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Directionality effects of pulse-like near field ground motions on seismic response of tall buildings Archila, Manuel 2014

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 DIRECTIONALITY EFFECTS OF PULSE-LIKE NEAR FIELD GROUND MOTIONS ON SEISMIC RESPONSE OF TALL BUILDINGS by Manuel Archila  M.A. Sc. (Civil Engineering), The University of British Columbia, 2011 B.Sc. (Civil Engineering), Universidad del Valle de Guatemala, 2006   A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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)  December 2014  © Manuel Archila, 2014ii Abstract  Earthquake ground shaking at near-field sites has produced severe damage to buildings and infrastructure, caused large economic and human losses. These earthquakes continue to pose a great threat to many populous urban centers around the world. Over the last 4 decades, major research efforts have been devoted to characterize near-field ground motions and their effects on the response of buildings.   A challenging engineering problem that remains unsolved in the seismic design of buildings is the directionality effect of near-field pulse-like ground motions on tall buildings. This dissertation presents a computationally efficient method to calculate the critical displacement response of tall buildings. In this method, the duration and amplitude of the ground velocity pulses contained in the input motion and the building’s first mode translational period are compared to determine the orientation of the Conditional Maximum Velocity (CMV), a new ground motion metric. Nonlinear response history analyses (NRHA) using the CMV ground motion provide a close estimate of the critical displacement response of a tall building along the structural axis.   The CMV Method is developed on the basis of a series of parametric studies. First synthetic pulse-like excitations and simple structures were used to systematically investigate the effects of pulse duration and amplitude on the dynamic elastic and inelastic response of structures. It is found that the critical displacement response is influenced significantly by the CMV ground motion.  iii This finding is further validated through series of NRHA of simple structures and several case study tall buildings to near-field ground motion records. It is shown that the NRHA using CMV ground motions result in approximate, but significantly accurate estimates of the critical displacement with small errors and moderate dispersion.   Ground motion pairs rotated to fault-normal (FN) orientation do not always result in the critical displacement response. Neither does the maximum direction (MD) ground motion at the fundamental period of the building. Analyses made using FN or MD may result in significant underestimates of critical displacement response compared to the proposed CMV.  The use of the CMV Method may lead to a better quantification of potential seismic demands on tall buildings.            iv Preface This research work was conducted under direct supervision of Prof. Carlos E. Ventura and Prof. Liam Finn in the Civil Engineering Department at The University of British Columbia (UBC). The author of the dissertation identified and designed the research program with close guidance from the supervisors. Important feedback regarding the scope of the research program and the research findings was also provided during the progress report meetings by other members of the PhD supervisory committee including Dr. Farzad Naeim, Prof. Tony Yang and Prof. Don Anderson, both at UBC.   The computer model used to analyze the 44-storey building was developed by the author and Prof. Carlos E. Ventura. The computer models of the 52-storey and 30-storey buildings were developed at the Earthquake Engineering Research Facility (EERF) by a former member of the research group at UBC under supervision of Prof. Carlos E. Ventura. Additional computer models for the 45-storey and 32-storey buildings were provided by a private consulting engineer.  Software development for the analysis of results and statistical analyses was carried out by the author. A program to generate animations of the effects of ground motion directionality on the response spectrum was developed by Prof. Carlos E. Ventura and used extensively throughout the course of this study to analyze and verify the results from the simulations.  Every chapter of this dissertation was written entirely by the author and benefited from editorial revisions, comments and suggestions from the research supervisors and other members of the PhD supervisory committee.  v The major findings from chapters 4, 5 and 6 were presented at the 2014 Conference on Advances in Structural Design for Seismic Regions organized by Los Angeles Tall Buildings Structural Design Council. The outline of the paper submitted to this conference was drafted by the author of the dissertation and Prof. Carlos Ventura. The author of this dissertation wrote the draft upon which Prof. Carlos Ventura and Prof. Liam Finn provided comments and feedback to improve the presentation of the material. The reference to this article is below.  Archila, M., Ventura, C. E. and Finn, L. W.D. (2014). New Insights on effects of directionality and duration of near fault ground motions on seismic response of tall buildings. Advances in Structural Design for Seismic Regions. Los Angeles Tall Buildings Structural Design Council.        vi Table of Contents Abstract ........................................................................................................................................................ ii Preface ......................................................................................................................................................... iv Table of Contents ....................................................................................................................................... vi List of Tables .............................................................................................................................................. ix List of Figures ............................................................................................................................................. xi Glossary ..................................................................................................................................................... xv Acknowledgements ................................................................................................................................ xviii Dedication ................................................................................................................................................. xxi Chapter 1: Introduction ......................................................................................................................... 1 1.1  Motivation ............................................................................................................................... 4 1.2  Hypothesis ............................................................................................................................... 4 1.3  Research objectives ................................................................................................................. 4 1.4  Scope of this study .................................................................................................................. 5 1.5  Thesis outline .......................................................................................................................... 6 Chapter 2: Directionality effects on seismic response of a tall building ............................................ 7 2.1  Description of case study ........................................................................................................ 7 2.2  Input motion for seismic analyses ........................................................................................... 8 2.3  Nonlinear displacement responses ........................................................................................ 11 Chapter 3: Previous studies ................................................................................................................. 14 3.1  Lessons learned from earthquakes ........................................................................................ 14 3.2  Effects of near-field pulse-like and far-field ground motions on buildings .......................... 17 3.3  Directionality effects of near-field ground motions on response spectrum .......................... 20 3.4  Directionality effects of ground motions on seismic response of buildings ......................... 24 vii 3.5  Recommendations and building code provisions concerning seismic input and ground motion directionality .......................................................................................................................... 25 3.6  Main findings from past studies ............................................................................................ 26 3.7  Gaps in state of knowledge ................................................................................................... 27 3.8  Problem Statement ................................................................................................................ 28 Chapter 4: Response of simple structures to pulse-like excitations ................................................. 29 4.1  Model definition and numerical solution of equation of motion ........................................... 29 4.2  Pulse duration and amplitude effects on unidirectional response of SDOF oscillators ......... 31 4.3  Discussion of results ............................................................................................................. 42 4.4  Recommended criteria to estimate critical response ............................................................. 44 Chapter 5: Response of simple structures to velocity pulses in ground motions ............................ 46 5.1  Selection of near-field ground motion records ...................................................................... 46 5.2  Analysis of velocity pulses in ground motions ..................................................................... 49 5.3  Discussion ............................................................................................................................. 58 5.4  Summary of analysis of velocity pulses ................................................................................ 61 5.5  Directionality effects of pulse-like ground motions on elastic response spectrum ............... 62 5.6  Remarks................................................................................................................................. 73 5.7  Directionality effects on nonlinear response of simple structures ......................................... 73 5.8  Input ground motions ............................................................................................................ 73 5.9  Simple structure models ........................................................................................................ 74 5.10  Dynamic responses ................................................................................................................ 76 5.11  Dependence of dynamic responses on ground motion orientation ........................................ 78 5.12  Comparison of critical displacement predictions .................................................................. 79 5.13  Remarks................................................................................................................................. 84 5.14  Chapter summary .................................................................................................................. 85 Chapter 6: Method proposed to estimate critical displacement response ....................................... 86 viii 6.1  Prediction of critical displacement response ......................................................................... 86 6.2  Tall building case 1 ............................................................................................................... 88 6.3  Tall building case 2 ............................................................................................................... 94 6.4  Tall building case 3 ............................................................................................................... 98 6.5  Tall building case 4 ............................................................................................................. 103 6.6  Tall building case 5 ............................................................................................................. 104 6.7  Discussion ........................................................................................................................... 113 6.8  Concluding remarks ............................................................................................................ 115 Chapter 7: Application to seismic design of tall buildings .............................................................. 117 7.1  Procedure for performance assessment ............................................................................... 117 7.2  Step 1 - Design target spectrum .......................................................................................... 119 7.3  Step 2 - Selection of ground motions .................................................................................. 121 7.4  Step 3 - Determine the CMV ............................................................................................... 122 7.5  Step 4 - Linearly scale ground motions ............................................................................... 122 7.6  Step 5 – Performing NRHA ................................................................................................ 126 7.7  Concluding remarks ............................................................................................................ 126 Chapter 8: Conclusions ...................................................................................................................... 128 8.1  Contribution of this research to the state of knowledge ...................................................... 129 8.2  Contribution of this research to the state of practice ........................................................... 130 8.3  Future work ......................................................................................................................... 130 Bibliography ............................................................................................................................................ 132 Appendices ............................................................................................................................................... 141  ix List of Tables Table 5.1 Set 1 strike-slip near field ground motion records at Rrup ≤ 10km .............................................. 48 Table 5.2 Set 2 of near field ground motion records at Rrup ≤ 10km ........................................................... 48 Table 5.3 Set 3 of near field ground motion records at 10 km < Rrup< 30 km. ........................................... 49 Table 5.4 List of ground velocity pulse durations and maximum amplitude. ............................................. 58 Table 5.5 Longest pulses present in the ground motion of minimum spectra among FN and FP. ............. 66 Table 5.6  List of angle from fault-parallel for maximum spectral response. ............................................. 68 Table 5.7 Different parameters evaluated for force-displacement relation of the simple structures .......... 76 Table 5.8 Median value of errors in estimating critical displacement for systems to unscaled motions .... 82 Table 5.9 Median value of errors in estimating critical displacement for systems to scaled motions ........ 82 Table 5.10 Cases which produced collapse using scaled motions .............................................................. 83 Table 6.1 Averaged errors of floor displacement over height of the 44-storey building. ........................... 92 Table 6.2 Errors of averaged interstorey drift ratio over height of the 44-storey building. ........................ 93 Table 6.3 Averaged errors of floor displacement over height of the 45-storey building. ........................... 96 Table 6.4 Errors of averaged interstorey drift ratio over height of the 45-storey building. ........................ 97 Table 6.5 Dynamic properties of 52-storey building. ............................................................................... 100 Table 6.6 Averaged errors of floor displacement over height of the 52-storey building. ......................... 101 Table 6.7 Errors of averaged interstorey drift ratio over height of the 52-storey building. ...................... 101 Table 6.8 Dependence of 50th percentile averaged errors of floor displacement over height on earthquake mechanisms and distance. ......................................................................................................................... 111 Table 6.9 Dependence of 84th percentile averaged errors of floor displacement over height on earthquake mechanisms and distance. ......................................................................................................................... 111 Table 6.10 Dependence of 50th percentile averaged errors of floor displacement over height on amplitude of roof displacement. ................................................................................................................................ 112 x Table 6.11 Dependence of 84th percentile averaged errors of floor displacement over height on amplitude of roof displacement. ................................................................................................................................ 112 Table 6.12 Dependence of 50th percentile for errors of average interstorey drift ratio on earthquake mechanisms and distance. ......................................................................................................................... 112 Table 6.13 Dependence of 84th percentile errors for average interstorey drift ratio on earthquake mechanisms and distance. ......................................................................................................................... 113 Table 6.14 Dependence of 50th percentile for errors of average interstorey drift ratio on amplitude of average interstorey drift ratio. ................................................................................................................... 113 Table 6.15 Dependence of 84th percentile for errors of average interstorey drift ratio on amplitude of average interstorey drift ratio. ................................................................................................................... 113 Table A.1 Mode periods from CANNY model and ambient vibration test. ............................................. 148 Table A.2 Modal damping ratios. ............................................................................................................. 148 Table D.1 Predictive models for pulse duration........................................................................................ 154 Table D.2 Predictive models for velocity pulse amplitude (peak ground velocity). ................................. 156  xi List of Figures Figure 1.1 Regional map showing main faults of southern California and northern Baja California.  Adapted by permission from Macmillan Publishers Ltd [Nature from Walls et. al. 1998]. ......................... 2 Figure 1.2 Trace of main shallow faults and their current stress regime in metro Los Angeles. Adapted by permission from Macmillan Publishers Ltd [Nature from Walls et. al. 1998] .............................................. 3 Figure 2.1 Building plan layouts/sections and 3D view of typical floor layout. ........................................... 8 Figure 2.2 FN/FP time series and response spectra (=5%) for Imperial Valley earthquake NGA185 motions. ......................................................................................................................................................... 9 Figure 2.3 Coordinate reference system for rotated ground motion pairs. ............................................. 10 Figure 2.4 Envelopes of nonlinear displacement responses along Y-axis for cases 1 and 2. ................... 11 Figure 2.5 Sensitivity of displacement response to ground motion orientation. ..................................... 12 Figure 3.1  Comparison near-field and far-field records from the 1979 Imperial Valley earthquake. ...... 18 Figure 3.2  Comparison of response spectra (=5%) along different orientations. ................................. 23 Figure 4.1 SDOF system and spring force-displacement relationship. .................................................. 30 Figure 4.2 Three-half-sine acceleration pulse and associated velocity and displacement pulses. ............ 32 Figure 4.3 Pulse acceleration, velocity and displacement time series for velocity pulse amplitude of 100cm/s. ......................................................................................................................................... 34 Figure 4.4 Force response to pulse with peak amplitude of 100 cm/s for system having Cy=0.20 and T=2s. .............................................................................................................................................. 35 Figure 4.5 Displacement ductility demands for system with T=2s due to pulse peak amplitude of 100 cm/s. ............................................................................................................................................... 36 Figure 4.6 Peak displacements for SDOF with T=1s for a pulse with peak amplitude of 40 cm/s. .......... 39 Figure 4.7 Peak displacements for SDOF with T=1s for a pulse with peak amplitude of 100 cm/s. ........ 39 Figure 4.8 Peak displacements for SDOF with T=1s for a pulse with peak amplitude of 150 cm/s. ........ 39 xii Figure 4.9 Peak displacements for SDOF with T=2s for a pulse with peak amplitude of 40cm/s. ........... 40 Figure 4.10 Peak displacements for SDOF with T=2s for a pulse with peak amplitude of 100cm/s. ....... 40 Figure 4.11 Peak displacements for SDOF with T=2s for a pulse with peak amplitude of 150cm/s. ....... 40 Figure 4.12 Peak displacements for SDOF with T=3s for a pulse with peak amplitude of 40cm/s. ......... 41 Figure 4.13 Peak displacements for SDOF with T=3s for a pulse with peak amplitude of 100cm/s. ....... 41 Figure 4.14 Peak displacements for SDOF with T=3s for a pulse with peak amplitude of 150cm/s. ....... 41 Figure 5.1 Flowchart of procedure to identify amplitude and duration characteristics of velocity pulses. 50 Figure 5.2 Coordinate reference systems and orientation of input motions FN/FP. ............................... 51 Figure 5.3 Orientation of maximum velocity of the ground motion records. ......................................... 52 Figure 5.4 Velocity time series of 1979 Imperial Valley record NGA181 and extracted velocity pulse. . 55 Figure 5.5 Pulse spectrum for duration and amplitude of velocity pulses from NGA1244. .................... 56 Figure 5.6 Pulse spectrum for duration and amplitude of velocity pulses from NGA181. ...................... 57 Figure 5.7 Maps of epicenter and surface projection of rupture for several strike slip (SS) earthquakes having multiple-segment ruptures. Reproduced from Shahi (2013). ..................................................... 60 Figure 5.8 Maps of epicenter and surface projection of rupture for several strike slip (SS) and reverse oblique (RO) earthquakes having multiple-segment ruptures. Reproduced from Shahi (2013). .............. 61 Figure 5.9 Representation of SDOF analyzed and input ground motion. .............................................. 63 Figure 5.10 Orientation dependence of spectral ordinates. .................................................................. 64 Figure 5.11 Displacement response spectra (=5%) for near-field records at distance Rrup ≤ 10km from strike slip events in Set 1. ................................................................................................................. 69 Figure 5.12 Displacement response spectra (=5%) for near-field records at distance Rrup ≤ 10km from reverse/normal/oblique events in Set 2. ............................................................................................. 70 Figure 5.13 Displacement response spectra (=5%) for near-field records at distance Rrup={10km..30km}from different earthquake mechanisms in Set 3. ................................................. 71 xiii Figure 5.14 Geometric mean spectrum for NGA879. .......................................................................... 72 Figure 5.15 Force deformation relationship for systems investigated. .................................................. 74 Figure 5.16 Force deformation relationship for systems investigated. .................................................. 75 Figure 5.17 Response history of system with Cy=0.2 and T=2s to CMV ground motion for NGA181. .. 77 Figure 5.18 Orientation dependence of peak displacement using NGA181 and system with Cy=0.2. ..... 79 Figure 5.19 Responses CMV method vs critical displacement from direct analysis for T=3s and Cy=0.15........................................................................................................................................................ 80 Figure 5.20 Responses  CMV method vs critical displacement from direct analysis for T=3s and Cy=0.25........................................................................................................................................................ 81 Figure 6.1 Steps to rotate ground motion pair NGA181 to CMV and calculate the respective nonlinear seismic response. ............................................................................................................................. 90 Figure 6.2 Envelopes of peak interstorey drift ratios for 44 storey. ...................................................... 93 Figure 6.3 Elevation and plan view of Perform 3D model for 45-storey building. ................................. 95 Figure 6.4 Critical displacement vs CMV response (Tp/T=1.36) for 45 storey building using NGA1244........................................................................................................................................................ 96 Figure 6.5 Envelopes of peak interstorey drift ratios for 45 storey building. ......................................... 97 Figure 6.6 Plan view of typical floor layout for 52 storey building. ..................................................... 99 Figure 6.7 Picture and computer model representation of 52-storey steel frame building ...................... 99 Figure 6.8 Critical displacement responses vs CMV Method response for 52 storey building. ............. 100 Figure 6.9 Envelopes of peak interstorey drift ratio responses for 52 storey building. ......................... 102 Figure 6.10 Elevation and 3D view of Perform 3D model for 32 storey building. ............................... 103 Figure 6.11 Critical displacement responses vs CMV Method response for 32 storey building. ........... 104 Figure 6.12 Plan layout and elevation of the 30 storey building. ........................................................ 105 xiv Figure 6.13 Comparison of geometric mean from 45 ground motions with UHS in Vancouver 2% in 50yr...................................................................................................................................................... 107 Figure 6.14 Critical displacement responses vs predicted critical displacement. ................................. 108 Figure 6.15 Histogram of errors with respect to roof critical displacements in Y direction. ................. 109 Figure 6.16 Summary of errors with respect critical displacements in Y direction. ............................. 110 Figure 7.1 Procedure to integrate the CMV Method in performance based design framework. ............ 118 Figure 7.2 Acceleration design spectrum. ........................................................................................ 121 Figure 7.3 Displacement design spectrum. ....................................................................................... 121 Figure 7.4 Response spectrum for scaled CMV ground motion and design spectrum. ......................... 125 Figure 7.5 Floor displacement response of 44-storey building to NGA316. ........................................ 126 Figure A.1  Elevation of 44 storey building. .................................................................................... 142 Figure A.2  Typical shear wall and columns sections of the building modeled in CANNY. ................. 143 Figure A.3  Forces in diagonal coupling beams. ............................................................................... 144 Figure A.4  Hysteretic model SS3. .................................................................................................. 146 Figure A.5 Hysteresis model CS3 for concrete. ................................................................................ 147 Figure B.1 Distance dependence of MV, P1max and pulse duration. ................................................. 149 Figure C.1 Envelopes of nonlinear responses along X-axis for NGA143 and NGA181. ...................... 151 Figure C.2 Hysteretic response of coupling beams in core wall along X-axis for NGA143 and NGA181...................................................................................................................................................... 152 Figure D.1 Dependence of proportion of pulses on epsilon and distance ............................................ 153 Figure D.2 Pulse duration as a function of earthquake moment magnitude. ........................................ 155    xv Glossary Coseismic displacement: Is the geodetic displacement that is produced by the ground movement during an earthquake rupture, and is associated with a fling-step motion. Conditional Maximum Velocity (CMV): Is the absolute maximum value of the ground velocity in the horizontal plane conditioned on having a minimum pulse duration T. Critical displacement response: Is the largest response along the structural axis for all possible incident angles of the ground motion. Direct analysis: A parametric analysis for nonlinear dynamic response of a building where a ground motion pair is applied at different angles of incidence and the response calculated for each one of the rotated motions. Fault-normal ground motion: Is the horizontal ground motion component that is aligned with the perpendicular direction to the fault-strike. For instance see map in next page. Fault-parallel ground motion: Is the horizontal ground motion component that is aligned with the parallel direction to the fault-strike. For instance see map in next page. Forward directivity ground motion: A motion due to the constructive interference of shear waves that radiate from the fault and propagate towards the direction of the slip. It occurs when the fault rupture occurs at a velocity almost as large as the shear wave velocity. Geometric mean spectrum: Is the square root of the product of spectral ordinates from orthogonal components of a horizontal ground motion pair, calculated period by period.  Maximum direction (MD) response: Is the maximum response an oscillator of period T will experience among all possible orientations of a ground motion pair. Maximum direction (MD) response spectrum: Is an envelope of the maximum direction spectral ordinates for a range of oscillators having different periods T. xvi  Velocity time series of pulse-like ground motion record (NGA181) from 1979 Imperial Valley earthquake at El Centro Array #6 station within a fault distance of 1.4km. Map adapted from Shahi (2013).   Fault normalFault parallel112 cm/s 65cm/s xvii Maximum ground velocity (MV): Is the absolute maximum horizontal ground velocity. A polar plot of the horizontal particle motion is shown in figure below for the pulse-like record from 1979 Imperial Valley earthquake above. The MV is indicated with a red dot.  Horizontal particle velocity motion 1979 Imperial Valley earthquake record NGA181.  Near-fault region: In this dissertation it refers to sites located at distances within 5km from the causative fault. Near-field region: In engineering applications it is defined as sites within 20km to 30km from the causative fault. Pulse-like ground motion: Is a ground motion which contains large motions with pulse waveforms distinguishable either in the acceleration, velocity or displacement time series.  Tall building: In this study it refers to buildings taller than 50m and with a first mode translational period longer than 1.5s.  cm/s cm/s Maximum velocity 117 cm/s xviii Acknowledgements I am thankful to my supervisor Prof. Carlos E. Ventura for his guidance, encouragement and support throughout my graduate studies at The University of British Columbia. It has been a great experience for me conducting research with him and his team at the Earthquake Engineering Research Facility. I gratefully appreciate the diverse opportunities he gave me to learn from different aspects of structural and earthquake engineering. His care for my well-being is very much appreciated.  The support and advice I received from Prof. Liam Finn through all my years at UBC has been instrumental for my research. The opportunities he gave to participate as teaching assistant in his courses helped me to broaden my knowledge on seismic design parameters. Working with him has been a valuable and enriching experience.  I am very thankful Dr. Farzad Naiem, Prof. Don Anderson and Prof. Tony Yang, both at UBC, for their insightful comments and feedback provided through the course of this research. Their dedication to work with me through the progress reports and the revision of this dissertation is very much appreciated. Suggestions received from Prof. Ricardo Foschi and Prof. Frank Lam at UBC, are gratefully acknowledged.  My gratitude goes also to Prof. Ricardo Medina at University of New Hampshire who provided thorough comments and suggestions on the final version of this dissertation.   This research was partially funded by the Ministry of Education of British Columbia and a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada xix (NSERC) through the UBC Civil Engineering Department. The economic support and resources provided are gratefully acknowledged.  I am thankful to my parents, Manuel and Martha, my siblings Tita, Danny and Pablo for their unconditional love and support. Since my first day at UBC they have not stopped cheering for me. Despite the distance they did not spare any effort to support me and provide words of encouragement. I had the blessing of enjoying the company of my brother Danny, his contagious enthusiasm, wit and dedication was a source of inspiration for the last years of my PhD program. I am thankful for Sarita, the newborn of the family who brought many blessings and happiness into our life.   During my time at UBC I had the opportunity to enjoy wonderful friendships. My friends Seku Catacoli, Hugon Juarez and Elsa Pelcastre, Myhie Park, Yann and Susana Dumont have been exceptional friends during my graduate studies, I am immensely thankful for their kindness and all the support they provided me. Very special thanks go to Freddy Pina and Claudia Reyes, Jose Centeno, Miguel Fraino and  Bishnu Pandey for their friendships. Thanks to all my colleagues at the EERF, Felix Yao, Sheri Molnar, Armin Bebamzadeh, Yavuz Kaya and Jason Dowling for helping me through different research tasks. I befriended many other people at UBC it has been a rewarding experience getting to know so many people.  Finally, I want to thank God for countless blessings I have received every day of my life and for giving me the strength to continue when times were difficult. I was blessed by having amazing xx friends from University Chapel and Kerrisdale Presbyterian Church who provided unconditional support, prayers and love. God bless you all.    xxi Dedication  This dissertation is dedicated to the Lord, the Rock of my salvation.   And to my lovely family who taught me to persevere and work hard to achieve my goals.1 Chapter 1: Introduction Development of populous urban centers in seismic regions has resulted in the exposure of buildings and infrastructure to a significant near field ground shaking seismic hazard. For instance in Southern California, the metropolitan area of Los Angeles with 18.2 million inhabitants is founded on numerous active faults; regional maps with the main active faults are shown in Figures 1.1 and 1.2.  There is evidence from past earthquake events that severe seismic damage to buildings has been caused by near-field ground motions containing strong long duration velocity pulses. In this context, a major concern for seismic design of buildings is to estimate the nonlinear displacement response of the structure to distinct earthquake scenarios with due consideration of the directionality effects of strong near-field ground motions. This is because the dynamic response a structure experiences depends on the orientation of the ground motion with respect to the principal directions of structural response (structural axes).   Directionality effects of a ground motion can in some instances cause the building to attain the critical displacement response. This occurs when the orientation of the ground motion is critical such that it maximizes the displacement response. Currently there is no computationally efficient method available to determine the critical displacement response of tall buildings. Instead the conventional approach, direct analysis, is a rather computationally expensive procedure where the ground motion is systematically rotated at discrete angle increments with respect to the structural axes and the response calculated for each rotated motion to define the maximum envelope of responses.2   Figure 1.1 Regional map showing main faults of southern California and northern Baja California.  Adapted by permission from Macmillan Publishers Ltd [Nature from Walls et. al. 1998].Inset shown in more detail in Figure 1.2 3    Figure 1.2 Trace of main shallow faults and their current stress regime in metro Los Angeles. Adapted by permission from Macmillan Publishers Ltd [Nature from Walls et. al. 1998] 4  1.1 Motivation Despite nonlinear response history analysis is progressively becoming a standard practice to assess the seismic performance of buildings, the use of direct analysis to estimate the critical displacement is computationally expensive and prohibitive. The motivation for this study is to overcome this limitation and provide a computationally efficient method to determine the critical displacement response along the structural axis of a tall building when nonlinear response history analysis is conducted.  1.2 Hypothesis It is proposed that the duration and amplitude of ground velocity pulses can be used to assess the directionality effects of near field ground motions on the seismic response of tall buildings, specifically to determine the critical displacement response along the structural axis.  1.3 Research objectives  The main goal of this research is to develop a better understanding of the directionality effect of pulse-like ground motions on the displacement response of tall buildings. Besides verifying the hypothesis, the objectives of this thesis are to answer the following questions:  • Which rotated ground motion is the most detrimental for the nonlinear displacement response along the structural axis of a tall building? Is it the fault-normal, the fault-parallel or the maximum direction (MD) ground motion? • If none of them provides a reliable estimate, can a computationally efficient method be developed to estimate the critical displacement response of a tall building?  5  1.4 Scope of this study This research is limited to investigate horizontal directionality effects of pulse-like near field ground motions on seismic response of tall buildings along the structural axis. The influence of the vertical ground motion is not considered.   This investigation is conducted using parametric studies on single-degree-of-freedom (SDOF) systems and several case studies of tall buildings. The SDOF systems having long period are useful to gain a better understanding of the directionality effects on inelastic response of tall buildings. Although these SDOF systems are a simplified representation of tall buildings, the insight from these parametric studies is important for response quantities that are contributed significantly by the first mode response in tall buildings, such as the horizontal displacement response.   Because resources were limited and conducting large number of NRHA is computationally expensive, only the critical displacements along the structural axis were investigated in detail for the SDOF and case studies. In addition, the directionality effects on the critical interstorey drift ratios were determined for the case study tall buildings. The evaluation of critical displacements was limited to tall buildings having first modes with translational mode shapes along orthogonal directions, which define a pair of structural axes.   The investigation of the directionality effects for other response quantities that are contributed by higher modes would require a selection of many more pulse-like ground motions and thousands 6  of additional simulations which was not affordable. In this context, the critical values for other response quantities such as overturning moments and storey shears were not investigated.  Although nonlinear models of reinforced concrete tall buildings and a steel building were used for the analyses, these models were not intended to investigate progressive collapse.   1.5 Thesis outline Chapter 2 introduces an example of ground motion directionality effects on seismic response of a tall building. Chapter 3 presents a review of past studies on pulse-like near-field ground motions and directionality effects on buildings. Chapter 4 presents parametric analyses to investigate the elastic and inelastic dynamic response of single-degree-of-freedom systems to pulse-like excitations. In these parametric analyses the pulse characteristics, duration and amplitude are systematically varied by making use of synthetic pulses representative of real earthquake records. The results from this study are used gain insight on the directionality effects of pulse-like ground motion records on seismic response of buildings. Chapter 5 shows the effect of velocity pulses contained in ground motion records on the linear and nonlinear response of simple structures. Parametric analyses are performed to determine the critical displacement response of SDOF systems to pulse-like ground motions. Chapter 6 presents the method proposed to determine the critical displacement response which is verified using several tall building case studies. In Chapter 7 applications of the proposed method to seismic design of tall buildings is discussed. Finally, Chapter 8 summarizes the findings and contributions of this research study. The appendices provide information on the computer model for the 44-storey building and additional information regarding pulse-like ground motions. 7  Chapter 2: Directionality effects on seismic response of a tall building This chapter investigates the directionality effects of a pulse-like near field ground motion record on the seismic response of a tall building using nonlinear response history analyses. The objective of this chapter is to introduce the problem of ground motion directionality that will be discussed throughout this dissertation.   2.1 Description of case study The case study is a reinforced concrete tower of 44 storey above ground level, including the machine rooms in the upper storeys. It was designed in accordance to the National Building Code of Canada 1995 (NRC, 1995). The design of reinforced concrete elements was carried out using the CSA A23 (1994).  The building is for residential occupancy. It is part of a complex that comprises a residential tower and a hotel tower. The plan layout of the residential building is non-symmetrical and the columns are arranged in a non-rectangular grid. The plan average dimensions are 25m and 31m along East-West and North-South. The typical floor layout distribution of the residential tower is shown in Figure 2.1.   Two building’s sections along East-West and North-South is presented in the same figure. The residential tower extends 130m above the podium level. The podium structure includes two storey above ground level and 5 underground parking storey. Additionally along the perimeter of the podium in the underground parking levels, thick continuous walls of 20 cm are provided to 8  carry gravity loads and work as retaining walls. The foundation for columns are spread footings. The foundation for the core is a mat raft, plan dimensions 15m by 12m and thickness of 2m.  The height-to-width ratio of the building is 5 along east-west and 4 along north-south. The building main seismic force resisting system (SFRS) is provided by reinforced concrete core shear walls, and the gravity load resisting system is provided by reinforced concrete columns and shear walls. The core for the elevator and staircase is 8.4m by 8.7m in plan and the shear walls have a constant thickness of 0.66m from the foundation to the roof level. A description of the computer nonlinear model created with program CANNY (Li, 2015) is described in Appendix A.         East-West North-South 3D View  Figure 2.1 Building plan layouts/sections and 3D view of typical floor layout.  2.2 Input motion for seismic analyses The input motions for analyses were near-field pulse-like ground motion recorded during the 1979 Imperial Valley earthquake with Mw 6.5 at Hotville Post Office station distant 7.7km from Columns Post-tensioned concrete slab 180mm thick (flat plate) C-Shaped Shear Walls 9  the rupture. The rotated ground motion pair into fault-normal (FN) and fault-parallel (FP) components for this record were obtained from the PEER Ground Motion Database as record sequence NGA185. The time series and response acceleration spectrum (=5%) are shown in Figure 2.2.   Figure 2.2 FN/FP time series and response spectra (=5%) for Imperial Valley earthquake NGA185 motions.  10  A distinct velocity pulse with an approximate duration of 4.8s is observed in the fault-normal ground motion, this strong pulse is likely due to forward directivity effects (Shahi, 2013), which is the constructive interference of the shear waves that radiate from the rupture towards the site. The amplitude of ground motion velocity and displacement along the fault-parallel direction are lower, this difference is also observed in the acceleration response spectra above.  Nonlinear response history analyses were performed using the computer model of the 44-storey building and a suite of rotated ground motion pairs of the Imperial Valley record NGA185. Input motions for case 1 was defined as the FN/FP pair applied along the structural axes and for case 2 with FN/FP applied with a 90° clockwise rotation. Additional input motions for other cases were calculated by systematically rotating the ground motion pair at different orientations relative to the structural axes of the building as shown in Figure 2.3. The angle was increased in the clockwise direction at discrete increments of 5 °and in the range 0°≤<180°. Equation 2.1 was used to calculate the different input motions in the form of acceleration time series along X and Y structural axes. These two axes coincide with the NS (Y-axis) and EW (X-axis) directions, respectively.  Figure 2.3 Coordinate reference system for rotated ground motion pairs.  X-axisY-axis11  ቈݑሷ௚௑ሺߠ, ݐሻݑሷ௚௒ሺߠ, ݐሻ቉ ൌ ቂcos ߠ sin ߠെ sin ߠ cos ߠቃ ቈݑሷ௚ிேሺݐሻݑሷ௚ி௉ሺݐሻ቉ (2.1) 2.3 Nonlinear displacement responses The peak displacement responses along Y-axis from cases 1 and 2 are compared in Figure 2.4b. The response obtained when the ground motion pair is oriented to apply the fault-parallel component along the Y-axis is shown in blue (case 1) and when the fault-normal is aligned with the Y-axis in red (case 2). It is evident that a large difference occurs between these two cases. This difference is expected since the motion in the fault-normal direction was stronger, as it features a strong velocity pulse of long duration. The period of vibration for translational mode in the Y-axis is 3.5s, which is shorter than the duration of the velocity pulse contained in the FN motion.     (a) Orientation ground motion cases 1 and 2 (b) Responses along Y-axis Figure 2.4 Envelopes of nonlinear displacement responses along Y-axis for cases 1 and 2.  CASE 1 CASE 2 12  An investigation of ground motion directionality effects warrants finding out how different the displacement response would be if the ground motion has different orientations from cases 1 and 2. Could it be larger than the response due to FN component applied along the Y-axis? To answer this question the response envelopes of floor lateral displacement along Y-axis for other orientations of the ground motions are shown in Figure 2.5 as displacement envelopes in gray.   Some of the responses are bound by cases 1 and 2, and others are below case 1 and above case 2. The upper bound of the displacement responses from all the different rotated motions shown with broken line in Figure 2.5 is defined herein as the critical displacement responses.     Figure 2.5 Sensitivity of displacement response to ground motion orientation.  Although for this particular ground motion the fault-normal component produces a fairly large response it is not close enough to the critical displacement response.  The evidence from this initial investigation suggests the fault-normal ground motion may not be the most severe for 13  displacement response of buildings. This issue requires further verifications, the results for other ground motions and tall buildings are presented in chapter 6.  Finding the critical displacement response is a challenging task which requires large number of nonlinear dynamic analyses, as presented for this example. In the course of this research and performing thousands of analysis it was found that computing time for a single analysis can range from 1 to 16 hours depending on the duration of the ground motion, the size of the model, computing efficiency of the hardware and software. Thus carrying out a systematic rotation of the ground motions to estimate the critical nonlinear displacement becomes prohibitively expensive for engineering practice.   14  Chapter 3: Previous studies This chapter presents an overview of past studies related to pulse-like near field ground motions and their directionality effects on buildings. Observations of seismic damage from post-earthquake evaluations and analytical studies on the response of structures to near field motions are discussed. Different research studies that dealt with the effects of ground motion directionality are also presented and compared. All these studies are helpful to identify gaps in the current state of knowledge.   3.1 Lessons learned from earthquakes Observations made during post-earthquake investigations and evaluations of seismic response of buildings indicate that buildings located near the causative fault are prone to experience severe damage due to ground shaking. A brief overview of major earthquake events and lessons learned from them are presented below.  3.1.1 Mw 6.6 San Fernando Earthquake 1971 The 1971 San Fernando earthquake in California caused 65 fatal casualties and more than 2,000 injuries, and caused damage property estimated at 505 million U.S. dollars (USGS, 2014).   In the aftermath of this event, studies were conducted to understand the effects of the ground shaking on the seismic response of the Olive View Hospital, which experienced severe structural damage. Upon studying the characteristics of ground motion records at the site and the damage on the building it was found that long duration acceleration pulses present in the ground motion were the cause of the severe structural damage observed (Bertero et al., 1978).  15  3.1.2 Mw 6.7 Northridge Earthquake 1994 This earthquake caused 60 fatalities and left more than 7,000 people injured. Over 40,000 buildings were damaged and the estimated economic losses exceeded 20 billion U.S. dollars. Severe damage was observed in the San Fernando Valley: including the Northridge region, its surroundings and the Sherman Oaks region (USGS, 2014).   The effects of near field pulse-like ground motions on buildings were noticeable in many regions of Southern California. After the earthquake an extensive survey of damaged buildings was carried out by Borchardt et al. (1996). The survey included low-rise and high-rise buildings. The data collected during this survey demonstrated that 59 buildings on the hanging wall (in this case North-East of the fault) were left with a residual displacement greater than 2.54 cm (1 inch). Out of these buildings it was observed that 92% were leaning towards the North, in the direction of the fault block movement and rupture propagation. The northward direction of the displacements was associated with the presence of strong velocity pulses along the North-South component of the ground motion records. The records at the Sylmar County Hospital and at the LA County Fire Station in Newhall contained strong velocity pulses. Roof level records of building’s response at Woodland Hills showed that buildings experienced its maximum velocity and displacement in the North-South direction during the first cycle of response (Paret et al.,1997). Subsequent analytical investigations used several of these near-field records to determine the seismic response of idealized buildings, and confirmed that the pulses which propagated from South to North may have caused some buildings to experience inelastic response with residual displacements toward the North direction (Atalla et al., 1998). 16  3.1.3 Mw 7.4 Kocaeli and Mw 7.2 Duzce Earthquakes 1999 Turkey was struck by two major earthquakes in 1999. These events occurred 86 days apart along the 1200 km North Anatolian Fault in two sequential ruptures. The Kocaeli earthquake Mw 7.4 happened on August 17. It ruptured 140 km on the western side of the fault and propagated eastward to stop 12km to the east of the city of Duzce. On November 12 the Mw 7.2 Duzce earthquake ruptured an additional segment of 40km, the trace of the fault passed 12km to the south of Duzce City. The strong near fault ground motions from the two earthquakes caused widespread damage across the city. The earthquake event of August 17 resulted in over 17,000 fatalities and the November 12 earthquake added another 452 fatalities.   A damage survey was conducted by the Directorate of Disaster Affairs after these events. The survey showed a strong correlation of cumulative damage with increasing number of storeys. It is important to note that the majority of buildings in Duzce did not comply with the most recent code requirements (Sucuoglu and Yilmaz, 2001). Post-earthquake observations also correlated severe damage to the buildings and infrastructure with near-field ground motions having long duration strong velocity pulses (Kalkan et al., 2004).  3.1.4 Other earthquake events with pulse-like near-field ground motion records Throughout the past 40 years many recordings of near field ground motions have been acquired that showed the presence of long duration strong pulses in ground shaking. The duration of these pulses is typically bound between 1.0s and 12.0s. Other events not discussed above include: 1979 Imperial Valley, 1989 Loma Prieta, 1992 Landers, 1995 Kobe, 1999 Chi-Chi and 2010-2011 Canterbury earthquakes.  17  3.2 Effects of near-field pulse-like and far-field ground motions on buildings Near field pulse-like ground motions occur within a distance of 20km to 30km from the causative fault and contain few cycles of large motions distinguishable either in the form of acceleration, velocity or displacement pulses. Ordinary far-field ground motions occur at distances greater than 30km from the source and have many more cycles of motion without any distinguishable pulses.  It has been observed that near field impulsive ground motions affect the seismic response of structures in a different manner than far-field ground motions (Bertero et al., 1978). This is because pulse-like near field ground motions have very distinct characteristics compared to ordinary far-field ground motions.  The arrival of long duration energetic pulses prompts a building structure to deform significantly under just a few cycles. This type of response prevents absorption of input energy in the form structural damping and results in large strain energy absorption, which ultimately produces structural damage. In contrast, far-field ground motions impose numerous cyclic loading reversals that progressively build up the building resonant response (Naeim, 1995; Kalkan and Kunnath, 2006).    Figure 3.1 compares the velocity time series and displacement spectra (=5%) for two horizontal components of ground motion at El Centro array #6 (NGA181) and Delta (NGA169) stations from the Mw6.5 1979 Imperial Valley earthquake, at distances close to the rupture (Rrup) of 1.4 km (near-field) and 22 km (far-field), respectively. The two components are in the perpendicular 18  direction and parallel direction to the strike, defined here as fault-normal (FN) and fault-parallel (FP), respectively. Spectral ordinates of the near-field motion are larger at periods longer than 2s and larger displacement demands can be expected on tall buildings when compared to far-field.   (a) (b)   (c) (d)   (e) (f) Figure 3.1  Comparison near-field and far-field records from the 1979 Imperial Valley earthquake.  The complex nature of near-field ground motions is one of the reasons why the effects of earthquake ground shaking on buildings located at close distance of the causative fault are not fully understood. The results from Anderson and Bertero (1987) indicate that the nonlinear response of steel frame buildings is sensitive to (1) the pulse duration (Tp) relative to the 19  fundamental building period (T), and (2) the relation between the amplitude of the acceleration pulse to the yield-resistance seismic coefficient of the structure. This coefficient can be expressed as the base shear capacity divided by the effective weight. Using pulse-like ground motion records and computer models they determined the nonlinear dynamic responses of two 10-storey steel frames, having first mode periods of 2.1s and 0.6s, respectively. It was observed that the building with T=0.6s experienced higher ductility demands at the base when compared to the building with T=2.1s, this was attributed to a larger ratio Tp/T. It was concluded that pulse-like ground motions that tend to concentrate deformations in the lower floors of buildings can be particularly damaging because this increases the P-delta effects.  Alavi and Krawinkler (2004) used near-fault pulse-like ground motion records to analyze the dynamic response of generic steel frame models representative of multi-degree-of-freedom structures. They found that for buildings with T>Tp, large elastic storey shear forces occur in the upper storeys. Under this scenario they identified two types of inelastic responses (1) when the structure is relatively strong, these large storey shear forces produce early yielding in the upper storeys, and (2) if the structure strength is lower the maximum story ductility demands occur at the base. Furthermore, they found that for buildings with T≤Tp the maximum storey ductility demands always occur in the bottom portion of the building irrespective of its strength.  Calugaru and Panagiotou (2011) investigated the inelastic response of shear wall reinforced concrete tall buildings to pulse-like ground motions. The findings from this study are that strong velocity pulses with durations close to the second mode period excite the first and second mode, producing large inelastic response at the base of the wall. Furthermore, if the pulse duration is 20  shorter than the second mode significant bending moments at the intermediate wall height that exceed the base moment yield strength can occur. Overall when the ratio Tp/T is less than one, the peak shear force at 75% of the building height is close or even larger than 50% of the peak base shear. The buildings subjected to ground motions with ratios Tp/T larger than one consistently experienced very large curvature ductility demands at the wall base.   3.3 Directionality effects of near-field ground motions on response spectrum Earlier studies have associated the presence of strong long duration pulses in the horizontal component perpendicular to the fault strike with forward directivity effects (Singh, 1981; Somerville et al. 1995; Somerville et al. 1997). These effects appear when three conditions are met: (i) the rupture front propagates towards the site; (ii) the slip direction and the rupture propagation are aligned; and (iii) the propagation of the rupture front at a velocity almost as large as the shear wave velocity (called subshear rupture) causes constructive interference of the seismic waves such that significant seismic energy radiated from the rupture arrives at the site as a strong pulse of long duration. Somerville et al. (1997) found that the response spectrum of the ground motion normal to the fault strike is greater than the spectrum of the parallel component at periods longer than 0.6s as a function of (1) earthquake magnitude (2) distance and (3) the angle between the fault plane and the path from the hypocenter to the site.  Subsequent studies have reassessed these early observations. Howard et al. (2005) analyzed 29 near-field records and identified the orientation of the ground motion that produced the maximum spectral ordinates (SAMAX) as the component that yielded the maximum spectral intensity (SI), defined as maximum area under the response spectra (SA) in the period range of 21  0.5s to 3.0s. For this study a total of 26 record pairs were retrieved from the PEER Ground Motion Database. The remaining records were obtained from the CSMIP, USGS and COSMOS databases. An analysis of ground motion records at distances Rrup less than 5km from strike-slip earthquakes in the magnitude range of Mw 6.5-7.3 revealed that the azimuth of SAMAX deviated on average 21° from fault-normal, the minimum deviation was found to be 0° for the Erzincan 1992 earthquake recording at the Erzincan station and the maximum deviation was 66° for Imperial Valley earthquake recording 1979 at the El Centro array #8 station. A similar evaluation was performed for reverse-faulting records at distances Rrup less than 10km and magnitudes in the range Mw 6.6-7.4. This evaluation showed that, on average, the difference between azimuths of SAMAX and fault-normal was 33°, the minimum 1° and the maximum deviation of 76°. The study from these records concludes that typically the SAMAX were closely aligned with strong horizontal velocity pulses.  Upon analyzing the ground motion records from the 1999 Mw 7.4 Kocaeli earthquake in Turkey (Yarimca station) and the 2002 Mw 7.9  Denali earthquake in Alaska (PS-10 pump station) the study by Howard et al. (2005) suggests that reliance on SAMAX being along fault-normal at near-field sites for large strike slip events (Mw >7.25) depends on the likelihood of superposition of fling and directivity motions, as could be the case of a supershear velocity rupture (at a fault that unzips at a velocity larger than shear-wave velocity).   Another study by Watson-Lamprey and Boore (2007) selected 3397 ground motions pairs from the PEER NGA database to investigate the use of conversion factors between spectral ordinates from the geometric mean of two orthogonal horizontal components into other ground motion 22  spectral intensity measures. Among these are maximum response spectrum over all rotation angles (SaMaxRot) and response spectrum for a random chosen component. They found that the rotation angle of the ground motion giving the maximum value of the spectral ordinate is period dependent. The circumstances where FN provided spectral ordinates close to SaMaxRot were investigated. Their observations are that rarely the FN will correspond to SaMaxRot, except for limited conditions of strike-slip earthquakes, at stations closer than 3 km from the causative fault and located off the end of the fault.  Other recent studies have systematically investigated the dependence of the elastic spectrum on ground motion orientations and they conclude that the orientation of maximum spectral ordinates at distances greater than 3km deviates significantly from FN orientation (Huang et al., 2008; Shahi and Baker, 2013). Huang et al. (2008) investigated the maximum spectral demands in the near field ground motions and compared them to geometric mean spectral demands defined by the Next Generation Attenuation (NGA) relationships (Boore and Atkinson, 2008; Campbell and Bozorgnia, 2008; Chiou and Youngs, 2008). Using a suite of 147 records obtained from the PEER NGA strong motion database, they found, an orientation along which the maximum spectral demand was attained for several period ranges of the elastic oscillator. The ground motions were recorded during events of moment magnitude Mw greater than 6.5 and a source to site distances Rrup less than 15km. The study showed that scaling factors could be developed to obtain maximum spectral demands from geometric mean spectral demands. It was found that spectral demands obtained from the strike normal ground motion may significantly underestimate the maximum spectral demands.  23  Figure 3.2 shows two examples of the variability in spectral ordinates when the near-fault ground motion is analyzed along the fault-parallel, fault-normal and for the resulting ground motion at an intermediate orientation. The ground motions correspond to the 1979 Mw 6.5 Imperial Valley earthquake at El Centro array #6 (NGA181) and 1999 Mw7.5 Imperial Valley earthquake at Yarimca (NGA1176). The records rotated into FN/FP were obtained from the PEER NGA Ground Motion Database (PEER, 2010). The spectra confirm the observations above that the fault-normal motions are not necessarily the component that produces the maximum response spectra. NGA181 at Rrup 1.4km NGA1176 at Rrup 4.8km Figure 3.2  Comparison of response spectra (=5%) along different orientations.  A recent study by Shahi (2013) provided an empirical framework to estimate the spectral ordinates along a single orientation from the fault strike. This is a design spectrum conditioned on orientation and period. The development of this spectrum is based on a method that uses empirical data from earthquake recordings, and be updated upon acquisition of new ground motion records.   24  3.4 Directionality effects of ground motions on seismic response of buildings The critical response of structures considering the effects of ground motion directionality has been investigated in several past studies. Earlier studies were related to response of structures to multi-component earthquake excitation and critical angle of incidence using elastic response spectrum analysis and elastic response history analysis (Menum and Der Kiureghian, 1998; Wilson et al., 1995; Lopez et al., 2000; Athanatopoulou, 2005). However, these studies provide limited insight for the performance evaluation of buildings subjected to strong ground shaking that produces nonlinear response.   The effects of ground motion directionality on seismic response of buildings are at the center of ongoing debate among earthquake engineering professionals and researchers. This is due to the modification of the horizontal design spectrum definition from geometric mean to maximum direction in the 2009 NHERP provisions. The new definition was included in the seismic maps of the standard ASCE 7-10.  Concerns were raised over the conservatism introduced by this new definition of design spectrum by Stewart et al. (2011).  It is considered that MD spectrum is adequate to design structures that have identical dynamic properties in all directions, have same lateral strength and stiffness along all directions and do not have preferred directions of response, e.g. flagpoles and circular tank, which have been named as azimuth-independent structures. On the other extreme, structures whose response is deemed to be sensitive to the incident angle of the ground motion are called azimuth-dependent. Conventional modern buildings, bridges and dams belong to this category.  It has been argued that the probability of the design MD ground motion acting along 25  the axis of structural response is unlikely, this is because (i) the orientations of a ground motion that would be compatible with the design MD motion are typically period dependent and (ii) the probability of the MD ground motion to be acting along the principal axis of a building is low.  Recent studies have undertaken the task of assessing the effects of ground motion directionality on the seismic response of case study buildings using nonlinear response history analysis and have found that response of the building is highly dependent on the orientation and intensity of the ground motion (Rigato and Medina, 2007; Lagaros, 2010; Archila, 2011; Archila and Ventura, 2012; Reyes and Kalkan; 2012a, 2012b). The latter study by Reyes and Kalkan (2012a) investigated the effect of rotating near field pulse-like ground motions to FN/FP and MD (at first mode period) on the seismic response of a 9-storey building. Their main finding is that neither FN/FP nor MD rotated ground motion result always in the critical response. Furthermore, they observe the critical drift seems to be polarized in the direction where the duration of the velocity pulse is close to the first mode period of the building.  3.5 Recommendations and building code provisions concerning seismic input and ground motion directionality The Council of Tall Buildings recommendations (Willford et al., 2008) are that seismic input ground motions be applied along the structural (principal) axes of the building model when conducting NRHA. A complementary NRHA should follow by applying the same ground motions but rotated 90 degrees when the target spectrum for scaling both horizontal components is the maximum and minimum spectrum. The response envelopes of both analyses are combined 26  to define an overall envelope. This approach attempts to address the influence of ground motion directionality on seismic response of tall buildings without large computational demands.  Chapter 16 of ASCE 7-10 provisions for seismic response history analysis of a building at a site within 3 miles (5 km) of the active fault that controls the hazard require the input motion to be aligned along FN and FP of the causative fault and scaled such that FN is not less than the design spectrum in the range of 0.2T to 1.5T. Therein T is the first mode translational period of the building. The provision does not require verification whether FN is actually the dominant component or not. Furthermore, there are no guidelines on how to deal with seismic input for sites located at distances farther than 3 miles (5km) from the causative fault. A recommendation by Haselton et al. (2012) for selecting and scaling near-fault ground motions for performing RHA is to apply the scaled motions using the same orientation of the seed ground motions had relative to the causative fault. Any further rotations of the scaled ground motion pair (typically rotated 90° from the original orientation) to perform additional simulations are deemed unnecessary.  3.6 Main findings from past studies Pulse-like ground motions affects significantly the seismic response of buildings. Strong pulse-like ground motions have produced severe structural damage and residual displacements of buildings. It has been shown that the seismic response to near-fault motions depends on the ratio Tp/T and the relation of the acceleration pulse amplitude to yield-resistant seismic coefficient. Whenever the ratio Tp/T is greater or equal to 1 the maximum storey ductility demands occur at the base of the building. Instead, when the ratio Tp/T is less than 1 the maximum storey ductility 27  demands can occur at the upper storeys if the building is relatively strong, otherwise they occur at the base.  Studies on near field pulse like ground motions have showed that there is not a fixed orientation with respect to the fault strike that will result in the maximum response spectra. The use of FN component from near field ground motions does not guarantee good estimates of the maximum response spectra.  Furthermore, studies on directionality effects of ground motions have shown that building response is quite sensitive to the ground motion angle of incidence. There is not a practical method that engineers can use to determine the critical response and the only option is conducting analysis applying the ground motion at multiple angles of incidence (direct analysis), a computationally expensive method. The critical response is defined as the envelope of the peak responses the from rotated ground motion pair.  3.7 Gaps in state of knowledge  Despite of all the crucial findings and recommendations above, several issues related to pulse-like near field ground motions and seismic response of buildings have not been widely investigated yet, including the effect of ground motion directionality. Understanding the potential demands that near field ground shaking can impose on buildings is a crucial task in order to make informed decisions regarding the seismic design of tall buildings. The following gaps in the current state of knowledge were found:  28  1. Given a pulse-like ground motion, the orientation that leads to the critical displacement response of a tall building cannot be determined a priori without systematically performing a nonlinear analysis of the building at multiple angles of incidence. 2. The nonlinear response of a tall building to ground motions rotated to FN/FP and its relation with the critical displacement has not been widely investigated yet, thus it is not clear whether these components are detrimental for the seismic response or not. 3. The effectiveness of the maximum direction ground motions to predict the critical displacement response of tall buildings has not been quantified yet.  3.8 Problem Statement Significant uncertainty exists for seismic design of tall buildings at near-field sites. A source of uncertainty is the strong dependence of the seismic response on the orientation of the ground motion with respect to the structural axes because such orientation cannot be predicted a priori for future earthquakes with the current state of knowledge. Under these circumstances, to obtain upper bound estimates of potential seismic demands for seismic design of tall buildings the critical displacement response must be estimated using pulse-like ground motions.  However a computationally efficient method to determine the critical displacement response of tall buildings when NRHA is conducted does not exist. The direct analysis approach is currently the only viable method to evaluate directionality effects on the nonlinear seismic response of buildings. This limitation poses difficulties for seismic design of tall buildings. 29  Chapter 4: Response of simple structures to pulse-like excitations The dynamic response of simple structures to double-sided velocity pulses is presented in this chapter. Simple pulse waveforms of constant peak velocity amplitude but varying duration (Tp) are used to systematically analyze the elastic and inelastic response of simple structures. These analyses are used (1) to evaluate the effect of the pulse duration on the displacement response, (2) to assess the effect of the pulse amplitude on the displacement response and (3) to find whether or not the maximum displacement response consistently occurs for particular characteristics of the velocity pulse duration and peak amplitude.  The rationale behind this approach is that if the pulse characteristics which lead to the maximum (critical) displacement are determined, bounds on pulse duration and amplitude may be proposed to identify the orientation of a pulse-like ground motion which will cause the critical displacement response of a tall building.  4.1 Model definition and numerical solution of equation of motion The simple structure analyzed in this study was idealized as a single-degree-of-freedom system. The model used for dynamic analyses is a mass attached to a weightless spring and viscous dashpot attached to a rigid base as shown in Figure 4.1a. The spring model has an elastic-perfectly plastic behavior, with equal yield resistance in tension and compression as shown in Figure 4.1b. There is no stiffness degradation during unloading and reloading.  30    (a) SDOF (b) Spring model Figure 4.1 SDOF system and spring force-displacement relationship.  The equation of motion of the system to a base excitation ݑሷ௚ሺݐሻ is given below: ݉ݑሷ ሺݐሻ ൅ ܿݑሶ ሺݐሻ ൅ ݇ሺݐሻݑሺݐሻ ൌ െ݉ݑሷ௚ሺݐሻ (4.1)   This differential equation of motion was solved numerically using Newmark’s average acceleration method with =0.25 and =0.5. A damping ratio of 5% was used and initial conditions at rest were ݑሺ0ሻ=0 and ݑሶ ሺ0ሻ=0. The response was traced while the base excitation was being applied and after the excitation was over, for a period equal to two times the elastic period to capture the free vibrations of the system. The peak displacement was determined as the maximum absolute value from the response history.   The calculation was performed using a program written by the dissertation author in Microsoft Visual Basic for Applications (VBA), a development platform built in Microsoft Excel® (2010). This program was developed because of the versatile solution it provides for parametric studies, which can easily accommodate a large number of systematic variations of the pulse-like input ݑሷ௚(t) ݑሺݐሻ ݉ ݇ሺݐሻ ܿ ݂݁ܿݎ݋݀݅ݐ݈݊݁݉݁ܿܽ݌ݏ	ሾݑሿFݕ݇ ݇u Fݕ݇ܽ݁݌ݑ ݕݑ 31  motions and system models in comparison to other commercial programs.  Specific analyses within the parametric studies were verified by independently performing elastic and inelastic dynamic analysis of SDOF using programs Bispec V2.2 (Hachem, 2012) and NONLIN V8.00 (Charney, 2011). Differences between peak displacements from the VBA program developed in this study and the other programs were within ±0.5%, indicating that the VBA program was reliable and sufficiently accurate. A total of 3 pulse-like synthetic excitations and 3 pulse-like ground motion records were used for the verification.  4.2 Pulse duration and amplitude effects on unidirectional response of SDOF oscillators A series of parametric studies were performed to evaluate the effect of varying the duration of velocity pulses on the displacement response of the SDOF systems, while keeping the peak velocity amplitude of the pulses constant. The condition of constant peak velocity was adopted to isolate to some extent the effect of pulse duration on the displacement response. In order to obtain more insight, this effect was explored for pulses of three different peak velocity amplitudes.   4.2.1 Definition of synthetic pulses The range of the peak velocity pulse amplitudes and durations for synthetic pulses was selected to encompass the characteristics of the velocity pulses typically contained in near-field ground motions. The synthetic pulse durations were set in the range of 0.5s to 10.0s, at increments of 0.1s. To account for the different peak ground velocity amplitudes, independent parametric studies were performed for pulses with constant peak amplitudes equal to 40 cm/s, 100 cm/s and 150 cm/s. These amplitudes are based on findings from Ambraseys (1969) and Brune (1970), 32  who determined a theoretical limit range for peak ground velocity at near-field, which should not exceed 100 to 150 cm/s.  4.2.1.1 Pulse waveforms The input excitations are given as acceleration time series. The acceleration pulse is a three half-sine pulse, shown in Figure 4.2 with normalized amplitude. It can be seen that the velocity time series is a double-sided pulse, while the displacement is a single-sided pulse. This pulse has been used in several previous studies to develop constant ductility spectra for SDOF systems (Newmark and Veletsos, 1964; Sasani and Bertero, 2000; Kalkan and Kunnath, 2006). This velocity waveform resembles the double-sided velocity pulses contained in pulse-like near-field ground motions. Besides this pulse provide a smooth increase in the initial velocity and displacement motions.  Figure 4.2 Three-half-sine acceleration pulse and associated velocity and displacement pulses.  	࢛ሶ ࢍ	࢛ሶ ࢍ	࢖ࢋࢇ࢑⁄ 	࢛ሶሷ ࢍ	࢛ሷ ࢍ	࢖ࢋࢇ࢑ൗ ࢛ ࢍ	࢛ ࢍ	࢖ࢋࢇ࢑⁄ 33  The acceleration time series for each pulse excitation were calculated using the Equation 4.2. This input motion is dependent on the peak acceleration (ݑሷ௚	௣௘௔௞) and the pulse duration.  To ensure that the excitation had a constant peak velocity amplitude (ݑሶ௚	௣௘௔௞), the peak acceleration was defined using Equation 4.3, where the peak velocity was set equal to a constant value while the duration Tp was varied. It can be observed that while the peak acceleration is directly proportional to the peak velocity it is inversely proportional to the pulse duration.  The excitation’s peak displacement (ݑ௚	௣௘௔௞) was calculated using Equation 4.4, dependent on two parameters, pulse duration and peak velocity amplitude. It follows from this equation that the ratio 	௨೒	೛೐ೌೖ௨ሶ ೒	೛೐ೌೖ ೛் is constant, and the peak displacement is directly proportional to the pulse duration and the peak velocity. The acceleration, velocity and displacement pulses are shown in Figure 4.3 for different values of Tp and peak velocity amplitude of 100 cm/s. The maximum values of peak acceleration occur for the pulse of shortest duration and the maximum values of peak displacement for the pulse of longest duration. ݑሷ௚ሺݐሻ ൌەۖۖۖ۔ۖۖۖۓ ݑሷ௚	௣௘௔௞ sin ൬4ݐߨ ௣ܶൗ ൰ ݐ ൑௣ܶ 4ൗെݑሷ௚	௣௘௔௞ sinቌ2ߨ ௣ܶൗ ൭ݐ െ௣ܶ 4ൗ ൱ቍ ௣ܶ 4ൗ ൑ ݐ ൑ 3 4ൗ ௣ܶݑሷ௚	௣௘௔௞ sin ቆ4ߨ ௣ܶൗ ൫ݐ െ 3 4ൗ ௣ܶ൯ቇ 3 4ൗ ௣ܶ ൑ ݐ ൑ ௣ܶ0 ௣ܶ ൑ ݐ ൑ ௣ܶ ൅ 2 ௡ܶ (4.2) Where ݑሷ௚	௣௘௔௞ and 	ݑ௚	௣௘௔௞ can be determined with the following expressions ݑሷ௚	௣௘௔௞ ൌ 2	ݑሶ௚	௣௘௔௞ ௣ܶ⁄  (4.3) 	ݑ௚	௣௘௔௞ ൌ 	ݑሶ௚	௣௘௔௞	 ௣ܶ ൤ 12 ൅18൨ (4.4) 34    Figure 4.3 Pulse acceleration, velocity and displacement time series for velocity pulse amplitude of 100cm/s.  4.2.2 Scope of parametric study  To perform the parametric study the dynamic analysis was repeated for the pulse excitations defined above and SDOF systems having different characteristics. In the first case the pulse excitations with peak velocity amplitudes of 40 cm/s, 100 cm/s and 150 cm/s were used to analyze the response of systems with elastic periods of 1s. Besides evaluating the elastic response, the inelastic response was evaluated for systems having normalized yield strengths ࢛ሶ ࢍሺࢉ࢓࢙⁄ሻ ࢛ሷ ࢍሺࢉ࢓࢙૛ ⁄ሻ ࢛ ࢍሺࢉ࢓ሻ 35  (Cy=lateral yield force/weight) equal to 0.15, 0.20 and 0.30. Second and third cases included the same analyses but for systems having elastic periods of 2s and 3s respectively.   4.2.3 Force response history as a function of Tp/T The force response histories of the system with T=2s and Cy=0.20 for different Tp/T ratios are compared in Figure 4.4. Yield forces are attained in all cases shown, however yielding is sustained for longer durations when Tp/T falls within 1 and 2.   Figure 4.4 Force response to pulse with peak amplitude of 100 cm/s for system having Cy=0.20 and T=2s.  Normalized spring force = F/[m g] 36  It is observed that when the ratio Tp/T is equal to 0.5 the system yields for a short duration while the pulse is applied, then after the pulse vanishes the system yields in the opposite direction and subsequently undergoes decaying motion in free vibration. At the other extreme when the ratio Tp/T is equal to 3 the system yields for short durations while the pulse is applied. The displacement ductility demands for all these cases are presented in the following section.  4.2.4 Displacement ductility demands as a function of Tp/T The displacement ductility demands for the system with T=2s as a function of the ratio Tp/T and pulse peak amplitude of 100 cm/s are shown in Figure 4.5. In these cases it is observed that irrespective of the normalized yield resistance the maximum ductility demands always occur when the ratio Tp/T is greater than unity, in the range of 1.1 to 1.6. At the extremes when the ratio Tp/T<<1 or Tp/T>>1 the trend is for the responses to become elastic. This observation is consistent throughout the responses calculated for other systems and pulse peak velocity amplitudes.  Figure 4.5 Displacement ductility demands for system with T=2s due to pulse peak amplitude of 100 cm/s. 37  4.2.5 Peak displacement responses The peak displacement responses as function of the pulse duration Tp and the ratio of the pulse duration to the system’s period (Tp/T) are shown for a system with T=1s in Figures 4.6 to 4.8. The pulse duration is shown in the upper horizontal axis and the ratio Tp/T in the lower horizontal axis. Each figure includes the elastic and inelastic responses to pulses having constant peak velocity amplitudes of 40 cm/s, 100 cm/s and 150 cm/s, respectively. It is observed that as the pulse peak velocity amplitude increases the peak response increases.   Comparing the pulse durations in Figure 4.3 that produce the maximum pulse peak acceleration and maximum pulse peak displacement with the durations that cause maximum (critical) peak responses in Figure 4.7, it is found that they differ significantly. Thus pulses having maximum peak acceleration or maximum peak displacement will not excite the SDOF system to the critical displacement response, unless the pulse duration matches Tp/T at critical response. The ratio Tp/T at critical responses is not constant, it increases as the pulse peak velocity amplitude increases and for the amplitudes considered it ranges from 1.1 to 3.  The responses of the system with T=2s are shown in Figures 4.9 to 4.11. The responses are displayed up to pulse durations of 10s. This range of periods is wide enough to capture the ratio Tp/T that produces the maximum peak response. In this case, the maximum peak displacements in each figure occurs instead for Tp/T ratios that fall in the range of 1.1 to 2.2, a smaller range compared to the system with T=1s.  38  Figures 4.12 to 4.14 show responses for the system with T=3s. In this case, the maximum peak displacement response occurs for ratios Tp/T between 1.1 and 1.7. This is a narrower range of ratios Tp/T compared to the systems with periods of vibration of 1s and 2s.   Several trends are observed from these figures (1) the maximum peak response occurs at smaller Tp/T ratios whenever the elastic period or the normalized yield strength Cy are increased (2) the maximum peak displacement occurs at larger Tp/T ratio when the peak amplitude of the velocity pulse increases.   Overall the maximum peak displacement occurs for Tp/T ratios within the range of 1.1 to 3. When the ratio Tp/T is much smaller or larger than unity the displacement responses are smaller.  It is observed throughout the simulations that the peak displacement increases whenever the peak velocity amplitude of the excitation increase, see Figures 4.6 to 4.8, Figures 4.9 to 4.11 and Figures 4.12 to 4.14. It is further observed that the peak displacement increases as the period of the system increases. See, for instance responses in Figures 4.8, 4.11 and 4.14.  It is evident that the maximum peak displacement for inelastic systems is similar to that of the elastic systems when the pulse peak amplitude is 40 cm/s, but for larger amplitudes the elastic response significantly differs from the inelastic response, see Figure 4.8. Therefore, elastic analysis will not predict the critical response of a system undergoing significant inelastic response.   39   Figure 4.6 Peak displacements for SDOF with T=1s for a pulse with peak amplitude of 40 cm/s.  Figure 4.7 Peak displacements for SDOF with T=1s for a pulse with peak amplitude of 100 cm/s.  Figure 4.8 Peak displacements for SDOF with T=1s for a pulse with peak amplitude of 150 cm/s. 40   Figure 4.9 Peak displacements for SDOF with T=2s for a pulse with peak amplitude of 40cm/s.  Figure 4.10 Peak displacements for SDOF with T=2s for a pulse with peak amplitude of 100cm/s.   Figure 4.11 Peak displacements for SDOF with T=2s for a pulse with peak amplitude of 150cm/s. 41   Figure 4.12 Peak displacements for SDOF with T=3s for a pulse with peak amplitude of 40cm/s.  Figure 4.13 Peak displacements for SDOF with T=3s for a pulse with peak amplitude of 100cm/s.  Figure 4.14 Peak displacements for SDOF with T=3s for a pulse with peak amplitude of 150cm/s.  42  4.3 Discussion of results The parametric study on the effect of pulse duration on peak displacement response of SDOF systems showed that the maximum peak response occurs at ratios Tp/T greater than 1. In all cases, when the response was elastic, the Tp/T ratio for maximum peak response was between 1.1 and 1.2. An upper bound of these ratios was found to be 4 for inelastic systems with an elastic period of 1s.    It must be recalled that although this observation is drawn from studies made using an idealized pulse waveform, the duration and amplitude of the synthetic velocity pulses were selected to fall within the typical range of durations and peak ground velocities from the ground motion records. Thus realistic limits on the pulses within which the maximum displacement occur were established.   The results of the parametric study are consistent with findings from earlier studies from Newmark and Veletsos (1964), Anderson and Bertero (1987) and Mavroeidis et al. (2004). However, the focus of these past studies was on the effects of pulse duration and amplitude without consideration of directionality effects of pulse-like ground motions on structures. The study by Newmark and Veletsos (1964) included extensive parametric analyses using different pulse-like excitations, to study the effects of earthquake and nuclear explosion ground motions on elastic and inelastic response of SDOF. From the report of Newmark and Veletsos, it was observed that the acceleration three-half-sine pulse excitation led to peak displacement responses of elastic undamped and damped (5%) systems, when the ratios of pulse duration (td) to elastic period were close to 1.  43  Anderson and Bertero (1987) addressed uncertainties in the selection of an adequate design earthquake for structures. In doing so, they performed nonlinear dynamic analyses of steel frame buildings to pulse-like earthquake ground motions. One of their key findings was that the seismic response of the buildings was particularly sensitive to the duration of the pulse acceleration with respect to the fundamental period of the structure.  The study by Mavroeidis et al. (2004) evaluated the elastic and inelastic responses of SDOF using a set of 25 near fault pulse-like records whose pulse durations were determined as the inverse of the prevailing frequency fp. Their results show that the normalized peak pseudo-velocity response occurs when the ratio T/Tp is between 0.7 and 1.0 for elastic spectra, and between 0.3 and 1.0 for inelastic response spectra at a displacement ductility ratio of =8. This corresponds to Tp/T ratios of 1.0 to 1.43 for elastic spectra and 1.0 to 3.33 for inelastic response spectra =8. In addition, they built a normalized design spectrum for near fault motions, where the normalized peak spectral displacement is bound by ratios T/Tp between 0.75 to 1.0, where the inverse gives Tp/T that range between 1.0 and 1.33.   There is evidence from the studies above and the present study that shows both the elastic and inelastic maximum peak displacement responses to pulse-like excitations that contain double-sided velocity waveforms occurs when the ratio Tp/T ranges between 1 and 3, see Figures 4.6 to 4.14.     One important characteristic of earthquake recorded ground motions that is not captured properly in the synthetic pulse is the presence of multiple velocity pulses, which for a given ground 44  motion can result in maximum peak response due to different Tp/T ratios. Thus establishing an upper bound for the Tp/T ratio for every structure and pulse ground motion record is not a straightforward evaluation. Although establishing an upper bound for Tp may be challenging and difficult to verify for every case, a lower bound for Tp longer than T can be relied upon for inelastic response.   However, the condition of Tp longer than T is necessary for maximum peak displacement response but not enough when angle of incidence of a ground motion is considered as well. The amplitude of the excitation must be considered as well to determine whether the critical response will be produced or not by a particular orientation of the pulse-like ground motion. This finding is consistent with the study from Anderson and Bertero (1987), who found that the seismic response of two steel frame buildings was also sensitive to the amplitude of the pulse acceleration relative to the yield seismic resistance coefficient.  The results from the parametric studies show that a pulse having either large peak acceleration or displacement may not produce the critical displacement response of a structure. Therefore, it is proposed herein to use the maximum ground velocity as the metric to quantify the intensity of the ground motion that produces the critical displacement response.  4.4 Recommended criteria to estimate critical response On the basis of the previous results a criteria is proposed to approximate the critical displacement response of a tall building to earthquake pulse-like ground motion records.  45   Duration criterion: Since the duration of the velocity pulse that produces the largest displacement response on SDOF is longer than T, the critical response can be expected along an orientation of the ground motion where the strong velocity pulse has duration longer than the translational first mode period of the building.  Amplitude criterion: It is proposed that the amplitude of the velocity pulse should be maximum among all the orientations of the ground motion to produce the critical response.  For a given record the ground motion component that satisfies this dual criterion is called the conditional maximum velocity (CMV) ground motion. In chapter 5 this dual criterion will be used to compare the response of simple structures to ground motions records rotated to the conditional maximum velocity with the critical displacement response from direct analysis.    46  Chapter 5: Response of simple structures to velocity pulses in ground motions  The duration and amplitude characteristics of velocity pulses contained in pulse-like near field ground motion records and the response of simple structures to these motions are studied in this chapter. The objectives are (1) to examine in detail the duration and amplitude of velocity pulses present in ground motion records along different orientations, (2) analyze directionality effects of the ground motions on the linear elastic response spectrum, (3) investigate the sensitivity of the nonlinear response of simple structures to the orientation of pulse-like ground motions and (4) verify the effectiveness of the CMV ground motion to estimate the critical displacement of simple structures.   5.1 Selection of near-field ground motion records Ground motions were selected from the PEER Ground Motion Database available at the Pacific Earthquake Engineering Research Center website (PEER, 2010).  Recent studies of the records contained therein have identified sets of pulse-like ground motions (NIST, 2011; Shahi, 2013). Ground motions in these pulse-like motion sets that met the following criteria were selected:  Crustal earthquake magnitude  Mw 5.5-7.9  Recorded within 30km of the rupture (Rrup). Two orthogonal pairs of horizontal ground motion should be available  Fault Normal (FN) and Fault Parallel (FP) orientations clearly identified  Velocity pulses in the range of 30 cm/s - 150 cm/s   Velocity pulse duration (Tp) preferably in the range of 1s to 8 s.  47  Near field ground motions are conventionally bound within a distance to the rupture of 20 km to 30 km. In the studies cited above velocity pulses of large amplitude were found in records up to 30 km from the fault and were included in the selection to investigate velocity pulses at near fault (less than 5 km) and farther away giving a broader application of the findings from this research.  The selection criteria resulted in a collection of 45 near-field ground motion pairs, which were separated into three sets of records and listed in Table 5.1, 5.2 and 5.3. Set 1 in Table 5.1 includes 15 records from strike slip earthquake mechanisms at Rrup≤10km. Set 2 in Table 5.2 comprises 15 records either from normal, reverse-oblique, or thrusting earthquake mechanisms at Rrup≤10km. Set 3 in Table 5.3 has 15 records from different faulting mechanisms at 10km<Rrup<30km, where the majority of records where obtained at distances less than 20km. These sets will allow investigate the orientation dependence of the velocity pulse amplitudes for different earthquake mechanisms.   The second column of the tables indicates the identification number (NGA index) that the ground motion has in the PEER database. The last column of the tables indicates an abbreviation for the faulting mechanism, where SS stands for Strike Slip Mechanism, RO for Reverse Oblique Mechanism, N for Normal Mechanism and TH for Thrusting Mechanism.   The archived records in the PEER Strong Motion Database have been rotated into fault-parallel (FP) and fault-normal (FN) components. Most of the ground motions provided by PEER have been processed with a high-pass filter and baseline correction (PEER, 2010), such that “fling 48  effects” in the form of residual coseismic displacements are not preserved. Within the three sets only the record from Landers earthquake at Lucerne station (NGA879) preserves the residual coseismic displacements. All the records in sets 1 to 3 were resampled to a uniform time step of 0.005s (sampling rate of 200 sps). Table 5.1 Set 1 strike-slip near field ground motion records at Rrup ≤ 10km No NGA Station Rrup (km) Earthquake Year Mw Mechanism 1 170 EC County Center FF 7.3 Imperial Valley 1979 6.5 SS 2 173 El Centro Array #10 8.6 3 181 El Centro Array #6 1.4 4 182 El Centro Array #7 0.6 5 183 El Centro Array #8 3.9 6 184 El Centro Diff. Array 5.1 7 185 Hotville Post Office 7.7 8 568 Geotech Investigation Center 6.3 San Salvador 1986 5.8 SS 9 569 National Geographical Inst 7.0 10 879 Lucerne 2.2 Landers 1992 7.3 SS 11 1119 Takarazuka 0.3 Kobe 1995 6.9 SS 12 1165 Izmit 7.2 Kocaeli 1999 7.4 SS 13 1176 Yarimca 4.8 14 1605 Duzce 6.8 Duzce 1999 7.2 SS 15 2114 TAPS Pump Station #10 2.7 Denali 2002 7.9 SS  Table 5.2 Set 2 of near field ground motion records at Rrup ≤ 10km No NGA Station Rrup (km) Earthquake Year Mw Mechanism 16 143 Tabas 2.0 Tabas Iran 1978 7.4 RO 17 285 Bagnoli Irpinio 8.1 Irpinia Italy-01 1980 6.9 N 18 802 Saratoga Aloha Ave. 8.5 Loma Prieta 1989 6.9 RO 19 803 Saratoga W Valley Coll. 9.3 20 825 Cape Mendocino 7.0 Cape Mendocino 1992 7.0 RO 21 828 Petrolia 8.2 22 983 Jensen Filter Plant Generator 5.4 Northridge 1994 6.7 TH 23 1013 LA Dam 5.9 24 1084 Sylmar Converter Sta 5.3 25 1085 Sylmar Converter Sta East 5.2 26 1244 CHY101 9.9 Chi-Chi 1999 7.6 RO 27 1494 TCU54 5.3 28 1510 TCU075 0.9 29 1511 TCU076 2.7 30 1528 TCU101 2.1  49  Table 5.3 Set 3 of near field ground motion records at 10 km < Rrup< 30 km. No NGA Station Rrup (km) Earthquake Year Mw Mechanism31 178 El Centro Array #3 12.9 Imperial 1979 6.5 SS 32 292 Sturno 10.8 Irpinia Italy-01 1980 6.9 N 33 316 Parachute Test Site 16.7 Westmorland 1981 5.9 SS 34 527 Morongo Valley Fire Station  12.1 N. Palms 1986 6.9 RO 35 721 El Centro Imp. Co. Cent 18.2 Superstition Hills 1987 6.5 SS 36 725 Poe Road (temp) 11.2 37 767 Gilroy Array #3 12.8 Loma Prieta 1989 6.9 RO 38 1148 Arcelik 13.5 Kocaeli 1999 7.4 SS 39 1158 Duzce 15.4 40 1476 TCU029 28.0 Chi-Chi 1999 7.6 RO 41 1480 TCU036 19.8 42 1481 TCU038 25.4 43 1482 TCU039 19.9 44 1502 TCU064 16.6 45 1602 Bolu 12.0 Duzce 1999 7.2 SS   5.2 Analysis of velocity pulses in ground motions  In this section the pulse-like ground motions selected above were used to investigate the orientation dependence of the maximum velocity pulse amplitude and duration.  A flowchart presented in Figure 5.1 outlines the procedure followed to analyze and compile the characteristics of the velocity pulses. The definition of the reference system and the method to determine the pulse amplitude and duration are explained in detail in subsequent sections.  The analysis of the velocity pulses contained in the selected ground motions will be used later on to determine the CMV ground motion using the dual criterion described at the end of chapter 4. Then, it will be investigated through dynamic analysis if this analysis provides goods estimates of the critical displacement response of a building. 50    Figure 5.1 Flowchart of procedure to identify amplitude and duration characteristics of velocity pulses.  5.2.1 Definition of reference system The calculation of the characteristics of the velocity pulses along different orientations of the horizontal ground motion requires first the definition of a coordinate reference system. The convention adopted for the orientation of ground motion pairs in this analysis was to have the FN component along the X’-axis and FP component along the Y’-axis of a reference X’-Y’ plane as shown in Figure 5.2.  The ray A-a will be used to study the ground motion along different orientations . This angle will be measured counterclockwise with respect to the Y’-axis and varied between 0° and 180°. The acceleration ground motion along ray A-a can be calculated using Equation 5.1, which resolves FN and FP ground motion components onto the orientation of ray A-a. ݑሷ௚஺ି௔ሺ, ݐሻ ൌ ሾെ sin  cos ሿ ቈݑሷ௚ிேሺݐሻݑሷ௚ி௉ሺݐሻ቉ (5.1)INPUT GROUND MOTIONS DATABASE OF VELOCITY PULSE PARAMETERS   PEAK GROUND VELOCITY OF EACH GROUND MOTION ALONG ORIENTATION   DURATION AND AMPLITUDE OF EXTRACTED PULSE ALONG ORIENTATION  REFERENCE SYSTEM DEFINITION TIME SERIES OF GROUND MOTION ALONG DIFFERENT ORIENTATIONS  IN THE RANGE OF 0° TO 180° EXTRACTION OF VELOCITY PULSE IN EACH GROUND MOTION 51   Figure 5.2 Coordinate reference systems and orientation of input motions FN/FP.  5.2.2 Determination of velocity pulse amplitude Past studies have shown that the amplitude of the velocity pulse in the time domain can be represented adequately by the peak ground velocity (Bray and Rodriguez-Marek, 2004). Therefore the amplitude of the velocity pulses along different orientations of the ground motion pairs was determined by tracing the (PGV) along rays A-a in the range of 0° to 180°, at clockwise increments of 5°.  From this analysis, the orientation of the ground maximum velocity (MV) with respect to the fault-parallel orientation was found.   The increment in rotation of rays A-a affects the accuracy of the MV orientation and the respective pulse amplitude. At increments of 5° the orientation of MV could have a maximum deviation of 2.5°; for practical engineering applications this deviation is negligible.  In Figure 5.3 polar plots show with a red dot the minimum angle with respect to the fault-parallel where the maximum velocity is achieved for each one of the 45 ground motions. In only a few cases the maximum velocity is perfectly oriented along the fault-normal or fault-parallel 52  directions. No dominant orientation is observed in these polar plots. The median MV from all the motions is 69 cm/s.  (a) Motions from set 1  (b) Motions from set 2 (c) Motions from set 3 (d) All motions Figure 5.3 Orientation of maximum velocity of the ground motion records.  5.2.3 Determination of velocity pulse duration Several methods are available to quantify the duration of velocity pulses in near-field ground motions. Bray and Rodriguez-Marek (2004) define the pulse duration using either the zero 53  crossings or the time at which the ground velocity amplitude is equal to 10% of PGV, this method requires use of judgment, for instance, to determine the zero crossing in the presence of noise.  Alavi and Krawinkler (2004) estimated the duration pulses contained in a near-field record as the period at the global peak in the velocity response spectrum. This procedure is based on the observation that the response spectrum of synthetic pulses show a peak when T/Tp=1. The drawbacks they recognized of following this procedure are that some records may produce more than one clear peak or that in some instances the global would be too wide requiring judgmental decisions to identify the period. Baker (2007) demonstrated that in some instances certain predominant peaks in the velocity response spectrum for near-field records are caused by high frequency motions and not by the presence of the velocity pulse (low frequency motion), thus a more robust measure of pulse duration is warranted.  Another method to quantify the characteristics of pulse-like ground motions was developed by Baker (2007) and implemented as a MatLab program, which is available online at http://www.stanford.edu/~bakerjw/pulse-classification_old.html. This program performs a continuous wavelet transform (CWT) on the time series of the velocity ground motion, which essentially decomposes the velocity time series into the frequency-time domain. In this wavelet analysis a Daubechies mother wavelet of order 4 dilated by a scale s and shifted to a location l is used to define the basis function for the CWT; this mother wavelet is preferred over others because of its resemblance with the waveform of velocity pulses found in ground motion 54  records. The coefficient (ܥ௦,௟) which corresponds to the wavelet having scale s and location l is calculated as follows:  ܥ௦,௟ ൌ න ݂ሺݐሻ 1√ݏ ൬ݐ െ ݈ݏ ൰ஶିஶ݀5.2) ݐ)  where ሺ௧ି௟௦ ) is the dilated mother wavelet at location l.  Baker’s program was developed to isolate and extract a waveform that closely resembles the transient features of the largest velocity pulse in a ground motion, and ultimately determine duration (Tp1) and amplitude (P1) characteristics of the extracted waveform. After the CWT is performed the largest wavelet coefficient is determined and used to identify the location of the velocity pulse. This wavelet is subtracted from the original ground motion to define the residual ground motion. Once the location and the duration of the pulse have been identified, a CWT analysis is performed again to obtain the next largest wavelet coefficients from the residual ground motion, such analysis is restricted for wavelets having the same duration and located within one half of the width of the wavelet of scale s from the first largest wavelet coefficient. The additional wavelet coefficients ensure an adequate representation of the velocity pulse.   The duration of the extracted pulse is defined as the inverse of the frequency at the maximum Fourier amplitude of the wavelet with the largest coefficient from the CWT of the original ground motion; this coefficient is associated with the wavelet of largest energy content (Baker, 2007). As shown in Figure 5.4 for the fault-normal component of the 1979 Imperial Valley 55  NGA181 record, the extracted pulse is an approximation to the waveform of the ground motion. The duration of the extracted pulse (Tp1) equal to 3.85s is a good estimate of the duration of the velocity pulse (Tp) determined as the time between zero crossings.  Figure 5.4 Velocity time series of 1979 Imperial Valley record NGA181 and extracted velocity pulse.  Baker verified the ability of his program to estimate the duration of velocity pulses using a set of 91 near-field ground motions. A comparison between the pulse duration determined from wavelet analysis and spectral-velocity-based Tp showed these are similar with the wavelet duration being slightly longer; when significant differences were found the estimated duration from wavelet analysis provided a more robust measure of the pulse duration. Therefore, the Tp1 can be regarded as a reasonable approximation of the duration (Tp) of the ground motion velocity pulse. Due to its versatility to systematically quantify the duration of velocity pulses, Baker’s program was used in this study to analyze the duration of the velocity pulses present along different orientations of the selected ground motions.  PGV P1  Tp ~ Tp1= 3.85s 56  5.2.4 Ground velocity pulse spectrum for amplitude and duration The ground velocity pulse spectrum, a new concept, is introduced herein to graphically represent the orientation dependence of amplitude and duration of velocity pulses contained in near-field ground motions. The orientation dependence of the pulse duration, extracted pulse amplitude and peak ground velocity is shown with dots in Figure 5.5 for the ground motion record at CHY101 (NGA1244) station from the Mw7.6 1999 Chi-Chi earthquake, at distance close to the rupture of 9.9km. The polar plots in the left and right panel show along the radial direction durations and amplitude of the pulses, respectively. The angular coordinate for each dot is the angle measured counterclockwise from the Y-axis as in Figure 5.2. (a) Velocity pulse spectrum for duration (b) Velocity pulse spectrum for amplitude Figure 5.5 Pulse spectrum for duration and amplitude of velocity pulses from NGA1244.  Overall the pulse spectrum for the remaining ground motions showed that both duration and amplitude of the velocity pulse vary significantly with orientation. Another pulse spectrum is shown in Figure 5.6 for ground motion recorded at El Centro array #6 (NGA181) station from the Mw6.5 1979 Imperial Valley earthquake, at distance close to the rupture of 1.4 km.  The sec cm/s 57  maximum velocity occurs at an angle of 70° with respect to the Y-axis, same as for the maximum extracted pulse amplitude (P1max) but not duration. In the orientations close to the maximum velocity both duration and amplitude have smaller orientation dependence.  (a) Pulse spectrum for duration (b) Pulse spectrum for amplitude Figure 5.6 Pulse spectrum for duration and amplitude of velocity pulses from NGA181.   Overall the velocity pulse spectrum of the ground motions showed that duration and amplitude of velocity pulses have significant orientation dependence with noticeable record-to-record variability. There is no trend about the orientation of maximum amplitude. The pulse spectrum developed for each record can be used to determine the orientation of the conditioned maximum velocity ground motion described in chapter 4 and investigate the directionality effects of these motions on the nonlinear dynamic response of buildings. For instance, in Figure 5.6 the orientation of maximum velocity conditioned on having pulse duration longer than 1s is shown with a broken line at an angle of 70°. A summary of the pulse duration in the orientation of maximum ground velocity is given in Table 5.4. An exploratory study is conducted in the cm/s sec MV Tp>1s 58  following sections to find the orientation of these pulse-like ground motions that would lead to the critical displacement response of elastic and inelastic simple structures.  Table 5.4 List of ground velocity pulse durations and maximum amplitude. No NGA Tp(s) MV(cm/s)  No NGA Tp(s) MV(cm/s) 1 170 4.23 68.77  24 1084 3.51 132.86 2 173 4.34 51.50  25 1085 3.26 117.91 3 181 3.72 117.34  26 1244 5.98 115.05 4 182 4.16 109.42  27 1494 32.42 61.71 5 183 5.93 70.21  28 1510 5.18 88.39 6 184 5.39 71.31  29 1511 4.95 88.81 7 185 4.75 69.31  30 1528 10.04 68.40 8 568 0.86 63.45  31 178 3.62 54.64 9 569 0.98 74.03  32 292 3.59 52.41 10 879 5.12 147.14  33 316 3.79 45.99 11 1119 1.98 89.63  34 527 2.42 41.15 12 1165 5.26 30.00  35 721 6.83 52.43 13 1176 4.56 72.92  36 725 2.94 35.90 14 1605 6.59 87.42  37 767 2.32 49.71 15 2114 13.08 146.50  38 1148 6.94 42.69 16 143 6.13 122.26  39 1158 6.76 62.24 17 285 8.27 40.39  40 1476 5.80 65.43 18 802 4.47 55.95  41 1480 5.64 69.30 19 803 4.77 74.69  42 1481 6.28 52.39 20 825 4.63 128.94  43 1482 7.15 60.77 21 828 2.98 98.10  44 1502 7.98 61.22 22 983 3.08 79.76  45 1602 0.91 62.98 23 1013 1.65 77.40       5.3 Discussion The variability observed in the pulse spectra makes it very difficult to draw conclusions a priori whether or not the fault-normal ground motion would be the most detrimental for the displacement response of tall buildings. However, often the fault-normal is referred as the critical 59  orientation. The pulse-like ground motions analyzed indicate that pulse amplitude and duration are orientation dependent. The analyses also demonstrated that in a limited number of records the maximum velocity was perfectly aligned to the fault-normal or the fault-parallel orientation.  Some practical limitations are related with the identification of the direction of fault-normal and fault-parallel components. The orientations of these components are defined with respect to the orientation of the strike of the causative fault, which is not always well defined. For instance, several earthquakes listed in Table 5.1 to Table 5.3 had multi-segment rupture strike for which the fault strike alignment cannot be modeled as a linear feature, as shown in Figures 5.7 and 5.8. Therefore, subjective decisions are made to define these FN and FP directions.   In some cases researchers have used their own convention to define FN/FP. For instance, Bray and Rodriguez-Marek (2004) defined the FN direction as the one which maximized the PGV for three of the fifty-four motions they analyzed. These are the Lucerne record from 1992 Landers earthquake and Chi-Chi earthquake records at stations TCU052 and TCU068 located near northern end of the rupture.   The developers of PEER NGA West 2 concluded that the strike-parallel direction at a recording station is ambiguous when the finite fault model is made up of multi-segment ruptures. A consensus among NGA developers is to define the local strike-parallel direction at the recording station as the averaged direction of the fault strike over a stretch of 20 km that extends from the closest point on the fault towards the epicenter (Ancheta et al., 2013).   60  Kobe, 1995 (SS) Landers, 1992 (SS)  Kocaeli, 1999 (SS)   Figure 5.7 Maps of epicenter and surface projection of rupture for several strike slip (SS) earthquakes having multiple-segment ruptures. Reproduced from Shahi (2013).    61  Duzce, 1999 (SS)  Chi Chi, 1999 (RO)   Denali, 2002 (SS)  Figure 5.8 Maps of epicenter and surface projection of rupture for several strike slip (SS) and reverse oblique (RO) earthquakes having multiple-segment ruptures. Reproduced from Shahi (2013).  5.4 Summary of analysis of velocity pulses The directionality characteristics of velocity pulses from near-field ground motions were studied in detail. This was achieved by analyzing ground motion parameters derived from a set of 45 pulse-like ground motion records. The major observations are summarized as follows: 62   The orientation of the maximum ground velocity was found to have a significant record-to-record variability and typically occurs along directions different from fault parallel and fault normal, with few exceptions.   Ground velocity pulse spectra developed for the velocity pulses in these ground motions showed that the pulse duration and amplitude have significant orientation dependence. The duration and amplitude of the velocity pulses at orientations close to the maximum velocity have smaller orientation dependence.  5.5 Directionality effects of pulse-like ground motions on elastic response spectrum Linear elastic analysis is still widely used by earthquake engineers so the study of directionality effects of pulse-like ground motions on the elastic damped response spectrum is warranted. Furthermore, observations drawn from this investigation will be useful for studying the alternatives of ground motion scaling to the design spectrum discussed in chapter 7.   As shown in the previous chapter the displacement response of long period systems can in some instances be quite sensitive to excitations having long duration strong velocity pulses. Therefore in the parametric study that follows the displacement response spectrum is investigated.   The displacement response spectra for multiple orientation of a ground motion are calculated through a parametric study where the angle of incidence of a ground motion pair with respect to the oscillator’s axes is varied systematically. Within the scope of this parametric study the spectra are analyzed to characterize the ground motion directionality effects for each one of the 45 ground motion pairs selected in this chapter. 63  For each record the differences between response spectra for FN/FP components and the envelope of response spectra for all the orientations are assessed over the period range of 0.01s to 7.0s. The ground motion orientation that yields the envelope of the elastic response spectra for an oscillator of period T is called maximum direction ground motion at period T.  5.5.1 Parametric study on single degree-of-freedom oscillators The parametric study is carried out on elastic damped oscillators having a single degree-of-freedom in the horizontal direction Y, as shown in Figure 5.9. The relative orientation of the FN and FP ground motion components with respect to the Y-axis was varied and measured clockwise by the  angle. P-delta effects were not included in these analyses.   Figure 5.9 Representation of SDOF analyzed and input ground motion.  The same equation of motion described in the previous chapter applies to the system above and the input motion is calculated using Equation 5.1 with the value of the parameter  equal to the angle . The solution for the equation motion within this parametric analysis for linear elastic systems can be computed efficiently using the principle of superposition. Thus to calculate the response of the system for any orientation  of the ground motion pair first the response to each g yy64  one of FN and FP components is determined, then a linear combination of both responses is performed as per Equation 5.3. The equation of motion was solved for each ground motion and oscillator using the program developed in the previous chapter and then the linear combination below used to calculate additional responses to other ground motion orientations.   ݕݑሺݐሻ ൌ െsinܰܨݑߠሺݐሻ ൅ cos	ߠ	ܲܨݑሺݐሻ (5.3)  5.5.2 Response spectrum dependence on ground motion orientation To gain insight on the sensitivity of the displacement spectra to the orientation of the ground motion, the spectral ordinates at periods of 2s, 3s and 4s are presented in Figure 5.10 for NGA181 and NGA1244. The maximum ordinate at period T from all 36 rotations of a ground motion pair is used to define the maximum direction spectrum. For instance, the peak value from the each on the curves shown in the figure defines the MD at 2s, 3s and 4s. It can be observed that the orientation which produces the MD ordinate may vary with period. Therefore, it follows that the spectral ordinates may not be maximized by a unique ground motion orientation. (a) NGA181 (b) NGA1244 Figure 5.10 Orientation dependence of spectral ordinates.  75° 65° MD(3s)=123cm MD(4s)=153cm 65  5.5.3 Comparison of elastic displacement spectra An effective inspection of directionality effects of a motion on the response spectrum requires quantification of variation of spectral ordinates and a simple graphical representation. Since presenting all the spectra in a single plot for each rotated motion would be difficult to understand, the spectra will be plotted for responses to FN, FP, and the MD. The FN/FP spectra are shown since they are commonly considered to be representative of pulse-like near-field motions. These displacement response spectra are shown in Figures 5.11 to 5.13.  5.5.4 Responses to ground motions in set 1 The spectra shown in Figure 5.11 do not appear to have a prevalent rotated ground motion that is associated to the largest spectral displacement (MD). In some instances, the spectrum from the FP component is close to the MD and in others is the spectrum from the FN. Few cases show that FP or FN is dominant over the entire spectrum. Often at periods shorter than 2 seconds the spectrum from the FN is far from MD. The peak amplitude of the response spectra for the different motions ranges from 40 cm to 150 cm. These peaks occur for SDOF oscillators having periods between 1 and 7 seconds.  The response spectra of ground motions NGA170, NGA173, NGA181 (Imperial Valley Earthquake 1979), NGA568, NGA569 (San Salvador Earthquake 1986) and NGA879 (Landers Earthquake 1992) shows a significant difference between the FN and FP components at periods longer than 3 seconds.  Furthermore, the minimum spectra among FN and FP, shows approximately constant amplitude at periods longer than 3 seconds. For these ground motions, an analysis of the velocity pulses contained in the rotated motions which produce minimum 66  ordinates indicates the durations are not longer than 3 sec. The FP ground motion for NGA879 had pulse duration of 15s but the amplitude of the extracted pulse is just 11cm/s, this confirms it is not a pulse-like motion. This explains the large difference among the two spectra. The summary of the pulse durations for these ground motions is shown in Table 5.4 below.  Table 5.5 Longest pulses present in the ground motion of minimum spectra among FN and FP. GM Minimum component Velocity pulse duration (s) 170 FP 2.2 173 FP 2.0 181 FP 2.6 568 FN 1.0 569 FN 2.9 879 FP -  5.5.5 Responses to ground motions in set 2 In general the spectra in Figure 5.12 do not appear to have a prevalent rotated component that is associated to the envelope of spectral displacement (MD). None of the responses shows FP or FN as dominant over the entire spectrum. In certain period range the spectrum from the FP component is close to the MD and in others is the spectrum from the FN. Often the spectrum from the FN is far from MD at periods shorter than 2 seconds. The peak amplitudes of the response spectra for the different motions range from 33 cm to 250 cm. These peaks occur for systems having periods between 2 and 7 seconds.  A significant difference between the spectral ordinates of FN and FP is observed for NGA143, NGA285, NGA825, NGA983, NGA1084 and NGA1085.  67  5.5.6 Responses to ground motions in set 3 The spectra in Figure 5.13 include motions of the set 3 that were recorded within Rrup distances 10km to 30km. Neither these spectra appear to have a prevalent component that is associated to the largest spectral displacement (MD). The peak amplitude of the response spectra for the different motions ranges from 22 cm to 200 cm. These peaks occur for systems having periods between 2 and 7 seconds.  The amplitudes of the response spectra for ground motions NGA1476, NGA1480, NGA1481 (Chi Chi Earthquake 1999) and NGA1148 (Kocaeli Earthquake 1999) are quite small at periods shorter than 2s when compared to the long period ordinates. The spectral ordinates at short periods are not greater than 15 cm, low values compared to other strong ground motions in sets 1 and 2. The difference could be due to the attenuation of the ground shaking with distance from the causative fault, which is larger for records in set 3. At periods longer than 3s the spectral ordinates increase rapidly as the period increases. These large ordinates are produced by the presence of velocity pulses with durations longer than 4s.  The spectra of NGA1502 show a rapid increase in spectral ordinates at periods longer than 5s. There is a large disparity between the FN and FP spectral ordinates for ground motions NGA527 and NGA1148 at periods longer than 2s and 3s, respectively. The FN spectrum of ground motion NGA527 has approximately constant amplitude at periods longer than 3s.   68  5.5.7 General observations from parametric study on response spectra The following trends were observed in the displacement spectra:  The spectral displacements corresponding to the fault-normal ground motion are usually smaller than fault-parallel at periods shorter than 2 sec.  As the distance from the rupture increases, the MD spectrum at longer periods differs significantly from fault-normal.  There is no evidence for a single orientation to define the MD spectrum, neither fault-normal nor fault-parallel. In some instances, the FP spectrum is close to the MD and in others is the FN the closest one to MD. Many cases show that neither FP nor FN is dominant. The results of a statistical analysis carried out at periods of 0.5s, 1.0s, 2.0s and 4.0s are listed in Table 5.6.  Table 5.6  List of angle from fault-parallel for maximum spectral response. Distance<5km Distance≥5km  Angle from fault strike Angle from fault strike Period (s) Mode   Median   Mean Mode Median  Mean 0.5 45 45 50 10 50 45 1 45 45 35 20 35 45 2 25 45 45 15 25 35 4 80 75 60 25 50 45 6 55 55 55 75 55 50  69      Figure 5.11 Displacement response spectra (=5%) for near-field records at distance Rrup ≤ 10km from strike slip events in Set 1. Spectral displacement at critical angle (MD) Spectral displacement using Fault Normal (SdFN) Spectral displacement using Fault Parallel (SdFP)70      Figure 5.12 Displacement response spectra (=5%) for near-field records at distance Rrup ≤ 10km from reverse/normal/oblique events in Set 2. Spectral displacement at critical angle (MD) Spectral displacement using Fault Normal (SdFN) Spectral displacement using Fault Parallel (SdFP)71    Figure 5.13 Displacement response spectra (=5%) for near-field records at distance Rrup={10km..30km}from different earthquake mechanisms in Set 3. Spectral displacement at critical angle (MD) Spectral displacement using Fault Normal (SdFN) Spectral displacement using Fault Parallel (SdFP)72  5.5.8 Dependence of geometric mean response spectrum on ground motion orientation The geometric mean spectrum is often used to quantify the intensity of bi-directional ground motions and to define a design spectrum. It is calculated period by period, as the square root of the product of spectral ordinates from two orthogonal horizontal components.   A relevant analysis which will be useful for ground motion scaling presented in chapter 7 is the orientation dependence of the geometric mean spectrum. To gain insight on this dependence, Figure 5.14 shows the geometric mean as a function of the orientation for the ground motion NGA879. This record obtained at Lucerne station during the 1992 Landers earthquake at a source-to-site distance of 2.2km had a maximum ground velocity of 147 cm/s. The dependence curves are shown for several periods representative of the long period range. It is observed that the geometric mean is quite sensitive to the orientation of this ground motion, the challenges this sensitivity presents for scaling of pulse-like near-field ground motions are discussed in chapter 7.   Figure 5.14 Geometric mean spectrum for NGA879.  73  5.6 Remarks The evaluation of the effects of the directionality of the set of 45 near fault ground motions on the elastic displacement spectra is consistent with earlier observations drawn from the properties of the velocity pulses. The FN is not a reliable proxy for the component that produces the maximum response (MD), despite having a significant number of records having forward directivity effects. The ground motion orientation that produces the maximum response deviates significantly from the FN at periods shorter than 2 seconds or distances greater than 5km.  5.7 Directionality effects on nonlinear response of simple structures In this section the directionality effects on nonlinear response are investigated using the selected pulse-like near-field ground motion records. The critical displacement response of 2D simple structures which exhibit strength and stiffness degradation is then determined and compared with the response estimates from the CMV ground motion identified using the proposed dual criterion in chapter 4. Parametric 2D analyses are performed to consider various systems within a range of fundamental periods and lateral force resisting capacities. These systems are representative of slender rectangular shear wall buildings whose response is governed by flexure.  5.8 Input ground motions The direct analysis is currently the only viable method to determine the displacement critical response when evaluating the inelastic seismic response of buildings. Thus the input motions for direct analyses were determined as per Equation 3.1 at orientation increments of 5° from 0° to 180°. These produced 36 motions determined along different orientations of each one of the 45 ground motion records, which adds up to 1620 input motions for dynamic analyses. 74  5.9 Simple structure models The simple structures are modeled using SDOF as in previous chapter 4. Instead of analyzing a system having elasto-plastic force-deformation relationship, a system that exhibits stiffness and strength degradation was considered to better approximate the load-deformation response of some structural systems typical for tall buildings, such as slender shear walls. While in chapter 4 the elasto-plastic system of elastic period T had a yield displacement proportional to the yield force, herein a system of elastic period T has a yield displacement which is constant irrespective of its lateral resistance. The model used for this evaluation is shown in Figure 5.15.  Figure 5.15 Force deformation relationship for systems investigated.  The force deformation relationship of the system investigated was defined to represent the general response features observed from experimental tests reported by Ibrahim (2000) and Orakcal et al. (2006). The adopted relationship is shown in Figure 5.16. The different parameters evaluated for the systems are summarized in Table 5.7. The parameter  was set equal to 1% for all the cases. The parameters for reverse loading were defined with the negative values of the ones presented in the table. The values for Ku and d0 were calculated during analysis since these are dependent on the response history. Systems having elastic periods of 1s, 2s and 3s were gY-axis 75  investigated, each one of these systems having a normalized yield resistance Cy equal to 0.15, 0.20 and 0.25. The P-delta effects were considered for the nonlinear response analysis.  The hysteretic behavior considers stiffness degradation for the unloading branch by directing unloading onto a point at the intersection of the elastic stiffness ray and an ordinate equal to  times the yield strength, where  is equal to 2. Effect of pinching is not considered in this model and reloading is directed towards the peak deformation from the previous inelastic excursion.  Figure 5.16 Force deformation relationship for systems investigated.   F'cr K0 Ku Fcr Fy K0 dy K'u Force d0 Deformation K0 'K0 'K0 'K0 Fy F'y dy dy' 'dy' F‘y K0 Collapse point 76  Table 5.7 Different parameters evaluated for force-displacement relation of the simple structures T (s) h (m) Ko (kN/m) Weight (kN) Fcr (kN) Cy  Fy (kN) dy (m) dy Stability Index 1 30 129220 32110 2408 0.15 0.10 4817 0.21 1.05 0.21 0.05 1 30 129220 32110 3211 0.20 0.13 6422 0.21 1.05 0.27 0.04 1 30 129220 32110 4014 0.25 0.17 8028 0.21 1.05 0.32 0.03 2 60 69041 68624 5147 0.15 0.22 10294 0.42 2.10 0.40 0.05 2 60 69041 68624 6862 0.20 0.31 13725 0.42 2.10 0.49 0.04 2 60 69041 68624 8578 0.25 0.42 17156 0.42 2.10 0.62 0.03 3 90 53657 120000 9000 0.15 0.36 18000 0.63 3.15 0.55 0.05 3 90 53657 120000 12000 0.20 0.55 24000 0.63 3.15 0.75 0.04 3 90 53657 120000 15000 0.25 0.80 30000 0.63 3.15 0.90 0.03  5.10 Dynamic responses To investigate the ability of the CMV ground motion to estimate the critical displacement two sets of analyses were performed. The first set with the unscaled ground motions and the second with ground motions with an amplitude scaling factor of 2. In total, the number of simulations performed was 29,160. All the calculations were performed using the program CANNY (Li, 2015). Few cases that resulted in convergence issues and exceeded the “collapse point” on the backbone curve were discarded because it was not possible to determine whether the CMV component would approximate well the critical displacement or not.   The response history of a system with period of 2s and Cy=0.2 to the CMV ground motion NGA181 is shown in Figure 5.17. The relative acceleration, velocity, displacement and energy (kinetic EK, damping ED and absorbed EA) indicate that the system undergoes few cycles of large inelastic oscillation after the velocity pulse arrives. This characteristic of pulse-like near-field ground motions result in a burst of energy which the structure must dissipate quickly (Naeim, 1995). 77   Figure 5.17 Response history of system with Cy=0.2 and T=2s to CMV ground motion for NGA181.   78  5.11 Dependence of dynamic responses on ground motion orientation The dependence of the peak displacement response for systems with periods of 1s, 2s and 3s with respect to the orientation of the ground motion NGA181 measured from Y-axis are shown in Figure 5.18. It can be observed that the magnitude of the displacement is much larger for the 2s and 3s systems but the orientation dependence is similar. As recalled from chapter 4, when Tp/T>>1 the displacement response is rather small, here Tp ranges between 2.4s and 4.2s.   We recall that the maximum peak displacement response among all the rotated ground motions is referred as the critical displacement and identified as DCRIT.  The displacements calculated for the CMV ground motion are indicated as DCMV. A comparison of DCRIT and DCMV suggests that CMV could provide good estimates of the critical displacement.   Furthermore, the displacements are quite large for a relatively narrow range of orientations around the critical displacement. This implies that when the input motion is oriented closely to the angle which produces critical displacement response, the responses still remain significantly large. This observation provides confidence that an approximate method can be developed to estimate the critical response without having to determine the exact orientation that produces it.   The CMV ground motion is proposed herein to approximate the critical displacement response. In the next section the critical responses determined from the parametric analyses will be compared with responses using CMV ground motion.   79  (a) Response for system with T=1s (Tp/T>>1)(b) Response for system with T=2s (Tp/T>1)(a) Response for system with T=3s (Tp/T>1)Figure 5.18 Orientation dependence of peak displacement using NGA181 and system with Cy=0.2.  5.12 Comparison of critical displacement predictions The maximum displacement response from direct analysis was determined by systematically rotating ground motions from 0° to 180° and compared in Figure 5.19 against the predicted displacement response determined using the CMV ground motion for a system with a period of DCRIT DCMV 70° 70° DCRIT DCMV 70° DCRIT DCMV 80  3s and capacity Cy=0.15. The critical response corresponds to the upper bound of the expected displacements from each ground motion, thus every point in this scatter plot cannot not lie below a 45° line. The scatter plot of the blue dots shows that the predicted displacement using CMV method is consistently a good approximation to the critical response even when the ground motions have been scaled with an increasing amplitude factor of 2. Ratios of the critical displacement to the yield displacement range from 0.3 to 5, which demonstrated that the CMV method works well within a wide range of nonlinear responses. Cases whose response was beyond the collapse point are not shown. (a) Unscaled ground motions (b) Scaled ground motions by factor of 2 Figure 5.19 Responses CMV method vs critical displacement from direct analysis for T=3s and Cy=0.15.   Figure 5.20 presents a similar comparison for systems having period of 3s and resistances Cy equal to 0.25. Again the CMV ground motion provides remarkably good estimates of the critical displacement response. The next section uses statistical analysis of the parametric data to analyze the differences between the CMV estimates and the critical response. 81  (a) Unscaled ground motions (b) Scaled ground motions by factor of 2 Figure 5.20 Responses  CMV method vs critical displacement from direct analysis for T=3s and Cy=0.25.    5.12.1 Statistical evaluation of errors in prediction of critical displacement Since the CMV ground motion provides a lower bound estimate of the critical displacement, differences are expected. The differences between the critical displacements determined using direct analyses (DCRIT) and the estimates using the CMV ground motion (DCMV) are calculated as errors by using the following equation: ݎ݋ݎݎܧሺ%ሻ ൌ ܦ஼ோூ் െ ܦ஼ெ௏ܦ஼ோூ் ∗ 100 (5.4) The calculation of the errors from the analyses leads to a distribution of errors. The median value of the errors calculated for the responses of all the systems are summarized in Table 5.8 and 5.9 for unscaled and scaled ground motions, respectively. Comparing the errors from both tables it is observed that the system subjected to the unscaled ground motion resulted in larger errors when compared to the system under scaled ground motions.  82  Table 5.8 Median value of errors in estimating critical displacement for systems to unscaled motions  Median value of errors (%) with respect to critical displacementPeriod (s) Cy = 0.15 Cy=0.20 Cy=0.25 1 12.8 12.1 16.02 6.1 7.3 6.43 6.8 5.9 3.5  Table 5.9 Median value of errors in estimating critical displacement for systems to scaled motions  Median value of errors (%) with respect to critical displacementPeriod (s) Cy = 0.15 Cy=0.20 Cy=0.25 1 5.5 10.3 15.22 5.2 6.1 6.13 6.6 3.8 4.7  The reduction is more noticeable for the system with period of 1s and Cy equal to 0.15. In many cases analyzed for this system, the unscaled ground motions produced critical displacements below the yield displacement and the response to the CMV ground motion significantly underestimated the critical value. Upon ground motion scaling, the errors obtained for this system were reduced for 20 of the 45 cases and increased for other 4 cases. It is observed that when these ground motions were scaled, critical displacements beyond the yield displacement occurred.  To justify the reduction in the error we refer to findings from the previous chapter where it was shown that the ratio Tp/T at the critical displacement for systems with linear elastic response are between 1.1 and 1.2. In many of the 20 cases the ratio Tp/T is larger than 3, therefore it is expected the effect of the CMV ground motion on the elastic displacement response to be rather small. It was found that when the peak amplitude of the velocity pulse is increased –as in ground motion scaling– inelastic critical displacement response takes place and occurs for larger ratios 83  Tp/T, see for instance the trend presented in Figures 4.6 to 4.8. This observation is consistent with the responses in this chapter and explains why the scaled CMV ground motion with Tp/T larger than 1.1 have a significant effect on the inelastic displacement response.   The amplitude scaling of the ground motions produced “collapse” for many ground motions. The errors were not quantified for a given ground motion when the collapse occurred over a range of orientations larger than 10º. A summary with the set of scaled ground motions that produced collapse is given in Table 5.10, for the majority of these cases the CMV ground motion exceeded the collapse point. The drastic reduction in errors for the system with period of 1s and Cy equal to 0.15 subjected to scaled ground motions cannot be due to the collapse cases which were disregarded, since the respective unscaled CMV motions half produced errors below 2%.  Table 5.10 Cases which produced collapse using scaled motions Period (s) Cy = 0.15 Cy=0.20 Cy=0.25 1  NGA143  NGA181  NGA182  NGA879  NGA1084  NGA1176  NGA1244  NGA1510  NGA2114  NGA143  NGA181  NGA879  NGA1084 NGA1176  NGA1244  NGA1510  NGA2114 NGA143  NGA181  NGA879 NGA1084  NBA1244  NGA2114 2 NGA143  NGA181  NGA879  NGA1244  NGA1510*  NGA143  NGA181*  NGA879  NGA181 3  NGA879  NGA879   Indicates that response to CMV ground motion exceeded the collapse point *     Indicates the ground motions orientations which produced collapse spanned less than 10º 84  The errors were analyzed per each different fault mechanism, as given in ground motion sets 1, 2 and 3. A slight decrease was observed for errors obtained with ground motions in set 1, which suggest the CMV ground motion may provide better estimates of critical displacement response for strike slip earthquake motions at distances less than 10km.  Overall, the median errors for long period systems of 2s and 3s are below 10%, which is considered to provide a reasonable approximation to the critical response. The errors for systems of 1s are larger and suggest the CMV ground motion may not be suitable to predict the critical displacement for mid-rise buildings.  5.13 Remarks The ability of the CMV ground motion to predict the critical displacement response of simple structures was verified for a wide range of systems. The verification was conducted by performing thousands of simulations using the pulse-like ground motions selected in this chapter.   To further verify the efficiency of the CMV ground motion, nonlinear response history analysis will be conducted on several case studies in the next chapter. The models for these case studies will be 3D to incorporate the influence bi-directional ground shaking on the seismic response, thus any orthogonal effects will captured. Using these models it will be possible to include the influence of higher modes on the displacement response.   85  5.14 Chapter summary This chapter developed a suite of pulse-like ground motions records. The duration and amplitude of the velocity pulses contained in each motion was determined. The orientation dependence was investigated for these parameters and resulted with the development of a graphical representation, the ground velocity pulse spectrum which allows a better interpretation of the ground motion pulse-like characteristics.  The directionality effects of these ground motions on elastic and inelastic systems were investigated. Overall it was found that fault-normal ground motion does not provide a reliable alternative to estimate the critical displacement response. On the other hand the use of the CMV ground motion proved to be a good approximation to predict the elastic and inelastic displacement response using unscaled and scaled motions.   86  Chapter 6: Method proposed to estimate critical displacement response A method to estimate the nonlinear critical displacement response of a tall building along the structural axis when the input ground motions has strong velocity pulses of long duration is presented in this chapter. A comparison between the critical displacement response obtained from a NRHA parametric analysis –direct analysis with motions rotated systematically at multiple orientations– and the response predicted using the method proposed demonstrates its computational efficiency. Ground motion records from chapter 5 are used for seismic input.  The method is applied to a range of different computer models compiled from different sources and created with different nonlinear analysis programs to assess the sensitivity of the proposed method to the different modeling assumptions and analysis methods implemented for each model.  6.1 Prediction of critical displacement response First of all the critical displacement response had to be determined for each ground motion record by doing parametric analysis. For this purpose each record (seed) ground motion pair was rotated systematically in the range of 0° to 180° and the nonlinear response calculated for each rotated pair to find the absolute maximum displacement at every floor. As explained in chapter 2 the critical response to a given record was taken as the maximum envelope of displacement responses from all rotated pairs at every floor.   87  6.1.1 Conditioned maximum velocity (CMV) method The criteria defined in chapter 4 to identify the CMV component and applied in chapter 5 to estimate the critical displacement of simple structures provided good results. Therefore, on the basis of these results a method is proposed (called CMV Method) to determine the critical displacement response of a tall building to earthquake near field pulse-like ground motion records.  Step 1. Determine the first mode translational period T of the structure along the structural axis where the critical displacement response is to be estimated. Step 2. From the ground motion pair determine the duration of the velocity pulses along orientations in the range of 0° to 180°.  Step 3. Consider only pulses that have duration Tp longer than T. From these pulses identify the orientation of maximum velocity, which becomes the orientation of conditioned maximum velocity (CMV). If none of the pulses satisfies this criterion, then the recommended alternative is to select a different ground motion record, see appendix C. Step 4. Rotate the pair to align the maximum velocity ground motion from step 3 with the structural axis in step 1. Step 5. Calculate the nonlinear seismic response to the rotated ground motion pair from Step 4.  Since the method imposes the condition that the duration of pulse Tp from the input motion must be longer than the first mode period of the building model and identifies the orientation associated with maximum velocity, the method is designated as the “Conditional Maximum Velocity (CMV) Method”.  88  6.1.2 Implementation steps 2 and 3 The duration of the velocity pulses and their amplitudes along different orientations required for steps 2 and 3 can be systematically quantified using the computer program developed by Baker (2007). This procedure implemented in this dissertation is a fully-automated analysis, the most reasonable approach due to the large number of records analyzed. However another procedure less sophisticated can be implemented in engineering practical applications with the horizontal particle motion to estimate maximum ground velocity (see second in figure in glossary) and counting zero crossings to estimate the pulse duration (Figure 5.4).  6.2 Tall building case 1 A step by step application of the CMV Method to determine the critical displacement response of the 44 storey building described in chapter 2 is presented in the following example. The method is applied using ground motion NGA181 to estimate the displacement response of the building along the Y axis.   Step 1. The first mode translational period of the building along the Y axis is T=3.5s.  Step 2. The orientation dependence of pulse duration is shown in Figure 6.1b, this is the ground velocity pulse spectrum representation of pulse duration. This information was retrieved from the database of velocity pulse parameters presented in chapter 5.  Step 3. The range of orientations where pulse durations are shorter than T is shown with a blue shade in Figure 6.1b. These orientations should be excluded from the identification of the CMV ground motion. The orientation dependence of the pulse amplitude is shown in Figure 6.1c, this is the ground velocity pulse spectrum for amplitude and the 89  information was retrieved from the database of velocity pulse parameters in chapter 5. The orientation of the ground motion that has the conditional maximum velocity is at 70° and is indicated with a red broken line in this Figure.  Step 4. The ground motion pair is rotated clockwise to an incident angle of 70° such that the CMV ground motion is oriented along the Y-axis. The horizontal particle velocity of the ground motions are shown for the seed and the rotated pairs in Figure 6.1d.  Step 5. The seismic response for an incident angle of 70° of the ground motion pair is shown in Figure 6.1e and compared with the critical displacement response obtained from direct analysis.   In this case a very good estimate of the critical displacement response was obtained at every floor of the building. The accuracy of this prediction suggests that the CMV Method may indeed provide an efficient way to circumvent the challenges that arise with the direct analysis, without the need of extensive computer calculations and processing of numerous outputs from multiple analyses.   From the suite of 45 ground motions a set of seven ground motions that met the criteria for the CMV Method were used to perform NRHA for this 44-storey building. Performing analyses for a larger set of motions on this building would be prohibitively expensive in terms of computational time. Using seven bi-directional ground motions it was found that the critical displacement prediction with the CMV method is remarkably good, this was achieved with a single nonlinear response history analysis for each ground motion record. 90     (a) T along Y axis (b) Exclude motions with Tp<T (c) Identification CMV    (d) Rotating seed ground motion to align CMV with structural axis Y (e) Response to CMV Figure 6.1 Steps to rotate ground motion pair NGA181 to CMV and calculate the respective nonlinear seismic response. cm/s sec cm/s cm/s TY-AXIS Y-AXIS 91  6.2.1 Quantifying the effectiveness of the CMV Method To quantify the effectiveness of the CMV Method for each analysis the differences (errors) with respect to the critical displacement along the Y-axis was calculated at every floor using the equation below. ݎ݋ݎݎܧ௙௟௢௢௥	௝	ሺ%ሻ ൌ ݐ݅ݎܿܦ௝ െ ݀݁ݐܿ݅݀݁ݎ݌ܦ௝ݐ݅ݎܿܦ௝ ∗ 100  (6.1) Where the term Dcrit is the critical displacement at floor j and Dpredicted is the predicted displacement at floor j. As other rotated ground motions are currently being used in practice, it was deemed necessary to calculate the error that other rotated ground motions will have with respect to the critical displacement response. Other rotated motions included the FN/FP components and the maximum direction at the fundamental period of the building MD(T).   The errors were then averaged over the height of the building using Equation 6.2 to have a single index representative of the ability each rotated ground motion has to predict the critical displacement response.  ݀݁݃ܽݎ݁ݒܣ	ݎ݋ݎݎ݁	ሺ%ሻ ൌ෍ݎ݋ݎݎܧ௙௟௢௢௥ ௝௡௝ୀଵ  (6.2) These averaged errors are summarized with their median, standard deviation and maximum deviations in Table 6.1. These results confirm for this case that the CMV method provides a better approximation to the critical displacements than the MD, FN and FP components, while the MD component provides better estimates than the FN and FP components. 92  Table 6.1 Averaged errors of floor displacement over height of the 44-storey building. Averaged error CMV MD FN FP Median 3% 3% 24% 26% Std. deviation 2% 16% 15% 17% Maximum 8% 43% 55% 65%  The source of these errors can be attributed to the following: (i) in some instances the critical displacements are obtained from the envelope of multiple rotated ground motions, in such cases it is not possible to accurately estimate the critical displacement using a single rotated ground motion pair and (ii) even if the critical response is obtained using a single ground motion pair it is possible that the prediction will not match the critical response due to the use of an inadequate rotated ground motion. Thus, in some instances the error will not be minimized unless envelopes from multiple rotated ground motions are used, a less efficient approach.    The efficiency of the CMV ground motion to estimate the critical value of peak interstorey drift ratios was also analyzed by comparing the peak interstorey drift ratios from different rotated ground motions with the critical interstorey drift ratio in the direction of the structural axis. To facilitate this comparison the average of the peak interstorey drift ratios over the height of the building was calculated to provide a global damage measure. This index obtained for the responses to each rotated ground motion and the critical responses was then used to determine the errors. A summary of these errors is presented in Table 6.2, which demonstrate that the CMV ground motion may also provide a good estimate of the critical interstorey drift ratios. Overall the MD ground motion gives better approximations then FN or FP ground motion. The profiles of peak interstorey drift ratios for different rotated ground motions and the critical response are 93  shown in Figure 6.2. In the case of NGA 181 the critical values are well captured by the CMV, MD and FN ground motions. Instead, for the NGA1605 only the CMV ground motion provides an acceptable approximation.  Table 6.2 Errors of averaged interstorey drift ratio over height of the 44-storey building. Error CMV MD FN FP Median 2% 1% 25% 23% Std. deviation 2% 18% 14% 16% Maximum 6% 44% 53% 61%    (a) Responses to NGA181 (b) Responses to NGA1605 Figure 6.2 Envelopes of peak interstorey drift ratios for 44 storey.  To further explore the efficiency of the CMV Method seismic response analyses were performed for computer models of different buildings created by other researchers and practicing structural engineers. 94  6.3 Tall building case 2 A second case study is based on a modern 45 storey reinforced concrete shear wall building located in an active seismic zone in California. The building has a core of shear walls 812 mm thick along the East-West direction and 712 mm thick along the North-South direction that surround the elevators and staircases. The core has plan dimensions of 8.5m North-South and 16m East-West. The typical floor plan of the building has average dimensions of 31m North-South and 39m East-West. The total building height is 162m above the lowest basement. Performance based seismic design was implemented for this building to ensure that it meets a minimum level of performance under the design earthquake loading. The seismic design of this building was reviewed by an expert peer-review panel.  The design requirements for this building warranted the use of pulse-like ground motions to verify the building performance under near-field motions, thus it is an ideal case study to investigate the efficiency of the CMV method.  A building model for this structure was created using the state of the art commercial program Perform 3D. The computer model was developed by the consulting firm that performed the structural design of the building and was shared to the dissertation author to with the sole purpose of verifying the CMV Method. The nonlinear 3D model includes core shear walls with fiber elements, coupling beams, columns, equivalent frame beams and the podium. An elevation and 3D view of the model is shown in Figure 6.3. The first mode period of the building model in the EW (X-axis) direction is 4.4s. Nonlinear response history analyses were conducted using a set of eight pulse-like ground motions from the 45 selected records. The computing time required to complete the analyses required between 7 and 16 hours depending on the input motion. Since 95  direct analysis at multiple orientations was required to define the critical displacement envelope, input motions were rotated at 15° increments instead of 5° to reduce the overall computing time.   (a) Elevation view (b) 3D View Figure 6.3 Elevation and plan view of Perform 3D model for 45-storey building.  The critical displacement response of the building to the ground motion NGA1244 was estimated quite well with the CMV method as shown in Figure 6.4. This ground motion was recorded 96  during 1999 Chi Chi Mw7.6 earthquake at a distance of 9.9 km from the rupture. The strongest velocity pulse contained in this motion had a duration of 5.98s and PGV of 115 cm/s.  Figure 6.4 Critical displacement vs CMV response (Tp/T=1.36) for 45 storey building using NGA1244.  The averaged errors for floor displacements obtained from the analyses are presented in Table 6.3. The median error from the CMV Method is relatively small in comparison to the other ground motions, besides the standard and maximum deviation is much smaller. This suggests the CMV Method is superior then other rotated ground motions to approximate the critical displacement. This verification is consistent with results obtained for the 44-storey building.  Table 6.3 Averaged errors of floor displacement over height of the 45-storey building. Averaged error CMV MD FN FP Median 3% 15% 14% 18% Std. deviation 5% 18% 22% 10% Maximum 14% 28% 68% 50% 97  The envelopes of peak interstorey drift ratios for a couple of the ground motions investigated are shown in Figure 6.5. Despite being an approximate method for critical displacement response, the CMV ground motion also provides close estimates of the critical peak interstorey drift ratios. The errors obtained from the averaged interstorey drift ratios are summarized in Table 6.4. This case study also confirms the ability of the CMV method to provide good estimates of the critical peak interstorey drift ratio in the direction of the structural axis when compared to other rotated ground motions.   (a) NGA1244 (b) NGA1502 Figure 6.5 Envelopes of peak interstorey drift ratios for 45 storey building.  Table 6.4 Errors of averaged interstorey drift ratio over height of the 45-storey building. Error CMV MD FN FP Median 4% 7% 11% 32% Std. deviation 3% 9% 16% 19% Maximum 9% 30% 70% 54%  98  6.4 Tall building case 3 The third building case study is a 52 storey steel frame building for office occupancy completed in 1990. A spine structure is the structural system used for this 218m building located in downtown Los Angeles. The plan layout of the typical floor is shown in Figure 6.6. The spine is comprised of three components:  (1)  A concentrically rectangular core that runs uninterrupted within the building and consists of four core steel columns that are tied together by beams and diagonal braces.   (2) Outrigger beams linking the core with perimeter columns which work as ductile moment resisting frames.  (3) Steel perimeter columns which provide additional overturning resistance to the building.  A picture of this building and a 3D view of the computer model are shown in Figure 6.7. This building’s floor plan has an octagon shape and exhibits many setbacks above the 36th floor. The computer model of this building was developed by Ventura and Ding (2000) in a previous study that investigated this building’s nonlinear response to near-field pulse-like motions.   The computer model was created using the program CANNY (Li, 2015) and does not include the setbacks. The dynamic properties of the computer model are listed in Table 6.5. For further details about this model please refer to Ventura and Ding (2000). 99   Figure 6.6 Plan view of typical floor layout for 52 storey building.   Figure 6.7 Picture and computer model representation of 52-storey steel frame building 100  Table 6.5 Dynamic properties of 52-storey building. Mode Natural Period (s) Mode shape Modal Damping (%) 1 5.89 Translational X-direction 2.00% 2 5.49 Translation Y-directional 1.93% 3 4.74 Torsional 1.82% 4 1.83 Translational X-direction 2.00% 5 1.74 Translation Y-directional 2.06% 6 1.71 Torsional 2.08% 7 1.04 Torsional 2.97% 8 0.97 Translational X-direction 3.14% 9 0.95 Translation Y-directional 3.19% 10 0.73 Torsional 4.01%  The CMV Method was applied using a set of seven ground motions. In Figure 6.8 the envelope of critical displacement response in X-direction is compared with the response from the CMV method for the ground NGA1158. The CMV ground motion had pulse duration of 6.8s and PGV of 62.2 cm/s.  Figure 6.8 Critical displacement responses vs CMV Method response for 52 storey building. 101  The median, standard deviation and maximum deviation obtained from this case study using a set of seven ground motions are summarized in Table 6.6 and 6.7 for displacements and interstorey drift ratios, respectively. Overall these results are consistent with the previous case studies, a noticeable increase in the median and maximum error was observed for interstorey drift ratios when the CMV ground motion was used.  The envelopes of interstorey drift ratios in the X-direction are shown in Figure 6.9 for several ground motions. In some cases where large interstorey drifts occurred at mid-height the efficiency of the CMV ground motion to approximate the critical interstorey drift ratios is reduced. Two additional case studies are investigated to confirm the observations made so far.  Table 6.6 Averaged errors of floor displacement over height of the 52-storey building. Average error CMV MD FN FP Median 5% 2% 14% 25% Std. Dev. 6% 7% 20% 11% Maximum 15% 21% 65% 37%  Table 6.7 Errors of averaged interstorey drift ratio over height of the 52-storey building. Error CMV MD FN FP Median 13% 10% 22% 28% Std. Dev. 8% 10% 20% 16% Maximum 24% 26% 67% 54%    102    (a) NGA1148 (b) NGA1502   (c) NGA143 (d) NGA1158 Figure 6.9 Envelopes of peak interstorey drift ratio responses for 52 storey building. 103  6.5 Tall building case 4 The fourth building is a 32 storey reinforced concrete building located in an active seismic zone in California. A model of this building created in Perform 3D was shared to the dissertation author to verify the CMV Method, although no blueprints were made available. Elevation and 3D views are presented in Figure 6.10. The nonlinear model includes core shear walls with fiber elements, coupling beams, columns, equivalent frame beams and the podium. The first mode period of the building model in the X-direction is 5.5s.   (a) Elevation view (b) 3D view Figure 6.10 Elevation and 3D view of Perform 3D model for 32 storey building. 104  The CMV method was applied using the ground motion NGA1148 with pulse duration of 6.94s and PGV of 42.7 cm/s. This ground motion was selected since it met the criteria required for the CMV Method. As shown in Figure 6.11, the CMV method provided similar approximates as reported for the previous case studies.   Figure 6.11 Critical displacement responses vs CMV Method response for 32 storey building.  6.6 Tall building case 5 The fifth case study building is a 30-storey reinforced concrete shear wall building for residential occupancy. The lateral force resisting system of this building is a core of coupled shear walls and flat plate slab, surrounded by gravity columns in the perimeter. This building is the archetype of tall buildings built in Vancouver, Canada.  On the east side of the tower is a two storey townhouse linked to the tower by two pedestrian bridges.  On the north side of the tower is a terrace extending from the second floor.  This terrace 105  is monolithically linked to the tower via the floor slab.  An underground parking garage, consisting of 6 staggered levels, is located beneath the tower and the townhouses.   The building is 85 m (279 ft.) in height from ground level.  The parking levels are located beneath ground level with a depth of 9.4 m (31 ft.).  The first storey is 4 m (13 ft.) in height while the other storeys are 2.6 m (8.7 ft.) high.  This building has a corner setback at floor level 10 and further reduces in upper stories. A plan layout and elevation of the building is shown in Figure 6.12.  Figure 6.12 Plan layout and elevation of the 30 storey building.  A computer nonlinear tridimensional model for this structure was developed by members of the UBC Earthquake Engineering Research Facility in a previous study (White and Ventura, 2004) using the program CANNY (Li, 2015). The model was calibrated against the dynamic properties 106  obtained from ambient vibration measurements of the actual building. The fundamental period of the building is 1.8s in the North-South direction.  The computer tridimensional model resembles the general features of this tall building. The model includes column, shear wall, slabs and beam elements. The core walls are comprised of slender shear walls coupled with beams. These walls have distributed longitudinal reinforcement with zones of concentrated reinforcement at the boundary elements, corners and intersection of walls. The walls were modeled using fiber model available in the program CANNY, where the rebar reinforcement provided in construction layouts for each section was included into the model. In a similar manner the columns were modeled with their respective section properties. For further details about this model please refer to White and Ventura (2004).  6.6.1 Parametric study Computing the critical displacement response for all the selected ground motions in chapter 5 using tall building case 1, 2, 3 and 4 was prohibitively expensive in terms of computational demands. Besides many of the selected ground motions did not have pulses of duration longer than first mode period T of these buildings as required per the CMV Method.   In order to better quantify the accuracy of the CMV Method a large number of simulations was required to develop a statistical analysis on the prediction errors. Therefore, the seismic responses for tall building case 5 were calculated using the entire set of 45 ground motions.  All these motions were appropriate since all of them met the requirements of the CMV Method. Furthermore, since these motions have a wide range of intensities it would be possible to 107  evaluate the efficiency of the CMV Method for different levels response. A total of 1665 simulations were performed to determine the critical displacement response which required a computation time of 2,065 hours. Instead, when using the CMV Method to determine the critical displacements for the 45 ground motions requires a total of 60 hours.  6.6.2 Seed ground motions The spectra shown in Figure 6.13 compare the geometric mean spectrum for all the ground motions with the Uniform Hazard Spectrum (2% in 50 yr) in Vancouver city. Evidently the ground motion set has a wide range of intensities higher and lower when compared to the UHS.   Since the interest of this study is to investigate the directionality effects at different intensities of ground shaking the selected ground motions were not scaled to any target spectrum to supply a wide range of intensities.  Figure 6.13 Comparison of geometric mean from 45 ground motions with UHS in Vancouver 2% in 50yr.  108  6.6.3 Errors from case 5 Several displacement responses using the CMV Method for tall building case 5 are compared with the critical displacement responses along the Y-axis in Figure 6.14. Although the amplitudes of the responses among the different motions are significantly different the method works well for all of them.  (a) NGA178 (b) NGA292 (c) NGA2114 (d) NGA1176 Figure 6.14 Critical displacement responses vs predicted critical displacement.  109  6.6.4 Statistical analysis of errors for estimates of critical displacements The errors calculated resulted in a distribution of values, for instance the distribution of errors at the roof level are shown in a histogram in Figure 6.15. It is observed that the errors from CMV ground motion are rather small when compared to other ground motions. The MD ground motion seems to provide a better prediction than the FN component. The FP component has a quite uniform error distribution. The 50th percentile and 84th percentile from these distributions were calculated to quantitatively compare the efficiency of each rotated ground motion.     Figure 6.15 Histogram of errors with respect to roof critical displacements in Y direction.  The variation of these percentiles over the height of the building is shown in Figure 6.16.The minimum and maximum values of the differences are shown with error bars. The observation 110  made from the histograms above holds valid for the remaining floors, the CMV Method provides a much better prediction of the critical response than the other rotated components. Furthermore, it is observed that the maximum error from CMV ground motions is smaller than the other motions.  Figure 6.16 Summary of errors with respect critical displacements in Y direction.  To develop an index of the differences in the displacement responses calculated to the ground motions FN, FP, MD and CMV, errors for each analysis were averaged over the height of the building. The 50th and the 84th percentile from the respective averaged errors were determined. 111  These are presented in Tables 6.8 to 6.11 where the dependence of average errors on earthquake mechanism, source-to-site distance and on peak roof displacement amplitude is indicated.  The CMV ground motion provides consistently small differences irrespective of the earthquake mechanism, distance or response amplitude. Significant differences are observed for the other rotated ground motions.  Comparatively the efficiency of the rotated ground motions can be ranked as follows: (1) CMV ground motion, (2) MD (T) ground motion, (3) FN ground motion and (4) FP ground motion. The 96% confidence interval for the errors from the CMV ground motion is between 3% and 8%.  Table 6.8 Dependence of 50th percentile averaged errors of floor displacement over height on earthquake mechanisms and distance.  R<30km R<=10km 10km<R<30km Motion All mechanisms (45 cases) Strike slip earthquake motions (15 cases) Motions from other earthquake mechanisms (15 cases) Motions from different mechanisms (15 cases) CMV 5% 4% 5% 7% MD 16% 18% 20% 6% FN 19% 21% 13% 26% FP 21% 28% 33% 21%  Table 6.9 Dependence of 84th percentile averaged errors of floor displacement over height on earthquake mechanisms and distance.  R<30km R<=10km 10km<R<30km Motion All mechanisms (45 cases) Strike slip earthquake motions (15 cases) Motions from other earthquake mechanisms (15 cases) Motions from different mechanisms (15 cases) CMV 15% 12% 9% 17% MD 41% 32% 50% 25% FN 39% 43% 25% 39% FP 54% 53% 54% 46%  112  Table 6.10 Dependence of 50th percentile averaged errors of floor displacement over height on amplitude of roof displacement. Motion All simulations          (45 cases) Roof disp>1.0m         (12 cases) 0.5m<Roof disp<1.0m    (23 cases) Roof disp<0.5m         (10 cases) CMV 5% 7% 2% 5% MD 16% 23% 18% 11% FN 19% 7% 16% 34% FP 21% 52% 25% 16%  Table 6.11 Dependence of 84th percentile averaged errors of floor displacement over height on amplitude of roof displacement. Motion All simulations          (45 cases) Roof disp>1.0m         (12 cases) 0.5m<Roof disp<1.0m    (23 cases) Roof disp<0.5m         (10 cases) CMV 15% 12% 8% 18% MD 41% 48% 35% 25% FN 39% 34% 32% 49% FP 54% 59% 50% 19%  6.6.5 Statistical analysis of errors for estimates of critical interstorey drift ratios The ability of the CMV method to estimate the critical peak interstorey drift ratios is summarized with errors on average interstorey drift ratios in Tables 6.12 to 6.15. The 50th and 84th percentile from the CMV ground motion are rather low compared to other rotated ground motions. The 96% confidence interval for the errors from CMV Method is between 3% and 9%.  Table 6.12 Dependence of 50th percentile for errors of average interstorey drift ratio on earthquake mechanisms and distance.  R<30km R<=10km 10km<R<30km Motion All mechanisms (45 cases) Strike slip earthquake motions (15 cases) Motions from other earthquake mechanisms (15 cases) Motions from different mechanisms (15 cases) CMV 5% 4% 5% 9% MD 16% 16% 26% 3% FN 21% 17% 22% 27% FP 27% 26% 31% 19%  113  Table 6.13 Dependence of 84th percentile errors for average interstorey drift ratio on earthquake mechanisms and distance.  R<30km R<=10km 10km<R<30km Motion All mechanisms (45 cases) Strike slip earthquake motions (15 cases) Motions from other earthquake mechanisms (15 cases) Motions from different mechanisms (15 cases) CMV 16% 15% 16% 16% MD 37% 45% 37% 23% FN 37% 24% 46% 37% FP 50% 51% 58% 46%  Table 6.14 Dependence of 50th percentile for errors of average interstorey drift ratio on amplitude of average interstorey drift ratio. Motion All simulations          (45 cases) Average interstorey drift ratio ≥ 1%    (19 cases) Average interstorey drift ratio < 1%    (26 cases) CMV 5% 4% 5% MD 16% 19% 10% FN 21% 17% 23% FP 27% 44% 18%  Table 6.15 Dependence of 84th percentile for errors of average interstorey drift ratio on amplitude of average interstorey drift ratio. Motion All simulations          (45 cases) Average interstorey drift ratio ≥ 1%    (19 cases) Average interstorey drift ratio < 1%    (26 cases) CMV 16% 19% 16% MD 37% 46% 26% FN 37% 35% 43% FP 50% 58% 35%  6.7 Discussion The large number of nonlinear building responses to the bi-directional ground motions provided quantitative measures of the ability of different rotated ground motions (FN, FP, MD and CMV) to predict the critical displacement responses along the structural axis. The comparison among these responses and the critical response confirmed that the CMV Method provides remarkably good estimates of the critical displacement response.  114  Calculating the differences of displacement responses to ground motions rotated to CMV against critical displacements from all the 45 ground motions provided small median error and dispersion. The same calculation was performed for building responses to the MD ground motion, the FN component and the FP component. It was found that the displacement estimates with MD (at first mode period), FN and FP result in significant errors and large dispersions. Similar results were obtained for estimates of the interstorey drift ratios; the CMV ground motion did provide better estimates of critical peak interstorey drift ratios when compared to MD, FN and FP. The observations regarding the underestimates of critical response from MD, FN and FP components are in agreement with a previous study by Reyes and Kalkan (2012a) which studied the nonlinear dynamic response of 9-storey symmetric and asymmetric buildings using 30 near-fault ground motions.  The method proposed has the big advantage that requires a single response history analysis to bi-directional ground motions rotated to CMV instead of the multiple orientations required for direct analysis. Another advantage of the method is that it does not depend on the identification of FN/FP orientations. Identifying the fault-normal and fault-parallel components for ground motions that originate from rupture segments of very different azimuths is not straightforward. In these cases a subjective decision must be made to rotate the ground motion pairs into the FN/FP orientations. The use of the proposed method eliminates any bias that could be introduced from such rotation.  It was confirmed that the duration and amplitude of velocity pulses present in ground motions are crucial to assess the potential seismic demands on tall buildings at sites near the causative 115  fault. The relation of the pulse duration with respect to the first mode period serves as an index whether a ground motion would drive the building into a first-mode dominated response or not. Building response quantities that are contributed significantly by the first-mode such as lateral displacements are then highly dependent on the presence of pulses of duration longer than the period of the building.   The CMV Method has been verified using nonlinear models of several tall buildings. However strength degradation leading to collapse was not captured by these models and such condition was not investigated in this dissertation. Furthermore, this method operates under the assumption that the building model will experience a similar response if the CMV ground motion is applied in the positive or negative direction.  6.8 Concluding remarks The seismic responses of the case studies presented in this chapter demonstrated the following:   The critical floor displacement response is not always obtained when the input motion are rotated along the FN components, hence using such component might lead to inadequate underestimates of seismic response. These observations made on the response of the case studies buildings were obtained for different records obtained at distances Rrup that range from 0 to 30 km.  The nonlinear critical floor displacement response can be predicted with reasonable accuracy and precision using a single simulation when the input motion is defined by the pair that corresponds to CMV and its orthogonal component.  116   The CMV Method computational efficiency is far superior than, the direct analyses, a computationally expensive procedure. In addition to its high efficiency it provides a good approximation to the critical displacement response having a 50th percentile error of 10% or less and 84th percentile error of 20%. It is far better than other methods investigated herein, whose 50th percentile errors ranged between 15%-30% and 84th percentile errors between 30%-60%.  The critical peak interstorey drift ratios from the nonlinear response were evaluated for several case studies, and the ability of the CMV ground motion to approximate these ratios was calculated. When the averaged interstorey drift ratios were compared, the median errors from all case studies fall within a range of 2% to 13% for the CMV method. Overall the error obtained using the CMV ground motion tends to have smaller dispersion when compared to MD, FN and FP.  The case studies were limited to reinforced concrete tall building having core shear walls and a steel building with a spine structure. Further verifications of the CMV method are required for other different structural systems.  117  Chapter 7: Application to seismic design of tall buildings The seismic performance assessment of tall buildings at the design level is based on nonlinear response history analyses and the evaluation of this response against performance acceptance criteria. The definition of the seismic input for these nonlinear analyses requires a process of selecting, scaling and orienting a suite of design ground motions compatible with the expected ground shaking.   Currently, limited guidance is available to earthquake engineers on how to carry out the process of seismic input definition for nonlinear response history analysis when dealing with near-field ground motions having strong velocity pulses. This view is shared by many members of the earthquake engineering community (Haselton et al., 2012).    This chapter outlines a procedure to integrate the CMV Method into the performance based assessment of tall buildings and provides recommendations on how to select, scale and orient pulse-like ground motions for nonlinear response history analyses. It is expected that this will facilitate the use of pulse-like ground motions in performance based seismic design.  7.1 Procedure for performance assessment The outline of the procedure proposed to integrate the CMV Method in the framework of performance based assessment of tall buildings is shown in Figure 7.1. Steps 1, 2 and 4 are discussed in detail in the following sections, while steps 3 and 5 are briefly reviewed as have been already addressed in previous chapters. Step 6 is outside the scope of this dissertation.   118                      Figure 7.1 Procedure to integrate the CMV Method in performance based design framework. 2. Select appropriate pulse-like ground motion records using the following criteria:  Tp>T (preferred range 1.1 ≤ Tp/T ≤ 2), where T is the first mode translational period of the building model along the structural axis where critical displacement will be estimated  30 cm/s ≤ PGV ≤ 150 cm/s  Recorded within Rrup of 20 km to 30 km  Earthquake magnitude 5.5 ≤ Mw ≤ 7.5  Spectral shape comparable to target spectrum within the target period range  4. Linearly scale CMV ground motions to the target spectrum. Apply same scaling factors to the orthogonal components. 3. Determine the orientation of the ground motion conditional maximum velocity (CMV) for each record. Then rotate the CMV motions to the structural axis where the critical displacement will be estimated. Rotate the orthogonal component to coincide with the perpendicular structural axis.  6. Verify building response against performance acceptance criteria from governing building code.  5. Apply scaled ground motion pairs to the building model and perform NRHA. 1. Obtain design target spectrum  119  7.2 Step 1 - Design target spectrum The CMV Method aims to estimate the critical displacement response of tall buildings and therefore the design target spectrum compatible with this goal should be based on displacement ordinates. Since there are several definitions commonly used for spectral ordinates, including geometric mean and maximum direction, it is necessary to identify the proper definition for the target spectrum.  7.2.1 Spectral ordinate definitions In many situations the geometric mean may appeal as the most convenient definition to employ due to its widespread use in ground motion prediction equations and seismic hazard analyses, which is ultimately used to define the design spectrum. However, despite of these advantages, the geometric mean (geomean) spectrum is not appropriate to characterize pulse-like ground motions. A major drawback is that the geometric mean is not invariant to the orientation of the sensors at the recording station (Boore et al., 2006).   Therefore, an alternative definition more appropriate for the target spectrum must be used. The maximum direction spectrum is preferable because it considers the directionality effects of ground motions on the elastic spectrum.  Baker and Cornell (2006) recommended that the procedure used to select and scale the ground motions must be consistent with the definition of the target spectrum ordinates.  Given that the conditioned maximum velocity ground motion has been introduced in this dissertation, a design target spectrum derived from CMV ground motions do not exist yet. Therefore the maximum 120  direction spectral ordinates are adopted for the target spectrum, regardless of the differences in the spectral ordinate definitions.  7.2.1.1 Development of target spectrum The following example uses the spectral acceleration uniform hazard spectra (UHS) for Vancouver city hazard level 2% in 50 years as the basis for developing a design spectrum. Since the Vancouver UHS is the geometric mean of the two orthogonal components, it is necessary to convert it into a maximum direction spectrum. This requires the use of approximate conversion factors (Watson-Lamprey and Boore, 2007). At periods shorter than 0.1s the ordinates are increased by a conversion factor of 1.2, at periods longer than 1.0s the conversion factor is 1.3 and the factor is interpolated for intermediate periods.  To account for the potential seismic demands due to the proximity to an active fault, near-source factors recommended in the 1997 Uniform Building Code (ICBO, 1997) are applied to amplify the spectral ordinates. A factor equal to 1.1 is applied at periods equal or shorter than 0.5s and 1.4 at longer periods.  The design target spectrum is shown for spectral accelerations in Figure 7.2 which includes maximum direction conversion factors and near-source factors.  The acceleration design spectrum is converted to a displacement design spectrum dividing each ordinate by ሺ2ߨ/ܶሻଶ, this calculation is repeated period by period. The displacement design spectrum obtained from this calculation is shown in Figure 7.3. Priestley and Kowalsky (2010) recommended the use of a corner period to limit the spectral displacements in the long period range. For this example the spectral ordinates reach a plateau at a corner period of 4s. 121   Figure 7.2 Acceleration design spectrum.  Figure 7.3 Displacement design spectrum.  7.3 Step 2 - Selection of ground motions As mentioned in chapter 5, a list of pulse-like ground motions has been identified and is available from the reports of NIST (2011) and the Shahi’s dissertation (2013). The recommended criteria for the selection of ground motions using the CMV Method is indicated in Figure 7.1. If a site specific probabilistic seismic hazard analysis (PSHA) is conducted as part of the seismic 122  input definition, the criteria can be further refined to specifically consider this hazard. Instead of using the prescribed range for the peak ground velocity and duration of the velocity pulses, these ground motion characteristics can be predicted using empirical relationships that depend on the earthquake scenarios that govern the seismic hazard.   The earthquake magnitude, the source-to-site distances and epsilon – the number of standard deviations above the median- that govern the hazard at the design level can be determined from a seismic hazard deaggregation. A summary of different empirical models that depend on these parameters and that are useful for selection of ground motions is presented in Appendix D (Alavi and Krawinkler, 2004; Rodriguez-Marek and Bray, 2004; Baker, 2007; NIST, 2011).  7.4 Step 3 - Determine the CMV For a detailed explanation on the steps required to find the orientation of the CMV ground motion refer to section 5.2. The CMV ground motion should be identified for each pulse-like ground motion within the suite of records. As explained in chapter 6, once the CMV ground motion has been identified, the pair must be rotated to make the CMV motion coincide with the structural axis where the critical displacement will be estimated.   7.5 Step 4 - Linearly scale ground motions Linear scaling of a ground motion to the target spectrum preserves the frequency content of the ground motion and the duration of the velocity pulse, affecting only the intensity of the motion. If the same scaling factor is applied to both orthogonal components the directionality of the motion remains unchanged and the CMV Method can be conveniently used. For this reason 123  linear scaling of pulse-like ground motions to closely approximate to the target spectrum is preferred over spectral matching techniques which may significantly change the frequency content of the motion and the duration of the velocity pulses.  There are no generally accepted limits on the amplitude of the scaling factors that can be applied to a ground motion record. However, when using pulse-like ground motions it must be ensured that peak ground velocity does not exceed 100 to 150 cm/s, a theoretical limit range determined by Ambraseys (1969) and Brune (1970). A large set of near-field ground motions analyzed by Mavroeidis et al. (2004) showed that strong velocity pulses usually fall in the range of 70 cm/s to 130 cm/s at close distances to the fault, except for Chi-Chi records at stations TCU052 and TCU068 which have much larger amplitudes. Shahi (2013) studied these Chi-Chi records using wavelet analyses and found that the strongest pulse at TCU052 had a PGV of 209 cm/s and Tp of 11.96s, while at TCU068 had a PGV equal to 342 cm/s and Tp of 12.29s. These pulses were oriented 45° with respect to the fault-parallel orientation. The suite of ground motions analyzed in chapter 5 had velocity pulses whose amplitudes ranged between 30 cm/s and 147 cm/s. If this limit of 150 cm/s is not observed unrealistic pulse-like ground motions with excessively large PGV may be provided for nonlinear analysis. The recommendation given in this dissertation is not to use CMV ground motions with PGV that exceed the 150 cm/s.  A maximum scaling factor of 2 is recommended; otherwise another pulse-like ground motion having larger peak ground velocity should be selected to minimize the amplitude scaling factor. At the lower bound a minimum scaling factor of 0.70 is recommended. These limits are 124  suggested by the dissertation author since there is evidence that large scaling factors introduce bias on the nonlinear dynamic response of buildings (Luco and Bazurro, 2005).  7.5.1.1 Target period range Different recommendations are available for the appropriate target period range. The standard ASCE/SEI 7-10 (ASCE, 2010) prescribe the target period range for scaling of ground motions from 0.2T to 1.5T, where T is the calculated first mode translational period. Haselton et al. (2012)  recommended to scale ground motions in the range of 0.2T1,min to 2T1,max for shear wall buildings and 0.2T1,min to 3T1,max for moment resisting frame buildings, where T1,min and T1,max correspond to the shorter and longer first mode translational periods along the two horizontal axes of the building. The recommended range extends to periods shorter than T to ensure the contribution of higher modes to the seismic response is properly considered.  The recommendation from the Los Angeles Tall Buildings Structural Design Council (LATBSDC, 2014) is to scale ground motions, such that the dynamic response of buildings in each significant mode is captured properly. The ground motions shall be scaled within the target period range of 0.1T to 1.5T.   Since the goal of the CMV Method is to estimate critical displacement response and the first mode is expected to contribute significantly to this response, the short period range is not considered for the target period range. The proposed period range for scaling of ground motions is from T to 2T. Herein, the period T corresponds to the translational first mode in the direction of the structural axis where the critical displacement response will be estimated. 125  7.5.1.2 Example Following the guidelines described above the CMV ground motion for record NGA316 was linearly scaled to the design spectrum. The period range for scaling was T (3.5s) to 2T (7.0s), where T is the fundamental period of the 44 storey building in direction of the Y-axis.  This motion had velocity pulse duration of 3.8s and PGV equal to 46 cm/s. The amplitude scaling factor applied to the ground motion was 1.29. Upon application of the scaling factor the PGV was increased to 59.3 cm/s.   Figure 7.4 compares the design spectrum and the response spectrum for the scaled ground motion. It can be observed that the response spectrum of the scaled motion provides a good match to the target spectrum. The response spectrum of the orthogonal component is not shown but in order to preserve the directionality, the same scaling factor determined for the CMV ground motion was applied to it.  Figure 7.4 Response spectrum for scaled CMV ground motion and design spectrum. 126  7.6 Step 5 – Performing NRHA The scaled ground motion pair was used for nonlinear response history analysis of the 44 storey tall building. In Figure 7.5 the displacement response obtained from the scaled ground motion pair is compared against the critical response obtained using direct analysis, which is shown for reference purpose only. Consistent with predictions in previous in chapter 6, where unscaled ground motions were used, scaled CMV motion provides a good estimate of the critical displacement response.   Figure 7.5 Floor displacement response of 44-storey building to NGA316.  7.7 Concluding remarks The CMV Method can be used for performance based assessment of tall buildings. The critical displacements can be estimated for a suite of design ground motions and provide an upper bound for the potential displacement responses that the building could undergo when subject to ground 127  shaking having strong velocity pulses of long duration. This can be required at sites where significant uncertainty prevails about the characterization of the near seismic sources and ground shaking. The CMV Method can also be applied in the seismic design of tall buildings which are required to exhibit seismic performance that exceeds building code requirements.  The CMV Method should be applied to estimate the critical displacement response along each one of the orthogonal axes, if necessary different ground motions should be used. This will ensure that the first mode translational responses in each direction are captured properly.  128  Chapter 8: Conclusions This dissertation presents a computationally efficient method for estimating the critical displacement along the structural axis of a tall building caused by near-field pulse-like motions.  Implementing the CMV Method entails analyzing the orientation dependence of the ground motion velocity pulse duration and amplitude to determine the CMV orientation prior to rotating the ground motion pair and performing nonlinear response history analysis of a tall building model. The effort and resources required to determine the CMV orientation and estimate the critical displacement response using a single response history analysis pale in comparison with the direct analysis. The later requires applying the same ground motion at different orientations relative to the structural axis of the building model, which results in a greater number of nonlinear response history analyses.  The conditional maximum velocity method can be used to efficiently estimate the critical displacement response, conduct performance assessment of a building and provide a basis to make informed decisions for performance based design without the complex processing of extensive response data.    Another advantage of the CMV Method is that it does not depend on the identification of fault-normal and fault-parallel orientations of near-field ground motions. In past studies such orientations have been identified to be relevant for seismic design of long period buildings. Identifying the FN/FP components for a ground motion that was produced by an earthquake rupture having multiple segments of very different azimuths is not straightforward though. In these cases a subjective decision must be made to rotate the ground motion pairs into the FN/FP 129  orientations. The use of the proposed CMV Method eliminates any bias that could be introduced from such identification.  The use of direct analysis is not recommended in the engineering practice. The disadvantages of the direct analysis are:  It escalates the number of seismic response analysis which requires extensive computational time. This is an undesirable feature for applications within the framework of performance based design when response history analysis using many different ground motions may be needed.  If the realization of critical values corresponds to different rotations of the ground motion, unrealistic seismic demands may be obtained.   After all, performing direct analysis as part of the conventional engineering practice does not promote a better understanding of the near-field ground motion problem and the response of tall buildings.  8.1 Contribution of this research to the state of knowledge Until now, the state of knowledge related to directionality effects of pulse-like near-field ground motions on seismic response of tall buildings lacked a computationally efficient method to reliably estimate the critical displacement response. To overcome this limitation this dissertation has:  Introduced the concept of ground velocity pulse spectrum, a graphical representation useful to characterize the orientation dependence of the duration and amplitude of 130  velocity pulses in crustal near-field pulse-like ground motions.  This representation serves to gain insight on the dominant orientations of ground shaking.  Identified the CMV ground motion as a suitable input to estimate the critical displacement response along the structural axis of a tall building.   Demonstrated that fault-normal, fault-parallel and the maximum direction ground motion at the fundamental period of the building can produce significant underestimates of the critical displacement response of tall buildings. In some instances the error in predicting the critical displacement can be as large as 70%.  8.2 Contribution of this research to the state of practice The contributions of this dissertation to the state of practice include:  Engineers engaged in seismic design of tall buildings now have at their disposal a computationally efficient method to estimate the critical displacement of tall buildings.  A program to generate the elastic displacement response spectrum for multiple orientations of a ground motion is now available for engineers who use Microsoft Excel https://drive.google.com/file/d/0Bx6L6P1KZlSGSF9yQWtGd0xVNTg/view?usp=sharing  8.3 Future work The ground motion records analyzed in this research were processed by the Pacific Earthquake Engineering Research Center (PEER) using baseline correction and band-pass filter processing of the frequency content. Such processing removed residual ground displacements. Therefore, 131  fling effects in the form of coseismic displacement were removed from the ground motions, and their influence on the seismic response of tall buildings was not investigated.  The CMV Method has been verified using several tridimensional building models with translational first mode shapes. The building models were for structural systems typical of reinforced concrete tall buildings having core shear wall with perimeter columns, and a spine steel frame building. Although the building models were nonlinear, strength degradation leading to progressive collapse was not captured by these models.   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A 3D building model was created using the program CANNY (Li, 2015) which resembles the general features of the tall building, although it is not an exact representation of it.  The layouts of the architectural design were not available, thus the height of each storey was measured with a scalimeter directly from the structural blueprints. The storey height for the lower lobby was measured to be approximately 4.8m, the upper lobby 3.96m and onwards the typical floor height 2.90m.  A.1 Columns and shear walls The columns and shear walls were modeled with the multi-axial spring model (MS model) available in the CANNY program which is able to simulate the interaction of bi-directional bending and axial forces on reinforced concrete components. A column idealized by the MS model has a line element and two MS elements at the column-end. Each MS element is made up of a number of uniaxial springs which correspond to a discretization of the section into steel and concrete fibers. The deformation in every spring in the MS model conforms to the plane section assumption.    142   Figure A.1  Elevation of 44 storey building.  Sections of every wall and column were defined in the CANNY program with the rebar layout and transverse reinforcement. An example of a shear wall and column sections of this building model are shown in Figure A.1. The material properties and stress-strain relationships of the concrete and rebar steel are described in the following sections. 143  The shear walls sections of the elevator core walls were created using rectangular column and wall sections. Boundary elements at the ends of shear walls with high density of longitudinal and transverse reinforcement were modeled as columns while low reinforced sections were modeled as wall sections. The transverse reinforcement of each section was introduced to allow the program CANNY to account for the confinement effect on concrete due to the combined action of longitudinal and transverse reinforcement.     Figure A.2  Typical shear wall and columns sections of the building modeled in CANNY.  A.2 Coupling beams The reinforced concrete coupling beams that connect the C-shape shear walls along the east-west direction are diagonally reinforced. The typical dimensions for the coupling beams are 660mm wide and 510mm height. The model of the coupling beams was defined using the parameters shown in Figure A.3, where Td is the tension force carried by the diagonal, Cd is the compression Wall Column in Boundary zone Column in Boundary zone Longitudinal Transverse Longitudinal Transverse 144  force carried by the diagonal and Cc is the concrete compression strut, zb is the lever arm between the diagonals.  Figure A.3  Forces in diagonal coupling beams.  Moment capacity were calculated from the following equation by Paulay and Priestley (1992) ݎܯ ൌ ሺܣ௦ௗ cos ∝ሻ	 ௬݂	ݖ௕ The effective stiffness EcIeff was set equal to 0.3EcIg. A one-component element was used to model flexural response of the coupling beams. This model has two perfectly rigid plastic hinges at the beam ends that are connected through an elastic beam of stiffness EcIeff and yielding resistance Mr.  A.3 Slabs The building has 178mm (7 in) thick flat plates with post-tension tendons. At ground level a transfer slab has slab-bands of varying thickness. The slabs bands at ground level are typically 2438 mm (8 ft) wide with varying depths in the range of 510 mm (20 in) – 1200 mm (4 ft), while the slabs were in general 203 mm (8 in) thick. In the parking levels the flat slab system is provided with 254 mm (10 in) thick slabs.  TdTdCd + CcMrAsdAsdMlThzbCd2ln145  Equivalent elastic frame beams were used to include the bending action of the slabs following recommendations from ATC 72 (PEER, 2010). A rigid diaphragm was considered for in plane slab deformations using the supernode element in CANNY.  A.4 Material properties The material properties for concrete and steel rebars were taken as specified in construction blueprints. The concrete was specified to be 50 MPa (7.25 ksi) compressive strength and steel reinforcing bars 400 MPa (58 ksi) yielding strength.   A.5 Material stress-strain models The steel rebars were modeled in CANNY using a hysteretic model SS3, which is a trilinear backbone curve. This model includes stiffness degradation before onset of yielding to account for rebar slippage at the joints. This hysteresis model is shown in Figure A.4. The onset of yielding in the skeleton curve was controlled by parameters  and . The stiffness of the unloading branch is calculated by the program using the equation below with = 0 (no stiffness degradation prior to yielding during load reversal): ܭ௨ ൌ ܭ ቀ೤ିᇱᇱ೤೘ିᇱ೘ ቁ  Remaining parameters that defined the unloading branch were = 0.75 (onset of yielding under a reverse load) and = 0.2 for post-yield stiffness degradation.  146   Figure A.4  Hysteretic model SS3.  The concrete material was defined using CS3 hysteretic model is shown in Figure A.5. It is a trilinear model with stiffness degradation after the cracking point. The skeleton curve loading in compression was defined with =0.5, f’c=50 MPa, elastic stiffness K=37,200 MPa for columns, walls, beams and slabs, and peak strain c=0.002. The ultimate stress was defined with the parameter =0.3 and =0.8 for unconfined and confined concrete, respectively. The ultimate strain was defined with the factor as 2.5 times the peak strain c for unconfined concrete. The peak strain cc for confined concrete was calculated by the program CANNY taking into account the longitudinal and transverse reinforcement. The unloading stiffness degradation was set as = K11StrainStressfyfyKf’yy yKu1Ku=f(K,,)fy=400 MPa K=200,000 MPa y=0.002 =0.01 =0.5 =1.8 =0.2=0.75 =0’’y 147  0.5 (stiffness degradation prior to peak stress), and = 0.4 for post-peak stiffness degradation. The tension maximum strain was set equal to =3.  Figure A.5 Hysteresis model CS3 for concrete.  A.6 Distribution of structural mass The masses were initially calculated as structure’s dead load. Then the masses were modified to closely match the first three natural periods of the actual building. The periods were obtained from an ambient vibration test. The corresponding dead loads were applied to the model.  A.7 Dynamic properties of building model The dynamic properties of the model calibrated to the ambient vibration test are presented in Table A.1. The translational modes match closely the ambient vibration results, and the largest differences occur for the torsional modes. 1cStressf’c fc Kf cKu 1Ku=f(Kcu,,) ft=5 MPa f’c=50 MPa c=0.002 =0.8 (confined) =0.3 (unconfined) =3 (confined) =2.5 (unconfined) =0.4 =0.5 t t Strain=0.5 =3 fc 148  Table A.1 Mode periods from CANNY model and ambient vibration test. Mode Description Period (s) from CANNY Model Period (s) from Ambient Vibration Test 1 Translational E-W 3.8 3.8 2 Translational N-S 3.5 3.5 3 Torsional 1.3 1.5 4 Translational E-W 0.9 0.8 5 Translational N-S 0.7 0.7 6 Torsional 0.4 0.5  A.8 Modeling of damping The damping effects from non-structural components and other energy dissipation sources were modeled using Rayleigh damping. It was set equal to 3% in the E-W translational modes 1 and 4. The damping ratios are summarized in Table A.2. Table A.2 Modal damping ratios. Mode Period (s) from CANNY Model Damping ratio (%) 1 3.8 3.00% 2 3.5 2.82% 3 1.3 2.55% 4 0.9 3.00% 5 0.7 3.74% 6 0.4 5.45%   Appendix B  Distance dependence of maximum velocity pulses The dependence of the pulse duration and amplitude on distance to the rupture was analyzed for all the 45 ground motions. Scatter plots in Figure B.1 show the distance dependence for maximum velocity of the ground motion, maximum amplitude of the extracted pulse (P1) and duration of the velocity pulse.   149  The extracted velocity pulses using wavelet analysis results in waveforms rich in low-frequency content. The amplitude of low-frequency motions usually decay at a smaller rate than motions of high-frequency. This can help to explain the presence of long period velocity pulses in ground motion records within the near field of the fault, up to 30km distances.    Figure B.1 Distance dependence of MV, P1max and pulse duration.   150  Appendix C  Effect of pulse duration on seismic response of tall buildings The effect of pulse duration on the dynamic response was investigated in chapter 4 using simple structures and synthetic pulse-excitations. A similar evaluation on the dynamic response of the 45-storey building to near field pulse-like ground motions is presented. Two ground motions were used for the analyses, NGA143 (1978 Tabas earthquake) and NGA181 (1979 Imperial Valley earthquake). The NGA143 ground motion meets the requirements of the CMV Method with a Tp equal to 6.1s and maximum ground velocity of 122 cm/s. In this case the ratio Tp/T for NGA143 is 1.39. The NGA181 instead, does not have any pulse of duration longer than the building period 4.4s, and the maximum ground velocity is 117 cm/s with pulse duration of 3.7s. The ratio Tp/T for NGA181 is 0.84. The ground motion NGA181 was linearly scaled for NRHA by a factor of 1.04 to have a maximum ground velocity equal to 122 cm/s.   The envelopes of calculated floor displacements, interstorey drift ratios, core wall overturning moment and shear in the X-direction are compared in Figure C.1. The ground motion NGA143 having Tp longer than T results in much larger displacements than the ground motion NGA181 with Tp shorter than T. This is consistent with the observations made in chapter 4 and confirms the influence of strong pulses with durations longer than the fundamental period in the displacement response.   Furthermore, the interstorey drift ratios are much larger for the ground motion NGA143 having Tp longer than T. The differences observed for shear forces in the core wall are also noticeable; however the overturning moments in the core wall are rather similar for both ground motions.  151      Figure C.1 Envelopes of nonlinear responses along X-axis for NGA143 and NGA181.  The force deformation responses for coupling beams inside the core wall in the X-direction are shown in Figure C.2, this beams were modeled in Perform 3D with a shear spring. The few large 152  inelastic excursions observed in these figures were caused by the strong velocity pulses. Again, the input motion NGA143 with the CMV method provides larger inelastic demands when compared to the demands imposed by NGA181 having Tp shorter than T. The observations drawn from this evaluation should be revisited in future studies considering cases where the duration Tp is close to the second mode or third mode period, such extreme was not investigated in this dissertation.   (a) Level 10 (b) Level 20   (c) Level 30 (d) Level 40 Figure C.2 Hysteretic response of coupling beams in core wall along X-axis for NGA143 and NGA181.  153  Appendix D  Empirical relationships for selection of ground motions Using the earthquake magnitude, source-to-site distance and epsilon parameters from a seismic hazard deaggregation, the model in Equation D.1 developed by NIST (2011) to estimate the proportion of pulse-like ground motions within a suite of design ground motions can be used. This proportion guides the selection process specifying the required number of pulse-like ground motions. The parameter epsilon is usually obtained for the spectral acceleration at the period of the structure at the design hazard level. ܲ݊݋݅ݐݎ݋݌݋ݎ	݂݋	݁ݏ݈ݑ݌	ݏ݊݋݅ݐ݋݉ ൌ ݁ሺ଴.ଽ଴ହି଴.ଵ଼଼ோାଵ.ଷଷ଻ఌሻ1 ൅ ݁ሺ଴.ଽ଴ହି଴.ଵ଼଼ோାଵ.ଷଷ଻ఌሻ (D.1) For instance at a site located within 5km from the active fault and a ground shaking hazard with an epsilon of 1.5 the proportion of pulse motions from Equation D.1  is 0.87. Thus the number of pulse-like motions should be 4 (0.87*7~6) for a suite of seven ground motions. An evaluation of how the proportion of pulse motions varies with distance and epsilon is shown in Figure D.1.   Figure D.1 Dependence of proportion of pulses on epsilon and distance  154  Once the number of pulse-like motions has been defined it remains to establish the expected duration and amplitude of the ground motion velocity pulses. Several researchers have developed empirical models to predict these using statistical regressions analysis on large sets of pulse-like ground motions. Since these models depend on the earthquake magnitude and source-to-site distance again these parameters obtained from seismic hazard deaggregation can be used to evaluate the empirical models.  Some of the models that predict the duration of the velocity pulse are summarized in Table D.1. The median prediction from these models depends only on the magnitude of the earthquake event. The model by Rodriguez-Marek and Bray (2004) provides the option to estimate the expected duration of the velocity pulse either for soil or rock conditions. A comparison of the different models is shown in Figure D.2 it can be observed that the pulse duration at soil sites is expected to be longer than at rock sites. Table D.1 Predictive models for pulse duration. Predictive model Standard deviation Author ݈݊ ௣ܶሺݏሻ ൌ െ4.05 ൅ 0.71ܯ - Alavi  and Krawinkler (2004) ݈݊ ௣ܶሺݏሻ ൌ െ5.6 ൅ 0.93ܯ [at soil] 0.58 Rodriguez-Marek and Bray (2004) ݈݊ ௣ܶሺݏሻ ൌ െ8.6 ൅ 1.32ܯ [at rock] 0.40 Rodriguez-Marek and Bray (2004) ݈݊ ௣ܶሺݏሻ ൌ െ5.78 ൅ 1.020.55 ܯ Baker (2007)   Although there are significant differences between these models the trend is for pulse duration to increase monotonically with the earthquake magnitude. An average value can be determined to define the minimum duration of the pulse for the selection of the ground motions. It is important to recall that the duration of the velocity pulse has an important effect on the response of tall 155  buildings. As shown in this dissertation if the critical displacement demands are to be evaluated the duration of the velocity pulse should be longer than the fundamental period of the building, preferably within the range required in chapter 7.   Figure D.2 Pulse duration as a function of earthquake moment magnitude.  Finally the amplitude of the expected ground motion velocity pulses should be estimated. Two models available to estimate the peak ground velocity are presented in Table D.2. In these models the pulse amplitude increases in proportion to the earthquake magnitude and decreases with increasing source-to-site distance. The model developed by Alavi and Krawinkler (2004) is valid for distances in the range of 3km to 10km only. Overall an important reference to use is that the maximum amplitudes of strong velocity pulses usually fall in the range of 70 cm/s to 130 156  cm/s at close distances to the fault, except for Chi-Chi records TCU052 and TCU068 which have much larger amplitudes (Mavroeidis et. al., 2004).  Table D.2 Predictive models for velocity pulse amplitude (peak ground velocity). Predictive model Standard deviation Author ݈ܸ݊ܲܩሺܿ݉ݏ ሻ ൌ െ4.67 ൅ 1.50ܯ െ 1.08lnܴ - Alavi  and Krawinkler (2004) ݈ܸ݊ܲܩሺ௖௠௦ ሻ ൌ 4.46 ൅ 0.34ܯ െ 0.58ln	ሺܴଶ ൅ 49ሻ [at soil] 0.49 Rodriguez-Marek and Bray (2004) ݈ܸ݊ܲܩሺ௖௠௦ ሻ ൌ 4.58 ൅ 0.34ܯ െ 0.58ln	ሺܴଶ ൅ 49ሻ – [at rock] 0.39 Rodriguez-Marek and Bray (2004)  The continuous development of probabilistic seismic hazard analysis (PSHA) has resulted in novel frameworks that can be used to estimate the probability of occurrence of a pulse-like ground motions at a given site (Tothong et al., 2007; Shahi, 2013). Furthermore the duration, amplitude and possible orientation are byproducts of these analyses. Although it is likely the use of such complex frameworks will become a standard practice for PSHA in the coming years a discussion of their applications is outside of the scope of the present dissertation.  

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