definition of the state of collapse similar to the definition for gravity load collapse, side-sway collapse could also be based on lateral capacity and demand. In this study, a subroutine is also integrated in the process (implemented in OpenSees) to detect possible side-sway collapse. The process is figuratively demonstrated in Figure 4-8. As shown in this figure, whenever the lateral resistance (defined by the story strength corresponding to the peak inter-story drift ratio) decreases below a pre-established minimum lateral capacity, the structure is considered to sustain a side-sway collapse. The minimum shear/flexural capacity can be estimated based on an approximation of residual column shear/flexural strengths or can be set to zero. As mentioned before, this type of collapse is not common in non-ductile frames. However, in the sensitivity analyses covered in the following chapters, varying parameters of the model could switch the governing failure mode from a gravity load collapse to a side-sway collapse. Therefore, the numerical model should also have this feature implemented in its process to cover all potential modes of collapse. Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 71  Figure 4-7 Side-sway collapse captured in an incremental dynamic analysis based on maximum inter-story drift ratio  Figure 4-8 Side-sway collapse captured in an incremental dynamic analysis based on lateral capacity and demand FloorPeak Interstory DriftPotential?Sidesway? collapseIncrease in spectral intesity0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Story Shear @ Peak Interstory Drift Peak Interstory DriftMinimum Shear Capacity?Side-sway?collapseIncrease in spectral intensityChapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 72 4.4. COMPARISON OF COLLAPSE DEFINITION TO EXPERIMENTAL RESULTS  In this section, an analytical model incorporating the collapse criteria discussed in section 4.3 is evaluated. In order to validate and compare the collapse criteria for non-ductile concrete frames, data from a shaking table test (Yavari, 2011) are compared with results from the analysis. It should be noted that the model provides a relatively simple representation of a very complex phenomenon at the onset of gravity load failure and hence, may not perfectly capture the behaviour of the building up to the point of collapse. 4.4.1. Details of Shaking Table Test Specimen The specimen was tested at the National Center for Research on Earthquake Engineering (NCREE) in Taiwan in 2009. The test structure studied here was part of a series of test structures investigating seismic behaviour of older-type reinforced concrete frames in Yavari (2011). It consisted of a 1/2.25 scaled model of a two-bay and two-story reinforced concrete frame and was tested under moderate gravity loads. The geometries and details (Figure 4-9) were selected to be representative of elements used in an existing six-story building in Taichung, Taiwan. Table 4-4 summarizes the critical properties of the frame. Complete details of the study, test setup, and the analyses can be found in Yavari (2011).   The specimen was subjected to one horizontal component from a scaled ground motion recorded at station TCU047 during the 1999 Chi-Chi earthquake. The specimen did not collapse under the first table motion; therefore, the intensity of the table motion was elevated to deliver a   Figure 4-9 Shaking table test specimens (units in mm) from Yavari (2011 ? Ph.D. thesis by permission) 200208 # 5#3 @ 150 mm 600#3 @ 150 mm Beam Section A-A (First Floor)602 # 32 # 3200300 4 # 4 #3 @ 150 mm 600Beam Section B-B (Second Floor)604 # 3300Varies at beam endsVaries at beam ends20200200178 # 45 mm @ 120 mm c/c Column Section C-C15501700C B A2000 20001000 1000 10005000500450 3001400600600600200Elevation Side View300200300200AAC CC CBB(Specimens MCFS, HCFS, MUFS)5 mm @ 40 mm c/c (Specimen MUF)MCFS & HCFS#3 @ 150 mm 1400Only in Specimens2 # 32 # 3NChapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 73 Table 4-4 Parameters and column material properties of tested specimen f?c  on Test Day (MPa) 34.0 Column Longitudinal Bars fyl (MPa) 439.0 Column Transverse Bars fyt (MPa) 469.0 Column Longitudinal Steel Ratio ?l (%) 2.6% Column Transverse Steel Ratio ?" (%) 0.16%          ??=Ast/bs where, Ast is the area of transverse reinforcement with spacing s,          and b is the column width perpendicular to the direction of shaking peak ground acceleration of 1.35g where collapse of the frame was observed due to shear and axial failure of all first- story columns. 4.4.2. Response to Recorded Table Input Motion The analytical model for the shaking table specimen is subjected to the unidirectional horizontal table acceleration recorded during testing of the specimen. The test data and the response of the analytical model are compared in Figures 4-11 through 4-13. To focus on the collapse behaviour, these figures show only data from 25 to 35 (sec) of the collapse test, which provides the most critical period of response, to assess the performance of the analytical model. It should be noted that the test data are terminated in all figures at collapse of the test specimen and the analysis results are terminated when gravity load collapse is detected. Despite the lack of agreement early  Figure 4-10 Analytical model for the shaking table specimen from Yavari (2011 ? Ph.D. thesis by permission) Axial Spring (Limit State Material with Axial Limit CurveShear Spring (Limit State Material with Shear Limit CurveTop Node (bottom of Rigid Element)Bottom Node (top of Column)Shear & Axial SpringsSlip SpringSlip SpringTop Node Bottom Node Slip SpringShear & AxialSpringsSlip Spring+Shear & AxialSpringsSlip Spring+Shear & AxialSpringsSlip Spring+Foundation ElementColumn ElementRigid ElementBeam ElementNodeRotational SpringSlip SpringShear & AxialSpringsSlip Spring+Rotational SpringSlip SpringRotational Spring Rotational SpringSlip SpringChapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 74 in the response histories, the analytical model provides a reasonable estimate during the critical periods of response (Figure 4-11). Similar to the sequence of failure observed during the test, the analytical model was able to first detect shear failure of column B1 (middle column of first story), followed by C1 (right column of first story), and finally A1 (left column of first story). Shear failure was first detected for column B1, followed by degradation of shear resistance during a positive displacement cycle at approximately 34.13 (sec). The initial shear resistance degradation decreased the stiffness of the center column in the analytical model, which is in a close agreement with the hysteretic response (Figure 4-12). The post-peak negative stiffness during that pulse was also estimated relatively accurately. However, as shown in                         Figure 4-11 Response histories for collapse test 25 26 27 28 29 30 31 32 33 34 35-240-180-120-60060120180240Base Shear (kN)25 26 27 28 29 30 31 32 33 34 35-2-1012Time(sec)First-StoryDrift Ratio(%)25 26 27 28 29 30 31 32 33 34 35-100-80-60-40-20020406080100Column A1Shear (kN)  Test DataAnalys is25 26 27 28 29 30 31 32 33 34 35-100-80-60-40-20020406080100Column B1Shear (kN)25 26 27 28 29 30 31 32 33 34 35-100-80-60-40-20020406080100Column C1Shear (kN)Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 75   Figure 4-12 Shear hysteretic response  Figure 4-12, the analytical model estimated the shear failure at a larger drift ratio for columns B1 and C1. Furthermore, the slope of shear resistance degradation was observed to be steeper for the columns during the test. Axial load hysteretic responses of first-story columns are shown in Figure 4-13. It is observed that the analytical model was capable of simulating the axial load loss recorded for first-story columns reasonably well. Figure 4-14 compares the gravity load demand and capacities and the point of gravity load failure. As explained in section 4.3, the point at which the axial load capacity of a single story drops below the gravity load demand is defined as gravity load collapse. In this example the initial gravity load is considered as the gravity load demand. It should be noted that during the   Figure 4-13 Axial load hysteretic response of first-story columns -4 -2 0 2 4-100-80-60-40-20020406080100Column A2 Shear (kN)  Test DataAnalysis-4 -2 0 2 4-100-80-60-40-20020406080100Column B2 Shear (kN)-4 -2 0 2 4-100-80-60-40-20020406080100Column C2 Shear (kN)-4 -2 0 2 4-100-80-60-40-20020406080100Story Drift Ratio (%)Column A1 Shear (kN)-4 -2 0 2 4-100-80-60-40-20020406080100Story Drift Ratio (%)Column B1 Shear (kN)-4 -2 0 2 4-100-80-60-40-20020406080100Story Drift Ratio (%)Column C1 Shear (kN)-4 -2 0 2 4-400-300-200-1000Story Drift Ratio (%)Column A1 Axial Load (kN)-4 -2 0 2 4-400-300-200-1000Story Drift Ratio (%)Column B1 Axial Load (kN)  Test DataAnalys is-4 -2 0 2 4-400-300-200-1000Story Drift Ratio (%)Column C1 Axial Load (kN)Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 76   Figure 4-14 Gravity load collapse captured explicitly in the numerical model compared with the test data shaking table test the pre-stressing used to apply gravity load on the frame showed moderate fluctuation, particularly at the time of column axial load failure. This resulted in a drop in the applied gravity loads at the time of collapse. To avoid a dependence on the fluctuations in the pre-stress load, the initial gravity load was used to detect collapse. Figure 4-15 shows the first story drift response from 33 to 34.4 sec (in the analysis). As shown in the plot, the analytical model with limit-state materials has the ability to capture the structural response after first column shear failure (approximately 2.5% inter-story drift). When the inter-story drift ratio reaches 5%, gravity load collapse is detected in the first floor. If the analysis is allowed to go past gravity load collapse (for a model without limit-state material springs for axial load failure), side-sway collapse is detected when the inter-story drift ratio   Figure 4-15 Side-sway and gravity load collapse captured explicitly in the numerical model 600800100012001400160034.11 34.12 34.13 34.14 34.15 34.16 34.17 34.18Vertical Load [kN]Time [s]First Floor CapacityFirst Floor Demand (initial gravity load)First Floor Demand (test data)?Gravity Load? collapse-6%-4%-2%0%2%4%6%8%33 33.2 33.4 33.6 33.8 34 34.2 34.4First Story Drift Ratio [%]Time [s]First Column Shear Failure@ t = 34.13 (s)"Gravity Load" Collapse@ t = 34.177 (s)"Side-Sway" Collapse@ t = 34.236 (s)Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 77 reaches 7%. It should be noted that for a ground motion without such a large pulse (resulting in all collapse criteria being detected in one cycle), it is possible that side-sway collapse may occur considerably later in the ground motion or not at all. This emphasizes the importance of considering gravity load collapse for existing concrete frames. 4.5. COLLAPSE FRAGILITIES The collapse criteria could be used to develop estimates of collapse fragilities. These collapse fragilities will be used as an essential tool in the following four chapters. Use of a suite of earthquake records, nonlinear dynamic analysis via incremental dynamic analysis (IDA) (Vamvatsikos and Cornell, 2002), and/or multiple stripe analysis (MSA) (Jalayer and Cornell, 2009), information about variability in ground motions can be directly incorporated into the collapse performance assessment. However, this process only captures the record-to-record variability and does not account for how well the nonlinear simulation model represents the collapse performance of the building; hence, model uncertainties should also be accounted in the collapse simulation. These modeling uncertainties are especially important in predicting collapse because of the high degree of empiricism and uncertainty in predicting deformation capacity and other critical parameters for modeling collapse (e.g., post-peak resistance degradation). The Los Angeles Tall Buildings Structural Design Council (LATBSDC) have also presented procedures which provide a performance-based earthquake engineering approach for seismic evaluation and strengthening of existing tall buildings in the Los Angeles region with predictable and safe performance when subjected to strong earthquake ground motions (Hart et al. 2012, Hart et al. 2013). These new procedures incorporate both record-to-record variability and modeling uncertainty to obtain relevant demands and capacities of the performance limit states of tall buildings. In this study, the uncertainty for each random variable is explicitly considered in the analysis and reflected in the final probabilities of collapse. The random variables selected with the respective probability distribution should have the capability of capturing the major uncertainties inherent in non-ductile reinforced concrete frames. Past research has indicated that modeling uncertainties associated with damping, mass, and material strengths have a relatively small effect on the overall uncertainty in seismic performance predictions and will primarily have an influence on pre-collapse performance of structures (Lee and Mosalam, 2005). Ibarra Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 78 and Krawinkler (2005) have shown that modeling variables related to component strength and deformation capacity are the dominant model parameters affecting collapse assessment. Therefore, uncertainty in the shear and axial failure models for non-ductile columns are considered as the main sources of model uncertainty in this study. Figure 4-16 presents the model variability considered in these non-ductile columns. The non-ductile model variability is represented by a lognormal distribution with a mean equivalent to the limit-state material failure models and a coefficient of variation of approximately 0.3 (Elwood and Moehle, 2005). In addition to model variability, record-to-record variability is considered in the process. For the example presented below, the MSA method has been adopted with 20 ground motions considered for each of the selected hazard levels. A site in Taichung, Taiwan, is used to select the ground motions. Simultaneous consideration of the effects of record-to-record and model uncertainties on the collapse capacity necessitates performing reliability analysis. The outcome of the collapse assessment procedure, when applied to a particular structure, is a probability distribution representing the cumulative probability of collapse, P[Collapse] = P[IMcollapse < IM] as a function of a hazard intensity measure, e.g., spectral acceleration or return period, as illustrated in Figure 4-17. This collapse fragility curve is obtained by creating nonlinear analysis models of the non-ductile reinforced concrete (RC) frame and conducting nonlinear dynamic analysis for each realization of the random variables. As shown, the collapse fragility curve can be idealized by a lognormal distribution, which is defined by a median value and dispersion (logarithmic standard deviation, ?ln).   As outlined in Liel et al. (2009), a variety of approaches have been used to study the effects of these modeling uncertainties on the fragilities for structural response. The reliability methods used to propagate modeling uncertainties vary from the simplest yet less computationally efficient Monte Carlo method (Porter et al., 2005) to the first-order-second-moment (FOSM) methods (Lee and Mosalam, 2005; Ibarra and Krawinkler, 2005) and, finally, to the more complicated first-order reliability method (FORM) and second-order reliability Chapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 79   (a) (b) Figure 4-16 Structural model uncertainty in (a) shear and (b) axial failure models method (SORM). Monte Carlo sampling with a response surface (Liel et al., 2009) is used in this study to incorporate the influence of modeling uncertainties on the collapse capacity. The response surface is a multivariate function representing the relationship between the modeling uncertainties and the collapse capacity. As explained in Liel et al. (2009), sensitivity analysis is first executed to build the response surface, then Monte Carlo methods are used to sample the modeling uncertainties, and finally the collapse capacity is predicted using the response surface. The advantage of this method is that it avoids time-consuming nonlinear simulations. The outcome of this process is collapse fragilities that incorporate both the modeling uncertainty and the record-to-record variability. The reader is referred to Liel et al. (2009) for more details of the procedure. The procedure is applied to the two-story shake table specimen explained in the previous section. For this two-story specimen, the quadratic response surface for the median collapse capacity [ = ,	,] are given by the following equation: ,	, = 2.04 ? 0.36 	? + 14.91 	? + 35.34 	?  	?        (4.5) In Figure 4-17, two collapse fragility plots are shown, the first of which includes only record-to-record (RTR) variability (aleatory uncertainty), which is obtained by the nonlinear time history analysis. The other collapse fragility curve is computed from the Monte Carlo sampling (10,000 realizations) with a response surface in which structural modeling uncertainty (epistemic uncertainty) is accounted for and has resulted in an increase in the dispersion. Shear-failure based on ASFI modeldriftV?sf (?s)Shear failuredriftDegradation based on shear-friction model?af (?a)AxialfailureChapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 80  Figure 4-17 Collapse fragility obtained for the shaking table specimen 4.6. CONCLUSIONS With the efforts to develop performance-based seismic design methodologies, and in order to strengthen susceptible reinforced concrete structures against seismic loading, it is important to understand the mechanisms causing collapse in such structures under both gravity and seismic loads. Shake table collapse experiments could be conducted to assess the capacity of a structure to resist earthquake induced collapse, however, experimental tests are often costly and time consuming, therefore developing analytical models that can reasonably predict the behaviour of structural elements up to the stage of collapse is of great interest. The main objective of this chapter is to overcome the shortcomings of existing collapse criteria for non-ductile concrete frames.  Collapse of most non-ductile concrete buildings will be controlled by the loss of support for gravity loads prior to the development of a side-sway collapse mechanism. This chapter presents the application of these collapse criteria to previously tested shaking table specimens to examine their accuracy, reliability, and practicality. Comparison of the analytical model with shake table test data indicated that the system-level collapse criteria implemented in the analytical model captured the collapse observed in  the test.    A column model developed to detect shear failure based on the Modified Compression Field Theory (MCFT) and Axial-Shear-Flexural Interaction (ASFI) model that degrades the 0%10%20%30%40%50%60%70%80%90%100%0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5P [Collapse]IM [Sa (T1 = 0.3 s)/g]RTR (neglecting model uncertainty)RTR + model uncertaintyChapter 4: System-Level Collapse Assessment of Non-Ductile Concrete Frames 81 shear resistance and initiates axial load failure based on shear-friction concepts was also validated in this study through comparisons of the results from nonlinear dynamic analyses with data from shaking table tests. Numerous sources of uncertainty complicate the ability to identify buildings which are vulnerable to collapse. For this reason, it is important to develop estimates of collapse probability which account for all significant sources of uncertainties. The collapse criteria are used to develop estimates of collapse probability of the previously tested shaking specimen, taking into consideration all significant sources of uncertainties.  Chapter 5: Collapse Indicators - Methodology 82 Chapter 5. COLLAPSE INDICATORS - METHODOLOGY 5.1. OVERVIEW In the mid-1990s new seismic rehabilitation guidelines (e.g., FEMA 273, 1997) were introduced to the world, providing the structural engineering profession with the first generation of 'performance-based' procedures for seismic assessment and rehabilitation design. These documents revolutionized the assessment of existing buildings by encouraging the use of nonlinear analysis, by enabling the engineer to select project-specific performance objectives, and perhaps most importantly, by recognizing that structural collapse was limited by deformation capacity as well as strength. The past 15 years has seen modest improvements to this first-generation performance-based design procedure as FEMA 273 has evolved into ASCE/SEI 41 (2006), including updated acceptance criteria for concrete components based on new experimental data (Elwood et al, 2007). However, the overall framework remains essentially deterministic and inconsistent conservatism in specified deformation capacities throughout the document make it difficult to reliably establish the expected performance of a building structure for a given earthquake hazard. Furthermore, the component-based assessment procedures (i.e., once one component is determined to have exceeded a performance level, the entire structure is deemed to have exceeded the performance level) ignore the ability of a structural system to redistribute loads as damage accumulates and will tend to lead to overly conservative assessments of collapse vulnerability. Numerous sources of uncertainty complicate the ability to identify collapse vulnerable buildings, including uncertainty in the deformation capacity of non-ductile components. For this reason in order to identify relative collapse risk of a building inventory it is important to develop estimates of collapse probability accounting for all significant sources of uncertainty. Non-ductile moment frames, constructed prior to the introduction of seismic provisions in modern building codes, are susceptible to irrecoverable damage during severe earthquake shaking. As stated before, regarding the huge inventory of these non-ductile buildings, the need for a procedure to identify the most vulnerable buildings has been recognized as a high priority (NIST GCR 10-917-7, 2010) and is one of the main motivations for this study. Chapter 5: Collapse Indicators - Methodology 83 The main objective of this chapter is to summarize and extend an outline for the development of possible guidelines for collapse assessment of older reinforced concrete frames, initiated in the ATC-76-5 project (NIST GCR 10-917-7, 2010). The proposed guideline should be readily accessible to engineers and will essentially provide recommendations for component- and system- level acceptance criteria for existing concrete buildings. These acceptance criteria could be based on various global and local design and response parameters. This research will introduce the concept of collapse indicators. This chapter summarizes a methodology for identifying collapse indicators. These indicators are selected based on results of comprehensive collapse simulations and estimation of collapse probabilities for a collection of building prototypes. Intended outcomes of this chapter include: ? Define the concept of design and response parameter collapse indicators; ? Select a potential suite of design and response parameters which have a high influence on the risk and probability of collapse; and ? Develop a general methodology to calculate relative risk of collapse, from hazard analysis to structural analysis in a probabilistic framework, for each collapse indicator. Chapter 6 will apply this methodology to example buildings to identify possible collapse indicator limits. 5.2. COLLAPSE INDICATORS The term collapse indicator was first introduced in the ATC-76-5 (NIST GCR 10-917-7, 2010) report titled ?Program Plan for the Development of Collapse Assessment and Mitigation Strategies for Existing Reinforced Concrete Buildings?. In this report, collapse indicators are defined as design and response parameters that are correlated with an elevated collapse probability of non-ductile concrete buildings. The report has envisioned that these collapse indicators are expected to influence the collapse vulnerability of existing and non-ductile concrete buildings and will be used to identify critical non-ductile buildings most in need of seismic retrofit.  Ideally there should be a variety of collapse indicators, ranging from those appropriate for rapid assessment to others used to identify collapse potential buildings based on results of detailed nonlinear analysis. Collapse indicators for rapid assessment must be very simple parameters which can be established based on basic information available from a quick survey of Chapter 5: Collapse Indicators - Methodology 84 the building or engineering drawings. Collapse indicators for detailed collapse prevention assessment can make use of the results from building analyses.  Table 5-1 provides a list of potential collapse indicators. They have been identified as local or global parameters, and categorized as follows: (1) Design parameter collapse indicators: These parameters are related to design characteristics of the building. They can be categorized as ?rapid assessment? parameters which can be determined from a quick survey of the building or engineering drawings; and ?engineering calculations? which require some calculation of capacities and demands based on engineering drawings, but do not require structural analysis results from computer modeling. (2) Response parameter collapse indicators: These parameters are related to the response of the building. These quantities are for detailed collapse prevention assessment using the results from building analyses, commonly nonlinear analysis. There is a higher level of effort involved in determining these collapse indicators compared to the design parameters. It is anticipated that relationships may exist among the indicators; and hence, may be hierarchical. Furthermore, vectors of indicators may be found to provide a better indication of collapse potential. However, the study reported herein only considers each collapse indicator individually. As seen in Table 5-1, for each type of collapse indicator there are global and local collapse indicators. As seen in this table, for rapid assessment (RA), a global indicator could be the minimum ratio of column area to wall area at each story and a local indicator could be the average minimum column transverse reinforcement ratio for each story (determined by taking the minimum, over all the stories, of the average column transverse reinforcement ratio of all columns in each story). For engineering calculations (EC), a global indicator could be the portion of story gravity loads supported by columns with a ratio of plastic shear demand to shear capacity greater than 0.7 and a local indicator could be the maximum ratio of plastic shear capacity (2Mp/L) to column shear strength, Vp/Vn. Finally, for building analysis (BA), a global indicator could be the maximum fraction of columns at a story experiencing shear and/or axial failures and a local indicator could be the maximum drift ratio. The collapse indicators are explained in more detail in Chapter 6. Chapter 5: Collapse Indicators - Methodology 85  Table 5-1 Collapse indicators Type of Collapse Indicator Global Local Design Parameters Rapid Assessment (RA) Quantities that can be determined from a quick survey of the building or engineering drawings. RA-G1. Maximum ratio of column-to-floor area ratios for two adjacent stories  RA-L1. Average minimum column transverse reinforcement ratio for each story  RA-G2. Maximum ratio of horizontal dimension of the SFRS in adjacent stories  RA-L2. Minimum column aspect ratio  RA-G3. Maximum ratio of in-plane offset of SFRS from one story to the next to the in-plane dimension of the SFRS  RA-L3. Misalignment of stories in adjacent buildings  RA-G4. Plan configuration (L or T shape versus rectangular)   RA-G5. Minimum ratio of column area to wall area at each story  Engineering Calculations (EC) Quantities that require some calculation of capacities and demands based on engineering drawings, but do not require structural analysis results from computer modeling. EC-G1. Maximum ratio of story stiffness for two adjacent stories   EC-L1. Maximum ratio of plastic shear capacity (2Mp/L) to column shear strength, Vp/Vn. EC-G2. Maximum ratio of story shear strength for two adjacent stories  EC-L2. Maximum axial load ratio for columns with Vp/Vn > 0.7 EC-G3. Maximum ratio of eccentricity (distance from center of mass to center of rigidity or center of strength) to the dimension of the building perpendicular to the direction of motion. EC-L3. Maximum ratio of axial load to strength of transverse reinforcement (45 deg truss model)  EC-G4. Portion of story gravity loads supported by columns with ratio of plastic shear demand to shear capacity > 0.7  EC-L4. Maximum ratio of joint shear demand (from column bar force at yield) to joint shear capacity for exterior joints. EC-G5. Minimum ratio of column to beam strengths (? /(?)  EC-L5. Maximum gravity shear ratio on slab?column connections  Response Parameters Building Analysis (BA) Quantities for detailed collapse prevention assessment using the results from building analyses, commonly nonlinear analysis. BA-G1. Maximum degradation in base or story shear resistance  BA-L1. Maximum interstory drift ratio   BA-G2. Maximum fraction of columns at a story experiencing shear failures  BA-L2. Maximum ratio of deformation demands to ASCE/SEI 41 limits for columns, joints, slab?column connections and walls BA-G3. Maximum fraction of columns at a storey experiencing axial failures    BA-G4. Minimum strength ratio (as defined in ASCE/SEI 41)      Chapter 5: Collapse Indicators - Methodology 86 5.3. METHODOLOGY TO IDENTIFY SUITABLE COLLAPSE INDICATORS The main objective for this section is to develop a methodology to first determine appropriate design and response parameters and if possible establish limits on these parameters. This section outlines the general framework of the methodology and describes the overall process.  In this methodology nonlinear analyses, capable of modeling system-level collapse, are coupled with probabilistic methods to establish limits on the collapse indicators. Detailed analytical models using research oriented software, e.g., OpenSees, are employed to model non-ductile concrete buildings. It is our goal that these collapse indicators could be integrated into an ASCE 31- and 41-type assessment procedure, enabling practicing engineers to consider the overall system response when evaluating the Collapse Prevention (CP) performance level.  The proposed procedure for identifying the limits on the collapse indicators is divided into six general steps: Identify potential collapse indicators, Build a detailed numerical model, Perform seismic hazard calculations and select ground motion records, Execute nonlinear response history analysis (record-to-record variability), Perform probabilistic analysis, and Assessment procedure and post-processing the results. These steps are shown in Figure 5-1 and described in the following sections. 5.3.1. Step 1 ? Identify a Suite of Potential Collapse Indicators The identification of collapse indicators appropriate for engineering practice, and establishing limits on these indicators, can only be done through detailed analytical studies. However, before embarking on the analytical studies it is essential to come up with a list of potential collapse indicators from which the recommended collapse indicators will be selected. Engineering judgement and experience with collapse analyses and post-earthquake observations are used to select the list of potential collapse indicators in Table 5-1. 5.3.2. Step 2 ? Numerical Model One of the main steps in this methodology is to perform a detailed and robust nonlinear analysis on the non-ductile reinforced concrete structures. Although the methodology is not limited to moment frames, as mentioned in Chapter 1, this research focuses on these types of structures. Accurate modeling of inelastic behaviour in beams, columns, joints and slab?column connections is an essential component of collapse modeling of these structures. Non-ductile Chapter 5: Collapse Indicators - Methodology 87  Figure 5-1 Methodology for quantitatively selecting and establishing collapse indicators behaviour originating from column shear, and subsequent axial failure, in addition to slab?column shear failure plays an important role in the collapse of these structures; and the analytical model must have the ability to capture this behaviour. Therefore, the nonlinear building models should incorporate elements capable of approximately capturing the loss of vertical load carrying capacity for critical gravity-load supporting components (e.g., columns, explained in detail in Chapter 3, and slab?column connections briefly outlined in Chapter 2 and further developed in Chapter 4) and they should definitely account for P-delta effects. It should be noted that future studies may need to consider axial load failure of joints and walls, but these are judged to generally be less critical and not considered in the current study. Although developed for analysis of two-dimensional frames, this modeling concept for gravity load failure can also be extended to three-dimensional frame element models of three dimensional models of concrete buildings (e.g., Barnes, 2013). It should be noted that these models provide a relatively simple representation of a very complex phenomenon at the point of gravity load failure, and hence, may lack some Step 1 ? Potential collapse  indicators Step 2 ?Numerical modelStep 3 - Seismic hazard calculations and ground motion records selectionStep 5 - Probabilistic analysis (ground motion and model uncertainty)Step 4 ? Record-to-Record VariabilityStep 6 - Assessment procedure and post processing the resultsChapter 5: Collapse Indicators - Methodology 88 sophistication required to accurately capture the behaviour of the building to the point of total collapse. In particular, given the lack of data available for validation, modeling of three- dimensional gravity load redistribution through a slab floor system after gravity load failure of a single component should be considered approximate at best. Despite possible inaccuracies, the failure models provide good estimates of observed collapse behaviour of simple two-dimensional frames (Elwood and Moehle, 2008; Yavari et al., 2012).  Furthermore, as described later on in this chapter, the current study will primarily be interested in relative changes in collapse probability which may not require a refinement in the model sophistication.  Overall the model should encompass the following key features: ? Behaviour of non-ductile columns before and after the onsets of shear and axial failure are modeled by a mechanical model (as described in Chapter 3). The model is an extension to the limit-state material model (Elwood, 2004) and is based on the Modified Compression Field Theory (MCFT) (Vecchio, 1990) and Axial-Shear-Flexural Interaction (ASFI) (Mostafaei et al. 2009). Properties of shear and axial springs in series (Figure 3-11) are defined using the mechanical model. Although developed for analysis of two-dimensional frames, the modeling concept for gravity-load failure can also be extended to model three-dimensional frame elements. ? The column model captures post-peak behaviour (in non-ductile columns, post-peak will be corresponding to post-shear failure) which will affect the collapse response of these structures (Figure 3-12). This feature is integrated in the mechanical model and degrades the shear resistance and initiates axial load failure based on shear-friction concepts (Chapter 3). ? In order to capture the degradation in shear strength resistance and stiffness originating from possible joint damage, a rotational spring is considered at the joints (Alath and Kunnath, 1995) which will account for non-linear shear deformations of the joint and bond-slip behaviour (Figure 4-2 and explained in Chapter 4). It is noted that a recent study on non-ductile reinforced concrete frames (Yavari et al., 2012), has shown that the likelihood of collapse of such frames due to joint failure is less than that due to column failure. ? In slab?column frames, zero-length torsional elements connect the slabs to the columns. The torsional elements use a limit-state material proposed by Kang et al (2009) to detect Chapter 5: Collapse Indicators - Methodology 89 and trigger punching failure based on the drift ratio demand and the gravity shear ratio. The torsional element limits moment transfer between slab and column after punching is detected. In addition to the moment, the axial interaction between the slab and the column should also be included in the numerical model and after punching failure the axial response should also degrade (Chapter 4). ? To account for the degradation of strength and stiffness associated with large deformations, the analysis utilizes suitable geometric transformations to take the P-delta effects into account. ? In order to capture the inherent randomness in the concrete and reinforcement characteristics (e.g., strength, placement of reinforcement, member sizes, etc.) throughout the structure, a random number generator is implemented in the OpenSees source code. This randomness is different from uncertainty in the limit state failure surface considered in section 5.3.5 later in this chapter. The random number generator generates normally distributed random numbers with a unit mean and 0.05 standard deviation. The random number is used to modify the concrete and reinforcement strength in the column shear failure model (explained in Chapter 3). In numerical simulation, there is always the challenge of defining the collapse state during the analysis and differentiating structural collapse from numerical non-convergence. In collapse analysis used in this methodology, the criterion is based on two types of global failure, gravity load collapse and side-sway collapse (explained in detail in Chapter 4). Side-sway collapse is based on lateral demand and capacity and there is an assumption that the structure is ductile enough to reach this state. However, because of the limited ductility of older reinforced concrete structures, the collapse state is also defined with different criteria, entitled here as gravity load collapse. Gravity load collapse may be precipitated by axial load failure of columns, punching shear failure of slab?column connections, failure of slab-diaphragm connections, or axial load failure of beam-column joints. Gravity load collapse is defined in this study by the point at which the axial load capacity at a single story drops below the gravity load demand (explained in detail in section 4.3.1). Both definitions of collapse will be adopted in the current study. It should be noted that the selection of the collapse criteria can significantly alter the final results, as will be discussed in more detail in Chapter 7. Chapter 5: Collapse Indicators - Methodology 90 5.3.3. Step 3 - Seismic Hazard Calculations and Ground Motion Records Selection There are many methods for ground motion selection and modification for nonlinear analysis of structures. Research by the Pacific Earthquake Engineering Research (PEER) Center?s Ground Motion Selection and Modification (GMSM) Program (Haselton et al., 2009) has identified 40 different methods. The two methods of interest in this study are: selection and scaling using Uniform Hazard Spectrum (UHS); and selection and scaling using Conditional Mean Spectrum (CMS). The commonly used Uniform Hazard Spectrum (UHS) as a target response spectrum is computed using probabilistic seismic hazard analysis (PSHA) (Bazzurro and Cornell, 1994; Bazzurro and Cornell, 1999) and is the envelope of spectral amplitudes at all periods which exceed a specific probability (e.g., 2%) in a specific time frame (e.g., 50 years). However, individual ground motion spectra will most likely have large amplitudes at limited number of periods and are unlikely to be equally above-average at all periods. Given that the Uniform Hazard Spectrum is not representative of individual ground motion spectra, it will be an unsatisfactory ground motion selection target due to its highly conservative nature (Baker, 2011). Baker and Cornell (2006) have proposed an alternative target spectrum named ?Conditional Mean Spectrum? (CMS) which provides the mean response spectrum conditioned on occurrence of a target spectral acceleration value at the period of interest, Sa(T*), using knowledge of the magnitude (M), distance (R) and ? value that caused occurrence of that target Sa. The parameter ? represents the number of standard deviations that the spectral acceleration is above its median value (estimated based on an attenuation model) for a given M and R (Haselton et al., 2009). It has been demonstrated that ground motions selected and scaled to match the Conditional Mean Spectrum (over the range of 0.2T1 ? 1.5T1) generate displacements in buildings comparable to deformations generated by unscaled ground motions and therefore it will likely not impact the resulting structural responses (Baker, 2005). In addition to this, because the CMS?s effect is more pronounced for rare ground motions, it is important to consider this spectrum when predicting the safety of buildings against collapse (Baker, 2011), which is the main performance level of interest in this research. A simple step-by-step procedure to compute the CMS is explained in detail in Baker (2011) and summarized in Figure 5-2. The entire CMS procedure starts from a design Sa value at the specified period T*, and the remaining spectrum is computed conditioned on that Sa(T*). Chapter 5: Collapse Indicators - Methodology 91 Typically, probabilistic performance-based assessments choose T* as the first-mode period of the structure for predicting peak displacements of first-mode dominated structures (Bazzurro and Cornell, 1994). If the target Sa(T*) is obtained from PSHA, then the M, R and ?(T*) values can be taken as the mean M, R and ?(T*) from deaggregation. Next, the mean and standard deviation of log spectral acceleration values at all periods, for the target M and R are computed using an attenuation relationship. Following this, the conditional mean ? is computed for many periods. Finally, the CMS is computed using the mean and standard deviation from Step 2 and the conditional mean ? values from Step 3. The whole procedure is described for an example building in Appendix A. The CMS for a site located in southern California is plotted in Figure 5-3. The UHS, in addition to the predicted median spectrum computed using Abrahamson and Silva 1997 attenuation model (Abrahamson and Silva, 1997), and the scaled predicted mean spectrum are illustrated in this figure for comparison. As seen in this figure, the CMS coincides with the UHS at the given period and is close to the UHS near this period (for the CMS the acceleration spectra values are highly correlated at the vicinity of the given period) but decays from the UHS for periods further apart. The CMS for different hazard levels (i.e. different return periods) is shown with the corresponding UHS, for comparison, in Figure 5-4. As seen in this figure, the ground motion probability level of interest affects the impact of the CMS, i.e., for higher return periods, such as 1% in 50 years, there is the largest gap between the UHS curve and the CMS curve. Once the CMS is computed, it can be used as the target spectrum to select and scale ground motions for use in nonlinear analysis. The selection and scaling will be for the given period range of interest. Different approaches are suggested for individual or group ground motion scaling which each have their own advantages. In this study, the approach is to individually scale each ground motion so that the average response spectrum over the periods of interest (0.2T1 to 1.5T1) is equal to the average of the target spectrum over the same periods (Baker, 2011). In this case, a given ground motion?s scale factor is,  	 =  .. ..        (5-1) where SaCMS (T) are spectra accelerations over the period range of 0.2T1 and 1.5T1, defined from the CMS, and Sa(T) are spectra accelerations for each ground motion and over the same period  Chapter 5: Collapse Indicators - Methodology 92   range. After scaling all ground motions available in the predefined set, the undesirable ground motions which do not provide a good match to the CMS are then removed.  If one is interested in multiple structural response parameters (where higher modes are important as well), it is suggested to construct conditional mean spectra conditioned on Sa values at multiple periods and to select ground motions for each CMS (Baker and Cornell, 2006). However, this approach is not considered in this study and it is recommended by Hamburger and Moehle (2010) to use the UHS curve as the target spectrum for cases where multiple conditional mean spectra are intended to be used.  Figure 5-3 Condition mean values of spectral acceleration at all periods, given Sa (T = 1 sec)  Step 1: Determine the target Sa at a given period, and the associated M, R and ? Step 2: Compute the mean and standard deviation of the response spectrum, given M and R Step 3: Compute ? at other periods, given ?(T*) Step 4: Compute Conditional Mean Spectrum 00.20.40.60.811.21.41.61.80 1 2 3 4 5Spectral Acceleration, Sa (g) for ?? ?? = 5%Period, T (sec)UHS for RT = 2475 [yrs]Median + 2*sigma spectrum, M=6.96, R=18.3 kmCMS for RT = 2475 [yrs]Predicted median spectrum, M=6.96, R=18.3 kmFigure 5-2 Steps to compute the Condition Mean Spectrum (Baker, 2011) Chapter 5: Collapse Indicators - Methodology 93  Figure 5-4 CMS vs. UHS for a specific site conditioned for Sa (T = 1 sec) 5.3.4. Step 4 ? Record-to-Record Variability   Multiple stripe analysis (MSA) and incremental dynamic analysis (IDA) are two methods suitable for making probabilistic assessments over a wide range of hazard levels (Jalayer and Cornell, 2009). IDA (Vamvatsikos and Cornell, 2002) consists of obtaining a suite of response curves for a suite of ground motion records. The IDA curve demonstrates the changes in a specific structural response parameter (e.g., maximum interstory drift ratio) when the structure is subjected to a particular ground motion record scaled to successively increasing levels of intensity. A stripe analysis consists of structural analyses for a suite of ground motion records that are scaled to a common intensity measure (e.g., scaled to have a specific return period or spectral acceleration at a selected period). MSA refers to a group of ?stripe? analyses performed at multiple intensity measure levels. The results of a MSA approach could be used to calculate the collapse fragility curve. The calculations to build the collapse fragility curve are based on the number of collapse and non-collapse cases, i.e., ?counted? statistics. In order to obtain the 00.20.40.60.811.21.41.61.820 0.5 1 1.5 2 2.5Spectral Acceleration, Sa (g) for ?? ??= 5%Period, T (sec)UHSCMS1% in 50 yrs2% in 50 yrs20% in 50 yrs5% in 50 yrs10% in 50 yrs50% in 50 yrsChapter 5: Collapse Indicators - Methodology 94 ?counted? statistical parameters of a data set, the collapse data are first sorted and then divided by the total number of cases.  Figure 5-5 illustrates the MSA results for a suite of ground motion records applied to a 7-story concrete moment frame (refer to Chapter 6 for full description of the building). The ?collapse? cases (full circles) are distinguished from the ?non-collapse? cases, where the state of ?collapse? for a ground motion record is detected using the collapse criteria explained in Chapter 4, i.e. gravity load collapse or side-sway collapse. Figure 5-6 illustrates the IDA results for the same suite of ground motion records used to perform the MSA (for RT = 2475 yrs) and applied to the same 7-story building. Figure 5-6.a shows typical IDA curves where the spectral acceleration at the first mode period, Sa(T1), is depicted versus the maximum interstory drift ratio reached during the nonlinear time history analysis. In Figure 5-6.b, the vertical axis is changed from spectral acceleration to return period in order to be comparable with Figure 5-5. By comparing these two figures, it could be concluded that the IDA covers a wider range of return periods but on the other hand it lacks the ability to demonstrate the variability of drift demands for different suite of ground motions at diverse return periods. Jalayer and Cornell (2009) indicate MSA are suitable for making  Figure 5-5 MSA for record-to-record (RTR) uncertainty 40040000.005 0.05IM [Return Period (yrs)]Maximum Interstory Drift Ratio (IDR)collapse casesChapter 5: Collapse Indicators - Methodology 95 probability-based assessments for a (wide) range of spectral acceleration values and IDA should be used to locate the onset of global dynamic instability (collapse capacity) in the structure. The IDA results would approximately provide the same estimates for structural demand responses as those presented for MSA. However, there are a few major differences between these two methods. Unlike the MSA method which is analyzed using different sets of records at different intensities, the IDA results pertain to a fixed set of ground motion records. In addition to this, IDA could introduce a conservative bias into the collapse capacity estimate, due to the effects generated by increasing the spectral accelerations (Haselton, 2007). Finally, as described in the next section, including the model variability (in the process of sampling explained in the next section) is much more straightforward in the MSA compared to the IDA. Due to these reasons, MSA is chosen as the method employed to perform nonlinear response history analysis in this study. However, it should be noted that the IDA has the advantage of simplicity and generality for multiple sites because it pertains to a fixed set of ground motion records, selected for one return period, and scaled afterwards.   (a) IDA curves for record-to-record (RTR) uncertainty for Sa (T1) 00.20.40.60.811.21.41.60 0.01 0.02 0.03 0.04 0.05 0.06Spectral Acceleration, Sa (T1) [g]Maximum Interstory Drift Ratio, MIDR [%]Chapter 5: Collapse Indicators - Methodology 96  (b) IDA curves for record-to-record (RTR) uncertainty for 2475 years return period Figure 5-6 IDA for record-to-record (RTR) uncertainty As a final note, fragility curves of the two methods, MSA and IDA, are shown in Figure 5-7 for comparison. As seen in this figure, the overall collapse capacity of the example building is different computed using the two methods. The probability of collapse changes from 40% to 80% at Sa(T1) = 1 [g] using the MSA and IDA approaches, respectively. The reason behind this difference is from the ground motion characteristics used to perform MSA and IDA approaches. FEMA P695 (2009) addresses this difference and recommends modification of the collapse capacity by a spectral shape factor (SSF) when using IDA to compile collapse fragilities. SSF is used to adjust the median of the collapse fragility and is based on the mean ? values of the ground motion sets used to perform IDA. SSF always has a value greater than one and applying it will decrease the probability of collapse (shift the collapse fragility to the right). This multiplier is only required when using a unique set of ground motions for each hazard level and therefore will not be used when using the MSA approach. It should be noted that the fragility curve obtained from the IDA results shown in Figure 5-7 has not been adjusted with the SSF.  2020020000.003 0.03IM [Return Period (yrs)]Maximum Interstory Drift Ratio (IDR)collapse casesChapter 5: Collapse Indicators - Methodology 97  Figure 5-7 Fragility curve for MSA and IDA 5.3.5. Step 5 - Probabilistic Analysis (Ground Motion and Model Uncertainty) Probabilistic analyses are utilized in order to determine the probability of collapse of a prescribed performance objective considering a limit-state function, uncertainties associated with each random variable, and using a suitable method to perform the analysis. The probabilistic method could be coupled with the numerical analysis to evaluate the probability of experiencing a selected structural performance state, such as collapse and the result is presented by collapse fragility curves. In this methodology, all essential uncertainties are explicitly accounted for in the probabilistic analysis.  Uncertainties are inherent in material properties, geometry, loading and modeling of the structure. Different reliability methods could be applied to perform this task, but due to technical difficulties arising when applying the first- and second-order (FORM and SORM) reliability methods to nonlinear softening models, the sampling method is considered a more appropriate approach in this methodology (Koduru et al., 2007). Using the distributions defined for each random variable (as described below), realizations of each random variable are generated and inputted into the numerical model. This process is repeated for several sets of realizations (explained in section 4.5) and the result for each simulation is used to obtain the collapse fragility 0%10%20%30%40%50%60%70%80%90%100%0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2P [Collapse|Sa(T1)]Spectral Acceleration, Sa (T1) [g]MSAIDAChapter 5: Collapse Indicators - Methodology 98 defined through a cumulative distribution function (CDF). These CDF curves relate the predefined intensity measure (e.g., spectral acceleration or hazard return period) to the probability of collapse, also known as collapse fragility curves. Performance-based earthquake engineering enables probabilistic prediction of structural response, incorporating key sources of uncertainty in the process. By using a suite of earthquake records, nonlinear dynamic analysis (via incremental dynamic analysis, IDA, and/or multiple stripe analysis, MSA) directly incorporates information about variability in ground motions in the collapse performance assessment. However, nonlinear dynamic analysis alone does not account for how well the nonlinear simulation model represents the collapse performance of the building; hence, model uncertainties should also be accounted for the duration of the collapse simulation. These modeling uncertainties are especially important in predicting collapse, because of the high degree of uncertainty in predicting deformation capacity and other critical parameters for modeling collapse. In this methodology the uncertainty for each random variable is explicitly considered in the analysis and reflected in the final probabilities of collapse. The random variables selected with the respective probability distribution should have the capability of capturing the major uncertainties inherent in non-ductile reinforced concrete frames. Uncertainty in the shear and axial failure models for non-ductile columns and punching shear failure for the slab?column connections are considered the main sources of modeling uncertainty in this study. In addition to this, record to record variability is considered in the process (see step 4).  Different sources of uncertainty are categorized as either aleatory or epistemic uncertainty. 1. Aleatory (randomness) The two source of aleatory uncertainty considered throughout this work are ground motion record-to-record variability and the randomness in the material properties and section layout. A random number generator is implemented in the process to account for the latter variability.   2. Epistemic (lack of knowledge) Modeling uncertainties incorporated in the numerical models are considered epistemic uncertainty. The distribution associated with epistemic uncertainty in this case may be obtained from test data or expert judgment. The assessment of modeling uncertainties focuses on uncertainties in the modeling parameters that define the limit curves used to Chapter 5: Collapse Indicators - Methodology 99 detect/predict the potential shear and axial failures in the non-ductile columns and punching shear failure in the slab?column connections. These parameters are assumed to be lognormally distributed, where the mean and standard deviation are obtained from previous research (Zhu et al., 2007; Kang, 2004). Past research has indicated that modeling uncertainties associated with damping, mass, and material strengths have a relatively small effect on the overall uncertainty in seismic performance predictions, and will primarily have an influence on pre-collapse performance of structures (Lee and Mosalam 2005). Ibarra and Krawinkler (2003) have shown that modeling variables related to component strength and deformation capacity are the dominant model parameters affecting collapse assessment. Simultaneous consideration of the effects of aleatory and epistemic uncertainties on the collapse capacity necessitates performing Monte Carlo simulation. Section 4.5 explains the approach used to incorporate the model uncertainty with the record-record variability.  The outcome of the collapse assessment procedure when applied to a particular structure is a probability distribution representing the cumulative probability of collapse, P[Collapse] = P[IMcollapse<IM] as a function of a hazard intensity measure, e.g., spectral acceleration or return period, as illustrated in Figure 5-8. This collapse fragility curve is obtained by creating nonlinear analysis models of non-ductile RC frames (explained in detail for sample structures in the following sections) and conducting nonlinear dynamic analysis following IDA or MSA procedures as described in Step 4. As shown, the collapse fragility curve can be idealized by a lognormal distribution, which is defined by a median value and dispersion (logarithmic standard deviation, ?ln).  The main source in structural modeling uncertainty which will have the highest effect on collapse behaviour is the variability inherent in shear and axial failure models. As shown in Figure 4-17, the shear and axial failure models could be represented with probabilistic distributions (e.g., lognormal distribution) which will cover the variability in these non-ductile failures.  Monte Carlo sampling with a response surface (Liel et al., 2009) is used in this study to incorporate modeling uncertainties on the collapse capacity. The response surface is a multivariate function representing the relationship between the modeling uncertainties and the Chapter 5: Collapse Indicators - Methodology 100 collapse capacity. As explained in Liel et al. (2009), sensitivity analysis is first executed to build the response surface, which represents the median collapse capacity as a function of model random variables. The response surface is approximated with a second-order polynomial functional form. Then, the Monte Carlo procedure is used to obtain a suite of sample realizations for the set of random variables under consideration. For each set of realizations, finally the median collapse capacity of the structure is computed from the response surface. The advantage of this method is that it avoids time consuming nonlinear simulations. The outcome of this process is collapse fragilities that incorporate both the aleatory and epistemic uncertainty. The reader is referred to Liel et al. (2009) for more details on the procedure. Table 5-2 summarizes the numerical parameters for which uncertainties are considered in the nonlinear analysis. In Figure 5-8, two collapse fragility plots are shown, the first of which includes only record-to-record (RTR) variability (aleatory uncertainty), which is obtained by the nonlinear response history analysis. The other collapse fragility curve is also computed from the nonlinear analysis but structural modeling uncertainty (epistemic uncertainty) is also accounted for and because of the addition of uncertainty, it has resulted in an increase in the dispersion. It should be noted that one could reduce the analysis effort by ignoring the model variability and only accounting for record-to-record variability.  Figure 5-8 Collapse fragility curve for record-to-record (RTR) uncertainty and record-to-record including model (RTR + Model) uncertainty  0%10%20%30%40%50%60%70%80%90%100%0 1000 2000 3000 4000 5000 6000P [Collapse]IM [Return Period (yrs)]RTRRTR + ModelChapter 5: Collapse Indicators - Methodology 101 Table 5-2 Uncertainty modeling of the random variables  Random Variables Material Non-Ductile Column Model Shear Failure Model Lognormal distribution with mean value corresponding to the limit-state material failure models and appropriate standard deviation (0.3 suggested by Zhu et al., 2007) Non-Ductile Column Model Axial Failure Model Lognormal distribution with mean value corresponding to the limit-state material failure models and appropriate standard deviation (0.3 suggested by Zhu et al., 2007) Slab-Column Connection Punching Shear Failure Lognormal distribution with mean value corresponding to the punching shear failure models and appropriate standard deviation(0.25 suggested by Kang, 2004) Load Record-to-record  Sufficient number of GMs (20 records were used to limit the number of analysis cases) 5.3.6. Step 6 - Assessment Procedure and Post ? Processing Results This section summarizes the proposed methodology for the selection of the collapse indicators and appropriate corresponding limits. The methodology slightly differs when obtaining limits for the design and response parameters shown in Table 5-1. Both cases are explained in the following sections with reference to two buildings: Building A and Building B. These buildings are only for illustrative purposes and details of these buildings are not important for the following descriptions. Example buildings are described in detail in Chapter 6 where an application of the methodology is presented. Assessment procedure for Design Parameters The goal of the response-history analysis will generally be for performance assessment. As explained in the introduction, collapse is the performance level of interest. In this section four different alternative approaches are identified to assess the correlation between design parameters and collapse performance. The alternatives used in this section are listed below: Chapter 5: Collapse Indicators - Methodology 102 Performance Assessment Risk-based 1) use limit on mean annual risk of collapse (the target 1% in 50 year collapse probability used in ASCE standard 7-10  (ASCE, 2010)  is an example of this approach); 2) use a ?good? existing building to define the acceptable risk of collapse; Intensity-based 3) use a specific probability of collapse for a particular hazard level (for example 10% probability of collapse for a RT = 2475 yrs); and 4) look at relative changes in the collapse fragilities instead of absolute values. In all these approaches, the first step will be to build the complete nonlinear model of a building and evaluate the building collapse fragility. The nonlinear model and the nonlinear analysis should comply with the criteria suggested in the previous sections. Two- and three-dimensional building models could be used. Although three-dimensional models provide a more reliable representation of the progressive collapse behaviour as gravity load is redistributed within a building frame, only two-dimensional models have been tested and validated, therefore, these models are only recommended for this study. Then a selected collapse indicator parameter will be altered (for a couple of values) in the building model (e.g., average minimum column transverse reinforcement ratio) and the collapse fragility will be reassessed for each realization of the collapse indicator (Figure 5-9.a). Figure 5-9.a shows the variation of the collapse fragilities for Building A with the selected collapse indicator. Collapse probability increases as the collapse indicator varies. The same figure can be represented by grouping the collapse fragility into different bins of hazard intensities as shown in Figure 5-9.b. In the first approach, the collapse fragilities and the hazard curve represented as the annual frequency of exceedance (Figure 5-9.c) are used to compute the mean annual frequency of collapse (?collapse) for each realization of the collapse indicator (Figure 5-9.d) using Eqn. (5.2).  	 =  	| |()|       (5-2) where CI is the collapse indicator and im represents the intensity measure. This assessment would be repeated for several different building types (Figure 5-9.e). Limits on the collapse indicators will be selected based on suitable mean annual frequency of collapse (?collapse) (Figure 5-9.e). The mean annual frequency of collapse is determined based Chapter 5: Collapse Indicators - Methodology 103  (a) Develop collapse fragilities for a range of the selected collapse indicator   (b) Develop collapse fragilities for a range of selected return periods (e.g., 475 yrs, ...) 0%10%20%30%40%50%60%70%80%90%100%0 0.2 0.4 0.6 0.8 1 1.2 1.4P [Collapse]IM [Sa (T1 )/g]Original buildingChange in Collapse Indicator0%10%20%30%40%50%60%70%80%90%100%0P col[CI>CI max|IM (Return Periods)]Collapse IndicatorChapter 5: Collapse Indicators - Methodology 104  (c) Hazard curve required to estimate the mean annual rate of collapse (?collapse)   (d) Estimate ?collapse integrating the collapse fragility curves with the hazard curve and seek trends in ?collapse for changes in collapse indicator  0.00020.0020.020.2 2Annual frequency of exceedance IM [Sa (T1=2 s)/g]RT = 2475 [yrs]RT = 475 [yrs]0.00E005.00E-051.00E-041.50E-042.00E-042.50E-043.00E-043.50E-044.00E-040?? ?? collapse(CI>CI max)Collapse Indicator Original BuildingChapter 5: Collapse Indicators - Methodology 105  (e) Repeat for ?several? building prototypes and choose an appropriate risk and determine the range for the collapse indicator Figure 5-9 Approach 1 for establishing collapse indicator limits on engineering and code related judgements. This measure is consistent with the new concept implemented in ASCE standard 7-10 (ASCE, 2010) entitled ?Probabilistic Risk-Targeted Ground Motions? (Luco et al. 2007). Such newly implemented measures would suggest that for a mean annual of frequency of 0.0002 (which corresponds to a probability of collapse of 1% in 50 years) the collapse indicators not be below specific values (where the P[collapse in 50 yrs]=1% crosses the curves for Building A and B in Figure 5-9.e). An ideal collapse indicator appropriate for application in engineering practice would have only limited variation in the collapse indicator limit suggested by the different building types. Results in Figure 5-9.e suggest a moderate difference in the value for the collapse indicator for the two buildings considered. More discussion regarding this issue is presented in Chapter 6. The second approach, illustrated in Figure 5-10 will be similar to the first approach with difference only in the last step, the criteria for suitable mean annual frequency of collapse. In this approach, the collapse probabilities for the prototype buildings will be compared with the collapse probability of a ?good? existing building, which is generally agreed that seismic rehabilitation is not required for a collapse prevention performance level. Prototype buildings would be evaluated considering a range of collapse indicator values and cases where the  0.00E+001.00E-042.00E-043.00E-044.00E-045.00E-046.00E-040?? ?? collapseCollapse IndicatorBuilding BBuilding AP[collapse in 50yrs] = 1%Chapter 5: Collapse Indicators - Methodology 106  Figure 5-10 Approach 2 for establishing collapse indicator limits (the ?good? existing building dictates the acceptable risk and the range for the collapse indicator is determined based on this risk) probability of collapse exceeded that of the ?good? existing building would indicate appropriate limits for the collapse indicator being investigated (Figure 5-10). The third approach, illustrated in Figure 5-11, establishes the limit on the collapse indicator based on a suitable probability of collapse at the intensity associated with the maximum considered earthquake, i.e., RT = 2475 years (for example, 10% as selected for the target probability of collapse in ASCE 7-10 (ASCE, 2010)). This assessment would be repeated for several different building prototypes and using the selected probability of collapse, one could indicate appropriate limits for the collapse indicator being investigated (Figure 5-11). The fourth approach, illustrated in Figure 5-12, will be also similar to the third approach with difference only in the last step, where the limits on the collapse indicators will be selected based on the relative changes in the collapse fragilities with changes in the collapse indicator parameter. Figure 5-12 shows example collapse fragilities (conjectured) for changes in a selected collapse indicator where the collapse indicator is varied for each subsequent collapse fragility.  00.0010.0020.0030.0040.0050.0060 1 2 3 4 5 6 7?? ?? collapse(CI>CI max)Collapse indicator Building ABuilding BBuilding CBuilding D"good" existing buildingChapter 5: Collapse Indicators - Methodology 107  Figure 5-11 Approach 3 for establishing collapse indicator limits Table 5-3 summarizes the four different approaches used to estimate appropriate limits on the collapse indicators. The advantages and disadvantages related to each approach are also reviewed in this table. It should be noted that the different approaches will be applied to sample buildings in Chapter 6 and the results will be discussed in that chapter.       Figure 5-12 Approach 4 for establishing collapse indicator limits (conjectured collapse fragilities for different collapse indicator values) 0%10%20%30%40%50%60%70%80%90%100%0 0.5 1 1.5 2 2.5 3 3.5 4P col[CI>CImax|IM(RT= 2475 yrs)]Collapse IndicatorBuilding ABuilding B0%10%20%30%40%50%60%70%80%90%100%0 1 2 3 4 5 6 7P [Collapse]Collapse indicator Building A0%10%20%30%40%50%60%70%80%90%100%0 0.2 0.4 0.6 0.8 1 1.2 1.4P [Collapse]IM [Sa (T1 )/g]IMmce CIlimitChapter 5: Collapse Indicators - Methodology 108 Table 5-3 Comparison of the different approaches used to indicate limits on the collapse indicators Approach Limit the collapse indicator based on Advantages Disadvantages 1 mean annual risk of collapse - All hazard levels are considered in the risk estimation - The approach is consistent with the current design practice based on uniform risk -The risk indicator (mean annual collapse of risk) results in very small values (1xe-5) and could be a great source for precision errors - The results will be site-dependant and will be difficult to generalize across many regions - Errors in the hazard curve (e.g., unknown earthquake sources) could result in inappropriately low ? 2 the ?good? existing building dictates the acceptable risk The risk of collapse is verified based on an existing building Determining a ?good? existing building could be subjective and not easily implemented in the procedure 3 10% probability of collapse at RT = 2475 yrs The procedure could be easily implemented in the current seismic codes Only one hazard level is used to indicate the risk of collapse and the results could significantly change in other hazard levels 4 on the relative changes in the collapse fragilities with changes in the collapse indicator parameter The results are based on relative changes and therefore results will be less sensitive to exact values of probabilities of collapse There is no guarantee that there will be a ?kink? in the relative changes in the collapse fragilities  It should be noted that these approaches are extremely computationally demanding. The number of nonlinear analysis required to build the collapse fragilities for each building will be approximately 3000. The best solution for this high demand is to use High Performance Computing (HPC) and taking advantage of cloud computing services. In this study, the cloud services provided by NEEShub (http://www.nees.org) were used to overcome this problem. More information on this issue is presented in Appendix C.   Chapter 5: Collapse Indicators - Methodology 109 Assessment procedure for Response Parameters The methodology for identifying response parameters is based on results of comprehensive collapse simulations. As explained before, this section presents a method to develop structural response fragility curves. The fragility curves are based on nonlinear dynamic analysis of a set of structural buildings. First, different methods to compile fragility functions is presented and next, similar to the final result for the design parameters, a procedure is explained to identify appropriate limits for the response parameters. Fragility Functions Formulation The variability in building responses can be accounted for using cumulative distribution functions (CDFs) to approximate the probability of each building response (e.g., maximum degradation in base or story shear resistance, maximum drift ratio) occurring. For each building response (BR) and intensity measure (in this study the return period, RT, is used to represent the intensity measure), cumulative distribution functions are developed based on the nonlinear dynamic analysis. The objective is to develop a fragility curve (i.e., CDF) to demonstrate the probability of collapse for a given BR and RT, P(Collapse| BR, RT). The probability of exceeding the collapse state conditioned on a particular building response and return period is modeled using a lognormal probability distribution, given by the following Eqn.: Collapse|, = ?       (5-3) Where P(Collapse| BR, RT) is the probability of exceeding the collapse state, BR is the median of the BRs at which the probability of collapse is observed, ? is the standard deviation of the natural logarithm of the BRs, and ? is the cumulative standard normal distribution (Gaussian distribution).  As suggested by Ramirez (2009), three different methods could be used to determine the statistical parameters of the lognormal distribution for the fragility functions, the least square methods, the maximum likelihood method, and the second method (?Method B?) for bounding BRs. Each of these methods is described briefly below. Least Squares Method The least squares method (Shinozuka et al, 2000), is a statistical approach that fits observed data to the values produced by a predicting function. This is accomplished by minimizing the sum of Chapter 5: Collapse Indicators - Methodology 110 the square of the differences between the observed data, g(BR), and the values predicted by the proposed function, F(BR). Mathematically, this can be expressed as: gBR, ? , BR = min? FBR? P(Collapse|BR)       (5-4) where N is the number of data points, BRi, is the peak BR observed for data point i, and P(Collapse|BRi) indicates whether damage state j has been exceeded by taking on a binary value of 1 when collapse has occurred and 0 when collapse has not occurred. The parameters BR and ?are varied until the sum of the distances FBR ? PCollapse|BR is minimized. Maximum Likelihood Method In the method of maximum likelihood (Shinozuka et al, 2000), the parameters of the lognormal distribution are estimated by the means of the following procedure. Shinozuka et al. (2000) presents the likelihood function as: , ? = ?  	!1 ?  "	        (5-5) where F(BRi) represents the fragility curve for probability of collapse and is the lognormal CDF as defined in Eqn. (1) defined by the parameters BR and ?; Collapse = 1 or 0 depending on whether or not the structure sustains collapse or not; and N = total number of nonlinear responses.   ?Method B? for bounding EDPs (BRs)  The bounding BRs method (Porter, 2000) determines the probability of a damage state occurring from observed data, by dividing the data set into discrete bins based on equal increments of BRs. For each subset of data in every bin, the probability of collapse occurring is calculated in each bin, according to: # ? $	| =           (5-6) where m is number of data points that experienced this level of damage and M is the total number of data points within the bin being considered. These probabilities are then plotted at the midpoints of the BR ranges in each bin. A lognormal distribution is then fitted to these points by varying the parameters BR and ?.   Chapter 5: Collapse Indicators - Methodology 111 Comparison of the different methods The result for each method is shown and compared in Figure 5-13. Ramirez (2009) states the main limitations in the methods which are also valid for the procedure presented in this chapter. These limitations are summarized in the following two points: ? When the range of BRs for the buildings that experienced collapse did not overlap with the range of BRs for the buildings that did not experience collapse, the least squares will not be able to find unique solutions for the median (BR) and standard deviation (?). ? ?Method B? can produce multiple solutions for estimated fragility function parameters. The plotting positions of the points that are used to fit the lognormal distribution are highly dependent on the number of data points and the distribution of those points along the range of BRs. Overall, based on the above limitations for the least squares method and Method B, the maximum likelihood method is chosen to compile the fragility functions in this research.  Figure 5-13 Fragility curve of the maximum interstory drift ratio obtained by three different methods  00.10.20.30.40.50.60.70.80.910 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08P [Collapse| ?? ??< IDR , RT = 2475 yrs]Maximum Interstory Drift (IDR)Collapse DataNon-Collapse DataFitted Curve - Least SquaresFitted Curve - Maximum LikelihoodFitted Curve - Method BChapter 5: Collapse Indicators - Methodology 112 Establish Response Parameter Collapse Indicator Limits Similar to the previous section for design parameters, different approaches are identified to assess the correlation between the response parameters and collapse risk. The alternatives used in this section are listed below: Performance Assessment Risk-based 1) use limit on mean annual risk of collapse (the target 1% in 50 year collapse probability used in ASCE standard 7-10  (ASCE, 2010)  is an example of this approach) Intensity-based 2) use a specific probability of collapse for a particular hazard level (for example 10% probability of collapse for a RT = 2475 yrs) The first alternative will be to use the mean annual frequency of collapse (?collapse), which combines the seismic hazard at the site and the probability of collapse for all intensities. For this approach, the first step is to compile the fragility functions demonstrating the probability of collapse for a given response parameter (e.g., maximum interstory drift) and for a specific hazard level (e.g., hazard level with a return period of 975 years). These curves are compiled using the maximum likelihood function (Figure 5-14.a). The process is repeated for a range of hazard intensity levels (e.g., RT = 475, 975, 2475 ... years) shown in Figure 5-14.b. The final product will be the mean annual probability of collapse as a function of the response parameter collapse indicator. However, the procedure used to derive the mean annual frequency of collapse for response parameters is slightly different with the method applied for design parameters. The difference between the response and design parameters is in the fragility functions built for the collapse indicators which for design parameters the cumulative distribution function (CDF) is a function of an appropriate intensity measure (e.g., return period, spectral acceleration), however, for response parameters the fragility curve is a function of the response parameter collapse indicator (Figure 5-14.b). The collapse fragilities developed for the response parameters for each intensity measure is multiplied with the corresponding annual frequency of exceedance represented by the hazard curve (Figure 5-14.c), the mean annual frequency of collapse (?collapse) is computed for each realization of the collapse indicator (Figure 5-14.d). Chapter 5: Collapse Indicators - Methodology 113 	 = ? 	| |()|       (5-7) This assessment would be repeated for several different building types (Figure 5-14.e). Limits on the collapse indicators will be selected based on suitable mean annual frequency of collapse (?collapse) (Figure 5-14.f).  (a) Use a method to fit the fragility curve  (b) Develop collapse fragilities for range of selected return periods (e.g., 475 yrs, 2475 yrs ...) 0.0 0.2 0.4 0.6 0.8 1.0 0 0.02 0.04 0.06 0.08 0.1P col[ ?? ??< IDR |IM (RT = 975 yrs)]Maximum Interstory Drift Ratio (IDR)Collapse DataNon-Collapse DataObserved dataFitted Fragility Function00.20.40.60.810 0.02 0.04 0.06 0.08 0.1P col[ ?? ??< IDR |IM (Return Periods)]Maximum Interstory Drift Ratio (IDR)2475 yrs975 yrs475 yrs224 yrsChapter 5: Collapse Indicators - Methodology 114  (c) Estimate mean annual of collapse (?collapse) for range of selected collapse indicator   (d) Seek trends in ?collapse for changes in collapse indicator 0.00020.0020.020.2 2Annual frequency of exceedance Spectral Acceleration, Sa (T1) [g]RT = 2475 [yrs]RT = 475 [yrs]00.00050.0010.00150.0020.00250.0030.00350.0040 0.02 0.04 0.06 0.08 0.1?? ?? collapse (IDR< idr)Collapse Indicator (e.g. IDR)Chapter 5: Collapse Indicators - Methodology 115  (e) Repeat for ?several? building prototypes   (f) Choose an ?appropriate risk? and determine the range for the collapse indicator Figure 5-14 Approach 1 for establishing collapse indicator limits  00.00050.0010.00150.0020.00250.0030.00350.0040 0.02 0.04 0.06 0.08 0.1?? ?? collapse (IDR< idr)Collapse Indicator (e.g. IDR)Building ABuilding B00.00050.0010.00150.0020.00250.0030.00350.0040 0.02 0.04 0.06 0.08 0.1?? ?? collapse (IDR< idr)Collapse Indicator (e.g. IDR)Building ABuilding BChapter 5: Collapse Indicators - Methodology 116 The second approach, illustrated in Figure 5-15, establishes the limit on the response parameter collapse indicator based on a suitable probability of collapse (e.g., 10%) at the intensity associated with the maximum considered earthquake (RT = 2475 years). This assessment would be repeated for several different building prototypes and using the suitable probability of collapse would indicate appropriate limits for the collapse indicator being investigated (Figure 5-15). It should be noted that the appropriate probability of collapse and intensity level of interest could both vary based on the tolerable risk level considered by society. The methodology presented in this study is independent from the values set for the probability of collapse and intensity level and the 10% probability and 2475 years return period are only used to illustrate the process and can easily be updated with any other values.   Figure 5-15 Approach 2 for establishing collapse indicator limits   00.20.40.60.810 0.02 0.04 0.06 0.08 0.1P col[ ?? ??< IDR|IM(RT= 2475 yrs)]Collapse Indicator (e.g. IDR)Building ABuilding BChapter 5: Collapse Indicators - Methodology 117 5.4. CONCLUSIONS The risk associated with older non-ductile concrete buildings during an earthquake is significant, and the development of improved technologies for mitigating that risk is a large and costly undertaking. Considering the limited funding available for seismic retrofit, to achieve a meaningful reduction in the collapse risk it is essential to be able to identify the very worst buildings and fix these first. A potential methodology for identifying collapse indicators based on results of comprehensive collapse simulations and estimation of collapse probabilities for a collection of building prototypes is described. The methodology presented here is general and can be applied to any building type. Eventually the probabilities of collapse must be considered for a broad cross section of building types to ensure the selected limits for the collapse indicators are appropriate for the large varied inventory of existing buildings. Different risk-based and intensity-based collapse assessment approaches can be used to determine the limits on the collapse indicators. For a risk-based approach, a suitable mean annual frequency of collapse (?collapse) is used to define the limit. In this chapter, ?collapse is selected based on either a target collapse risk or the ?collapse of a ?good? existing building for which seismic rehabilitation is not required to achieve a collapse prevention performance level. For an intensity-based approach, either a particular probability of collapse at a specific hazard level is used to define the limit on the collapse indicator or the relative changes in the collapse fragilities (at a specific hazard level) will be used to determine the limit. In Chapter 6, the proposed methodology will be applied to a cross section of concrete frames. The examples will be used to first outline the procedure and next to ensure that both the collapse indicators and the selected limits for the collapse indicators are suitable for a relatively broad range of different frames. Chapter 6: Collapse Indicators - Implementation 118 Chapter 6. COLLAPSE INDICATORS - IMPLEMENTATION 6.1. OVERVIEW  As explained in Chapter 5, numerous sources of uncertainty complicate the ability to identify collapse vulnerable buildings, including uncertainty in the deformation capacity of non-ductile components (e.g. shear and axial failure models for columns and punching shear failure model for slab-column connections).  For this reason in order to identify relative collapse risk of a building inventory it is important to develop estimates of collapse probability accounting for all significant sources of uncertainty. The methodology for the identification of collapse indicators and their limits is outlined in Chapter 5. The main objective of this chapter is to implement the methodology for a matrix of pre-1970 buildings. A predominant system among these existing buildings is reinforced concrete moment frames. A broad range of two-dimensional buildings is selected to construct a preliminary database of collapse indicators and to illustrate the proposed procedure. Both design and response parameter collapse indicators are studied in this chapter and if proven suitable, limits are suggested for each building case and collapse indicator. Intended outcomes of this chapter include: ? Establish preliminary values for limits on the collapse indicators, identified in Chapter 5, which engineers could use individually and/or as a group to assess the seismic risk of non-ductile frame buildings; and ? Improve the understanding of the seismic risk involved in existing concrete buildings and to compare them with structures designed according to the provisions of recently developed codes and guidelines, e.g. ASCE 7 2010 edition and ACI 318-11 for concrete buildings. The proposed methodology (outlined in Chapter 5) must be considered for a cross section of building types to ensure first the collapse indicators are suitable and the selected limits for the collapse indicators are appropriate for a relatively broad range of different buildings.  For completeness, this illustration example follows all the steps and sub-steps described in the methodology in Chapter 5. Steps 1 ? 5 are common for both collapse indicators, design and response parameters, and will be only presented once.  Chapter 6: Collapse Indicators - Implementation 119 6.2. EXAMPLE CASE STUDIES To access the procedure, a set of archetypical structural building systems are identified that are representative of practice for existing RC moment frames in high (Los Angeles, California) and moderate (Vancouver, British Columbia) seismic regions. Liel (2008) designed a set of 26 typical structures (archetypes), using the 1967 Uniform Building Code, to be representative of older reinforced concrete moment frame buildings in California built between 1950 and 1975. For the purposes of this study, the 4-story space frame is chosen from the Liel database and is summarized in Table 6-1. The 4-story building is designed as a three-bay, two-dimensional frame. The primary features of the design of this building are summarized in Appendix B. A 7-story non-ductile concrete moment frame structure in Los Angeles (Holiday Inn in Van Nuys, details of this building can be found in Krawinkler, 2005) is also selected as a comprehensive case study to demonstrate the procedure outlined in this chapter. The last building considered in this study is a 12-story building designed using the 1965 National Building Code of Canada (NRCC, 1965) and the Building Code Requirement for Structural Concrete (CSA A23.3 1966). The design details for this building can be found in Appendix B.  The three buildings encompass key structural design parameters including building heights from four to twelve stories, space and perimeter frame systems, and bay widths of 19 and 25 feet. Story heights are typically 15 ft. (13.5 ft in the 7-story building) in the first story and 13 ft. (8.5 ft in the 7-story building) in all other stories.  The spectral acceleration at the MCE hazard level for the 4- and 7-story buildings in Los Angeles were extracted from the USGS website (http://eqint.cr.usgs.gov/deaggint/2008/) and the 12-story building in Vancouver was retrieved from the program EZ-Frisk (Risk Engineering, 2009). 6.2.1. Numerical Model Finite element models developed using OpenSees (PEER 2009) are used to simulate the seismic response of the buildings. A fixed-base model is used in the analysis; as a result soil-structure foundation interaction is neglected. The frame elements are modelled using the force-based beam?column model with distributed nonlinear fibre sections. The joints are modeled using the scissor model (Alath and Kunnath, 1995), which includes the pinching hysteric behaviour to account for the degradation usually seen in these non-ductile elements (explained in Chapter 4). Chapter 6: Collapse Indicators - Implementation 120 Shear and axial failures in the columns are modeled using the Limit State material (explained in Chapter 3). Mass are assigned to the model using lump mass in the nodes and the fundamental period of structures are summarized in Table 6-1. It should be noted that the 4-story space frame has a large period compared to the 7- and 12-story buildings. The main reason for this unusual high period is because this space frame building was designed for a minimum column size and high vertical rebar percentage (~4%). In addition to this, the base shear design based on the 1967 Uniform Building Code for this building (0.068W) does not govern the design and this building was basically designed for gravity loads only. As observed from static pushover analysis results shown in Table 6-1, space frames have higher static overstrength (?) as compared to perimeter frames, where overstrength is defined by the ratio of ultimate strength from pushover analysis to the design strength.  Table 6-1 Archetype non-ductile RC frame structures Num. of Stories Num. of Bay Story Height (ft) Bay  Width (ft) Framing  System Period (s) Sa(T1) @ MCE [g]  Location  Overstrength 4 3 13 25 Space 1.98 0.58 LA 1.4 7 8 8.5 18.5 Perimeter 1 1.04 LA 1.3 12 4 13 25 Perimeter 2.8 0.13 Vancouver 1.35    Chapter 6: Collapse Indicators - Implementation 121 6.2.2. Seismic Hazard Calculations and Ground Motion Records Selection This section includes a short summary of the process taken to select and scale ground motion time histories. These records are required to perform nonlinear time history analysis. In general, the process should address the following aspects of the hazard analysis performed for each building: ? Site and structure conditions ? Uniform hazard spectra ? Deaggregation of the hazard ? Conditional Mean Spectrum ? Selection of Ground Motion Recordings ? Scaling of the Ground Motion Recordings The steps aforementioned are explained in detail in Appendix A.  Figure 6-1 displays the target spectra for each building in addition to the UHS for a return period of 2475 years for both high and moderate seismic regions. It should be noted that the target spectra used to scale and select the ground motions for the 4- and 7-story buildings located in the high seismic region is the conditional mean spectrum, and for the 12-story building located in the moderate seismic region the uniform hazard spectrum is used. The reason for this difference is because higher mode effects are important in the 12-story building and it is recommended by Hamburger and Moehle (2010) to use the UHS curve as the target spectrum.   6.2.3. Record-to-Record (RTR) Variability As explained, multiple strip analysis (MSA) and incremental dynamic analysis (IDA) are two methods suitable for making probabilistic assessments (record-to-record variability) over a wide range of tolerable probability levels. In order to avoid the uncertainties with spectra shape factors associated with the IDA method, the MSA method is used for the analysis. This is possible since a specific location has been selected for each of the example buildings. 6.2.4. Probabilistic Analysis (Ground Motion and Model Uncertainty) Uncertainty in the shear and axial failure models for non-ductile columns are considered the main sources of material uncertainty in this study. In addition to this, record-to-record variability   Chapter 6: Collapse Indicators - Implementation 122  Figure 6-1 Target spectra in addition to the UHSs for the three example buildings (RT = 2475 yrs) is considered in the process. Table 5-2 summarizes the numerical parameters for which uncertainties are considered in the nonlinear analysis. The non-ductile model variability is represented by a lognormal distribution with a mean equivalent to the limit-state material failure models and a coefficient of variation. Zhu et al. (2007) identifies a coefficient of variation of 0.35 for both for the shear and axial failure models used to model shear critical columns. The sampling process explained in section 4.5 was used to incorporate the model uncertainty in the collapse fragilities. The outcome of the collapse assessment procedure (explained in steps 2-5) is a probability distribution representing the cumulative probability of collapse, PCollapse = P[Sacollapse,norm < Sanorm], as a function of the ground motion intensity (normalized spectral intensity). This is illustrated in Figure 6-2 for the three sample buildings. Important metrics for quantifying collapse resistance of structures are defined and illustrated in Figure 6-2, including the median collapse capacity, and the conditional probability of collapse at an intensity level of interest, the code-defined Maximum Considered Earthquake (MCE). The ground motion intensity of interest depends on the ground-shaking hazard at the location of the structure. The code-defined Maximum Considered Earthquake (MCE) is typically consistent with the ground motion that has a return period of 2475 years at a particular site.  0.110.1 1Spectra Acceleration [g]Period [s]UHS, RT = 2475 yrs, LA RegionUHS, RT = 2475 yrs, Vancouver RegionCMS - 7 StoryCMS - 4 StoryChapter 6: Collapse Indicators - Implementation 123 As seen in this figure, the 12-story moment frame designed for a site in Vancouver has a lower probability of collapse at the MCE hazard level compared to the other two buildings and the collapse margin ration (defined as CMR =  @[]%@	 ) (FEMA, 2009) is approximately 1.6 which is larger than 0.6 and 0.8  for the 4- and 7-story buildings, respectively. The main reason behind this comparative better collapse behaviour is that the gravity load usually governs the design of the main components of the older moment frames designed for moderate hazard regions (e.g., Vancouver, Seattle...) and therefore these moment frames have more overstrength compared to the frames designed for high hazard regions (e.g., southern California). Comparing the 4-story and the 7-story buildings located in the same hazard region, Figure 6-2 indicates that the 7-story building has a better collapse behaviour compared to the 4-story building. The reason for this behaviour is due to lack of redistribution of gravity load in the 4-story case after one column has an axial failure; i.e., collapse will occur with one column axial failure in the 4-story building rather than several for the 7-story case. The results in the following sections will indicate the same trend with the collapse indicators in these three buildings.  Figure 6-2 Collapse fragility curve for three non-ductile RC buildings, illustrating key metrics for collapse performance 0%10%20%30%40%50%60%70%80%90%100%0 0.5 1 1.5 2 2.5 3 3.5 4P[Collapse|Sa norm]IM [Sanorm= Sa (T1)/SaMCE]4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)MedianP[Collapse|SaMCE]Chapter 6: Collapse Indicators - Implementation 124 6.3. SIMPLIFIED MODEL OF 2-D FRAMES ? DESIGN PARAMETERS 6.3.1. Collapse Indicators The full list of collapse indicators considered in this study (which could be considered by the two dimensional building model) is presented in Table 5-1. In this section, only the design parameters that can be captured by two-dimensional models are considered.  The following design parameters are addressed in this section using two-dimensional models: ? RA-L1: Average minimum column transverse reinforcement ratio for each story ? EC-G1: Maximum ratio of story stiffness for two adjacent stories  ? EC-G2: Maximum ratio of story shear strength for two adjacent stories  ? EC-G4: Portion of story gravity loads supported by columns with ratio of plastic shear demand to shear capacity > 0.7  ? EC-L1: Maximum ratio of plastic shear capacity (2Mp/L) to column shear strength, Vp/Vn ? EC-L3: Maximum ratio of axial load to strength of transverse reinforcement (45 deg truss model)  6.3.2. Assessment Procedure and Post ? Processing  The design parameters are explained below and the process explained in Chapter 5 will be applied to each collapse indicator and the results will be discussed in the next section. Indicators for Rapid Assessment Those categorized as ?rapid assessment? parameters can be determined from a quick survey of the building or engineering drawings. Average minimum column transverse reinforcement ratio for each story This collapse indicator could be used for rapid assessment and is categorized as a local variable. This indicator is determined by taking the minimum, over all the stories, of the average column transverse reinforcement ratio of all columns in each story (min{?average@each story}). The importance of this indicator is due to this fact that shear and subsequent axial failures of non-Chapter 6: Collapse Indicators - Implementation 125 ductile columns are highly related to column transverse reinforcement. This collapse indicator is also highlighted in ASCE 31 (ASCE, 2003) as:   4.4.1.4.6 NO SHEAR FAILURES: The shear capacity of frame members shall be able to develop the moment capacity at the ends of the members. For this collapse indicator, the transverse reinforcement ratio for all the columns in the building have been uniformly increased and decreased.  Indicators Based on Engineering Calculations These parameters are relevant to the design criteria employed to design the building. Those categorized as ?engineering calculations? require some calculation of capacities and demands based on engineering drawings, but do not require structural analysis results from computer modeling. Maximum ratio of story stiffness for two adjacent stories This collapse indicator could be used as engineering calculations and is categorized as a global variable. This collapse indicator is highlighted in ASCE 31 as:  4.3.2.2 SOFT STORY: The stiffness of the lateral-force-resisting-system in any story shall not be less than 70 percent of the lateral-force-resisting-system stiffness in an adjacent story above or below, or less than 80 percent of the average lateral-force-resisting-system stiffness of the three stories above or below for Life Safety and Immediate Occupancy. The stiffness of each story is calculated as the summation of the stiffness of all columns in that story, where the stiffness of each column in these models is defined as 12EI/L3. The stiffness of the first story (by changing the height of the first story) has been changed in each case for the three buildings studied. Maximum ratio of story shear strength for two adjacent stories This collapse indicator could be used based on engineering calculations and is categorized as a global variable. This deficiency can lead to weak-story mechanisms and failure of the building concentrating in one story. This collapse indicator is also highlighted in ASCE 31 as:  Chapter 6: Collapse Indicators - Implementation 126 4.3.2.1 WEAK STORY: The strength of the lateral-force-resisting-system in any story shall not be less than 80 percent of the strength in an adjacent story, above or below, for Life Safety and Immediate Occupancy. The shear strength of each story is calculated as the summation of the strength of all columns in that story, where the strength of each column in these models is defined as the summation of the moment strength at its two ends, divided by the height of a column. There is an assumption made here that although most of the columns in the three example buildings are shear critical, but the flexural strength is used to calculate the story shear strength.  This assumption is appropriate for these buildings since the columns are likely to experience flexure-shear failures. The strength of the first story (by changing the longitudinal reinforcement ratio) has been changed in each case for the three buildings studied. Portion of story gravity loads supported by columns with ratio of plastic shear demand to shear capacity > 0.7 This collapse indicator could be used as engineering calculations and is categorized as a global variable. This design parameter refers to robustness of the gravity system and is not well captured in current guidelines.  The plastic shear demand of each column is defined as the moment strength at its two ends, divided by the height of a column. The shear capacity of each column is defined using the following ASCE 41 (2006) equation: () =  + 6	  	  0.8           (6-1) where Av  is the area of transverse reinforcement, P is the axial load, M/Vd (also represented as a/d where a is the shear span and is usually L/2) is the span-depth ratio and Ag is the cross section area of the column.  Maximum ratio of plastic shear capacity (2Mp/L) to column shear strength, Vp/Vn This collapse indicator could be used as engineering calculations and is categorized as a local variable. This collapse indicator is also highlighted in ASCE 31 as:  4.4.1.4.6 NO SHEAR FAILURES: The shear capacity of frame members shall be able to develop the moment capacity at the ends of the members. Chapter 6: Collapse Indicators - Implementation 127 The plastic shear demand of each column is defined as the moment strength at its two ends, divided by the height of a column. The shear capacity of each column is defined using Eqn. (6-1). There are different approaches to vary this collapse indicator in the example buildings. For the results presented in this chapter, the approach taken is to uniformly increase the shear strength of all columns in the numerical model. Maximum ratio of axial load to strength of transverse reinforcement (45 deg truss model) This collapse indicator could be used as engineering calculations and is categorized as a local variable. This collapse indicator refers to axial failure in the frame members and is not well captured in ASCE 31.The ratio of axial load to strength of transverse reinforcement is defined using the following equation:  =             (6-2)  Referring back to the axial failure model for non-ductile columns in Chapter 3, Eqn. (6-2) is in the denominator of Eqn. (3-13) and an increase of this design parameter will increase the probability of axial failure in the frame members, therefore, there should be a maximum limit for this local variable. 6.3.3. Observations The mean annual frequency of collapse (?collapse) for the six collapse indicators and for the three different buildings are depicted in Figure 6-3, 5, 7, 9, 11, and 13. The probability of collapse equivalent to 2 percent in 50 years is also shown in all these figures. It should be noted that the 2% is chosen above the 1% in 50 years collapse risk target in ASCE 7-10 (2010) and assumes and allows a higher collapse risk for existing buildings. However, this value is only chosen to demonstrate the methodology and can be easily replaced with any other collapse risk target.  The results imply that structural characteristics, e.g. the number of stories and number of bays, in addition to the hazard level will control the limits proposed from this methodology and the limits for these collapse indicators representing a probability of collapse of 2% in 50 years for the three buildings vary for each building and collapse indicator but in general will result in lower values for the 12-story building located in moderate seismic hazard region.  Chapter 6: Collapse Indicators - Implementation 128 Probability of collapse for these six collapse indicators for the three buildings and at the 2475 return period hazard level are also compared in Figure 6-4, 6, 8, 10, 12, and 14. The limits for these collapse indicators representing a probability of collapse of 10% for a 2475-year return (the 10% corresponds with the limit considered in FEMA P695 (2009)) are also highlighted in these figures. Due to a relatively smaller probability of collapse for the 12-story building at the 2475 year return period (~20% from Figure 6-2) compared to the four (~85%) and seven (~70%) story buildings, in general, the limits for the 12-story building will result in larger values. Table 6-2 summarizes the suggested collapse indicator limits for all three buildings and for the design parameters using both approaches. The 12-story building demonstrates markedly superior seismic collapse performance when compared to the 4- and 7-story buildings. In addition to the proposed limits for these example buildings, the following observations can be made: ? Extrapolating the curve for the 4-story building in Figure 6-3 will result in a limit approximately around 0.013 (it would translate to a transverse reinforcement spacing around 5 in). This high number for the 4-story building essentially indicates that something other than just transverse reinforcement needs to change to get an acceptable performance.  While for the 12-story building perhaps acceptable performance can be achieved with only changing transverse reinforcement (ranging from 0.002 to 0.005 for this building using both methods, Table 6-2). ? As shown in Figure 6-5, the limit for this collapse indicator (ratio of stiffness of two adjacent stories), representing a probability of collapse of 2% in 50 years, is not satisfied for any of the buildings. However, the trends in all three buildings clearly demonstrate that decreasing the ratio of stiffness of two adjacent stories decreases the collapse risk as it reaches the value of one. A ratio smaller than one will shift the soft story from the first floor to floors above and will replicate the effect when the story stiffness ratio are higher than one. The results for the 4- and 7- story buildings imply that changing stiffness alone cannot decrease the collapse risk to an acceptable limit and based on these limited examples, the results suggest that this collapse indicator could not be used to improve the collapse performance of existing buildings. In addition to this, the 7-story results suggest that the collapse risk does not change over Chapter 6: Collapse Indicators - Implementation 129 a large variation of the story stiffness ratio (from 1 ? 4.5) and as a result is not an acceptable collapse indicator.   ? In Figure 6-7, the results for the design parameter strength ratio are depicted, and similar to the stiffness ratio, the probability of collapse of 2% in 50 years, is only satisfied for the 12-story building around a ratio approximately at 1.0 using both methods (Table 6-2). The results for the strength and stiffness ratio for two adjacent stories imply that varying a combination of these two collapse indicators might lead to a structure with an acceptable risk of collapse, particularly for buildings located in a high seismic hazard level. Additional studies are required to investigate the combination of changes in these collapse indicators that will lead to a ?safe? structure.   ? Figure 6-9 and 10 emphasize the impact of the robustness of the gravity system on the collapse safety of buildings. Both figures clearly indicate that for buildings in a moderate seismic hazard level (i.e., the 12-story building), a range from 40% of the columns in one story up to 84% of them can have a flexure-shear behavior and the building will still maintain an acceptable risk of collapse. The reason behind this acceptable high percentage of columns with flexure-shear behavior is that the deformation capacity is rarely exceeded in this building located in the lower hazard demand region. In the higher seismic region, the limit for acceptable collapse risk Table 6-2 Limits on the collapse indicators ? Design Parameters applying both approaches Num. of Stories Num. of Bays T1 [s] Hazard Level Limits for the Different Collapse Indicators for both approaches RA-L1 EC-G1 EC-G2 11 22 1 2 1 2 4 3 1.98 High - 0.0125 - - - - 7 8 1.0 High - 0.015 - - - - 12 4 2.8 Moderate 0.005 0.0016 1 - 1.0 1.05 Num. of Stories Num. of Bays T1 [s] Hazard Level Limits for the Different Collapse Indicators for both approaches EC-G4 EC-L1 EC-L3 1 2 1 2 1 2 4 3 1.98 High - 0.05 - 0.25 - 2 7 8 1.0 High - 0.05 - 0.25 - 1 12 4 2.8 Moderate 0.4 0.84 0.95 1.11 4.0 8.0 1 Approach 1, 2% probability of collapse in 50 years 2 Approach 2, 10% probability of collapse for a 2475 year return period hazard Chapter 6: Collapse Indicators - Implementation 130 will be approximately 5% of the columns could have a flexure-shear response. Regarding the limit around zero for the 4- and 7-story buildings located in LA, this result is consistent with the concept behind the rehabilitation standards (e.g. ASCE 41, 2006) which set strict limits for the collapse prevention performance level for buildings with very limited deformation capacity for flexure-shear columns. This also points out the need to consider multiple changes in the collapse indicators to achieve structures with acceptable collapse performance.   ? Figure 6-11 and 12 focus on the relationship between the maximum ratio of columns with shear and flexure-shear behavior (Vp/Vn > 0.7) and the collapse risk. The figures indicate that for the 12-story building the limit is slightly less than one, suggesting that as long as the columns are not pure shear critical, the building will be safe. However, for the 4- and 7- story buildings the limit will be for about a quarter of the columns in one story. In Figure 6-11, the shaded area indicates the region with flexure-shear behavior and the results show a gradual decrease in the collapse risk as the columns shift from shear dominant to flexural dominant behavior.   ? In Figure 6-13 and 14 the results indicate that the limit for this collapse indicator (P/Vs), in the 12-story building is approximately four times (ranging from 4 - 8) the value for the 4- and 7-story buildings (ranging from 1-2) and implying the need for higher transverse reinforcement ratio in columns located in a higher hazard level.  Chapter 6: Collapse Indicators - Implementation 131   Figure 6-3 Mean annual frequency of collapse for the average minimum column transverse reinforcement ratio Figure 6-4 Probability of collapse for the average minimum column transverse reinforcement ratio for a return period of 2475 years   Figure 6-5 Mean annual frequency of collapse for the maximum ratio of story stiffness for two adjacent stories Figure 6-6 Probability of collapse for the maximum ratio of story stiffness for two adjacent stories (CI) for a return period of 2475 years 0.00E+001.00E-032.00E-033.00E-034.00E-035.00E-036.00E-030 0.002 0.004 0.006 0.008 0.01 0.012?? ??collapse(?? ??< ?? ??ave.)Collapse indicator (min{?average @ each floor})4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[collapse in 50yrs] = 2%Base Building0%10%20%30%40%50%60%70%80%90%100%0 0.005 0.01 0.015 0.02 0.025 0.03 0.035Pcol[?? ??< ?? ??ave.|IM(RT= 2475 yrs)]Collapse Indicator (min{?average @ each floor})4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[col | RT = 2475 yrs] = 10%0.00E+001.00E-032.00E-033.00E-034.00E-035.00E-036.00E-030 1 2 3 4 5?? ??collapse(CI > CImax) Collapse indicator (maximum ratio of story stiffness for two adjacent stories)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[collapse in 50yrs] = 2%Base Building0%10%20%30%40%50%60%70%80%90%100%0 1 2 3 4 5Pcol[CI > CImax|IM(RT= 2475 yrs)]Collapse Indicator (maximum ratio of story stiffness for two adjacent stories)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[col | RT = 2475 yrs] = 10%Chapter 6: Collapse Indicators - Implementation 132   Figure 6-7 Mean annual frequency of collapse for the maximum ratio of story strength for two adjacent stories  Figure 6-8 Probability of collapse for the maximum ratio of story strength for two adjacent stories (CI) for a return period of 2475 years   Figure 6-9 Mean annual frequency of collapse for the portion of story gravity loads supported by columns with ratio of plastic shear demand to shear capacity > 0.7 Figure 6-10 Probability of collapse for the portion of story gravity loads supported by columns with ratio of plastic shear demand to shear capacity > 0.7 0.00E+001.00E-032.00E-033.00E-034.00E-035.00E-036.00E-030.5 1 1.5 2 2.5?? ??collapse (CI > CImax) Collapse indicator (maximum ratio of story strength for two adjacent stories)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[collapse in 50yrs] = 2%Base Building0%10%20%30%40%50%60%70%80%90%100%0.5 1 1.5 2 2.5 3Pcol[CI > CImax|IM(RT= 2475 yrs)]Collapse Indicator (maximum ratio of story strength for two adjacent stories)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[col | RT = 2475 yrs] = 10%0.00E+001.00E-032.00E-033.00E-034.00E-035.00E-036.00E-030 0.2 0.4 0.6 0.8 1 1.2?? ??collapse (CI > CImax)Collapse indicator (Portion of story gravity loads supported by columns with ratio of plastic shear demand to shear capacity > 0.7)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[collapse in 50yrs] = 2%Base Building0%10%20%30%40%50%60%70%80%90%100%0 0.2 0.4 0.6 0.8 1 1.2Pcol[CI > CImax|IM(RT= 2475 yrs)]Collapse Indicator (Portion of story gravity loads supported by columns with ratio of plastic shear demand to shear capacity > 0.7)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[col | RT = 2475 yrs] = 10%Chapter 6: Collapse Indicators - Implementation 133   Figure 6-11 Mean annual frequency of collapse for the maximum ratio of plastic shear capacity (2Mp/L) to column shear strength (Vp/Vn) Figure 6-12 Probability of collapse for the maximum ratio of plastic shear capacity (2Mp/L) to column shear strength (Vp/Vn)   Figure 6-13 Mean annual frequency of collapse for the maximum ratio of axial load to strength of transverse reinforcement (45 deg truss model) Figure 6-14 Probability of collapse for the maximum ratio of axial load to strength of transverse reinforcement (45 deg truss model) 0.00E+001.00E-032.00E-033.00E-034.00E-035.00E-036.00E-037.00E-030 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6?? ??collapse (CI > CImax)Collapse indicator (Maximum ratio of plastic shear capacity (2Mp/L) to column shear strength, Vp/Vn)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[collapse in 50yrs] = 2%Base Building0%10%20%30%40%50%60%70%80%90%100%0 0.2 0.4 0.6 0.8 1 1.2Pcol[CI > CImax|IM(RT= 2475 yrs)]Collapse Indicator (Maximum ratio of plastic shear capacity (2Mp/L) to column shear strength, Vp/Vn)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[col | RT = 2475 yrs] = 10%0.00E+001.00E-032.00E-033.00E-034.00E-035.00E-036.00E-037.00E-030 2 4 6 8 10 12 14 16?? ??collapse (CI > CImax)Collapse indicator (Maximum ratio of axial load to strength of transverse reinforcement, 45 deg truss model)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[collapse in 50yrs] = 2%Base Building0%10%20%30%40%50%60%70%80%90%100%0 2 4 6 8 10 12 14 16 18Pcol[CI > CImax|IM(RT= 2475 yrs)]Collapse Indicator (Maximum ratio of axial load to strength of transverse reinforcement, 45 deg truss model)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965,Vancouver)P[col | RT = 2475 yrs] = 10%Chapter 6: Collapse Indicators - Implementation 134 6.4. SIMPLIFIED MODEL OF 2-D FRAMES - RESPONSE PARAMETERS The proposed methodology for response parameters (outlined in Chapter 5) must be considered for a cross section of building types to ensure first the collapse indicators are suitable and selected limits for the collapse indicators are appropriate for a relatively broad range of different buildings. In this chapter, examples for modest range of two-dimensional frames are presented to demonstrate the framework delivered in Chapter 5 and further study is required to determine collapse indicators applicable for a broad class of buildings, appropriate for implementation in ASCE 41 or other guidelines.  6.4.1. Collapse indicators The list of collapse indicators estimated in this section by the two-dimensional building model is presented in Table 5-1. These parameters are relevant to the response parameters of the building. The following design parameters are addressed in this section using two-dimensional models:  ? BA-G1: Maximum degradation in base or story shear resistance  ? BA-G2: Maximum fraction of columns at a story experiencing shear failures ? BA-G3: Maximum fraction of columns at a story experiencing axial failures ? BA-L1: Maximum drift ratio 6.4.2. Assessment Procedure and Post ? Processing the Results Indicators for Building Analysis (BA) These quantities are for detailed collapse prevention assessment using the results from building analyses, commonly nonlinear analysis. Since these collapse indicators are related to the response of each analysis, in contrast to the design parameters where they are altered by the user and are known for each analysis, a procedure is required to extract the response. This procedure, wherever necessary, is explained for each collapse indicator.  Maximum degradation in base or story shear resistance This response could be used to investigate the correlation of collapse and story shear degradation. Story shear is estimated as the sum of shear force in all columns in one story. Chapter 4 demonstrates the importance of shear degradation for a side-sway collapse mechanism, however this section intends to examine the link between shear degradation and gravity load Chapter 6: Collapse Indicators - Implementation 135 induced collapse mechanism. The value used to represent story shear degradation is defined in Figure 6-15. As seen in this figure, the maximum shear degradation does not necessarily reach 100% when gravity load collapse occurs (in contrast to side-sway collapse where the story shear usually degrades by ~100%). It should be noted that story shear degradation, for the story where gravity load or side-sway collapse has initiated, will be used to define this parameter, not the maximum story shear degradation over the height of the building. In cases where the structure has not collapsed, the maximum story shear degradation over the height of the building will be used to define this engineering demand parameter.  Maximum fraction of columns at a story experiencing shear failures This response could be used to investigate the correlation of collapse and maximum percentage of columns (in a single story) sustaining a shear failure. Chapter 3 introduces the mechanical model used to detect shear failure in existing columns.  Maximum fraction of columns at a storey experiencing axial failures This response could be used to investigate the correlation of collapse and maximum percentage of columns (in a single story) sustaining an axial failure. Chapter 3 explains the mechanical model used to detect axial failure in existing columns. As explained in Chapter 2, the definition of collapse in a recent study performed by ATC-78 [ATC78, 2012] defines collapse when more   Figure 6-15 Definition of story shear degradation (4 story building) -200-150-100-50050100150200250300-2.00% -1.00% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00%Story Shear [kips]Interstory Drift [%]maximum story sheardegradation= 42%gravityloadcollapseChapter 6: Collapse Indicators - Implementation 136 than half of the columns in a particular story have sustained an axial failure. Maximum drift ratio This response could be used to investigate the correlation of collapse and interstory drift. It should be noted that the interstory drift for the story where gravity load or side-sway collapse has initiated is used to define this parameter. 6.4.3. Observations The mean annual frequency of collapse (?collapse) for the four collapse indicators and for the three different buildings are depicted in Figure 6-16, 18, 20, and 22. The probability of collapse equivalent to 2 percent in 50 years (similar to the design parameters) is also shown in all these figures. The results for the response parameters imply that primarily the hazard level will control the limits proposed from this methodology and the limits for these collapse indicators representing a probability of collapse of 2% in 50 years in general will result in lower values for the 12-story building located in moderate seismic hazard region.  Probability of collapse for these four collapse indicators for the three buildings and at the 2475 return period hazard level are also compared in Figure 6-17, 19, 21 and 23. The limits for these collapse indicators representing a probability of collapse of 10% for a 2475-year return period (similar to the design parameters) are also highlighted in these figures. Table 6-3 summarizes the suggested collapse indicator limits for all three buildings and for the selected response parameters using both approaches. The 12-story building demonstrates markedly superior seismic collapse performance when compared to the 4- and 7-story buildings. In addition to the proposed limits for these example buildings, the following observations can be made: ? The correlation of maximum degradation in story shear resistance with the collapse risk for the three different buildings is illustrated in Figure 6-16 and 17. As seen in these figures, for the 4- and 7- story located in a high seismic region (Los Angeles, California) the limit for story shear resistance degradation for a probability of collapse of 2% in 50 years and probability of collapse of 10% for the 2475 years return period is around 5-15%. However, the 12-story building designed for a moderate seismic region reaches the probability of collapse margin of both approaches around 35-40%. Lower demand/capacity ratios for the 12-story building compared to the 4- and 7-buildings Chapter 6: Collapse Indicators - Implementation 137 could be the main reason behind this finding. As explained in the previous section, the 12-story building was designed for a load combination case with gravity loads dominating and therefore this building has a higher lateral load resistance capacity relative to demand compared to the 4- and 7-story buildings. This will result in the larger values in the maximum degradation in story shear resistance for the 12-story building. The results imply that this limit could be simply related to the hazard level, although future studies considering a range of different buildings designed for different regions are needed to generalize this finding and develop a global criterion for implementation in rehabilitation standards. ? Figure 6-18 and 19 demonstrate the relationship between the percentage of columns experiencing shear failure and the collapse metrics for the three different buildings. As seen in both figures, for the 4- and 7-story located in a high seismic region the limit for percentage of columns sustaining shear failure for the pre-defined collapse performance level is around 25-35% (one third to one fourth of the columns in a floor can sustain shear failure). However, the 12-story building designed for a moderate seismic region reaches collapse margins ranging from 40-60%. The results shown in both figures suggest that the hazard level will dictate the limit for this collapse indicator and this limit is relatively independent of the number of floors and bays/floor. ? The mean annual frequency of collapse (?collapse) for percentage of columns experiencing axial failure for the three different buildings is illustrated in Figure 6-20. As seen in this figure, for the 4- and 7-story located in a high seismic region the limit for percentage of columns sustaining axial failure for a probability of collapse of 2% in 50 years is 20% and 30%, respectively. However, the 12-story building designed for a moderate seismic region reaches the probability of collapse margin of 2% in 50 years around 45% of columns experiencing axial failure. Probability of collapse for percentage of columns sustaining axial failure for the three different buildings and at the 2475 return period hazard level are compared in Figure 6-21. As seen in this figure, for the 4- and 7-story located in a high seismic region the limit for percentage of columns sustaining axial failure for a probability of collapse of 10% for a 2475 year return period is 14% and 27%, respectively, and for the 12-story building this value will be 36%. The results shown in Figure 6-20 and Figure 6-21 clearly show that the number of floors, number of bays/floor Chapter 6: Collapse Indicators - Implementation 138 and hazard level will dictate the limit for this collapse indicator. Comparing the buildings located in the high seismic region, the 4-story building has a lower limit from both approaches which is due to the fact that this building has a lower capacity to redistribute axial loads after failure of each column and therefore will have a higher collapse risk. Probability of collapse dependency on the number of columns per floor was also demonstrated in Chapter 4. ? The mean annual frequency of collapse (?collapse) and the probability of collapse at the 2475 return period hazard level for maximum interstory drift for the three different buildings is illustrated in Figure 6-22 and 23. As seen in these figures, for the 4- and 7-story located in a high seismic region and the 12-story building designed for a moderate seismic region, the limit for interstory drift for both approaches is around 3.0%. The results shown in Figure 6-22 and Figure 6-23 clearly show that the limit for this collapse indicator is independent from the vulnerability and hazard characteristics of the buildings and this collapse indicator could be used to distinguish buildings with a high collapse risk regardless of the site and building characteristics. However, it should be noted that this result is only valid for the range of collapse risk considered in this study and reexamining Figure 6-22 and Figure 6-23 clearly indicates that for higher collapse risk levels, the three curves representing the three example buildings start to follow different trends.  Table 6-3 Limits on the collapse indicators - Response Parameters applying both approaches Num. of Stories Num. of Bays T1 [s] Hazard Level Limits for the Different Collapse Indicators for both approaches BA-G1 BA-G2 BA-G3 BA-L1 11 22 1 2 1 2 1 2 4 3 1.98 High 10% 3% 33% 25% 20% 14% 3.2% 2.6% 7 8 1.0 High 13% 11% 27% 25% 30% 27% 3.5% 3.4% 12 4 2.8 Moderate 36% 38% 58% 39% 45% 36% 3.1% 2.8%       1 Approach 1, 2% probability of collapse in 50 years       2 Approach 2, 10% probability of collapse for a 2475 year return period hazard Chapter 6: Collapse Indicators - Implementation 139 .   Figure 6-16 Mean annual frequency of collapse for story shear degradation Figure 6-17 Probability of collapse for story shear degradation for return period of 2475 years   Figure 6-18 Mean annual frequency of collapse for shear failure in columns in one story Figure 6-19 Probability of collapse for percentage of columns with shear failures for return period of 2475 years 00.00050.0010.00150.0020.00250.0030.00350.0040 0.2 0.4 0.6 0.8 1?? ??collapse (EDP< edp)Maximum Degradation in Story Shear Resistance (EDP)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story( designed for 1965, Vancouver)P[col in 50yrs] = 2%00.20.40.60.810% 20% 40% 60% 80% 100%Pcol[?? ??< IDR|IM(RT= 2475 yrs)]Maximum Degradation in Story Shear Resistance (EDP)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story( designed for 1965, Vancouver)P[col | RT = 2475 yrs] = 10%00.00050.0010.00150.0020.00250 0.2 0.4 0.6 0.8 1?? ??collapse (EDP< edp)% of columns in one story which experience shear failure (EDP)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965, Vancouver)P[col in 50yrs] = 2%00.20.40.60.810% 20% 40% 60% 80% 100%Pcol[?? ??< IDR|IM(RT= 2475 yrs)]% of columns in one story which experience shear failure (EDP)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story( designed for 1965, Vancouver)P[col | RT = 2475 yrs] = 10%Chapter 6: Collapse Indicators - Implementation 140   Figure 6-20 Mean annual frequency of collapse for axial failure in columns in one story Figure 6-21 Probability of collapse for percentage of columns with axial failures for return period of 2475 years   Figure 6-22 Mean annual frequency of collapse for interstory drift Figure 6-23 Probability of collapse for interstory drift for return period of 2475 years 00.00050.0010.00150.0020.00250.0030.00350.0040 0.2 0.4 0.6 0.8 1?? ??collapse (EDP< edp)% of columns in one story which experience axial failure (EDP)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965, Vancouver)P[col in 50yrs] = 2%00.20.40.60.810% 20% 40% 60% 80% 100%Pcol[?? ??< IDR|IM(RT= 2475 yrs)]% of columns in one story which experience axial failure (EDP)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story( designed for 1965, Vancouver)P[col | RT = 2475 yrs] = 10%00.0010.0020.0030.0040.0050 0.02 0.04 0.06 0.08 0.1?? ??collapse (IDR< idr)Maximum Interstory Drift Ratio (IDR)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story (designed for 1965, Vancouver)P[col in 50yrs] = 2%00.20.40.60.810 0.02 0.04 0.06 0.08 0.1Pcol[?? ??< IDR|IM(RT= 2475 yrs)]Maximum Interstory Drift Ratio (IDR)4 Story (designed for 1967, LA)7 Story (designed for 1965, LA)12 Story( designed for 1965, Vancouver)P[col | RT = 2475 yrs] = 10%Chapter 6: Collapse Indicators - Implementation 141 6.5. CONCLUSIONS The methodology introduced in Chapter 5 for identifying collapse indicators based on results of comprehensive collapse simulations and estimation of collapse probabilities for a collection of building prototypes is evaluated and validated in this chapter. The details of the procedure are illustrated in this chapter for a 4-, 7- and 12-story moment frame designed for high and moderate seismicity and with different number of bays. Although only demonstrated here for frames, the probabilities of collapse must be considered for a broad cross section of building types to ensure the selected limits for the collapse indicators are appropriate for the large varied inventory of existing buildings.  Limits on the collapse indicators can be selected based on a suitable mean annual frequency of collapse (?collapse) or with a specific probability of collapse (e.g. 10%) at a specific hazard level (e.g. hazard level with a return period of 2475 years). In this chapter, ?collapse is selected based on a target collapse risk (2% probability of collapse in 50 years). One of the main conclusions from the results of the limited buildings considered in this chapter is that it is nearly impossible to achieve an acceptable collapse risk by only varying an individual design parameter. The buildings will most likely reach an acceptable collapse risk only by varying a vector of design parameters. On the other hand the response parameters have indicated that they could be used in future rehabilitation standards to define limits for system-level engineering demand parameters with specific collapse risk levels. The two most promising response parameters are the maximum number of columns with shear failures and the maximum interstory drift ratio. Table 6-2 and 3 report the suggested collapse indicator limits for all three buildings and for the design and response parameters using both approaches. The 12-story building demonstrates markedly superior seismic collapse performance when compared to the 4- and 7-story buildings Table 6-2 and 3 and Figure 6-24  suggest that some of the collapse indicator limits are dependent on the number of stories, period of the buildings and to the hazard level for each building. However, it should be noted that based on only three buildings the following results could not be generalized for a larger inventory of structures. Chapter 6: Collapse Indicators - Implementation 142 ? Maximum degradation in story shear resistance and maximum fraction of columns at a story experiencing axial failure (both response parameters) have a direct correlation with the number of stories. ? Maximum ratio of axial load to strength of transverse reinforcement (design parameter) and maximum fraction of columns at a story experiencing shear failures (response parameter) have a direct correlation with the period of the building and as a result have a reverse correlation with the spectral acceleration of the first mode of the building. On the contrary, the average minimum column transverse reinforcement ratio (design parameter) has an inverse correlation with the period of the structure and accordingly a direct correlation with the spectral acceleration of the first mode of the building. This suggests that stiffer buildings would require more transverse reinforcement to achieve a specific limit of collapse risk.  ? As expected, response parameters for the buildings in higher hazard levels result in lower limits with the exception of the limit for the interstory drift ratio. The limit for this collapse indicator (using both approaches) specifies a value around 3%.  Figure 6-24 Collapse indicator limit trends with Building characteristics 4 6 8 10 120.10.20.30.40.5Num. of StoriesBA-G1 Approach #14 6 8 10 1200.10.20.30.4Num. of StoriesBA-G1 Approach #24 6 8 10 120.20.250.30.350.40.45Num. of StoriesBA-G3 Approach #14 6 8 10 120.10.20.30.40.5Num. of StoriesBA-G3 Approach #21 1.5 2 2.5 300.0050.010.015Period (s)RA-L1 Approach #21 1.5 2 2.5 302468Period (s)EC-L3 Approach #21 1.5 2 2.5 30.20.30.40.50.60.7Period (s)BA-G2 Approach #11 1.5 2 2.5 30.0310.0320.0330.0340.0350.036Period (s)BA-L1 Approach #10 0.5 1 1.500.0050.010.015Sa(T1) @ MCE (g)RA-L1 Approach #20 0.5 1 1.502468Sa(T1) @ MCE (g)EC-L3 Approach #20 0.5 1 1.50.20.30.40.50.60.7Sa(T1) @ MCE (g)BA-G2 Approach #10 0.5 1 1.50.0310.0320.0330.0340.0350.036Sa(T1) @ MCE (g)BA-L1 Approach #1Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 143 Chapter 7. SEISMIC LOSS ESTIMATION OF NON-DUCTILE REINFORCED CONCRETE BUILDINGS4 7.1. INTRODUCTION On the basis of detailed surveys and extrapolation across California, the Concrete Coalition (Comartin et al., 2008) estimates there are approximately 1,500 pre-1980 non-ductile reinforced concrete buildings in the City of Los Angeles, 3,000 in San Francisco, and 20,000 more in the 33 most seismically active counties state-wide. These structures are susceptible to severe damage including collapse during severe earthquake shaking. To assess the seismic vulnerability of these structural systems, the performance-based assessment framework established by the Pacific Earthquake Engineering Research Center (PEER) is used. The framework (shown in Figure 7-1) divides the performance assessment into four analysis phases, including seismic hazard analysis, response analysis, damage analysis, and loss analysis. The outcome of each analysis is then integrated using a total probability theorem as shown in Eqn. (7-1).    ( > ) =    |	||	|(	)|  (7-1) where ( > ) represents the mean annual frequency when a decision variable exceeds a threshold value; | =  < | =  denotes the conditional complementary cumulative distribution function of a random variable X given the random variable Y = y; The notation dv, dm, edp and im denotes the random variable for decision variables (for example, the total repair cost), damage measures (for example, the amount of cracks in the shear wall), engineering demand parameters (for example, the inter-story drift ratio), and the intensity measure (for example, the peak ground acceleration), respectively.  The PEER performance assessment methodology was further developed by Yang et al. (2009), and later adopted by the ATC-58 research team (Applied Technology Council 58, 2008), to use a Monte Carlo simulation procedure to quantify the performance of different structural facilities. In this chapter, the performance assessment methodology is further extended to quantify the seismic loss of non-ductile reinforced concrete buildings, most importantly including the risk of collapse.                                                   4 A version of chapter 7 has been published. Baradaran Shoraka, M., Yang, T. Y. and Elwood, K. J. (2013). ?Seismic loss estimation of non-ductile reinforced concrete buildings.? Earthquake Engng. Struct. Dyn., 42: 297?310. doi: 10.1002/eqe.2213. Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 144  Figure 7-1 Performance-assessment framework from Cornell and Krawinkler  (2000 ? PEER by permission) There are several definitions of collapse used in the literature. Most seismic rehabilitation methodologies such as ASCE/SEI 41 (ASCE, 2006) deem a building to have exceeded the Collapse Prevention (CP) performance level when the first structural component reaches the CP limit. This ignores the structure?s ability to redistribute seismic and gravity load demands after first component failure. Other documents, FEMA P695A (2009), define collapse based on the formation of a side-sway mechanism (i.e., unrestrained increase in lateral displacements for a small incremental increase in seismic demand) detected using incremental dynamic analysis. With recent progress in simulating global collapse of non-ductile reinforced concrete frames (Yavari et al., 2009b), a new definition of collapse was introduced in Chapter 4 to capture the point at which a structure losses the ability to sustain gravity load carrying capacity. This collapse criterion was named the gravity load collapse. A detailed loss simulation methodology for non-ductile reinforced concrete frames, including all three collapse criteria, is presented in this chapter. The presented methodology provides engineers, insurance industry and other stake holders a robust and transparent procedure to quantify the seismic losses of non-ductile reinforced concrete buildings, including the possibility of collapse.  Although business interruption due to structural and non-structural damage can significantly impact the total loss after an earthquake, this is considered beyond the scope of the current study which focuses on the direct capital losses. It should be noted that the presented methodology can also be adapted to other non-ductile structural systems, such as unreinforced masonry structures.  7.2. SEISMIC LOSS ASSESSMENT PROCEDURE To address the seismic loss of a facility after earthquake events, the ATC-58 performance assessment approach  is adopted in this study. Based on the PEER framework (Figure 7-1), the approach is comprised of four analysis steps: quantification of the seismic hazard, response  Seismic Hazard Analysis ( )im?  IM: intensity measure Response Analysis ( )|G edp im  EDP: Engineering Demand Parameter Damage Analysis ( )|G dm edp  DM: Damage Measure Loss  Analysis ( )|G dv dm  DV: Decision Variable Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 145 analysis, definition of the performance groups and damage states, and finally loss analysis.  Each of the four steps is described briefly below; further details can be found elsewhere (Yang et al., 2009; Bradley and Lee, 2010; Goulet et al., 2007; Solberg et al., 2008). In subsequent sections, this seismic loss assessment procedure will be used to specifically investigate seismic losses for non-ductile concrete buildings. Step 1: Quantification of the seismic hazard  The first step of the performance-assessment framework is to quantify the seismic hazard at the site. This is typically done using the probabilistic seismic hazard analysis (PSHA), first proposed by Cornell (1968). The method takes into account the earthquake sources, the distance to the fault, uncertainties in earthquake size, location and ground motion intensity and using a total probability theorem to quantify the probabilistically distribution of the shaking intensity of the site. Using the results of the PSHA, ground motions can be selected to represent the seismic hazard at the site. For example, a suite of several ground motion records representing the seismic hazard with a 2% probability of exceedance in 50 years at the site may be selected. Details of the ground motion selection and scaling procedures can be identified in many leading research articles and discussion of this topic is beyond the scope of this chapter; for additional discussion see Abrahamson (2006) and Haselton et al. (2009).  Step 2: Response analysis With ground motion records selected in step 1, nonlinear dynamic response analyses are used to quantify the statistical distribution of the structural response (such as interstory drift ratios, IDR, and peak floor accelerations, PFA) at different levels of earthquake shaking intensities. Multiple strip analysis (MSA) (Jalayer and Cornell, 2009) and incremental dynamic analysis (IDA) (Vamvatsikos and Cornell, 2002) are two commonly used methods to quantify the distribution of structural response quantities over a range of earthquake shaking intensities. MSA procedure selects bins of ground motions to match a range of levels of earthquake shaking intensity. The IDA method, selected for the current study, is similar to the MSA method, except only a single bin of ground motions are selected and the ground motions are then amplitude scaled to reflect the range of the earthquake shaking intensities. Haselton et al. (2011) indicates that it is important to consider the shape of the response spectra for ground motions used to estimate collapse probabilities and outlines two methods to reflect the changes of the spectra shape at different shaking intensities. The most direct approach is to select ground motions that have Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 146 ?(T) values similar to the target ?(T). The target ?(T)  is obtained from a hazard analysis and measured at the fundamental period of the structure. The second approach modifies the structural response determined using IDA using a spectral shape factor (Haselton et al., 2011). For the current study, the first approach is used to account for the ground motion spectra shape characteristics to ensure an unbiased estimate of the collapse probabilities.  Step 3: Define the performance groups, damage states and corresponding repair actions Key structural and non-structural components of the building are identified and grouped into different performance groups (PG). Each performance group consists of one or more building components (structural and non-structural) whose performance is similarly affected by a particular engineering demand parameter. A sufficient number of damage states (DS) should be defined for each performance group to describe the range of damage for the components at different levels of structural response. In addition, detailed repair action and the associated repair cost for each performance group at each damage states must be identified. Damage states are typically defined using fragility curves. The horizontal axis of the fragility curve represents the engineering demand parameter that affects the performance group (e.g., interstory drift) and the vertical axis represents the probability that the performance group will experience each of the damage states. The values of the fragility curves are usually derived from past experimental data, expert judgment, and post-earthquake reconnaissance reports.  Step 4: Loss analysis Using the results of the response analysis (step 2) and the fragility data (step 3), a unique damage state is determined for each performance group. The damage state is obtained by identifying the probability of the performance group experiencing each damage state at the structural response obtained from the response analysis. A uniform random number generator, with a distribution over the interval (0, 1), is used to select the damage state (Yang et al., 2009). Once the damage state for a performance group is identified, the repair action and the associate repair cost for that performance group is obtained from the repair actions presented in step 3. The process is repeated for all performance groups and the repair cost for the building is determined by summing overall performance groups. Because the repair items could be similar between different performance groups (for example, the repair for the first floor partition walls can be repaired at the same time as the second floor partition walls), the total repair cost is calculated by summing the total repair quantities of similar items and multiplying the total repair quantities by Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 147 a unit repair cost. The unit repair cost typically reduces as the quantities increases. The unit repair cost used in this study reduces as a tri-linear function as shown in Figure 7-2. Note the unit repair cost is not known exactly due to uncertainty in the availability of resources following an earthquake. If structural collapse was detected, the total repair cost is calculated using the replacement value of the building plus additional costs related to demolition and debris removal (10% of the replacement value of the building). It should be recognized that there may be additional costs associated with loss use of the facility and possibly litigation; however, data for these additional costs are not readily available and not included in the current study. The total repair cost accounting both the collapse and non-collapse cases can be calculated using total probability theory as shown in Eqn. (7-2).   (|	) = (|, 	)| = 	 + (|, 	)| = 	  =  (|, 	)(1 ? | = 	) + (|, 	)| = 	   (7-2) where PNC|IM = im is the probability of non-collapse given the intensity measure = im; PC|IM = im is the probability of collapse given the intensity measure = im.  The conditional complementary cumulative distribution function, G(dv|im), determined in Eqn. (7-2) can be converted to the annualized loss by integrating with the contribution from all hazard levels, as shown in Eqn. (7-3).   ( > ) =  |	 |(	)|      (7-3) where the absolute value is used to account for the negative value of the derivative of the hazard curve.      Total repair quantitiesMax quantitiesMin quantitiesMin costMax costQi$iUnit repair costUncertaintyFigure 7-2 An example of the unit repair cost function from Yang et al., (2009 ? ASCE by permission) Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 148 7.3. MODELING THE COLLAPSE OF NON-DUCTILE REINFORCED CONCRETE FRAMES As shown in Chapter 4, the ability to accurately model the collapse of a structural system is crucial in estimating the seismic response of non-ductile structures. For non-ductile reinforced concrete frames, the ability to capture column shear distress and subsequent axial failure are particularly important. To achieve this goal, the limit-state material models introduced in chapter 3 are used in this chapter.  Three definitions of collapse are considered in this chapter ?first component failure?, ?side-sway?, and ?gravity load? collapse. The first and simplest definition is to consider the building to be at the collapse limit when ?failure? is detected in one structural component. For non-ductile concrete columns, ?failure? may be defined as drift at which column shear strength degradation is initiated or at the loss of column axial load capacity. This criterion is consistent with the approach used in ASCE/SEI 41 (ASCE, 2006), where ?Collapse Prevention? acceptance criteria are based on loss of lateral load resistance and axial load capacity for primary and secondary components, respectively. While straight forward to implement, this criterion does not account for the redistribution of lateral loads and any reserve axial load capacity after shear failure, and hence results in a very conservative estimate for the point of global collapse. In this study, the ?first component failure? criterion is defined as the point at which the first column experiences shear failure. Global collapse for ductile structures with robust gravity systems is frequently defined by side-sway collapse, inevitably due to significant P-delta effects and reduction in lateral capacity. This definition is adopted from FEMA P695 (FEMA, 2009). As explained in detail in Chapter 4, in this type of failure, side sway instability can occur in one or more storeys. Side-sway collapse is typically defined as the point in an IDA analysis when the maximum inter-story drift increases rapidly for a very small increase ground shaking intensity. Convergence problems are frequently encountered when conducting nonlinear analysis close to this point of dynamic instability. In order to overcome this issue, side-sway collapse is defined in the current study based on lateral capacity and demand.  Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 149   (a) Shear failure model    (b) Axial failure model Figure 7-3 Shear and axial failure definitions (adapted from Elwood, 2004) Collapse mechanisms due to loss of vertical load carrying capacity (e.g., column axial failure), however, are likely to control the global collapse probability in non-ductile concrete structures and should be accounted for when determining expected seismic losses for such structures. As explained in detail in Chapter 4, when a non-ductile reinforced concrete structure is laterally deformed, local failure is likely to occur in vertical load-carrying components (e.g., non-ductile columns or slab?column connections) as a result of deterioration of force transfer mechanisms with lateral deformation demands. As the gravity loads carried by these elements transfer to neighbouring elements, the local failure can propagate until the structure reaches a state where it loses its ability to support the imposed gravity loads. This chapter will focus on non-ductile concrete moment frames where collapse is governed by axial load failure of columns. The progression of damage with failure of each subsequent component can be tracked throughout the analysis by comparing floor-level gravity load demands and capacities (adjusted at each time step to account for degradation in column axial load capacities with increasing drift demands using the limit-state material models). 7.4. CASE STUDY FOR THE LOSS ASSESSMENT OF NON-DUCTILE CONCRETE BUILDINGS A 7-story non-ductile reinforced concrete moment frame structure in Los Angeles is selected for the case study in this chapter (the same building used in Chapter 6). Figure 7-4 illustrates the elevation view of the building and the cross sectional dimension of a typical column. The building is located on site class D with the seismic hazard dominated by nearby blind thrust faults and the San Andreas Fault. column lateralload demandLateralLoad?horizshear capacity curveshear failurecolumn axialload demandVerticalLoad?horizaxial capacity curveaxial failureChapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 150 7.4.1. Model, Ground Motions, and Fragility Curves Due to the relatively small stiffness contributed by the gravity system, only the longitudinal moment resisting frames are considered in the numerical model. It is assumed for the case study that the gravity system deformation capacity exceeds that of the longitudinal moment resisting frames. A fixed base finite element model developed using OpenSees (PEER, 2009) is used to simulate the nonlinear dynamic response of the building. The beams and columns are modelled using the force-based nonlinear beam?column element in OpenSees (de Souza, 2000). The joints are modeled using rotational spring elements (Alath and Kunnath, 1995) which includes a pinching hysteric behaviour to account for the nonlinear shear deformation of the joint. Shear and axial failure in the columns are captured using the model presented in chapter 3. A schematic view of a portion of the numerical model is shown in Figure 7-4.b. Based on the OpenSees model, the first three modes of the structure exhibit the following periods of vibration: 1.0, 0.45 and 0.18 seconds. A damping of 2% is assigned to the first and third modes using Rayleigh damping. Deaggregation of the seismic hazard for the site is performed based on Bazzurro and Cornell (1999) for the return period 2475 years. For this deaggregation, a shear wave velocity of 218 m/s and a first-mode period of 1 second are assumed. Based on the results of the seismic hazard deaggregation, 31 ground motions are selected from the PEER Next Generation Attenuation (NGA) database (Pacific Earthquake Engineering Research Center (PEER), 2010) and summarized in Table 7-1. As mentioned in Section 2, to reflect differences in the spectra shape at different shaking intensities, ground motions have been selected with 	() values similar to the target 	(). The ground motion set presented in Table 1 has a mean 	 = 1.15. Hazard deaggregation for this site provides a target epsilon of 1.18 for a 2475 years return period, using Abrahamson and Silva 1997 attenuation model (Abrahamson and Silva, 1997). Only uncertainty from the ground motions has been considered in this example since this source of uncertainty will typically dominate loss estimates (Porter et al., 2002). Major structural and non-structural components are identified and categorized into performance groups. Table 7-2 summaries the performance groups included in this study. For each performance group, the damage states associated with different repair actions are identified based on data provided by the ATC-58 project and from Aslani (2005). Figure 7-5 shows the fragility curves for each of the performance groups. It should be noted that ATC-58 does not  Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 151 (a) (b)   (c)  Figure 7-4 (a) South frame elevation (b) column, beam, and joint models (c) sample column cross section (Krawinkler, 2005)  specifically include axial failure of the column as an individual damage state and therefore cannot capture the losses due to extreme damage to the structural system. The current study addresses this deficiency by incorporating the probability of collapse in the loss estimation. To illustrate the information contained in the fragility curves, consider the fragility curves shown in Figure 7-5.b for joints. If the peak interstory drift equals 3%, the conditional probability that a joint is considered to have experienced DS2 (cracking) or worse is 0.91, to have a DS3 (spalling) or worse is 0.76, and DS4 (lateral load failure) or worse is 0.50. Therefore, by the mathematics of set theory the probability of being in DS1 (no damage), DS2, DS3, and DS4 is 0.09, 0.15, 0.26, and 0.50, respectively. As noted previously, a uniform random number generator between zero and one is used to determine the damage state. For the example described above a random number less than 0.5 would result in DS4, between 0.5 and 0.76 would result in DS3, between 0.76 and 0.91 would result in DS2, and above 0.91 would result in DS1.  8@ 5.72 m = 45.76 m4.12 m2.59 mA A2.59 m2.59 m2.59 m2.59 m2.60 mShear response springAxial response springZero length elementJoint modelNonlinear beam-column element8 #9#3@305 mm356 mm508 mmSection A-AChapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 152 Table 7-1 Ground motions selected for case study Earthquake Mw Station Record Distance (km) PGA (g) () Chi-Chi, Taiwan 1999/09/20 7.6 TCU042 TCU042-N 23.34 0.199 0.9 Chi-Chi, Taiwan 1999/09/20 7.6 CHY035 CHY035-N 18.12 0.246 1.1 Chi-Chi, Taiwan 1999/09/20 7.6 TCU123 TCU123-W 15.12 0.164 1.7 Chi-Chi, Taiwan 1999/09/20 7.6 CHY006 CHY006-E 14.93 0.364 1.2 Duzce, Turkey 1999/11/12 7.1 Bolu BOL090 17.6 0.822 1.6 Imperial Valley 1979/10/15 6.5 6617 Cucapah H-QKP085 23.6 0.309 1.3 Imperial Valley 1979/10/15 6.5 5059 El Centro Array #13 H-E13230 21.9 0.139 0.9 Imperial Valley 1979/10/15 6.5 6621 Chihuahua H-CHI012 17.7 0.27 1.2 Imperial Valley 1979/10/15 6.5 5115 El Centro Array #2 H-E02140 10.4 0.315 1.1 Landers 1992/06/28 7.3 22074 Yermo Fire Station YER270 24.9 0.245 1.4 Loma Prieta 1989/10/18 6.9 57425 Gilroy Array #7 GMR090 24.2 0.323 0.6 Loma Prieta 1989/10/18 6.9 57382 Gilroy Array #4 G04000 16.1 0.417 0.8 Loma Prieta 1989/10/18 6.9 47125 Capitola CAP000 14.5 0.529 1.4 N. Palm Springs 1986/07/08 6 12025 Palm Springs Airport PSA090 16.6 0.187 1.1 Northridge 1994/01/17 6.7 90054 LA - Centinela St CEN155 30.9 0.465 1.4 Northridge 1994/01/17 6.7 90091 LA - Saturn St STN020 30 0.474 1 Northridge 1994/01/17 6.7 24303 LA - Hollywood Stor FF HOL360 25.5 0.358 1 Northridge 1994/01/17 6.7 90063 Glendale - Las Palmas GLP177 25.4 0.357 1.4 Northridge 1994/01/17 6.7 90053 Canoga Park - Topanga Can CNP196 15.8 0.42 0.6 Northridge 1994/01/17 6.7 90003 Northridge - 17645 Saticoy St STC180 13.3 0.477 0.7 Northridge 1994/01/17 6.7 90057 Canyon Country - W Lost Cany LOS270 13 0.482 1 Northridge 1994/01/17 6.7 24279 Newhall - Fire Sta NWH360 7.1 0.59 0.9 San Fernando 1971/02/09 6.6 94 Gormon - Oso Pump Plant OPP270 48.1 0.105 1.4 San Fernando 1971/02/09 6.6 135 LA - Hollywood Stor Lot PEL090 21.2 0.21 0.9 Whittier Narrows 1987/10/01 6 24303 LA - Hollywood Stor FF A-HOL000 25.2 0.221 1.6 Whittier Narrows 1987/10/01 6 90012 Burbank - N Buena Vista A-BUE250 23.7 0.233 1 Whittier Narrows 1987/10/01 6 90084 Lakewood - Del Amo Blvd A-DEL000 20.9 0.277 1.1 Whittier Narrows 1987/10/01 6 90063 Glendale - Las Palmas A-GLP177 19 0.296 1.7 Whittier Narrows 1987/10/01 6 14368 Downey - Co Maint Bldg A-DWN180 18.3 0.221 1.6 Whittier Narrows 1987/10/01 6 90078 Compton - Castlegate St A-CAS270 16.9 0.333 1 Whittier Narrows 1987/10/01 6 90077 Santa Fe Springs - E Joslin A-EJS318 10.8 0.443 1.1     Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 153 Table 7-2 Summary of Performance Group assignment No. Location edp Components 1 ? 7 Floor 1 - 7 Interstory drift Columns 8 ? 14 Floor 1 - 7 Interstory drift joints 15 ? 21 Floor 1 - 7 Interstory drift Non-structural Drift-sensitive 22 ? 28 Floor 1 - 7 Floor acceleration Non-structural Acceleration-sensitive         a) Non-ductile columns used for PG 1-7 (Applied Technology Council 58, 2008) (b) Non-ductile joints used for PGs 8-14 (Applied Technology Council 58, 2008)   (c) Non-structural drift sensitive components for PG 15-21 (Aslani, 2005) d) Non-structural acceleration sensitive components for PG 22-28 (Aslani, 2005) Figure 7-5 Summary of the fragility curves used for the study 00.10.20.30.40.50.60.70.80.910 2 4 6P (DS > DS i)edp - Peak interstory drift ratio [%]DS1: crackingDS2: spallingDS3: failure00.10.20.30.40.50.60.70.80.910 1 2 3 4 5P (DS > DS i)edp - Peak interstory drift ratio [%]DS2: crackingDS3: spallingDS4: failure00.10.20.30.40.50.60.70.80.910 2 4 6 8 10P (DS > DS i)edp - Peak interstory drift ratio [%]DS1: slight damageDS2:  moderate damageDS3:  extensive damageDS4:  complete damage00.10.20.30.40.50.60.70.80.910 1 2 3 4 5P (DS > DS i)edp - Peak floor acceleration [g]DS1: slight damageDS2:  moderate damageDS3:  extensive damageDS4:  complete damageChapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 154 7.4.2. Response of the Building Figure 7-6 shows example results obtained from one of the ground motions to highlight the sequence of failures for the prototype building. As shown in the plot, the analytical model with limit-state materials has the ability to capture the structural response after first column shear failure (approximately 2% inter-story drift) and axial load failure (approximately 4% inter-story drift). When the inter-story drift ratio reaches 4.5%, gravity load collapse is detected in the first floor. If the analysis is allowed to go past gravity load collapse (for a model without limit-state material springs for axial load failure), side-sway collapse is detected when the inter-story drift ratio reaches 5.9%.  Examples of the two modes of collapse, side-sway collapse and gravity load collapse for the case study building are illustrated in Figure 7-7 and Figure 7-8. As shown in Figure 7-7, whenever the lateral shear resistance (defined by the story shear corresponding to the peak inter-story drift ratio) decreases below a pre-established minimum shear capacity the structure is considered to sustain a side-sway collapse. The minimum shear capacity can be estimated based on an approximation of residual column shear strengths or can be set to zero. Figure 7-8 shows an example of the vertical load carrying capacity changes in the first floor of a multi-story building as the structure is subjected to earthquake excitation. The total gravity loads are typically not varying with time (without considering significant excitation due to the vertical component of the ground motion). When the vertical load carrying capacity of the system goes below the vertical load demand, gravity loads can no longer be supported.   Figure 7-9 shows the IDA analysis results for the 31 selected ground motions considering the three collapse criteria described previously. In Figure 7-9, the maximum inter-story drift ratio (MIDR) for the structure at shear failure of the first structural component, gravity load collapse and side-sway collapse are indicated by the diamond, circle and square symbols, respectively. As shown, the maximum inter-story drift ratio for the prototype building reaches between 2.8% to 3.4% for first component failure, 4.2% to 4.8% for gravity load collapse, and 5.3% to 8.9% for side-sway collapse. It should be noted that the IDA results (shown in Figure 7-9) indicate that for this specific building and ground motions, gravity load collapse always happens before side-sway collapse; however, this result does not necessarily apply for other buildings where column axial load failure may occur at larger drifts.   Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 155  Figure 7-6 Side-sway and gravity load collapse example for case study building (Record CAP000 in Table 1)   Figure 7-7 Example of Side - sway collapse captured in an IDA analysis for the case study structure -8%-6%-4%-2%0%2%4%6%8%0 1 2 3 4 5 6 7 8 9 10 11 12First Story Drift Ratio [%]Time [s]First Column Shear Failure@ t = 9.2 (s)Second Column Shear Failure@ t = 9.41 (s) First ColumnAxial Failure@ t = 9.55 (s)"Gravity Load" Collapse@ t = 9.595 (s)"Side-Sway" Collapse@ t = 11.17 (s)050100150200250300350Story Shear corresponding to Peak IDR00.10.20.30.40.50.60.70.80.910 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Spectral Acceleration, Sa (T1) [g]IDRMinimum Shear Capacity?Side-sway?collapseChapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 156  Figure 7-8 Gravity load collapse captured explicitly in the numerical model for the case study structure Figure 7-10 shows the lognormal fit of the probability of collapse (determined from the IDA results in Figure 7-9) as a function of the shaking intensity for the three collapse criteria considered. The solid line represents the probability of collapse of side-sway or gravity load collapse, whichever comes first. For the same shaking intensity (for example Sa(T1) = 1.0 g), the ?first component failure? has the highest probability of occurrence, while the ?side-sway only? collapse scenario has the lowest probability of collapse. The ?side-sway? or ?gravity load? collapse curve is expected to be the most accurate estimation of collapse, since it considers multiple possible collapse modes. Figure 7-10 suggests that defining collapse based on ?first component failure? may be too conservative, since it ignores the ability for the structure to redistribute forces to other members. On the other hand, the ?side-sway only? mechanism may be un-conservative since ?gravity load? collapse is likely to occur before the ?side-sway? collapse mechanism can develop.      01000020000300004000050000600000 2 4 6 8Vertical Load [kN]Time [s]First Floor DemandFirst Floor Capacity ?Gravity Load? collapseChapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 157  Figure 7-9 Results of the IDA for different collapse criteria  Figure 7-10 Collapse fragilities for seven-story non-ductile RC building, illustrating the effect of different collapse criteria 00.20.40.60.811.21.41.61.80 0.02 0.04 0.06 0.08Spectral Acceleration, Sa (T1) [g]Maximum Interstory Drift Ratio, MIDR [%]First Component FailureGravity Load CollapseSide-Sway collapse00.10.20.30.40.50.60.70.80.910.4 0.6 0.8 1 1.2 1.4 1.6Probability of Collapse P(C|IM=im)Spectral Acceleration, Sa (T1) [g]First Component Failure"Side-sway" or  "Gravity Load" Collapse"Side-sway" Only CollapseChapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 158 7.4.3. Loss Estimation The seismic loss assessment procedure described in section 7.2 is used to carry out the loss simulation for the prototype building. The cumulative distribution functions (CDFs) of the total repair cost normalized by the total value of the building (non?collapse cases only) for the example building at a hazard level representing the return period of 72, 224, 475, 975, and 2475 years return period are plotted in Figure 7-11.a. Figure 7-11.b-f show the disaggregation of the total repair cost by each PGs at each hazard level, respectively. It is observed that there is a general trend that the loss is dominated by non?structural components (PGs 15 ? 28) in all five hazard levels. For higher hazard levels (e.g. RT = 975 and 2475 yrs) the drift?sensitive components, both structural and non ? structural, located on the first floor (PG = 1, 8, and 15) have a higher proportion of loss compared to the other components.  Figure 7-12 shows the cumulative distribution function of the total repair cost of the building normalized using the building replacement cost (including expected demolition costs). At the lower shaking intensities (earthquakes with return periods, RT, of 72 and 224 years) collapse was not detected, hence, only non-collapse cases contribute to the predicted loss. At higher earthquake shaking intensities (earthquakes with return periods, RT, of 475, 975 and 2475 years) collapse was sometimes detected depending on the ground motion and collapse criteria considered. The step in the CDF curves at a Normalized cost of unity (when full demolition and building replacement is required) is equal to the probability of collapse for the selected return period and collapse criterion.  Using the ?gravity load collapse? or ?side-sway? collapse criterion, the prototype structure has median repair costs (50% probability) of 3%, 13%, 28%, 53% and 92% of the total replacement value for at the 72 and 224, 475, 975 and 2475 years, respectively. If a constant replacement cost is used as the decision variable, the results clearly indicate that the ?first component failure? criterion has the most conservative repair cost estimation followed by the ?side-sway? or ?gravity load? collapse criterion and finally the ?side-sway only? criterion. In other words, if the decision to proceed with seismic rehabilitation is made based on the cost of repair exceeding 50% of the replacement value of the building at the 975 year return period, the ?first component failure? criterion will estimate that there is 60% probability that the building needs retrofit, while the ?side-sway? or ?gravity load? collapse and the ?side-sway only? criterion will estimate 50% and 35% probability, respectively. The ?first component failure? Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 159 criterion overestimates by 20% the confidence in cost exceeding the selected limit, while the ?side-sway only? criterion underestimates the confidence by 30%. The seismic loss can also be presented as an annualized loss by considering the hazard curve for the building site (Eqn. (7-3)). Figure 7-13 indicates that for the case study building the annual probability of experiencing losses greater than approximately 22% of the replacement value (plus demolition costs) increases when collapse is considered in the determination of seismic loss. In fact, for losses greater than 50% of the replacement cost, the consideration of collapse in the loss assessment will be more than double the estimated annual rate of exceedance, regardless of the collapse criterion selected. The lack of sensitivity to the collapse criterion is due to the fact that collapse is only observed for large shaking intensities (RT = 475, 975 and 2475 years), where the annual probabilities of such intensities (from the hazard curve) are relatively low. When the cumulative distribution functions shown in Figure 7-12 are integrated with the hazard curve to determine the annualized loss, the impact of variations in the CDF for different collapse criterion is reduced. Also shown in Figure 7-13 are the mean cumulative annual losses of the prototype building calculated by integrating the annualized loss curves.  Several aspects should be considered when interpreting the annualized repair costs shown in Figure 7-13.  First, the difference between the annualized losses for the three collapse criteria will increase for locations where high intensity ground motions are expected at lower return periods.  Secondly, it may not be appropriate to make decisions regarding retrofit of a non-ductile concrete building based on annualized losses alone since the impact of collapse is masked in the integration with the hazard curve. Due to the difficulties of assessing value for human life, fatalities have not been considered in this study. Clearly collapse would result in additional losses from increased fatality rates not reflected in Figure 7-13.        Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 160 (a) (b)  (c) (d)   (e) (f)   Figure 7-11 Repair cost (a) probability distribution (CDF) for 5 hazard levels and PGs for hazard level (b) RT = 72 [yrs], (c) RT = 224 [yrs], (d) RT = 475 [yrs], (e) RT = 975 [yrs], and (f) RT = 2475 [yrs] 00.10.20.30.40.50.60.70.80.910 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8P (Total Repair cost < C) Normalized Cost, CRT = 72 [yrs]RT = 224 [yrs]RT = 475 [yrs]RT = 975 [yrs]RT = 2475 [yrs]Chapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 161  Figure 7-12 Cumulative Distribution Functions for normalized cost at 5 different hazard levels considering different collapse criteria  [normalized cost = repair cost / (replacement cost + demolition costs)]   Figure 7-13 Annualized total repair cost 00.10.20.30.40.50.60.70.80.910 0.2 0.4 0.6 0.8 1P (Total normalized repair cost ? c) Normalized cost , c"Side-Sway" Only Collapse"Side-Sway" or "Gravity Load" CollapseFirst Component FailureRT = 2475 yrsRT = 975 yrsRT = 72 yrsRT = 224 yrsRT = 475 yrsP(C|RT = 975 yrs, collapse criteria = "Side-Sway" or"Gravity Load" collapse) = 0.200.0050.010.0150.020.0250.030.0350.040 0.2 0.4 0.6 0.8 1Annual rate of exceedingtotal repair cost = CNormalized Cost, C"Side-sway" Only Collapse"Side-sway" or  "Gravity Load" CollapseFirst Component FailureNon-Collapse1.52% 1.57% 1.63% 1.36%Mean cumulative annual total repair cost (normalized cost, c)  "Side-sway" Only Collapse "Side-sway" or  "Gravity Load" CollapseFirst Component FailureNon-CollapseChapter 7: Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings 162 7.5. SUMMARY AND CONCLUSIONS Performance-based earthquake engineering, which aims to describe the seismic performance of facilities, is been used in this chapter to estimate expected losses for non-ductile reinforced concrete buildings for a wide range of earthquake shaking intensities. The state-of-the-art loss simulation procedure developed for new buildings is extended in this chapter to estimate the expected losses of non-ductile concrete buildings considering their vulnerability to collapse.  Three collapse criteria have been included and compared in this study: 1) failure of the first structural component, adopted from ASCE/SEI 41 (ASCE, 2006); 2) formation of a ?side-sway? collapse mechanism, adopted from FEMA P695 (FEMA P695A, 2009); 3) failure of side-sway or gravity load collapse. Gravity load collapse is directly captured in the analysis using elements that are capable to modeling shear and axial load failure of concrete columns, followed by redistribution of loads, and finally collapse of building frame when the gravity load demands can no longer be supported.  A 7-story non-ductile reinforced concrete frame building located in Los Angeles, California, is used as an example to illustrate the loss simulation of non-ductile reinforced concrete structures. The results show that collapse does not occur in low earthquake shaking intensities and losses are dominated by non-structural damage. At higher shaking intensities, the first structural component failure has the highest collapse probability and overestimates the financial loss because it ignores the structure?s ability to redistribute the forces. On the other hand, the side-sway only collapse criterion underestimates the loss due to the likelihood that gravity load collapse will happen in a non-ductile concrete building before the side-sway mechanism develops. For the example structure, the annual rate of experiencing losses less than 22% of the replacement cost is not dependent on whether collapse is considered in the loss assessment.  In contrast, for losses greater than 50% of the replacement cost, the consideration of collapse in the loss assessment will more than double the estimated annual rate of exceedance.  If the annualized loss is used as the decision variable to decide if seismic rehabilitation is necessary, the results indicate that while the consideration of collapse in determining the annualized loss is important, the choice of collapse criteria is less significant. The presented methodology can be easily adapted to other non-ductile structural systems where the seismic vulnerability can be systematically analyzed.  Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 163 Chapter 8. COLLAPSE ASSESSMENT OF NON-DUCTILE, RETROFITTED AND DUCTILE REINFORCED CONCRETE MOMENT FRAMES5 8.1. INTRODUCTION With recent earthquakes worldwide, rehabilitation plays an important role in mitigating the seismic risk for older RC structures. Repair and upgrading of existing structures is becoming increasingly important and its economic impact has grown. Seismic rehabilitation of an existing building will either involve the design of new components connected to the existing structure, or increasing the deformation capacity of existing components. In order to decide on the most appropriate and economical rehabilitation strategy, it is necessary to assess the lateral load resistance, deformation capacity, and potential modes of collapse. Chapter 1 presents the evolution of the seismic rehabilitation guidelines. As currently formulated, ASCE 41 (ASCE, 2006) is not capable of identifying the risk of the building from collapse during an earthquake shaking. This is because the component-based evaluation criteria do not take into account the structure?s ability to redistribute gravity and lateral forces after one component fails. Hence, the evaluation tends to error on the conservative side. The performance of reinforced concrete moment frames retrofitted to the different performance objectives as specified in ASCE 41 is studied in this chapter. Specifically, the collapse vulnerability of the different retrofitting schemes is studied using the system-level assessment procedure introduced in Chapter 4.   The collapse vulnerability of existing non-ductile and retrofitted archetype structures has also been studied by Liel (2008). In her study, 4- and 8-story non-ductile RC space and perimeter moment frame structures were chosen. Three retrofit techniques (1) jacketing of the RC columns with reinforced concrete, (2) carbon fiber-wrapping of RC columns, and (3) construction of ?super column shear walls? around existing columns were included in that study. Liel (2008) did not consider the performance in the design of the retrofits and hence it is not known how these retrofits relate to ASCE 41 criteria. For each type of retrofit, both ?modest? and ?significant?                                                  5 A version of chapter 8 plan to be submitted. Baradaran Shoraka, M., Elwood, K. J. and Yang, T. Y. ?Collapse Assessment of Non-Ductile, Retrofitted and Ductile Reinforced Concrete Moment Frames.?   Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 164 retrofits were considered. In addition to this, the only collapse mode simulated and captured by Liel (2008) is related to the side-sway collapse mechanism (more details in section 4.3.2). Significant non-ductile collapse modes such as column shear, and subsequent axial, failures were considered as non-simulated collapse modes. The study assumes that when the first occurrence of this non- simulated failure mode occurs, it will lead to collapse of the structure, thus ignoring the system?s ability to redistribute the load after failure. Figure 8-1 shows the computed collapse fragilities for all the 4-story perimeter structures presented in Liel?s study: existing (un-retrofitted), retrofitted (using the three rehabilitation  techniques) and modern (with 20 ft. and 30 ft. bay widths). As shown in Figure 8-1, the performance of the retrofitted structures has a large variation in the probability of collapse.   8.2. CASE STUDY STRUCTURE To compare the collapse performance of existing and retrofitted (using the ASCE 41 standard) RC moment frames, a non-ductile perimeter concrete moment resisting frame building designed according to the 1967 UBC (ICBO 1967) code was selected as the prototype structure for this study. The 8-story perimeter frame is chosen from the Liel database (Liel, 2008) and is shown in Figure 8-2. The prototype building was retrofitted using three retrofitting techniques, each designed to satisfy two performance objectives (LS and CP) as specified in the ASCE 41 Supplement 1 document (Elwood et al., 2007).  In addition, the performance of a ductile             Figure 8-1 Collapse fragility functions for 4-story space frames Liel (2008 ? Ph.D. thesis by permission) Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 165  perimeter moment frame building designed according to the International Building Code (ICC 2003) was also included in this study.  The prototype building is assumed to be located in Los Angeles California, with an importance factor of 1.0, site class D, with SS = 1.5g and S1 = 0.6g, where SS is the earthquake spectral response acceleration at short periods and S1 is the earthquake spectral response acceleration at 1-second periods. The earthquake hazard level defined for the selected performance objectives has a 2% probability of exceedance in 50 years. 8.2.1. Non-ductile Perimeter Concrete Moment Resisting Frame Building Figure 8-2 illustrates the elevation view of the prototype building. This structure is designed according to 1967 UBC for Zone 3 (the highest seismic zoning criteria). Seismic design requirements such as the maximum and minimum steel reinforcement ratios, maximum stirrup spacing, and requirements on hooks, bar spacing and anchorage were included. It should be noted that there is no transverse reinforcement in the joints.  Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 166  h = height of beam-column ?tot = ratio of longitudinal reinforcement f?c = compressive strength of concrete  b = width of beam-column  ?sh =  ratio of transverse reinforcement fy = yield strength of reinforcement s = spacing of transverse reinforcement\ ? & ?? = ratio of longitudinal tension and                  compression reinforcement Figure 8-2 Design documentation for 8-story non-ductile perimeter frame structure from Liel (2008 ? Ph.D. thesis by permission) Floor 9Floor 8Floor 7Floor 6Floor 5Floor 4Floor 3feetFloor 2feetGrade beam column height (in) = Basement column height (in) =feetDesign base shear = g, kf'c beams = ksi f'c,cols,upper = ksify,rebar,nom. = ksi f'c,cols,lower = ksi60 4.024 364.0 4.02516.80.0031s (in) = 14.0 18.0 18.0 14.026 0.00750.01030.00150.001516.80.00403636 26 0.00750.01030.018 0.022   ?sh = 0.003136 26 0.00750.01030.001516.836 28b (in) = 26 30 30 26Story 3 h (in) = 28 36   ?tot = 0.022 0.0180.004016.80.0031s (in) = 14.0 18.0 18.0 14.026 0.00750.010.00150.001516.80.00363636 26 0.00750.010.018 0.022   ?sh = 0.003136 26 0.00750.010.001516.836 28b (in) = 26 30 30 26Story 4 h (in) = 28 36   ?tot = 0.022 0.0180.003616.80.0027s (in) = 10.0 10.5 10.5 10.026 0.0070.00930.00150.001516.80.00363636 26 0.0070.00930.00930.001516.80.00360.002736 26 0.0070.022 0.025 0.025 0.02226 26 26 260.0130.001514.0Story 5 h (in) = 28 28 28 28b (in) = 13.8s (in) = 14.0 14.0 14.026 0.00780.01130.00150.00150.00363030 26 0.00780.01130.0250.001530 26 0.00780.01130.001513.80.003613.82826 26 262826Story 6 h (in) = 28 28   ?tot = 0.013 0.025b (in) =    ?sh = s (in) = 14.0262824h (in) =    ?tot = 131526?sh = s (in) = h (in) = b (in) = b (in) = ? = ?' =    ?sh =    ?sh = s (in) = h (in) = b (in) = b (in) =    ?tot =    ?sh = s (in) =    ?sh =    ?tot = h (in) = b (in) =    ?tot =    ?sh = s (in) = h (in) = b (in) =    ?tot = 24 26 0.00330.0100.00630.001510.80.001510.824 2628260.00330.00630.0100.001514.02428260.0100.001514.02410.826 0.00330.00630.001528260.0100.001528 28 2814.00.0040.00950.001510.80.001526 26 26 260.013 0.017 0.017 0.0130.0015 0.0015 0.0015 0.00150.001513.80.00630.010014.0 14.028 36 3614.030 26 0.00630.01000.001513.826 30 30 260.022 0.018 0.018 0.0220.0031 0.0040 0.0040 0.003136 26 0.00750.01000.001516.814.0 18.0 18.0 14.036 26 0.00750.01000.001516.8h (in) = b (in) = ? = ?' = ? = ?' = ?sh = s (in) = 26 30 30 26?sh = s (in) = 28 36 36h (in) = 0.0050 0.00360.030 0.033 0.033 0.03014.0 15.5 15.5 14.00.001513.826 0.0040.00950.001530 26 0.00630.01000.0040.009510.82814.0280.001516.828Story 1Story 236 26 0.00750.01000.0036 0.0050Story 7Story 80.001510.830 2624 263910.054Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 167 Numerical Model Due to the relative small stiffness contributed from the gravity system, only the perimeter moment resisting frames are modeled to represent the seismic response of the building. The numerical analyses were conducted using a two-dimensional frame modeled in OpenSees (PEER 2009). The analytical model incorporates all the important features required to model the collapse of the structure (more details regarding the numerical model can be found in Chapter 4). ? The beam and column elements were modeled using the lumped plastic hinge model developed by Ibarra et al. (2005) to account for the strength and stiffness degradation under cyclic loads. Strain-softening behaviour associated with concrete crushing, rebar buckling and fracture, and/or bond failure is accounted for and including it in the model has an important effect on the collapse response of these structures (Figure 2-6). The backbone and cyclic response parameters used in the numerical model were developed by Haselton et al. (2008). ? The model includes large geometry transformations which take into account P-delta effects. ? To ensure the numerical model is capable of capturing joint failure, a two-dimensional joint model was used. The numerical model developed by Lowes and Altoontash (2003) was used to define the shear deformations of the joint and bond-slip behaviour (Figure 2-7d). ? Shear and axial failures in columns were modeled using the limit-state material developed by Elwood (2004). These models define the shear and axial failures of concrete columns as a function of the deformation capacity, as well as the geometric, material and design parameters (explained in detail in Chapters 2 and 3). It should be noted that the column model introduced in Chapter 3 was not used in the numerical models because it is not compatible with the lumped plastic hinge model. Mass were modeled using lump masses in the nodes. The first fundamental period of structure is about 2.4 second. It should be noted that the eight story perimeter frame has a large period. The main reason for this high period is because this frame was designed for a minimum column size and high vertical reinforcement percentage.  Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 168 8.2.2.  Retrofitted Buildings The non-ductile perimeter concrete moment resisting frame presented in the previous section is evaluated using ASCE 41. A pushover analysis, using the first mode distribution and the target roof displacement calculated using Eqn. (8-1), was performed. At such roof drift, the component deformation demands were checked using the acceptance criteria defined in ASCE 41. The prototype building was retrofitted to the Life Safety (LS) and Collapse Prevention (CP) objectives. For the Collapse Prevention performance level, the structural components can be severely damaged but the structure must be able to continue carrying gravity loads without collapse. Because the strength and stiffness degradation is modeled in all the numerical components, the ?Secondary Components? acceptance criteria as specified in ASCE 41 are used to assess the performance criteria for all the components.  Eqn. (8-1) shows the deformation limit used in the pushover analysis.  =        (8-1) where ? is defined as the roof drift.  S  represents the elastic spectral displacement of an equivalent single degree of freedom (SDOF) oscillatory at the earthquake hazard level of interest; chosen MCE (equivalent to a probability of occurrence of 2% in 50 years) shaking intensity for this study. Te is the fundamental period of the existing structure. C0 is characterized as a dimensionless coefficient that relates the spectral displacement of an equivalent SDOF to the roof displacement. C1 is a dimensionless coefficient that relates expected maximum inelastic displacements to the displacements calculated using the linear elastic response and finally C2 is a dimensionless coefficient that adjusts the roof drift ratio to account for the pinched hysteretic shape, stiffness degradation and strength deterioration effect on the structure. There are various retrofitting schemes available. Using the recommendations presented in FEMA 547 (FEMA, 2006), three retrofitting schemes were identified and included in this study:  1) Strengthening the existing columns, beams and joints by steel jacketing, or adding new reinforced concrete, steel, or fiber-reinforced polymer wrap overlays. This method enhances the strength and deformation capacity of the existing non-ductile components. 2) Weakening a certain portion of the structure by removing a portion of the existing structure. This technique, usually applied to weaken the beams, promotes formation of a strong-column weak-beam mechanism. Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 169 3) Adding supplementary lateral force-resisting systems, such as adding ductile reinforced concrete shear walls, to reduce demands on the existing elements.  Retrofitted Building ? Columns Modified (Strengthening Technique) The first approach to retrofit non-ductile RC frames is by increasing the strength and deformation capacity of the critical non-ductile columns. The critical non-ductile columns are identified based on the acceptance criteria provided in ASCE 41. The critical columns are jacketed with reinforced concrete. It is assumed that the critical columns which are retrofitted will no longer sustain shear/axial failures. The final retrofit design for the selected performance objective is illustrated in Table 8.3a. The highlighted section in this figure shows the retrofitted columns for each of the performance objectives. The highlighted numbers in the table also show the changes in the column dimensions and transverse reinforcement for the structures upgraded to the CP and LS performance objectives. Retrofitted Building ? Beams Modified (Weakening Technique) The second approach is to retrofit the non-ductile RC frames by weakening the existing beams such that the system can form a strong-column, weak-beam mechanism. This technique is applied by cutting the longitudinal reinforcement in the beams. The longitudinal reinforcement were cut floor by floor and the demand to the system was recalculated, after each iteration, until the component demands for the critical columns fall within the selected performance level. This rehabilitation measure will result in a decrease in strength and stiffness of the beams. For this technique, it was not possible to reach the LS level as the beams could not be weakened anymore. The final design (for the CP level) is illustrated in Table 8.3b. The highlighted section in this figure shows the weakened beams. The highlighted numbers in the table also show the changes in the table dimensions and longitudinal reinforcement for the structure upgraded to the CP performance objectives. Retrofitted Building ? Wall Added (Strengthening Technique) The third approach to retrofit the non-ductile RC frame is by adding shear walls to decrease the demand on the existing moment frame. The shear wall is designed with the same cross sectional area and reinforcement ratio over the full height of the wall. The strength of the shear wall is Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 170 increased by the means of expanding the depth and longitudinal steel reinforcement until the component demands for the critical columns fall into the performance target (only added enough stiffness to get rotations below selected performance target limits ? not a real wall). The final retrofit design for the selected performance objective is illustrated in Table8.3c. The highlighted section in this figure shows the wall added to this building. The numbers in the table also show the wall designs for the structures upgraded to the CP and LS performance objectives. It should be noted that the shear wall is modeled in OpenSees using the Nonlinear Beam-Column element (de Souza, 2000). The shear wall was connected to the frame using elastic truss elements at each floor.    Columns (CP) Columns (LS) Floor Columns Existing Columns (CP) Columns (LS) ?sh  b h ?sh  b h ?sh  b h 1 Exterior 0.0036 26 28 0.006 30 32 0.012 30 32 Interior 0.005 30 36 0.012 34 40 0.012 34 40 2 Exterior 0.0031 26 28 0.006 30 32 0.012 30 32 Interior 0.004 30 36 0.012 34 40 0.012 34 40 3 Exterior 0.0031 26 28 0.0031 26 28 0.012 30 32 Interior 0.004 30 36 0.004 30 36 0.012 34 40 4 Exterior 0.0031 26 28 0.0031 26 28 0.012 30 32 Interior 0.0036 30 36 0.0036 30 36 0.012 34 40 h = height of beam-column; b = width of beam-column;  ?sh =  ratio of transverse reinforcement  (a) Retrofitted Building - Columns Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 171  Beams (CP)  Beams of  Floor Existing Beams (CP) b h ? ?' b h ? ?' 1 26 36 0.0075 0.01 26 30 0.0075 0.0075 2 26 36 0.0075 0.01 26 30 0.0075 0.0075 3 26 36 0.0075 0.01 26 30 0.0075 0.0075 4 26 36 0.007 0.0093 26 30 0.007 0.007 ? & ?? = ratio of longitudinal tension and compression reinforcement (b) Retrofitted Building - Beams   Walls (CP) Walls (LS) Wall (CP) bf tf L Lw tw ?f ?w 12 5 50 40 8 0.05 0.0025        Wall (LS) bf tf L Lw tw ?f ?w 15 10 75 55 8 0.025 0.0025 ?f & ?w = ratio of longitudinal reinforcement in the flange and web; bf & tf = width and thickness of the flange (c) Retrofitted Building - Walls Figure 8-3 Design documentation for 8-story retrofitted perimeter frame structure (members highlighted in this picture are modified for the different retrofitting measures) Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 172 8.3. PERFORMANCE ASSESSMENT BASED ON ASCE 41 The performance assessments for the three types of retrofitted buildings conducted using the ASCE 41 document are presented in the following sections. 8.3.1. Pushover Results ASCE 41 uses pushover analysis to assess the deformation demand in the component.  The pushover analysis will represent an idealized force-deformation curve. The force?deformation is idealized using a bilinear curve to compute the yield and ultimate drift. The target roof deformation, calculated using Eqn. (8-1), for all buildings are presented in Table 8-1. As seen in this table, the target roof drift ratio varies from 0.9% to 1.9%. The spectral accelerations used in this study are based on a typical location in the Los Angeles area (downtown LA).  Table 8-2 summarizes the results of the static pushover analyses.  Table 8-3 summarizes the modeling and acceptance criteria of reinforced concrete columns specified in ASCE41, Supplement 1 (2007). The parameters corresponding to condition ii (columns with a flexure-shear failure) were chosen for this building (more details regarding the different conditions and the corresponding modeling parameters can be found in Elwood et al. (2007)). Table 8-1 Target drift according to ASCE 41  Table 8-2 Results of static pushover analysis  ?y/ht: yield drift; Vy: shear yield strength; ?d/ht: drift at maximum base shear; Vd: maximum base shear;  ?T: displacement ductility  T1 (s) Sa(T1) for 2% in 50 yrs (g) C0 C1 C2 ?t (in) ?t/L (%)2.4 0.28 1.3 1 1 19.8 1.56%Columns (Collapse Prevention) 2.2 0.31 1.3 1 1 19.3 1.52%Columns (Life Safety) 2.1 0.34 1.3 1 1 17.9 1.41%Beams 3.2 0.18 1.3 1 1 23.7 1.87%Walls (Collapse Prevention) 2.0 0.34 1.3 1 1 17.4 1.37%Wall (Life Safety) 1.1 0.71 1.3 1 1 11.1 0.87%RC typeExistingRetrofitt?y/ht Vy (kips) ?d/ht Vd (kips) ?T Vd/Vy T1 (s)0.42% 563.6 1.00% 714.0 2.40 1.27 2.4Columns (Collapse Prevention) 0.36% 598.7 1.01% 758.4 2.77 1.27 2.2Columns (Life Safety) 0.36% 587.4 1.06% 834.4 2.97 1.42 2.1Beams 0.29% 252.1 1.44% 348.7 5.04 1.38 3.2Walls (Collapse Prevention) 0.40% 650.5 1.09% 699.1 2.72 1.07 2.0Wall (Life Safety) 0.34% 1477.2 1.21% 1565.8 3.56 1.06 1.10.16% 360.2 0.76% 455.2 4.66 1.26 1.7RC typeExistingRetrofittDuctileChapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 173 Table 8-3 Modeling parameters of numerical acceptance criteria for nonlinear procedures?reinforced concrete columns (for condition ii) from ASCE 41, Supplement 1 (2007 ? ASCE by permission) Conditions Modeling parameters Acceptance criteria Plastic rotations angle, radians Residual strength ratio Plastic rotations angle, radians Performance level IO Component type Primary Secondary a b c LS CP LS CP Condition ii              ? 0.1 ? 0.006 ? 3 0.032 0.060 0.2 0.005 0.024 0.032 0.045 0.060 ? 0.1 ? 0.006 ? 6 0.025 0.060 0.2 0.005 0.019 0.025 0.045 0.060 ? 0.6 ? 0.006 ? 3 0.010 0.010 0.0 0.003 0.008 0.009 0.009 0.010 ? 0.6 ? 0.006 ? 6 0.008 0.008 0.0 0.003 0.006 0.007 0.007 0.008 ? 0.1 ? 0.0005 ? 3 0.012 0.012 0.2 0.005 0.009 0.010 0.010 0.012 ? 0.1 ? 0.0005 ? 6 0.006 0.006 0.2 0.004 0.005 0.005 0.005 0.006 ? 0.6 ? 0.0005 ? 3 0.004 0.004 0.0 0.002 0.003 0.003 0.003 0.004 ? 0.6 ? 0.0005 ? 6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  Figure 8-4 presents the pushover curve for the non-ductile prototype building. This figure also highlights the sequence of the critical component failures. In this building, yielding first started in the beams, then the joints and followed by the columns. Next, the deformation in the non-ductile columns increased and resulted in the shear failure in these critical columns. Axial failure in the columns leads to the collapse of the structure under gravity load (more details in section 4.3.1); this failure pre-empts side-sway collapse. It should be noted that this building was not able to reach the target roof drift as specified by Eqn. (8-1). Figure 8-5a, 7c and 7e present the pushover curves for the retrofitted building using the strengthening, weakening and adding techniques, respectively. As explained before, the strengthening and adding techniques were selected to achieve both the CP and LS levels, while using the weakening technique, the structure could only achieve the CP level. The ASCE 41 target drifts in these figures refer to targets for retrofitted buildings. Figure 8-5b, 7d and 7f present the component demands for the non-ductile columns, the critical component, for the strengthening, weakening and adding technique, respectively. The acceptance criteria specified by ASCE 41 for the non-ductile columns are also presented in these figures.  PAg fc?----------- ? Avbws-------=Vbwd fc?------------------Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 174 Figure 8-4 Force-deformation response of the non-ductile frame 01002003004005006007008000.00% 0.20% 0.40% 0.60% 0.80% 1.00% 1.20% 1.40% 1.60% 1.80%Base Shear [kips]Roof Drift [%]ASCE 41 Target Roof DriftNon-Ductile FrameFirst Beam Yielding First Joint Yielding First Column Yielding First and Second Shear Failure in Column Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 175   Base shear vs. roof drift ratio of the non-ductile frame vs. retrofitted building using the strengthening technique. Force-deformation response of the first story column in the non-ductile frame vs. retrofitted building using the strengthening technique.   Base shear vs. roof drift ratio of the non-ductile frame vs. retrofitted building using the weakening technique. Force-deformation response of the first story column in the non-ductile frame vs. retrofitted building using the weakening technique. 01002003004005006007008009000.00% 0.20% 0.40% 0.60% 0.80% 1.00% 1.20% 1.40% 1.60% 1.80%Base Shear [kips]Roof Drift [%]ASCE 41 Target Roof DriftCPLevelRetrofitted FrameLS LevelNon-Ductile FrameRetrofitted FrameCPLevelASCE 41 Target Roof DriftLS Level0100002000030000400005000060000700000 0.02 0.04 0.06 0.08Moment [kips.in]Plastic Rotation Angle [radian]ASCE 41 CPAcceptance CriteriaRetrofitted ColumnRetrofitted Frame LSLevelNon-Ductile FrameASCE 41 CP Acceptance CriteriaUnRetrofitted ColumnASCE 41 LSAcceptance CriteriaRetrofitted ColumnRetrofitted FrameCPLevel01002003004005006007008000.00%0.20%0.40%0.60%0.80%1.00%1.20%1.40%1.60%1.80%2.00%Base Shear [kips]Roof Drift [%]ASCE 41 Target Roof DriftCPLevelNon-Ductile FrameRetrofitted FrameCPLevel0100002000030000400005000060000700000 0.02 0.04 0.06 0.08Moment [kips.in]Plastic Rotation Angle [radian]Non-Ductile FrameASCE 41 CPAcceptance CriteriaUnRetrofitted ColumnRetrofitted FrameCPLevelChapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 176   Base shear vs. roof drift ratio of the non-ductile frame vs. retrofitted building using the adding technique. Force-deformation response of the first story column in the non-ductile frame vs. retrofitted building using the adding technique. Figure 8-5 Pushover results for the retrofitted buildings020040060080010001200140016000.00% 0.20% 0.40% 0.60% 0.80% 1.00% 1.20% 1.40% 1.60%Base Shear [kips]Roof Drift [%]ASCE 41 Target Roof DriftCPLevelRetrofitted FrameLS LevelNon-Ductile FrameRetrofitted FrameCPLevelASCE 41 Target Roof DriftLS Level0100002000030000400005000060000700000 0.02 0.04 0.06 0.08Moment [kips.in]Plastic Rotation Angle [radian]Retrofitted Frame LS LevelNon-Ductile FrameASCE 41 CPAcceptance CriteriaUnRetrofitted ColumnASCE 41 LSAcceptance CriteriaUnRetrofitted ColumnRetrofitted FrameCPLevelChapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 177 8.4. GROUND MOTION SELECTION To account for the uncertainties of the structural response under different range of earthquake shaking intensities, a suite of earthquake records was used in the nonlinear dynamic analysis. In this study, incremental dynamic analysis (IDA) (Vamvatsikos and Cornell, 2002) is chosen as the method to quantify the maximum structural response as the earthquake shaking intensity increases.  Typically a site-specific probability seismic hazard analysis (PSHA) is used for the selection and scaling of the ground motion. However, due to the fact that nonlinear analyses are performed by IDA, ground motions are not scaled to match the specific target spectrum at each hazard level. Instead, the ground motions are gradually scaled until global collapse (either gravity load collapse or side-sway collapse presented in Chapter 4) takes place in the structure.  In this study, the ground motions used for the nonlinear dynamic analyses are selected from earthquakes with moment magnitude between 6.5 to 7.6 and fault rupture distances between 10 to 45 km. A total of 39 ground motion records selected by Haselton and Deierlein (2007) are adopted in this study. This set of ground motions represent an expanded version of the far-field ground motion set which was used in the FEMA P695 document (FEMA, 2009).  Table 8-4 summarizes the characteristic of the selected ground motions. To ensure the selected suite of ground motion is appropriate for the range of structural systems included in this study (non-ductile, retrofitted and ductile moment frames), appropriate spectral shape factors has been included in study. Detail procedure in adjusting the ground motion response based on the spectra shape can be found in FEMA P695 document (FEMA 2009).  Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 178 Table 8-4 Far Field Ground Motion Set (adapted from Haselton and Deierlein, 2007)    EQ Index  Magnitude  Year  Event  Fault Type  Station Name  Vs30 (m/s)  Campell Distance (km)1 6.7 1994 Northridge Blind thrust Beverly Hills - 14145 Mulhol 356 17.22 6.7 1994 Northridge Blind thrust Canyon Country - W Lost Cany 309 12.43 6.7 1994 Northridge Blind thrust LA - Saturn St 309 274 6.7 1994 Northridge Blind thrust Santa Monica City Hall 336 275 6.7 1994 Northridge Blind thrust Beverly Hills - 12520 Mulhol 546 18.46 7.1 1999 Duzce, Turkey Strike-slip Bolu 326 12.47 7.1 1999 Hector Mine Strike-slip Hector 685 128 6.5 1979 Imperial Valley Strike-slip Delta 275 22.59 6.5 1979 Imperial Valley Strike-slip El Centro Array 11 196 13.510 6.5 1979 Imperial Valley Strike-slip Calexico Fire Station 231 11.611 6.5 1979 Imperial Valley Strike-slip SAHOP Casa Flores 339 10.812 6.9 1995 Kobe, Japan Strike-slip Nishi-Akashi 609 25.213 6.9 1995 Kobe, Japan Strike-slip Shin-Osaka 256 28.514 6.9 1995 Kobe, Japan Strike-slip Kakogawa 312 3.215 6.9 1995 Kobe, Japan Strike-slip KJMA 312 95.816 7.5 1999 Kocaeli, Turkey Strike-slip Duzce 276 15.417 7.5 1999 Kocaeli, Turkey Strike-slip Arcelik 523 13.518 7.3 1992 Landers Strike-slip Yermo Fire Station 354 23.819 7.3 1992 Landers Strike-slip Coolwater 271 2020 7.3 1992 Landers Strike-slip Joshua Tree 379 11.421 6.9 1989 Loma Prieta Strike-slip Capitola 289 35.522 6.9 1989 Loma Prieta Strike-slip Gilroy Array 3 350 12.823 6.9 1989 Loma Prieta Strike-slip Oakland - Outer Harbor Wharf 249 74.324 6.9 1989 Loma Prieta Strike-slip Hollister - South - Pine 371 27.925 6.9 1989 Loma Prieta Strike-slip Hollister City Hall 199 27.626 6.9 1989 Loma Prieta Strike-slip Hollister Diff. Array 216 24.827 7.4 1990 Manjil, Iran Strike-slip Abbar 724 1328 6.5 1987 Superstition Hills Strike-slip El Centro Imp. Co. Cent 192 18.529 6.5 1987 Superstition Hills Strike-slip Poe Road (temp) 208 11.730 6.5 1987 Superstition Hills Strike-slip Westmorland Fire Sta 194 13.531 7 1992 Cape Mendocino Thrust Rio Dell Overpass - FF 312 14.332 7.6 1999 Chi-Chi, Taiwan Thrust CHY101 259 15.533 7.6 1999 Chi-Chi, Taiwan Thrust TCU045 705 26.834 7.6 1999 Chi-Chi, Taiwan Thrust TCU095 447 45.335 7.6 1999 Chi-Chi, Taiwan Thrust TCU070 401 24.436 7.6 1999 Chi-Chi, Taiwan Thrust WGK 259 15.437 7.6 1999 Chi-Chi, Taiwan Thrust CHY006 438 13.238 6.6 1971 San Fernando Thrust LA - Hollywood Stor FF 316 25.939 6.5 1976 Friuli, Italy Thrust (part blind) Tolmezzo 425 15.8Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 179 8.5. DYNAMIC RESULTS, FRAGILITY ANALYSIS AND SYSTEM PERFORMANCE The incremental dynamic analysis (IDA) is chosen as the method to quantify the structural response as the shaking intensity increases. The outcome of the IDA is used to identify the earthquake shaking intensities when the structure collapses. The spectral acceleration at the first mode period was recorded when the structure collapsed and plotted using a log-normal distribution. Figure 8-6 illustrates the system collapse fragility curve for the existing non-ductile building. This figure also demonstrates that the relationship between the Sa(T1) and probability of collapse fits well with the log-normal distribution.  It should be noted that the ground motion records were selected and scaled without considering the distinctive spectral shape of rare (extreme) ground motions, due to difficulties in selecting and scaling a different set of records for a large set of buildings having a wide range of first mode periods. To account for the important impact of spectral shape on collapse assessment, shown by Baker and Cornell (2006), the collapse predictions made using the general set of ground motions are modified using a method proposed by Haselton et al. (2009). The expected spectral shape of rare (large) California ground motions is accounted for through a statistical parameter referred to as epsilon, which is a measure of the difference between the spectral acceleration of a recorded ground motion and the median value predicted by ground motion attenuation Eqn.. A target value of ?=1.5 is used to approximately represent the expected spectral shape of severe ground motions that can lead to collapse of code-conforming buildings (Appendix B of FEMA P695, 2009; Haselton et al., 2010). The spectral shape factor, SSF, is computed using the following equation:  = 	 ? 	()            (8-2) where ?1 depends on building inelastic deformation capacity; ?0 depends on the Seismic Design Category (SDC) and is equal to 1.0 for SDC B/C, 1.5 for SDC D, and 1.2 for SDC E; and ?(?) is the mean value of the Far-Field record set listed in Table 8-4. In all cases, the buildings were designed for a single level of high seismic ground motions representing Seismic Design Category (SDC) D buildings, therefore, ?0=1.5. ?1 and ?(?) are computer using the following equations:  = 0.14	 ? 1.                  (8-3) 	() = 0.6(1.5 ? )         (8-4)  Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 180  Figure 8-6 Probability of collapse vs. Sa(T1) for non-ductile building Table 8-5 summarizes the SSF values for the different buildings. Figure 8-7a and b demonstrate the fragility curves before and after the SSF modifications. The SSF modification will result in a shift in the CDF. Figure 8-7b clearly shows the significant variability in the probability of collapse of the retrofitted structures, which fill the spectrum between the non-ductile and modern design buildings. It should be noted that the retrofitted building using shear walls and upgraded to the LS performance level has a collapse performance better than the modern designed building (building similar to the 8-story non-ductile perimeter frame but designed using modern building codes), while the weakening retrofit approach only provides a slight improvement on the probability collapse over the existing building.  Table 8-5 Spectral shape factor  0%10%20%30%40%50%60%70%80%90%100%0 0.05 0.1 0.15 0.2 0.25 0.3 0.35P [Collapse]Sa (T1 = 2.36 s) [g]b1 ? 0 ? SSF0.16 1.5 0 1.27Columns (Collapse Prevention) 0.18 1.5 0 1.31Columns (Life Safety) 0.19 1.5 0 1.32Beams 0.25 1.5 0 1.46Walls (Collapse Prevention) 0.18 1.5 0 1.30Wall (Life Safety) 0.21 1.5 0 1.370.24 1.5 0 1.59RC typeExistingRetrofittDuctileChapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 181  (a) before spectral shape factor applied   (b) after spectral shape factor applied Figure 8-7 IDA results for the retrofitted buildings 0%10%20%30%40%50%60%70%80%90%100%0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2P [Collapse]Sa (T1) [g]ExistingColumns (Collapse Prevention)Columns (Life Safety)Beams (Collapse Prevention)Walls (Collapse Prevention)Walls (Life Safety)Ductile0%10%20%30%40%50%60%70%80%90%100%0 0.5 1 1.5 2 2.5P [Collapse]Sa (T1) [g]ExistingColumns (Collapse Prevention)Columns (Life Safety)Beams (Collapse Prevention)Walls (Collapse Prevention)Walls (Life Safety)DuctileChapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 182 The probability of collapse at the MCE (2% in 50 yr return period) intensity is shown with red circles in Figure 8-8. The circles indicate a variety of collapse ratios at this intensity level for the different retrofitting techniques and performance objectives. The first mode period of the existing, retrofitted and modern designed buildings are all considered as being in the intermediate period range and therefore the dominant period has an inverse relationship with the spectral acceleration. This variety in collapse performance can be compared by normalizing the collapse fragility curves with the spectral acceleration at the MCE intensity. The result is shown in Figure 8-9.  Figure 8-9 clearly indicates when the spectral acceleration is normalized to the MCE intensity, the collapse fragilities for the different retrofitted buildings has a smaller variability compared to the original collapse fragility curve (Figure 8-8). In addition to this, the acceptable probability of collapse at the MCE level which is defined as 10% by FEMA P695 is also shown in Figure 8-9. The non-ductile existing building does not meet this criteria and the modern code-conforming structure clearly passes this criteria. However, all the retrofitted buildings are very close to this criterion.    Figure 8-8 Collapse fragility curves for different buildings 0%10%20%30%40%50%60%70%80%90%100%0 0.5 1 1.5 2 2.5P [Collapse]Sa (T1) [g]ExistingColumns (Collapse Prevention)Columns (Life Safety)Beams (Collapse Prevention)Walls (Collapse Prevention)Walls (Life Safety)DuctileP[Collapse | Sa 2% in 50yrs]Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 183  Figure 8-9 Collapse fragility curves normalized by Sa 2% in 50 yrs Different collapse performance metrics are reported in Table 8-6. The predicted median collapse capacity, MedianSacol(T1), of the retrofitted structures is approximately 1.5 to 6 times larger than the existing non-ductile building. The probability of collapse in 50 years shows a higher difference, this performance metric decreases from 52% for the non-ductile building down to a range of 5% - 11% for the retrofitted buildings which is comparable to the 4% probability of collapse for the modern design structure.   Table 8-6 Collapse performance metrics    0%10%20%30%40%50%60%70%80%90%100%0 1 2 3 4 5P [Collapse]Sa (T1)/Sa2%in50yrs [%]ExistingColumns (Collapse Prevention)Columns (Life Safety)Beams (Collapse Prevention)Walls (Collapse Prevention)Walls (Life Safety)Ductile10% acceptable Pcol|MCE specified in FEMA P695Sa(T1) for P[col] P[col] 2% in 50 yrs for Sa(T1) in 50 yrs0.28 80.8% 0.19 0.47 1.5E-02 51.7%Columns (Collapse Prevention) 0.31 19.6% 0.46 0.47 2.3E-03 10.8%Columns (Life Safety) 0.34 12.0% 0.59 0.50 1.4E-03 6.9%Beams 0.18 10.9% 0.30 0.42 2.3E-03 10.9%Walls (Collapse Prevention) 0.34 8.9% 0.55 0.36 1.2E-03 5.6%Wall (Life Safety) 0.71 12.1% 1.12 0.40 1.1E-03 5.2%0.47 2.6% 1 0.40 8.0E-04 3.9%Ductile?LN[Sa,col(T1)] ?collapseExistingRetrofittRC type MedianSa,col(T1)Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 184 An overall summary of the three sets of buildings is presented in Table 8-7. This table clearly demonstrates an approximate estimation of the change in the material required to rehabilitate the existing non-ductile building. Modifying the beams would probably cost the least to retrofit (with the assumption that the labor cost to perform all three techniques would approximately be the same) and has the the least beneficial effect regarding the seismic collapse safety. On the other hand, adding shear walls and retrofitting to the LS performance level would cost the most but has the highest collapse performance (based on P[col] in 50 years).   Table 8-7 System performance of the three set of buildings  * Obtained from Haselton (2007)   Concrete Steel Concrete SteelExisting (Non-Ductile) - - 0.043 0.07 786 48 - - 51.7%Enhance Columns Collapse Prevention 0.06 0.04 810 49 3% 2% 10.8%Enhance Columns Life Safety 0.035 0.02 836 50 6% 4% 6.9%Weaken BeamsCollapse Prevention 0.043 0.03 786 48 0% 0% 10.9%Add ShearWallCollapse Prevention 0.043 0.019 835 50 6% 4% 5.6%Add ShearWall Life Safety 0.0161 0.005 868 56 10% 18% 5.2%New (Ductile) * - - - - - - - - 3.9%RetrofittBuilding Type Retrofit TechniquePerformance LevelASCE 41 Table 4.2 (Reinforced Concrete Columns)Material ListCollapse Margin (P[Col] in 50 yrs)Acceptance Criteria DemandWeight [kips] Change (compared to existing building)Chapter 8: Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted and Ductile Moment Frames 185 8.6. CONCLUSIONS Advancements in nonlinear dynamic analysis, seismic hazard analysis, and performance-based earthquake engineering are enabling more scientific assessment of structural collapse risk. The main objective of this study is to quantify and compare the seismic safety of non-ductile, retrofitted, and ductile RC buildings. At present, the general assumption when retrofitting a non-ductile building is that their seismic performance is enhanced such that it can reach the performance corresponding to ductile buildings designed based on current seismic codes. However, as shown in this study, different rehabilitation measures do not exhibit the same performance metrics as ductile buildings. This chapter has shown, by means of detailed nonlinear dynamic analysis, several seismic performance metrics (probability of collapse at Sa2/50, annual mean of probability of collapse, collapse metric ratio) of non-ductile, retrofitted, and ductile concrete frames. Based on the evaluation of the concrete frames presented in this study, the following observations are made: ? Retrofit provides an intermediate level of seismic performance, between non-ductile structures and modern code-conforming structures. ? The study finds that retrofitting schemes by the means of modifying the columns or beams and up to only the CP level has the least beneficial effect regarding the seismic collapse safety and conversely adding a shear wall will significantly improve the collapse performance. ? The non-ductile building does not meet the acceptable probability of collapse at the MCE level defined by FEMA P695 (10%) and the modern code-conforming structure clearly passes this criteria. However, all the retrofitted buildings are very close to this criterion. ? The predicted median collapse capacity (MedianSa,col(T1) in Table 8-6) of the retrofitted structures is approximately 1.5 to 6 times larger than the existing non-ductile building.  The probability of collapse in 50 years decreases from 52% (for the non-ductile building) down to a range of 5% - 11% for the retrofitted buildings, which is comparable to the 4% probability of collapse for a modern code-conforming structure. Notation List 186 Chapter 9. SUMMARY, CONCLUSIONS, AND FUTURE WORK 9.1. SUMMARY One of the greatest risks to life and property in seismically active regions around the world results directly from ground shaking associated with major urban earthquakes. These risks and correlated losses are varied in the affected regions, based on the vulnerability of individual buildings and ground conditions. Engineers need tools to accurately and efficiently assess the risk of individual buildings and to differentiate between those that are safe and those that are potential collapse hazards. Public and private stakeholders and decision makers all need better information on the extent of the risks so that effective policy measures and action plans can be put in place to mitigate the risks. Constructed on the framework of performance-based earthquake engineering, this dissertation focuses on advancement and application of collapse probabilities to the seismic assessment of existing concrete moment frames.  This dissertation includes important contributions to (1) modeling techniques for components in existing concrete frames through the development of a mechanical model for existing concrete columns, (2) development of system-level collapse criteria, and (3) application of collapse fragilities in defining collapse indicators, improving loss estimation of existing concrete frames, and differentiating the expected performance of existing and retrofitted concrete frames. The following sections highlight the important findings, limitations and future work related to this dissertation. 9.2. FINDINGS The findings and contributions for each chapter are summarized hereafter. 9.2.1. Mechanical Model for Non-Ductile Reinforced Concrete Columns (Chapter 3) Chapter 3 presents a mechanical model developed to simulate the non-ductile behaviour of concrete columns. Models for flexural, bar slip, and shear deformations are combined to obtain the total lateral drift response of a column. Most notably, the pre-peak shear behaviour, point of shear failure, and post-peak shear behaviour are determined based on mechanics principles, making the proposed column model applicable to a broader range of columns than similar empirical models (e.g., Elwood, 2004). Shear failure is determined based on the shear Notation List 187 deformations exceeding deformation limits based on the Modified Compression Field Theory (MCFT), in the plastic hinge zones, while the post peak response is determined based on shear-friction concepts. The two types of shear failure, diagonal tension and compression failure, are numerically represented in the mechanical model and are used to detect shear failure. The mechanical model is implemented in nonlinear analysis software (OpenSees) to demonstrate its capability of representing with acceptable accuracy the hysteretic response in columns with different failure modes.  The proposed model is verified using experimental data where the columns experienced shear failure either before or after flexural yielding. The comparison of the analytical model with the experimental data indicates that the numerical model adequately captures the pre-peak response and the point of shear failure. The model is found to provide a more accurate estimation of the test results for columns sustaining diagonal compression failures compared with columns experiencing diagonal tension failures.  9.2.2. System-Level Collapse Assessment of Non-Ductile Concrete Frames (Chapter 4) Collapse of most non-ductile concrete buildings will be controlled by the loss of support for gravity loads prior to the development of a side-sway collapse mechanism. System-level collapse criteria, namely gravity-load and side-sway collapse, are numerically defined in this chapter. A systematic approach for capturing gravity load collapse in beam?column and slab?column frames are presented and implemented in nonlinear analysis software (OpenSees). Chapter 4 presents the application of these collapse criteria to previously tested shaking table specimens to examine their accuracy, reliability, and practicality. Comparison of the analytical model with shake table test data indicates that the system-level collapse criteria implemented in the analytical model is capable of capturing the collapse observed in the test.  Numerous sources of uncertainty complicate the ability to identify buildings that are vulnerable to collapse. For this reason, it is important to develop estimates of collapse probability which account for all significant sources of uncertainties. The collapse criteria are used to develop estimates of collapse probability of the previously tested shaking specimen, taking into consideration uncertainties in failure models and ground motions. The results indicate that both the median and the dispersion increases when uncertainties in failure models are also considered in the collapse fragilities compared to the collapse probabilities only from record-to-record variability. Notation List 188 9.2.3. Collapse Indicators ? Methodology (Chapter 5) The risk associated with older non-ductile concrete buildings during an earthquake is significant, and the development of improved technologies to mitigate this risk is a large and costly undertaking. Because of the limited funding available to retrofit all older non-ductile concrete buildings, it is crucial to identify the most likely to collapse buildings and fix those first. A methodology for identifying collapse indicators, design and response parameters that are correlated with ?elevated? collapse probability, based on results of comprehensive collapse simulations and estimation of collapse probabilities for a collection of building prototypes is described in Chapter 5. The proposed procedure for identifying the limits on the collapse indicators is divided into six general steps: Identify potential collapse indicators, Build a detailed numerical model, Perform seismic hazard calculations and select ground motion records, Execute nonlinear response history analysis (record-to-record variability), Perform probabilistic analysis, and Assessment procedure and post-processing the results. Each step is explained in detail in this chapter. Different risk-based and intensity-based collapse assessment approaches are used to determine the limits on the collapse indicators. For a risk-based approach, a suitable mean annual frequency of collapse (?collapse) is used to define the limit. For the intensity-based approach, either a particular probability of collapse at a specific hazard level is used to define the limit on the collapse indicator or the relative changes in the collapse fragilities (at a specific hazard level) are used to determine the limit.  9.2.4. Collapse Indicators ? Implementation (Chapter 6) The methodology introduced in Chapter 5 for identifying collapse indicators based on results of comprehensive collapse simulations and estimation of collapse probabilities for a collection of building prototypes is evaluated and validated in Chapter 6. The details of the procedure are illustrated for 4-, 7-, and 12-story moment frames designed for high and moderate seismicity and with different number of bays. The 12-story building demonstrates markedly superior seismic collapse performance when compared with the 4- and 7-story buildings.  Six design parameters and four response parameters are chosen to evaluate the methodology. The results suggest that some of the collapse indicator limits are somehow relevant to the number of stories, period of the buildings, and the hazard level for each building.  Notation List 189 One of the main conclusions from the results of the limited buildings considered in this chapter is that it is nearly impossible to set limits for the design parameter collapse indicators. The buildings will most likely reach an acceptable collapse risk only by varying a vector of design parameters; typically multiple vulnerabilities contribute to making a building a collapse hazard.  On the other hand the response parameters have indicated that they could be used in future rehabilitation standards to define limits for system-level engineering demand parameters with specific collapse risk levels. The two most promising response parameters are the maximum number of columns with shear failures and the maximum interstory drift ratio. The results for the maximum number of columns with shear failures suggest that the hazard level will dictate the limit for this collapse indicator and this limit is relatively independent of the number of floors and bays/floor. The results for the maximum interstory drift ratio clearly show that the limit for this collapse indicator is independent from the vulnerability and hazard characteristics of the buildings and this collapse indicator could be used to distinguish buildings with a high collapse risk regardless of the site and building characteristics. 9.2.5. Seismic Loss Estimation of Non-Ductile Reinforced Concrete Buildings (Chapter 7) Performance-based earthquake engineering, which aims to describe the seismic performance of facilities, has been used in Chapter 7 to estimate the expected losses for non-ductile reinforced concrete buildings for a wide range of earthquake shaking intensities. Three collapse criteria have been included and compared in this chapter: (1) failure of the first structural component, adopted from ASCE/SEI 41 (ASCE, 2006); (2) formation of a ?side-sway? collapse mechanism, adopted from FEMA P695 (FEMA P695A, 2009); and (3) failure of side-sway or gravity-load collapse. A seven-story non-ductile concrete frame building located in Los Angeles, California, is used as an example to illustrate the loss simulation of non-ductile reinforced concrete structures. The results show that collapse does not occur in low earthquake shaking intensities and losses are dominated by non-structural damage. At higher shaking intensities, the first structural component failure criteria, has the highest collapse probability and overestimates the financial loss because it ignores the structure?s ability to redistribute the forces. On the other hand, the side-sway only collapse criterion, underestimates the loss due to the likelihood that gravity-load collapse will happen in a non-ductile concrete building before the side-sway mechanism develops. For the example structure, the annual rate of experiencing losses less than Notation List 190 22% of the replacement cost is not dependent on whether collapse is considered in the loss assessment.  In contrast, for losses greater than 50% of the replacement cost, the consideration of collapse in the loss assessment will more than double the estimated annual rate of exceedance.  If the annualized loss is used as the decision variable to decide if seismic rehabilitation is necessary, the results indicate that while the consideration of collapse in determining the annualized loss is important, the choice of collapse criteria is less significant.  9.2.6. Collapse Assessment of Reinforced Concrete Buildings: Comparison of Non-Ductile, Retrofitted, and Ductile Moment Frames (Chapter 8) The main objective of Chapter 8 is to quantify and compare the seismic safety of a non-ductile, retrofitted, and ductile RC building. At present, the general interpretation of retrofitting non-ductile buildings is to enhance their seismic performance such that it can reach the level corresponding to ductile buildings designed based on current seismic codes. However, as shown in this chapter, different rehabilitation measures do not result in the same performance metrics as those of ductile buildings. The purpose of this chapter was to use detailed nonlinear dynamic analysis to assess the probability of collapse and to extract seismic performance metrics (probability of collapse at Sa2/50, annual mean of probability of collapse, collapse metric ratio) of non-ductile, retrofitted, and ductile concrete frames. Based on the evaluation of the concrete frames presented in this chapter, retrofit provides an intermediate level of seismic performance between non-ductile structures and modern code-conforming structures. The study finds that retrofitting schemes where the columns or beams are modified only up to the CP level have the least beneficial effect regarding seismic collapse safety, and conversely, adding a shear wall will significantly improve this structural performance parameter. The non-ductile building does not meet the acceptable probability of collapse at the MCE level defined by FEMA P695, and the modern code-conforming structure clearly passes this criteria. However, the retrofitted buildings fall very closely meet the values of the probability of collapse of 10% at Sa2/50 and it could be concluded that the performances of these buildings have been, at minimum, enhanced to meet this criteria. The predicted median collapse capacity of the retrofitted structures is approximately 1.5 to 6 times larger than that of the existing non-ductile building. The probability of collapse in 50 years decreases from 52% (for the non-ductile building) down to a range of 5% to 11% for the Notation List 191 retrofitted buildings, which is comparable to the 4% probability of collapse for a modern code-conforming structure. 9.3. FUTURE RESEARCH The basic ingredient of the framework presented in this dissertation is a collapse fragility curve which presents the probability of collapse as a function of an intensity measure. In this study, many assumptions are made in the development of the fragility curve and its application for quantifying the collapse potential of existing concrete frames. Several limitations and future work are presented below: ? In Chapter 3, the mechanical model introduced to simulate the non-ductile behaviour of existing columns, divides shear failure into diagonal compression and tension failure. The column model is found to provide a more reasonable estimation of the test results for the post-peak response of columns sustaining diagonal compression failures compared with columns experiencing diagonal tension failures. Additional research is required to improve the post-peak behaviour for columns which experience diagonal tension failure. In addition, the cyclic parameters used to simulate the degrading behaviour should be calibrated based on test data. ? The two collapse criteria developed in Chapter 4 are meant to be general and applied to any building type; however, only moment frames have been analyzed and verified. It would be useful to conduct similar studies for other existing building types, especially those with concrete shear walls. The results of such a study would help the profession to better understand expected collapse performance for various types of building systems. ? All structural models used in this study are two-dimensional. It should be noted that these two-dimensional models provide a relatively simple representation of a very complex phenomenon at the point of gravity load failure, and hence, may lack some sophistication required to accurately capture the behaviour of the building to the point of total collapse. In a real building, the irregularities necessitate a three-dimensional model and the modeling concept for gravity load failure should also be extended to three-dimensional models of concrete buildings. ? Additional research is required to establish limits for use in design practice and to improve the methodology to address the interaction of multiple collapse indicators Notation List 192 introduced and examined in Chapters 5 and 6, respectively. There is a need to consider a broad range of buildings, varying the number of stories, bays, hazard region, lateral-load-resisting system, etc. Ongoing studies funded by FEMA and NIST through ATC-78 and ATC-95, respectively, are expected to result in specific guidance for practicing engineers, based in part on some of the concepts presented in this dissertation. ? The scope of the study conducted in Chapter 7 focuses only on the direct capital loss of existing concrete frames. 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(2009b). "Collapse of a non-ductile concrete frame: Evaluation of analytical models." Earthquake Engineering and Structural Dynamics 38, 225-241.  Youssef M. and Ghobarah A. (2001). "Modelling of RC beam-column joints and structural walls." Journal of Earthquake Engineering 5, 93-111. Zhu L., Elwood K. and Haukaas T. (2007). "Classification and seismic safety evaluation of existing reinforced concrete columns." Journal of Structural Engineering 133, 1316. Appendix A: Ground Motion Selection and Scaling 204 Appendix A. GROUND MOTION SELECTION AND SCALING A.1. INTRODUCTION This section includes a short summary of the process taken to select and scale proper ground motion time histories. These records are required to perform nonlinear time history analysis on a two-dimensional OpenSees model representing the perimeter frame of the Holiday Inn structure (the 7-story building used in chapters 6 and 7). This short summary addresses the following aspects of the hazard analysis performed for the two-dimensional Van Nuys building: ? Site and structure conditions ? Uniform hazard spectra ? Deaggregation of the hazard ? Process of Selecting Ground Motion Recordings ? Compute the Conditional Mean Spectrum ? Scaling of the Ground Motion Recordings A.2. SITE AND STRUCTURE CONDITIONS The Van Nuys Holiday Inn is located on a soil site (NEHRP Class D) in the San Fernando Valley which has both a variety of faults lying beneath it and the large San Andreas Fault passing some 50 kilometres to the northeast. This site class, Class D, can have a shear wave velocity, Vs30, ranging from 180-270 m/s. For this specific site, 218 (m/s) is used in all hazard analysis. The Van Nuys Holiday Inn model simulated in OpenSees has a first-mode period of 1 second. Therefore this value is used in all the analysis when T1 is referred. A.3. UNIFORM HAZARD SPECTRA The Uniform Hazard Spectrum is constructed by enveloping the spectral amplitudes at all periods that are exceeded with a particular probability (such as 1%, 2% ?) in given years (e.g. 50 years), as computed using probabilistic seismic hazard analysis (PSHA). PSHA also provides information about the earthquake events most likely to cause occurrence of the target spectral amplitude at a given period.  Uniform hazard spectra for any site (in the U.S.) can be derived from the USGS probabilistic ground motion maps available online in http://eqint.cr.usgs.gov/deaggint/2008/. A  Appendix A: Ground Motion Selection and Scaling 205  Figure A-1 USGS probabilistic ground motion map (http://eqint.cr.usgs.gov/deaggint/2008/) snapshot of the website is shown in Figure A-1. The parameters inserted as inputs to derive the UHS curve are the address of the site of interest and also the return period, fundamental period and site conditions (shown in Table A-1). The result will be the spectral acceleration for the initial period of interest, Sa(T1). Figure A-2 shows the U.S. Geological Survey Uniform Hazard Spectrum (UHS) with 1%, 2%, 5%, 10%, 20% and 50% probability of exceedance in 50 years design spectrum for a site in Van Nuys, California (latitude/longitude = 34.199 N/118.498 W).  Figure A-2 Probabilistic Uniform Hazard Spectra for the site in Van Nuys, California for six levels of annual exceedence probability modified for local site conditions (NEHRP class D soil) 00.511.522.50 0.5 1 1.5 2 2.5Spectral Acceleration (g)Period (sec)Soil UHS for Van Nuys Site 1%/50 yrs UHS2%/50 yrs UHS5%/50 yrs UHS10%/50 yrs UHS20%/50 yrs UHS50%/50 yrs UHSAppendix A: Ground Motion Selection and Scaling 206 Table A-1 Input parameters required to derive a UHS curve Address Return Period (% in 50 yrs) Spectral Acceleration Period of Interest (sec) Vs30 (m/s) 91406 1%, 2%, 5%, 10%, 20% & 50% PGA, 0.1, 0.2, 0.3, 0.5, 1 & 2 218  Ground motion hazard curves for a typical highly seismic site in Los Angeles (not near field) are illustrated in Figure A-3. The hazard curve defines the mean annual frequency of exceeding specified ground motion intensity at the site. When the collapse fragility curve and the hazard curve are integrated together, another possible metric for collapse performance, the mean annual frequency of collapse, is obtained, which describes how likely collapses are to occur. The USGS estimate of the hazard curve for Sa(T1) (i.e., 1 sec) is shown in Figure A-3.  A.4. DEAGGREGATION OF THE HAZARD Deaggregation is the process of decomposing the hazard (i.e., the annual probability of exceedance) into its various additive components. This can be done by faults, or it can be done  Figure A-3 Hazard curve for the Sa (T1 = 1 sec) for the Van Nuys Site, based on the USGS website results and modified for local site conditions (NEHRP class D soil) 00.0050.010.0150.020.0250 0.2 0.4 0.6 0.8 1 1.2 1.4Frequency of exceedance per yearSa (g)Appendix A: Ground Motion Selection and Scaling 207 by magnitude and distance. This procedure can be executed using online deaggregation programs (e.g., http://eqint.cr.usgs.gov/deaggint/2008/). Figure A-4 shows the deaggregation distribution of magnitudes, distances, and ??s (?epsilons?) that will cause the occurrence of four different Sa(T1 = 1 sec) with various hazard levels at the Van Nuys site. The deaggregation of the hazard shows that the hazard at this site is dominated by nearby earthquakes. The higher ground motions for the 2% in 50 year probability level than for the 10% in 50 year level will reflect not larger magnitudes, but higher ground motion levels for the same magnitude (larger number of standard deviations above the mean). Figure A-4 confirms that the largest contributions to this hazard level arise from very close events of magnitude 6.5 to 7.5. This can be summarized by the mean (or modal) M and R values of this deaggregation distribution. These mean M and R values are provided by USGS and appear in Table A-2. The mean magnitude is 6.95 and the mean distance becomes smaller for smaller probabilities (or larger ground motions). Appendix A: Ground Motion Selection and Scaling 208  (a) Given exceedance of the Sa value with 2475 year return period  (b) Given exceedance of the Sa value with 975 year return period Appendix A: Ground Motion Selection and Scaling 209  (c) Given exceedance of the Sa value with 475 year return period  (d) Given exceedance of the Sa value with 72 year return period Figure A-4 PSHA deaggregation for Van Nuys, given different return periods for Sa (T1 = 1 sec) (Figure from USGS Custom Mapping and Analysis Tools, http://eqint.cr.usgs.gov/deaggint/2008/.)Appendix A: Ground Motion Selection and Scaling 210    Table A-2 Deaggregation of Uniform Hazard Spectra for different hazard levels IM [Sa (T1 = 1 sec)] (g) level Hazard Level Return Period Range for  R  (km) Range for M (magnitude) Rmean (km) Mmean ?0,mean Rmodal (km) Mmodal ?0,modal Rmode (km) Mmode 1.25 4975 12.9 - 14.3 6.6 - 7.35 16.9 6.95 1.8 14.3 6.76 1.72 13.9 6.8 1.06 2475 14.1 - 14.2 6.6 - 6.98 18.3 6.96 1.65 14.1 6.6 1.76 13.8 6.6 0.82 975 14 6.6 - 6.77 20.7 6.96 1.41 14 6.6 1.4 14.1 6.6 0.65 475 13.9 - 14.1 6.61 - 6.77 23.2 6.97 1.18 13.9 6.61 1.11 13.8 6.6 0.49 224 13.6 - 14.1 6.6 - 6.77 26.6 6.97 0.91 14.1 6.6 0.64 14 6.6 0.29 72 14 6.61 34.7 6.97 0.4 14 6.61 -0.1 13.9 6.6 Appendix A: Ground Motion Selection and Scaling 211 A.5. CONDITIONAL MEAN SPECTRUM A CMS estimates the median geometric-mean spectral acceleration response of a pair of ground motions given an M, R pair and a target spectral ordinate, Sa(T1), for which ?(T1) is back calculated using an appropriate attenuation relationship. A simple four step procedure for computing the Conditional Mean Spectrum (CMS) is presented in Baker and Cornell (2006). The CMS for this site, is plotted for all six hazard levels in Figure A-5. A set of Matlab scripts is used to build the six conditional mean spectrums and to select 40 ground motions for each hazard level    (http://www.stanford.edu/~bakerjw/gm_ selection.html). The inputs for the Matlab scripts are: ? nGm (number of ground motions)  ? T1 (period of interest) ? outputFile ? M_bar (mean magnitude) ? R_bar (mean distance) ? eps_bar (mean epsilon), this value depends on the attenuation relationship used to build the CMS and should be modified if using a different attenuation relationship. ? Vs30 (soil type) A.6. PROCESS OF SELECTING GROUND MOTION RECORDINGS The output of the program will be a list of 40 ground motions which based on the number of ground motions required for each project a selection procedure could be used. For this project the following criteria were used: ? Select 1 record per earthquake ? Scale factor should be less than 2 ? The average of the spectra of selected records should be above the 90% of the target spectra in the scaling range of periods 10 ground motions are selected for each hazard level and listed in Table A-3.  Appendix A: Ground Motion Selection and Scaling 212  Figure A-5 Conditional mean values of spectral acceleration at all periods, given Sa(1s) and six hazard levels A.7. SCALING OF THE GROUND MOTION RECORDINGS For each set of recordings, a scaling factor is found by matching the time history to the conditional mean spectrum (CMS) for a period range of 0.2T1 to 1.5T1 (for T1 = 1 sec). For each ground motion record, the linear spectrum is generated by the computer program Bispec (http://eqsols.com/Bispec.aspx).  The ground motions are scaled in two consecutive steps. In the first step, each record is scaled to have the same area under the CMS curve (in the range of 0.2T1 to 1.5T1). In the second step, the average of all scaled records (shown as ?Average? in Figure A-6) is scaled to have the minimum sum of squared errors (SSE) between the logarithms of the ground motion?s spectrum and the target spectrum  = ? 	? 	     (A-1) Ground motions scaled to the CMS for a 4975 years return period hazard level is illustrated in Figure A-6. The scale factors for all the selected GMs (shown in Table A-3) corresponding to the hazard level of interest is demonstrated in Table A-4. 00.20.40.60.811.21.41.61.820 0.5 1 1.5 2 2.5 3Spectral Acceleration, Sa (g) for x = 5%Period, T (sec)1%/50 yrs CMS2%/50 yrs CMS5%/50 yrs CMS10%/50 yrs CMS20%/50 yrs CMS50%/50 yrs CMS1%/50 yrs UHS2%/50 yrs UHS5%/50 yrs UHS10%/50 yrs UHS20%/50 yrs UHS50%/50 yrs UHSAppendix A: Ground Motion Selection and Scaling 213  Table A-3 GMs representing six hazard levels suitable for the Van Nuys Building site Earthquake Mw Station Record Distance Site PGA (g)San Fernando 1971/02/09 6.6 94 Gormon - Oso Pump Plant OPP270 48.1 soil 0.105Northridge 1994/01/17 6.7 90054 LA - Centinela St CEN155 30.9 soil 0.465Northridge 1994/01/17 6.7 90091 LA - Saturn St STN020 30 soil 0.474Northridge 1994/01/17 6.7 24303 LA - Hollywood Stor FF HOL360 25.5 soil 0.358Northridge 1994/01/17 6.7 90063 Glendale - Las Palmas GLP177 25.4 soil 0.357Whittier Narrows 1987/10/01 6 24303 LA - Hollywood Stor FF A-HOL000 25.2 soil 0.221Landers 1992/06/28 7.3 22074 Yermo Fire Station YER270 24.9 soil 0.245Loma Prieta 1989/10/18 6.9 57425 Gilroy Array #7 GMR090 24.2 soil 0.323Whittier Narrows 1987/10/01 6 90012 Burbank - N Buena Vista A-BUE250 23.7 soil 0.233Imperial Valley 1979/10/15 6.5 6617 Cucapah H-QKP085 23.6 soil 0.309Chi-Chi, Taiwan 1999/09/20 7.6 TCU042 TCU042-N 23.34 soil 0.199Imperial Valley 1979/10/15 6.5 5059 El Centro Array #13 H-E13230 21.9 soil 0.139San Fernando 1971/02/09 6.6 135 LA - Hollywood Stor Lot PEL090 21.2 soil 0.21Whittier Narrows 1987/10/01 6 90084 Lakewood - Del Amo Blvd A-DEL000 20.9 soil 0.277Whittier Narrows 1987/10/01 6 90063 Glendale - Las Palmas A-GLP177 19 soil 0.296Whittier Narrows 1987/10/01 6 14368 Downey - Co Maint Bldg A-DWN180 18.3 soil 0.221Chi-Chi, Taiwan 1999/09/20 7.6 CHY035 CHY035-N 18.12 soil 0.246Imperial Valley 1979/10/15 6.5 6621 Chihuahua H-CHI012 17.7 soil 0.27Duzce, Turkey 1999/11/12 7.1 Bolu BOL090 17.6 soil 0.822Whittier Narrows 1987/10/01 6 90078 Compton - Castlegate St A-CAS270 16.9 soil 0.333N. Palm Springs 1986/07/08 6 12025 Palm Springs Airport PSA090 16.6 soil 0.187Loma Prieta 1989/10/18 6.9 57382 Gilroy Array #4 G04000 16.1 soil 0.417Northridge 1994/01/17 6.7 90053 Canoga Park - Topanga Can CNP196 15.8 soil 0.42Chi-Chi, Taiwan 1999/09/20 7.6 TCU123 TCU123-W 15.12 soil 0.164Chi-Chi, Taiwan 1999/09/20 7.6 CHY006 CHY006-E 14.93 soil 0.364Loma Prieta 1989/10/18 6.9 47125 Capitola CAP000 14.5 soil 0.529Northridge 1994/01/17 6.7 90003 Northridge - 17645 Saticoy St STC180 13.3 soil 0.477Northridge 1994/01/17 6.7 90057 Canyon Country - W Lost Cany LOS270 13 soil 0.482Whittier Narrows 1987/10/01 6 90077 Santa Fe Springs - E Joslin A-EJS318 10.8 soil 0.443Imperial Valley 1979/10/15 6.5 5115 El Centro Array #2 H-E02140 10.4 soil 0.315Northridge 1994/01/17 6.7 24279 Newhall - Fire Sta NWH360 7.1 soil 0.5972 yrsHazard Level Return Period4975 yrs2475 yrs975 yrs475 yrs224 yrsAppendix A: Ground Motion Selection and Scaling 214  Table A-4 Scale Factor of GMs for six different hazard levels      Earthquake Record PGA (g) 4975 2475 975 475 224 72San Fernando 1971/02/09 OPP270 0.105 3.1529Northridge 1994/01/17 CEN155 0.465 1.0879Northridge 1994/01/17 STN020 0.474 0.5759Northridge 1994/01/17 HOL360 0.358 0.8885 0.5799Northridge 1994/01/17 GLP177 0.357 1.9063 1.2441Whittier Narrows 1987/10/01 A-HOL000 0.221 2.8727 1.8749Landers 1992/06/28 YER270 0.245 0.86 0.5613Loma Prieta 1989/10/18 GMR090 0.323 1.5474 1.0099Whittier Narrows 1987/10/01 A-BUE250 0.233 1.8675 1.2188Imperial Valley 1979/10/15 H-QKP085 0.309 0.9856 0.6433Chi-Chi, Taiwan 1999/09/20 TCU042-N 0.199 1.5378 1.185 0.7734Imperial Valley 1979/10/15 H-E13230 0.139 1.2853 2.6694San Fernando 1971/02/09 PEL090 0.21 1.801 1.3879Whittier Narrows 1987/10/01 A-DEL000 0.277 1.49 1.2205 0.9405Whittier Narrows 1987/10/01 A-GLP177 0.296 3.056 2.5032 1.929Whittier Narrows 1987/10/01 A-DWN180 0.221 2.3074 1.8123 1.4845 1.144Chi-Chi, Taiwan 1999/09/20 CHY035-N 0.246 1.4438 1.134 0.9289 0.7158Imperial Valley 1979/10/15 H-CHI012 0.27 2.074 1.629 1.3343Duzce, Turkey 1999/11/12 BOL090 0.822 0.8551 0.6716 0.5501Whittier Narrows 1987/10/01 A-CAS270 0.333 4.1707 3.6632 2.8773 2.3568N. Palm Springs 1986/07/08 PSA090 0.187 5.6653 4.976 3.9084 3.2014Loma Prieta 1989/10/18 G04000 0.417 1.9529 1.7153 1.3473 1.1035Northridge 1994/01/17 CNP196 0.42 1.2936 1.1362 0.8925Chi-Chi, Taiwan 1999/09/20 TCU123-W 0.164 2.551 1.8283 1.4361Chi-Chi, Taiwan 1999/09/20 CHY006-E 0.364 1.2712 1.1165Loma Prieta 1989/10/18 CAP000 0.529 1.1689 1.0267Northridge 1994/01/17 STC180 0.477 0.8568Northridge 1994/01/17 LOS270 0.482 1.3426Whittier Narrows 1987/10/01 A-EJS318 0.443 2.5271Imperial Valley 1979/10/15 H-E02140 0.315 2.2745Northridge 1994/01/17 NWH360 0.59 0.7307Hazard Level Return PeriodScale Factor of GMsAppendix A: Ground Motion Selection and Scaling 215    Figure A-6 GMs scaled to CMS for Sa (T1 = 1 sec) and 4975 yrs return period hazard level0.0010.010.11100.01 0.1 1 10Sa(T) [g]Period [s]CMSmedian response spectraAppendix B: Example Buildings 216 Appendix B. EXAMPLE BUILDINGS B.1. 4- AND 7- STORY BUILDINGS  h = height of beam-column ?tot = ratio of longitudinal reinforcement f?c = compressive strength of concrete  b = width of beam-column  ?sh =  ratio of transverse reinforcement fy = yield strength of reinforcement s = spacing of transverse reinforcement\ ? & ?? = ratio of longitudinal tension and                  compression reinforcement Figure B-1 Design documentation for 4-story space frame from Liel (2008 ? Ph.D. thesis by permission)Floor 9Floor 8Floor 7feetFloor 2feetGrade beam column height (in) = Basement column height (in) =feeth (in) = 28Story 214.80.00150.00930.0065Story 7Story 80.001510.832 2424 24Story 10.0046 0.005324s (in) = h (in) = b (in) = 24?sh = 10.82814.0280.00470.01030.001514.824 0.00470.010332 24 0.0060.009014.00.0053 0.00460.032 0.033 0.033 0.03114.0 14.0 14.0s (in) = 28 28 28h (in) = ? = ?' = ?sh = ? = ?' = 14.0 14.024 2414.0 14.032 24 0.00650.00930.001514.832 240.0020 0.0033 0.0033 0.002032 24 0.00650.00930.001514.80.015 0.022 0.022 0.01524 24 24 2428 28 2814.032 24 0.0060.00900.001514.80.0015 0.0018 0.0018 0.00150.001514.80.0060.009014.0 14.00.015 0.022 0.022 0.01524 24 24 2428 28 2814.00.00470.01030.001510.80.00150.001528240.0100.001510.824 0.00330.00700.00150.0110.001514.02428240.0110.001514.02428240.00330.00700.00700.001510.80.001510.824 2424 24 0.00330.010h (in) = b (in) =    ?tot =    ?sh = s (in) = h (in) = b (in) =    ?tot =    ?sh =    ?sh = s (in) = h (in) = b (in) = b (in) =    ?tot =    ?sh = s (in) =    ?tot = 131524?sh = s (in) = h (in) = b (in) = b (in) = ? = ?' = 14.0242824s (in) = 24 025Appendix B: Example Buildings 217  h = height of beam-column  ?tot = ratio of longitudinal reinforcement f?c = compressive strength of concrete  b = width of beam-column  ?sh =  ratio of transverse reinforcement fy = yield strength of reinforcement s = spacing of transverse reinforcement\ ? & ?? = ratio of longitudinal tension and                                  compression reinforcement Figure B-2 South frame elevation (adapted from Krawinkler, 2005) Floor 8Floor 7Floor 6Floor 5Floor 4Floor 3feetFloor 2feetfeet0.00270.00590.001410.00.00160.001414 140.00270.00480.001410.022160.00270.00480.00142216   ?sh = Story 7h (in) = 14 14 14 1422160.00270.00590.0130 0.0130 0.013010.00.001622160.00330.00440.00270.00480.001410.00.001614b (in) = 20 20 20 20   ?sh = 0.001622160.00330.00330.001410.0   ?tot = 0.0130s (in) = 12.0 12.0 12.0 12.00.001410.00.0016Story 6h (in) = 14 14 14 1410.00.00660.001410.0   ?tot = 0.0130 0.0130 0.0130 0.0130b (in) = 20 20 20 200.00560.001410.00.00160.0016200.01300.001622160.0033s (in) = 12.0 12.0 12.0 12.00.00270.00590.001410.00.00270.00590.001410.010.00.00140.00590.0027160.00330.00560.001410.00.00160.00162212.0 12.00.001614 14162212.00.001620   ?tot = 0.0130 0.0130 0.0130 0.0130b (in) = 20 20 20 2010.00.00560.00140.001410.00.002712.0   ?sh = 0.001622160.0033s (in) = 12.0 12.0Story 5h (in) = 14 142216160.00440.00560.001410.012.010.00.001622160.00270.00660.001410.00.0016220.00270.00660.00141612.00.00140.00270.00162216Story 4h (in) = 14 14 14 140.00270.00830.001410.0   ?tot = 0.0130 0.0130 0.0130 0.0130b (in) = 20 20 20 2010.00.00660.001410.012.01414200.00162216   ?sh = 0.001622160.0044s (in) = 12.0 12.00.01300.0016160.00560.00660.001410.00.00270.00830.001410.00.00270.00660.001410.00.0016220.00270.00660.001410.00.00140.00660.0027162222160.00660.001410.012.0b (in) = 20 20 20 20Story 3h (in) = 14 14 14 14   ?tot = 0.0130 0.0130 0.0130 0.0130   ?sh = 0.001622160.00440.00660.001410.0h (in) = b (in) = ? = ?' = 30160.00210.00440.001622160.00270.00830.001420s (in) = ?sh = ?' = ? = b (in) = 22140.001622Story 2h (in) = 14 14 14 14s (in) = 12.0 12.0 12.0 12.00.001410.00.001610.00.001622160.00560.00660.00270.00830.001410.0   ?tot = 0.0130 0.0130 0.0130 0.0130b (in) = 20 20 20 20200.0130200.013012.00.00420.001410.00.00160.001410.00.001630160.00330.00210.00440.001410.010.00.001412.00.00163010.012.00.001410.0b (in) = ? = ?' = ?sh = s (in) = s (in) = 12.0 12.0 12.010.00.001630160.00210.0044   ?sh = 0.001630160.00330.00420.0014s (in) = b (in) = 20 20 20 2014Story 1h (in) = 14 14 14h (in) = b (in) = ? = ?' = ?sh = s (in) = h (in) = b (in) = ? = ?' = ?sh = s (in) = h (in) = h (in) = 12.0   ?sh = 0.0024 0.0024 0.0024 0.00240.0024 0.0024   ?tot = 0.0290 0.0290 0.0290 0.02900.029012.012.019s (in) = 12.0 12.0 12.00.01302012.00.00240.0290 14s (in) = ?sh = ?' = ? = b (in) = h (in) = 914202210.00.00140.00830.00271614 1410.00.00140.00590.0027162212.00.0016200.0130201410.00.00140.00660.002716220.001612.014200.01300.0016221610.00.00140.00480.002716220.0027162210.00.00140.0066162212.0200.01300.0016221612.00.002412.014200.01300.001612.014200.01300.001614200.029012.0142014200.01300.001612.0h (in) = b (in) = ? = ?' = ?sh = s (in) = 3016200.01300.001612.01420221614200.01300.01300.0016221612.014200.01300.0016220.01300.001612.014200.01300.02900.002412.012.0142012.014200.01300.00160.001612.014200.001612.014200.0130160.00210.004412.014200.0130 0.01300.0130 0.01300.0016 0.0016140.01300.0130201422160.00270.004812.00.0016200.00270.006612.00.001622160.00270.006612.01410.00.00140.00480.00271622141612.0 12.00.01300.0016221610.00.00140.00830.0027162010.00.00140.00660.002714 14200.02900.013030160.00210.00440.001410.00.001612.014200.01300.0016221612.0?sh = s (in) = 10.00.001410.012.014200.01300.0016221612.014200.01300.0016h (in) = b (in) = ? = ?' = ?sh = 142030160.00210.00440.00140.001622160.00330.00330.001410.00.00480.001410.0160.002710.022160.001410.0160.00270.00480.001410.0220.00480.001410.022160.00270.004822160.00270.00480.001410.022160.00330.00440.00140.00270.00480.001410.02222160.0027Appendix B: Example Buildings 218 B.2. 12- STORY BUILDING Project Information ? Description The building is designed for the Vancouver area. Structural Framing Systems: ? Gravity: o Floor system: Flat slab and perimeter beam ? Vertical support: Columns  ? Lateral: o North-South direction: Perimeter moment frame o East-West direction:  Perimeter moment frame ? Foundation: Mat foundation Building Geometry Height: ? Overall: 12 stories, 158 ft 0 in. o First story: 15 ft 0 in. o Typical floor-to-floor: 13 ft 0 in. Plan: ? North-South direction: Overall, 125 ft 0 in. o Column spacing: 5 spans @ 25 ft 0 in.  ? East-West direction: Overall, 125 ft 0 in. o Column spacing: 5 spans @ 25 ft 0 in.  Project Information ? Building Codes and Standards General Building Codes Governing Building Code: ? Adopted: 1965 National Building Code of Canada (NRCC, 1965)) ? Building Code Requirements for Structural Concrete (CSA A23.3 1966) Appendix B: Example Buildings 219 Occupancy / Building Use Occupancy classification, Office buildings Project Information ? Loads and Effects ? Load Combinations and Patterns Loads Structural members must be designed to resist effects from all applicable loads which in this case are dead, live, and seismic loads. Gravity Loads Dead Load (Service Load): ? Self-weight was calculated by the computer analysis program,  assuming concrete with a unit weight of 150 pcf (8 in slab). ? Superimposed dead load (all floors): 8 psf Live Load (Service Load): ? Live loads are a function of occupancy or use.  The minimum live loads are given in Table 4.1.3.A of the 1965 NBCC.  Table B-1 Live load for office occupancy  AREA LIVE LOAD (psf) Typical floors (upper floors for office use) 50 Lateral Load ? Strength-level seismic forces acting on the building in both the N-S and E-W directions. o Distributed seismic forces over the height of the building in both directions. Design parameters: ? North-South and East-West Direction: o Building frame system - ordinary reinforced concrete moment frame o Base Shear,  V = RCIFSW         (B-1)  ? Seismic base shear:  V (N-S direction) = 1100 kips Appendix B: Example Buildings 220 Table B-2 1965 NBCC lateral load parameters Seismic Regionalization R 4 R is the seismic regionalization factor with values of 0, 1, 2, and 4 for seismic intensity zones 0, 1, 2, and 3, respectively. The same seismic zoning map as the 1953 NBCC. Construction factor C 0.75 C is the type of construction factor with values of 0.75 for moment resisting frames and reinforced concrete shear walls that are adequately reinforced for ductile behaviour, and 1.25 for other types of buildings Importance factor I 1 I is the importance factor with values of 1.0 and 1.3 (buildings with large assemblies of people, hospitals, and power stations) Foundation factor F 1 F is the foundation factor with values of 1.5 for highly compressible soils and 1.0 for other soil conditions Structural flexibility factor S 0.012 S is the structural flexibility factor of 0.25/(N + 9), where N is the number of storeys Weight W(kips) 30804 W is the total weight (dead load plus 25% snow plus live load for storage areas) ? The table below shows the factored seismic forces acting on the building in the N-S direction. These forces were determined in accordance with NBCC 1965,   =?           (B-2) Table B-3 Lateral load distribution along the height of the building Story hx  (ft) wx  (kips) hx*wx Fx  (kips) 12 158 2567 405586 167 11 145 2567 372215 154 10 132 2567 338844 140 9 119 2567 305473 126 8 106 2567 272102 112 7 93 2567 238731 99 6 80 2567 205360 85 5 67 2567 171989 71 4 54 2567 138618 57 3 41 2567 105247 43 2 28 2567 71876 30 1 15 2567 38505 16  Appendix B: Example Buildings 221 Load Combinations Factored load combinations: ? These combinations are used when computing the required strength of members. The computer program identifies the critical case from all combinations.  ? The 1965 NBCC ultimate load, U, for gravity and earthquake design is given by  U = 1. 5D +1.8 L         (B-3) U = 1.35(D + L + E)        (B-4) where D, L, and E are the effects from dead, live, and earthquake loads, respectively.    Appendix B: Example Buildings 222  Figure B-3 Design documentation for 12-story Floor 13Floor 12Floor 11Floor 10Floor 9Floor 8Floor 7Floor 6Floor 5Floor 4Floor 3feetFloor 2feetfeetDesign base shear = g, kf'c beams = ksi f'c,cols,upper ksify,rebar,nom. = ksi f'c,cols,lower ksi38320.04100.004212.0260.02250.003212.032260.03000.003212.00.01500.001912.032260.01880.001912.0320.001912.032260.01500.001912.0322612.032260.0150.001912.032260.015b (in) = ? = ?' = ?sh = s (in) = 32260.0150.001912.032260.0150.001912.032260.0150.001912.032260.0150.001926 0.01130.01130.001312.032 26 0.01130.01130.001312.026 0.01130.01130.001312.032 26 0.01130.01130.001312.026 0.01130.01130.001312.032 26 0.01130.01130.001312.026 0.01130.01130.001312.032 26 0.01130.01130.001312.026 0.01130.01130.001312.032 26 0.01130.01130.001312.00.01130.001312.00.03000.003212.032260.03750.003212.032323232320.001912.032260.02250.003212.0322612.032260.01500.001912.032260.018832260.01500.001912.032260.01500.001932 26 0.01130.01130.001312.032260.015032 26 0.01130.01130.001312.032 26 0.01130.01130.001312.032 26 0.01130.01130.001312.032 26 0.01130.01130.001312.032 26 0.01130.01130.001312.032 26 0.01130.01130.001312.012.03232 26 0.01130.01130.001312.032 26 0.01130.01130.001312.032260.01500.001912.032260.01500.001912.032260.01500.001912.032260.01500.001932 26 0.01130.01130.001312.032 26 0.01130.01130.001312.026 0.01130.01130.001312.00.001932 26 0.011312.032b (in) = 26 26 26 2612.00.003s (in) = 12.0 12.0 12.0 1226 0.01130.011332 26 0.01130.01130.00130.001312.00.00320.023   ?sh = 0.003232 26 0.01130.01130.001312.032Story 3 h (in) = 32 32   ?tot = 0.0225 0.02250.003212.0Story 40.022532s (in) = 12.0 12.0 12.026 0.01130.011332 26 0.01130.01130.001332 32b (in) = 26 26 26 26h (in) = 320.0188 0.019   ?sh = 0.001932 26 0.01130.01130.001312.00.002120.001312.00.001932Story 5 h (in) = 32 32 32 32b (in) = 12.0s (in) = 12.0 12.0 12.026 0.01130.01130.00130.001312.00.001912.00.00190.00226 0.0113Story 6 h (in) = 32 32   ?tot = 0.0150 0.0150b (in) =    ?sh = 0.001326 260.001932 26 0.01130.01131315?sh = s (in) = h (in) = b (in) = b (in) = ? =    ?sh = b (in) =    ?tot = h (in) = b (in) = ? = ?' = ?sh = s (in) = 32 26 0.01130.01130.001312.0h (in) = 260.0150 0.01500.0300 0.0300s (in) = 32h (in) =    ?tot =    ?sh = s (in) = h (in) = b (in) =    ?sh = s (in) =    ?sh = 26   ?tot = h (in) = b (in) =    ?tot =    ?sh = h (in) = b (in) = 12.00.001312.032 2632 26 0.0113320.01130.01130.01130.0013   ?tot = 0.0150s (in) = 12.00.0019   ?tot = 0.018832 3226 2632 26 0.01130.01130.00130.002320.01130.001312.00.0020.01130.01130.001312.00.001932 26 0.01130.01130.00131232 26 0.0113320.00232322632260.01512.026 0.01130.01130.00130.001312.00.00232 26 0.01130.01130.001312.00.00232 32 3212.032 26 0.01130.01130.001312.0260.0150 0.015012.0 12.00.001312.00.001912.00.001932 26 0.01130.01130.01526 26 26 260.0019 0.0019 0.0019 0.0020.001312.00.01130.011312.0 12.0260.0032 0.0032 0.0032 0.00332 26 0.01130.01130.001312.0 12.0 1232 26 0.01130.01130.001312.012.0 12.026?sh = s (in) = 38 32 32h (in) = ?' = ? = 32120.001312.032s (in) = h (in) = b (in) = ? = ?' = 0.0300 0.033226260.0150.0150.00212120.01130.001332 12.0120.0032 0.0030.0410 0.0375 0.0375 0.03812.0 12.00.01500.0019Story 1032 26 0.01130.01130.001312.0s (in) = 12.0Story 1Story 232 26 0.01130.01130.0042 0.0032Story 732 2632 26?' = ?sh = 32 26 0.01130.01130.00132626 26 2660###0.0364.04.0 4.00.01500.00192612.0 12.012.012.00.015025320.01880.00190.01130.01130.001312.00.00190.015032 26 0.01130.01130.001312.00.001932 0.01130.01130.001312.032323232 26 0.01130.011332260.0150b (in) = 26 26 26 26   ?tot = 0.0150 0.0150 0.0150 0.01526 0.01130.01130.001312.00.001932 26Story 9 h (in) = 32 32   ?sh = 0.001932 26 0.01130.01130.001312.00.001932 26s (in) = 12.0 12.0 12.0 12Story 8 h (in) = 32 32 32 32b (in) = 26 26 26 26   ?tot = 0.0150 0.0150 0.0150 0.015   ?sh = 0.001932 12.0s (in) = 12.0 12.0 12.0 12Story 12h (in) = 32 32 32 32b (in) = 26 26 26 26   ?tot = 0.0150 0.0150 0.0150 0.015   ?sh = 0.01130.0113s (in) = 12.0 12.0 12.0 120.001932 26 0.01130.01130.001312.00.001932 26 0.01130.01130.001312.00.001932 26Story 11h (in) = 32 32 32 32b (in) = 26 26 26 26   ?tot = 0.0150 0.0150 0.0150 0.015   ?sh = 0.001932 26 0.01130.01130.001312.0s (in) = 12.0 12.0 12.0 120.001932 26 0.01130.01130.001312.00.0019Appendix B: Example Buildings 223  Figure B-4 Design documentation for 12-story (beam and column layout) F lo or 13Floor 12Floor 11Floor 10Floor 9Floor 8Floor 7Floor 6Floor 5Floor 4Floor 3feetFloor 2feetfeetDesign base shear = g, kf 'c beams = ksi f 'c,col s, upper  = ksif y, r ebar ,nom.  = ksi f 'c,col s, l ower  = ksiB1 B1 B1 B1 B1B1 B1 B1 B1 B1B1 B1 B1 B1 B1B1 B1 B1 B1 B1B1 B1 B1 B1 B1B1 B1 B1 B1 B1B1 B1 B1 B1 B1B1 B1 B1 B1 B1B1 B1 B1 B1 B160 4.0C1 C2 C2 C2250.036 11004.0 4.015C2Story 1C1B1B1 B1 B1 B113C3Story 2C3 C3C3 C3 C3B1 B1 B1 B1 B1C4Story 3C4 C4 C4 C4 C4C5Story 4C5 C5 C5 C5 C5C6Story 5C6 C6 C6 C6 C6C6Story 6C6 C6 C6 C6 C6C6Story 7C6 C6 C6 C6 C6C6Story 8C6 C6 C6 C6 C6C6Story 9C6 C6 C6 C6 C6C6Story 10C6 C6 C6 C6 C6Story 12C6Story 11C6 C6 C6 C6 C6C6 C6 C6 C6 C6B1B1B1B1B1C6Appendix B: Example Buildings 224    C1 C2   C3 C4   C5 C6 Figure B-5 Cross section of columns (all units are in inches)  32.038.02 layers of 6 - #11#4 @ 12.00 in2 layers of 2 - #11#4 @ 12.00 in#4 @ 12.00 in#4 @ 12.00 in#4 @ 12.00 in2 layers of 2 - #112 layers of 6 - #1126.032.05 - #114 - #11#4 @ 12.00 in#4 @ 12.00 in#4 @ 12.00 in2 - #11#4 @ 12.00 in4 - #115 - #1126.032.04 - #113 - #11#4 @ 12.00 in#4 @ 12.00 in#4 @ 12.00 in2 - #11#4 @ 12.00 in3 - #114 - #1126.032.03 - #112 - #11#4 @ 12.00 in#4 @ 12.00 in#4 @ 12.00 in2 - #11#4 @ 12.00 in2 - #113 - #1126.032.04 - #11#4 @ 12.00 in#4 @ 12.00 in2 - #11#4 @ 12.00 in4 - #1126.032.03 - #11#4 @ 12.00 in#4 @ 12.00 in2 - #11#4 @ 12.00 in3 - #11Appendix B: Example Buildings 225  B1 Figure B- 6 Cross section of beams (all units are in inches)   26.032.06 - #11#4 @ 12.00 in6 - #11Appendix B: Example Buildings 226 Appendix C. HIGH VOLUME PARALLEL ANALYSIS USING NEESHUB C.1. INTRODUCTION Parallel computation has become increasingly useful for conducting nonlinear analysis of structural systems. Taking advantage of high-performance computing resources has made parallel computation possible. The number of nonlinear analysis required to build the collapse fragilities for each building in Chapter 6 is approximately 3000. The best solution for this high demand is to use High Performance Computing (HPC) and taking advantage of cloud computing services. In this study, the cloud services provided by NEEShub (http://www.nees.org) were used to overcome this problem. The objective of this Appendix is to provide readers with an overall description of the computational services provided by NEEShub and an example how to run nonlinear analysis on multi-core supercomputers. C.2. NEESHUB NEEShub is a sophisticated platform and cyber-infrastructure for Earthquake Engineering research supported by the NEEScomm IT team. The facility is powered by HUBzero technology developed at Purdue University and provides access to multiple High Performance Computing (HPC) venues. The available venues in addition to their computational capacity which define the appropriate job size are summarized in Table C-1.  Table C-1 Available venues on NEEShub (Rodgers et al., 2013) Venue Max number of processors Number of nodes Number of processor per node Job size Ranger 4096 256 16 Large Kraken 512 - 12 Large Hansen 96 103 4 Medium Steele 4232 529 8 Medium OSG ~60000 NA NA Large Local (NEES) 24 3 8 Small  Appendix B: Example Buildings 227 C.3. OPENSEESMP The OpenSees framework provides three different versions for various applications, OpenSees, OpenSeesSP, and OpenSeesMP. OpenSeesSP is for the analysis of very large models which takes a very long time to run on a single processor; OpenSeesMP is for parameter type studies of small to moderate sized models (McKenna and Fenves 2008). In this study OpenSeesMP was used to execute the parametric studies required to build the collapse fragilities for each building. When running OpenSeesMP on a parallel machine (e.g. the supercomputers on NEEShub), each processor is executing the same main input script with each set of parameters being processed only once.  The following calculations demonstrate the significant effort of analysis to build the collapse fragilities for each building in Chapter 6. The final result clearly displays the advantage of using OpenSeesMP and the venues provided by NEEShub. ? Random Variables: ?  GMs (20) ?  Model Uncertainty (2 RVs & 3 realizations) ? 20 (GMs) x 3 (RVs) x 3 (realizations) = 180 RHA @ Sa ? Build Collapse Fragility (using MSA) @ 4 Sa intensities    Collapse fragility for each realization of collapse indicator (e.g. ?= 0.002)       = 180 x 4 = 720 RHA  ? Find limits for range of collapse indicators (e.g. ? ~ 0.002 ? 0.01)  720 x 4 = 2880 RHA  ? Execution time for a set of one collapse indicator   2880 (RHA) x 30 (min) = 60 days ? Execution time for a table of collapse indicators (CI) for one building   60 (days) x 10 (CI) = 600 days ? Execution time for a table of collapse indicators (CI) for a number of buildings(one processor)  600 (days) x 3 (buildings) = 5 years ? Execution time for a table of collapse indicators (CI) for a number of buildings(one processor)  5 (years) / 2000 (processors) = 1 day Appendix B: Example Buildings 228  C.4. INSTRUCTIONS TO WORK WITH NEESHUB The step-by-step procedure to start working on NEEShub is listed below: 1. Register at http://nees.org and get a NEEShub account and login. 2. Get access to NEEShub workspace by opening a support ticket.  Click on Support button on top right of screen near "?" (request for access to the Linux workspace). This will give you access to the local (NEES) venue. 3. Request increase in quota to 10GB by opening another support ticket.  4. Setup a WebDAV connection (for windows users) for file sharing between your local computer and the workspace. 5. Submit a ticket to get access to remote venues (e.g. Kraken). The step-by-step procedure to start running OpenSees analysis on NEEShub is listed below: 1. Login your NEEShub account at http://nees.org. 2. Start a NEEShub workspace by clicking "Launch Tool" at (https://nees.org/tools/ workspace). 3. Make a directory for your analysis in the workspace (> mkdir ?analysis folder?). 4. Using WebDAV, copy your OpenSees tcl files from your local folder to the analysis folder on your workspace. 5. In the workspace, change directory to the analysis folder (> cd $HOME/?analysis folder?). 6. If you need to edit a tcl file use the program emacs (> emacs ?file name?.tcl). 7. Execute OpenSees on local NEEShub computer (> opensees ?main file?.tcl). 8. Execute OpenSees on remote venues (e.g. Kraken) (> batchsubmit ?venue kraken ?appdir /apps/share64/opensees/kraken ?rcopyindir OpenSees $HOME/?folder directory?/?main file?.tcl). 9. Whenever the analysis starts, if you have submitted your analysis to a remote venue, you will get an email with the subject ?JOB ?your nees username? opensees job00x STARTED?. 10. If you have submitted your analysis to a remote venue, you can check the status of your analysis (> batchstatus). Appendix B: Example Buildings 229 11. Whenever the analysis ends, if you have submitted your analysis to a remote venue, you will get an email with the subject ?JOB ?your nees username? opensees job00x COMPLETED?. The step-by-step procedure to start running OpenSeesMP analysis (parallel simulations for parametric studies) on NEEShub is listed below: 1. Login your NEEShub account at http://nees.org. 2. Start a NEEShub workspace by clicking "Launch Tool" at (https://nees.org/tools/ workspace). 3. Make a directory for your analysis in the workspace (> mkdir ?analysis folder?). 4. Using WebDAV, copy your OpenSees tcl files from your local folder to the analysis folder on your workspace. 5. In the workspace, change directory to the analysis folder (> cd $HOME/?analysis folder?). 6. You need to edit your main tcl file for parallel analysis (> emacs ?main file?.tcl). 7. You need to add the following lines to your main tcl file: set pid [getPID] set numP [getNP] set count 0; # initial settings # open  for loop for parametric analysis if {[expr $count % $numP] == $pid} {      # analysis for each realization of the for loop } incr count 1; # close for loop 8. Execute OpenSeesMP on local NEEShub computer (> mpirun ?np ?number of processors? OpenSeesMP ?main file?.tcl). 9. Execute OpenSeesMP on remote venues (e.g. Kraken) (> batchsubmit ?venue kraken --ncpus ?number of processors? ?appdir /apps/share64/opensees/kraken ?rcopyindir OpenSeesMP $HOME/?folder directory?/?main file?.tcl). Appendix B: Example Buildings 230 10. Whenever the analysis starts, if you have submitted your analysis to a remote venue, you will get an email with the subject ?JOB ?your nees username? openseesmp job00x STARTED?. 11. You can check the status of your analysis, if you have submitted your analysis to a remote venue, (> batchstatus). 12. Whenever the analysis ends, if you have submitted your analysis to a remote venue, you will get an email with the subject ?JOB ?your nees username? openseesmp job00x COMPLETED?. The step-by-step procedure to build your own version of OpenSees on NEEShub is listed below: 1. Login your NEEShub account at http://nees.org. 2. Start a NEEShub workspace by clicking "Launch Tool" at (https://nees.org/tools/ workspace). 3. Make a directory for svn (> mkdir svn) and change directory to this folder (> cd svn). 4. Checkout the official version of the OpenSees source code in the directory openseesbuild. The svn checkout command may ask for your hub password and it will take several minutes to download the entire source (> svn checkout https://nees.org/tools/opensees build/svn/trunk openseesbuild). 5. You may want to make a backup copy of this directory because you will be making changes to this directory (> cp ?rp   $HOME/svn/openseesbuild $HOME/svn/opensees build.backup). 6. Build an appdir (Application Directory) so that the binaries you build can be run on Linux supercomputers available to NEESHub (> mkdir $HOME/myappdir). 7. Modify the source code for OpenSees per your requirements  in the svn directory (> emacs $HOME/svn/openseesbuild/SRC/?source file?.cpp). 8. Change to the root directory of the build tree (> cd $HOME/svn/openseesbuild). 9. Use Linux "make" command to build the executables for NEEShub (> make OpenSees). This command will take a long time the first time you do this.   10. Now copy the version you just built to your appdir and rename to OpenSees_Updated (>cp   $HOME/svn/openseesbuild/bin/OpenSees $HOME/myappdir/bin/OpenSees_ Updated). Appendix B: Example Buildings 231 11. Test your updated OpenSees on local NEEShub computer or on remote venues (e.g. Hansen) (>batchsubmit ?appdir $HOME/myappdir ?venue hansen ?rcopyindir OpenSees_Updated $HOME/?folder directory?/?main file?.tcl).  

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