NONLINEAR RESPONSE OF HIGH-RISE BUILDINGS: EFFECT OF DIRECTIONALITY OF GROUND MOTIONS by Manuel Archila B.Sc. Universidad del Valle de Guatemala, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April, 2011 © Manuel Archila, 2011 ii Abstract The response of high-rise buildings to strong ground shaking depends on ground motion parameters namely: intensity, frequency content, duration and horizontal ground motion directionality. The latter has been a concern to engineers for several decades in seismic design. The prediction of the direction where ground motion will hit the building is rendered difficult because in many regions faults are not mapped to a great extent, and for regions were fault locations are known accurate prediction of ground motion directionality is impeded because ground motions have unique wave propagation characteristics along its path. The purpose of this study was to evaluate the influence of ground motion directionality on the nonlinear dynamic response of a high-rise building. The influence of ground motion directionality was evaluated for a building case study. The building was 44 storey, and resembled general features of structural configuration commonly provided to reinforced concrete high-rise in Vancouver city. The nonlinear time history analysis (NLTHA) method was used to estimate seismic response of the building model to bi-directional ground shaking. This method was systematically applied for 40 ground motion component angles of incidence, which accounted for different ground motion directionalities ranging from 0 to 360 degrees. A suite of 3 pairs of horizontal ground motion representative of seismic hazard 2% in 50 years in Vancouver was considered for analysis. The ground motion directionality had significant effect over the calculated building seismic response. In some scenarios at critical angle of incidence the calculated floor displacements and interstorey drifts were 4 times as large as the displacements and drifts calculated for ground motion at 0 degrees angle of incidence. The largest building response envelope was obtained for several critical angles of incidence of the ground motion components. Critical angles of incidence were distributed over the entire building’s height. The relevance of ground motion directionality for seismic design of high-rise buildings was clearly demonstrated. The NLTHA used in conventional design practice still ignores ground motion directionality. It is concluded there is a need to develop the tools engineers can readily use to consider ground motion directionality in seismic design of modern high-rise buildings. iii Preface This research was conducted as collaboration between Manuel Archila, Professor Carlos Ventura and Professor Liam Finn. The contributions of the thesis author to this research project were: (i) preparation of research proposal, including identification and design of the research program (ii) evaluation of previous research related, (iii) establishment of the methodology to address the study, (iv) development of the case study including the computer mathematical model and calculations, (v) assessment of results obtained and preparation of this manuscript. Professor Carlos Ventura approved the research proposal and provided feedback over the project during the different stages of the research, and the different versions of the thesis. He also provided the reference material for the case study, and granted the thesis author access to hardware and software available in the Earthquake Engineering Research Facility (EERF) at The University of British Columbia to conduct the numerical modeling and analyses of the tall building. Fruitful discussions over the presentation of results were provided by the thesis authors and the Professors, who provided feedback, advice and constructive criticism about the results. The results from the seismic hazard assessment presented in Chapter 4 and partial results from the tall building analysis in Chapter 5 of this thesis were presented at two Conferences, the papers included in the Conference Proceedings were: Ventura C.E., Archila M., Pina F., Centeno J., 2010. Sensitivity of Design Spectrum for British Columbia to Different Levels of Probability of Exceedance. Proceedings of 9th U.S. National and 10th Canadian Conference on Earthquake Engineering, Toronto, ON, Canada. Archila M., Finn L. W.D., Ventura C.E., 2010. UHS Spectra for Different Types of Earthquakes and Hazard Levels. Proceedings of 14h European Earthquake Engineering, Ohrid, Republic of Macedonia. Archila, M., Ventura C.E., Finn L. W.D., 2010. Influence of Ground Motion Directionality on Seismic Response of a Tall Building. Proceedings of 14h European Earthquake Engineering, Ohrid, Republic of Macedonia. These publications were obtained after extensive work and were the outcome of all the research contributions of the thesis author and collaborators described above. The contributions of the thesis author to these publications were: (i) elaboration of all the calculations of the tall building dynamic analyses through computer simulations, (ii) preparation of the draft version of the manuscripts. iv Table of contents Abstract ....................................................................................................................................................................... ii Preface ........................................................................................................................................................................ iii Table of contents ......................................................................................................................................................... iv List of tables .............................................................................................................................................................. vii List of figures ........................................................................................................................................................... viii Acknowledgements ..................................................................................................................................................... xi Dedication ................................................................................................................................................................ xiii 1. Introduction ......................................................................................................................................................... 1 1.1 Problem statement ...................................................................................................................................... 1 1.2 Motivation .................................................................................................................................................. 1 1.3 Goal and objectives ..................................................................................................................................... 1 1.4 Scope of study ............................................................................................................................................ 2 1.5 Organization of this thesis .......................................................................................................................... 2 1.6 Definitions .................................................................................................................................................. 3 2. Background ......................................................................................................................................................... 6 2.1 Overview .................................................................................................................................................... 6 2.2 Building codes seismic design provisions .................................................................................................. 6 2.2.1 National Building Code of Canada (2005) ......................................................................................... 6 2.2.2 International Building Code (2006) ................................................................................................... 7 2.2.3 EuroCode 8 (prEN 1998-1:2003) ....................................................................................................... 8 2.2.4 Chile Official Normative (1996) ........................................................................................................ 9 2.2.5 Mexico Federal District Code (2004) .............................................................................................. 10 2.2.6 Recommendations for the seismic design of high-rise buildings (2008) ......................................... 10 2.3 Response spectrum analysis ...................................................................................................................... 10 2.3.1 Characteristics of 3-dimensional earthquake ground motions ......................................................... 11 2.3.2 Modal combination rules for multi-component earthquake excitation ............................................ 11 2.3.3 A clarification of the orthogonal effects in a three-dimensional seismic analysis ........................... 12 2.3.4 A replacement for the 30%, 40%, and SRSS rules for multicomponent seismic analysis ............... 13 2.3.5 Critical response of structures to multi-component earthquake excitation ...................................... 14 2.4 Time history analysis ................................................................................................................................ 14 2.4.1 Critical orientation of three correlated seismic components ............................................................ 15 2.4.2 Variation of response with incident angle under two horizontal correlated seismic components .... 15 2.4.3 Incremental dynamic analysis .......................................................................................................... 16 2.4.4 Influence of angle of incidence on seismic demands for inelastic single-storey structures subjected to bi-directional ground motions ....................................................................................................................... 17 2.4.5 Incremental dynamic analysis with two components of motion for a 3D steel structure ................. 18 v 2.4.6 Multicomponent incremental dynamic analysis considering variable incident angle ...................... 19 2.5 Summary ................................................................................................................................................... 20 3. Analysis methodology ....................................................................................................................................... 21 3.1 Overview .................................................................................................................................................. 21 3.2 Mathematical model ................................................................................................................................. 23 3.2.1 Columns and shear walls .................................................................................................................. 23 3.2.2 Coupling beams ............................................................................................................................... 24 3.2.3 Flat slabs .......................................................................................................................................... 26 3.2.4 P-delta effects................................................................................................................................... 27 3.2.5 Gravity load effects .......................................................................................................................... 27 3.2.6 Mass source for dynamic response .................................................................................................. 28 3.2.7 Damping and energy dissipation ...................................................................................................... 28 3.3 Multi-directional nonlinear static analysis ................................................................................................ 29 3.4 Seismic input ............................................................................................................................................ 29 3.4.1 Seismic hazard analysis ................................................................................................................... 29 3.4.2 Site location ..................................................................................................................................... 30 3.4.3 Treatment of uncertainty .................................................................................................................. 31 3.4.4 Magnitude recurrence relations ........................................................................................................ 31 3.4.5 Upper and lower bound for magnitude ............................................................................................ 31 3.4.6 Seismic sources in southwestern BC ................................................................................................ 31 3.4.7 Ground motion prediction equations ................................................................................................ 32 3.4.8 Selection of ground motion records ................................................................................................. 33 3.4.9 Scaling of horizontal bi-directional ground motion records ............................................................. 33 3.4.10 Ground motion component angle of incidence ................................................................................ 34 3.5 Multi-directional incremental dynamic analysis (MIDA) ......................................................................... 35 3.6 Assessment of ground motion directionality effect on calculated building nonlinear dynamic response and critical angle of incidence ............................................................................................................................... 36 4. Case study ......................................................................................................................................................... 37 4.1 Description of case study .......................................................................................................................... 37 4.2 Building description .................................................................................................................................. 37 4.3 Structural system ...................................................................................................................................... 41 4.4 Computer building model ......................................................................................................................... 42 4.4.1 Columns and shear walls .................................................................................................................. 42 4.4.2 Coupling beams ............................................................................................................................... 43 4.4.3 Slabs ................................................................................................................................................. 43 4.4.4 Frame reference system to define the model .................................................................................... 43 4.4.5 Degrees of freedom .......................................................................................................................... 44 vi 4.4.6 Mathematical model calibration ....................................................................................................... 44 4.4.7 Material properties ........................................................................................................................... 45 4.4.8 Material stress-strain models ........................................................................................................... 46 4.4.9 Dynamic characteristics of calibrated model ................................................................................... 47 4.4.10 Gravity load criteria ......................................................................................................................... 50 4.4.11 Dynamic characteristics of the building for analysis ....................................................................... 51 4.5 Seismic input ............................................................................................................................................ 52 4.5.1 Seismic hazard analysis ................................................................................................................... 52 4.5.2 Uniform hazard spectra .................................................................................................................... 52 4.5.3 Contribution to hazard level in UHS ................................................................................................ 53 4.5.4 Seismic hazard deaggregation .......................................................................................................... 54 4.5.5 Selection of ground motions for time history analysis ..................................................................... 55 4.5.6 Scaling of ground motions for time history analysis ........................................................................ 59 4.5.7 Ground motions for different angles of incidence ............................................................................ 64 4.5.8 Ground motion directionality ........................................................................................................... 68 4.6 Time step and integration method ............................................................................................................ 73 4.7 Computation time ..................................................................................................................................... 73 5. Results ............................................................................................................................................................... 74 5.1 Description ................................................................................................................................................ 74 5.2 Multi-directional nonlinear static analysis ................................................................................................ 74 5.3 Directional dynamic analysis .................................................................................................................... 76 5.4 Sensitivity of response parameters to ground motion component angle of incidence .............................. 94 5.5 Sensitivity of a response parameter to the ground motion ........................................................................ 96 5.6 Multi-directional incremental dynamic analysis for critical angle of incidence ....................................... 97 5.7 Sensitivity of column axial force to the ground motion component angle of incidence ......................... 106 6. Interpretation of results.................................................................................................................................... 108 6.1 Seismic hazard analysis .......................................................................................................................... 108 6.2 Directional analysis ................................................................................................................................ 108 7. Conclusions and recommendations ................................................................................................................. 110 7.1 Summary ................................................................................................................................................. 110 7.2 Conclusions ............................................................................................................................................ 110 7.3 Recommendations for design practice .................................................................................................... 112 7.4 Recommendations for future studies ...................................................................................................... 112 References ................................................................................................................................................................ 114 Appendix A – Building calculated response to Loma Prieta ground motion ........................................................... 117 Appendix B – Building calculated response to Chi Chi ground motion .................................................................. 182 Appendix C – Building calculated response to Imperial Valley ground motion ...................................................... 199 vii List of tables Table 3.1 Ground motion prediction equations used in analysis. ............................................................................... 32 Table 4.1 Concrete material properties. .................................................................................................................... 45 Table 4.2 Steel material properties. ........................................................................................................................... 46 Table 4.3 Comparison of building model dynamic characteristics and modal model. .............................................. 47 Table 4.4 Nominal values for superimposed dead and live load. .............................................................................. 50 Table 4.5 Periods of vibration and Rayleigh modal damping ratios. ........................................................................ 51 Table 4.6 Earthquakes selected to obtain ground motion records. ............................................................................ 55 Table 4.7 Ground motion records selected. ................................................................................................................ 55 Table 4.8 Intensity scaling factors for IDA in Vancouver city. .................................................................................. 59 Table 4.9 Maximum amplification factors of spectral ordinates. ............................................................................... 64 Table 4.10 Loma Prieta amplification factors of spectral ordinates. .......................................................................... 65 Table 4.11 Chi Chi amplification factors of spectral ordinates. ................................................................................. 66 Table 4.12 Imperial Valley amplification factors of spectral ordinates. .................................................................... 67 Table 4.13 Vectors representative of Loma Prieta displacement orbit. ...................................................................... 68 viii List of figures Figure 1.1 Horizontal ground motion orbit and displacement vector ........................................................................... 3 Figure 2.1 IDA curve from a single-record of a T1=4.27s, 44 storey RC building .................................................... 17 Figure 3.1 Structure of methodology. ....................................................................................................................... 21 Figure 3.2 Uniaxial spring model for flexural response. ........................................................................................... 24 Figure 3.3 Coupling beam forces. ............................................................................................................................. 24 Figure 3.4 Degrading bilinear model BL2 in CANNY program. .............................................................................. 25 Figure 3.5 Interaction frame-shear wall under lateral forces. ..................................................................................... 26 Figure 3.6 Transformation of ground motion to different coordinate system. .......................................................... 35 Figure 4.1 Plan layout of the residential tower and podium. ..................................................................................... 38 Figure 4.2 Plan layout of the residential tower upper levels. .................................................................................... 38 Figure 4.3 Building section A-A (East West). .......................................................................................................... 39 Figure 4.4 Building section B-B (North South). ....................................................................................................... 40 Figure 4.5 3D view of structural elements in upper levels of tower. ......................................................................... 41 Figure 4.6 Typical shear wall and columns sections of the building modeled in CANNY. ...................................... 42 Figure 4.7 View of frame reference system for building model in CANNY. ........................................................... 44 Figure 4.8 Hysteresis model SS3 for steel rebars (Li, 2010). .................................................................................... 46 Figure 4.9 Hysteresis model CS3 for concrete (Li, 2010). ........................................................................................ 47 Figure 4.10 Mode shapes of building. ....................................................................................................................... 48 Figure 4.11 Distribution of tributary areas. ............................................................................................................... 50 Figure 4.12 Uniform hazard spectra for Vancouver city (5% damping ratio). .......................................................... 52 Figure 4.13 Crustal and subcrustal relative contribution to the uniform hazard spectrum in Vancouver city for a hazard level of 2% in 50 year. .................................................................................................................................... 53 Figure 4.14 Magnitude-distance deaggregation of hazard level 2% in 50 year at Vancouver. .................................. 54 Figure 4.15 Loma Prieta STG000 component 1 acceleration, velocity, displacement motion. .................................. 56 Figure 4.16 Loma Prieta STG090 component 2 acceleration, velocity, displacement motion. .................................. 56 Figure 4.17 Chi Chi TCU 122E component 1 acceleration, velocity, displacement motion. ..................................... 57 Figure 4.18 Chi Chi TCU 122N component 2 acceleration, velocity, displacement motion. .................................... 57 Figure 4.19 Imperial Valley H-DLT262 component 1 acceleration, velocity, displacement motion. ........................ 58 Figure 4.20 Imperial Valley H-DLT352 component 2 acceleration, velocity, displacement motion. ........................ 58 Figure 4.21 Spectral demand level ratio for UHS in Vancouver. ............................................................................... 60 Figure 4.22 Loma Prieta as recorded geomean response spectrum amplitude scaled to Vancouver UHS 2% in 50 year. ............................................................................................................................................................................ 61 Figure 4.23 Chi Chi as recorded geomean response spectrum amplitude scaled to Vancouver UHS 2% in 50 year. 62 Figure 4.24 Imperial Valley as recorded geomean response spectrum amplitude scaled to Vancouver UHS 2% in 50 year. ............................................................................................................................................................................ 63 Figure 4.25 Loma Prieta geomean response spectra (5% damping). ........................................................................ 65 Figure 4.26 Chi Chi geomean response spectra (5% damping). ................................................................................ 66 ix Figure 4.27 Imperial Valley geomean response spectra (5% damping). ................................................................... 67 Figure 4.28 Loma Prieta ground motion displacement orbit. ..................................................................................... 69 Figure 4.29 Loma Prieta displacement trace 4.6s – 15.0s. ......................................................................................... 70 Figure 4.30 Chi Chi ground motion displacement orbit. ............................................................................................ 71 Figure 4.31 Imperial Valley ground motion displacement orbit. ............................................................................... 72 Figure 5.1 Loading directions for pushover. .............................................................................................................. 75 Figure 5.2 Directional pushover. ................................................................................................................................ 75 Figure 5.3 Ground motion component angle of incidence. ........................................................................................ 76 Figure 5.4 Effect of angle of incidence on envelope of overturning moment EW to ground motion. ....................... 78 Figure 5.5 Conditional probability of exceedance for overturning moment envelope along EW to motion at 0°. .... 79 Figure 5.6 Effect of angle of incidence on envelope of overturning moment NS to ground motion. ........................ 80 Figure 5.7 Conditional probability of exceedance for overturning moment envelope along NS to motion at 0°. ..... 81 Figure 5.8 Effect of angle of incidence on envelope of storey shear EW to ground motion. ..................................... 82 Figure 5.9 Conditional probability of exceedance for storey shear envelope along EW to motion at 0°. .................. 83 Figure 5.10 Effect of angle of incidence on envelope of storey shear NS to ground motion. .................................... 84 Figure 5.11 Conditional probability of exceedance for storey shear envelope along NS to motion at 0°. ................. 85 Figure 5.12 Effect of angle of incidence on envelope of displacement EW to ground motion. ................................. 86 Figure 5.13 Conditional probability of exceedance for displacement envelope along EW to motion at 0°. .............. 87 Figure 5.14 Effect of angle of incidence on envelope of displacement NS to ground motion. .................................. 88 Figure 5.15 Conditional probability of exceedance for displacement envelope along NS to motion at 0°. ............... 89 Figure 5.16 Effect of angle of incidence on envelope of interstorey drift EW to ground motion. ............................. 90 Figure 5.17 Conditional probability of exceedance for interstorey drift envelope along EW to motion at 0°. .......... 91 Figure 5.18 Effect of angle of incidence on envelope of interstorey drift NS to ground motion. .............................. 92 Figure 5.19 Conditional probability of exceedance for interstorey drift envelope along NS to motion at 0°. ........... 93 Figure 5.20 Sensitivity of lower lobby overturning moment to angle of incidence. .................................................. 94 Figure 5.21 Sensitivity of lower lobby storey shear to angle of incidence. ................................................................ 95 Figure 5.22 Sensitivity of roof displacement to angle of incidence. .......................................................................... 95 Figure 5.23 Sensitivity of interstorey drift at uppermost storey to angle of incidence. .............................................. 95 Figure 5.24 Sensitivity of overturning moment to different ground motion 0° angle of incidence. ........................... 96 Figure 5.25 Incremental dynamic analysis for overturning moment at lower lobby level. ........................................ 97 Figure 5.26 Incremental dynamic analysis for largest overturning moment EW to Loma Prieta motion with different hazard levels. .............................................................................................................................................................. 98 Figure 5.27 Incremental dynamic analysis for largest overturning moment NS to Loma Prieta motion with different hazard levels. .............................................................................................................................................................. 99 Figure 5.28 Incremental dynamic analysis for largest storey shear EW to Loma Prieta motion with different hazard level. ......................................................................................................................................................................... 100 Figure 5.29 Incremental dynamic analysis for largest storey shear NS to Loma Prieta motion with different hazard level. ......................................................................................................................................................................... 101 x Figure 5.30 Incremental dynamic analysis for largest displacement EW to Loma Prieta motion with different hazard level. ......................................................................................................................................................................... 102 Figure 5.31 Incremental dynamic analysis for largest displacement NS to Loma Prieta motion with different hazard level. ......................................................................................................................................................................... 103 Figure 5.32 Incremental dynamic analysis for largest interstorey drift EW to Loma Prieta motion with different hazard level. ............................................................................................................................................................. 104 Figure 5.33 Incremental dynamic analysis for largest interstorey drift NS to Loma Prieta motion with different hazard level. ............................................................................................................................................................. 105 Figure 5.34 Sensitivity of axial force in lower lobby columns to ground motion component angle of incidence. .. 107 xi Acknowledgements I owe special thanks to Professor Carlos Ventura, my supervisor at The University of British Columbia (UBC), whose constant guidance and support throughout my studies were instrumental to complete this research and achieve other goals I never considered feasible. His enthusiasm to explore new topics, state problems and find practical solutions in earthquake engineering is one of a kind and the best lesson I have learned at UBC. Professor Carlos Ventura had confidence in my person and helped me to secure the financial support for my studies, which I deeply appreciate. I am proud of the opportunity I had to be member of his research team in the Earthquake Engineering Research Facility at UBC. I thank Professor Liam Finn for his advice and thoughtful commentaries on how to address this research. Several discussions over recent developments on earthquake engineering helped me in my studies. His kindness and joyfulness always present in every meeting are one of the memories I will cherish the most from my student life at UBC, it has been an honor to work with him and learn from him. I thank Professor Tony Yang for his support and advice during the review process of this manuscript and the insightful discussions over the final results. I appreciate his comments to improve this work. The support provided by Dr. Perry Adebar during the mathematical modeling process of some structural elements is very much appreciated and his interest shown into the results of my study. I am grateful to the Organization of American States (OAS) who partially funded my MASc. studies at UBC and to Latin American Scholarship of American Universities (LASPAU) who administered my scholarship. Also the support received from NSERC is appreciated. Special thanks to my parents Manuel and Martha who always listened to me and shared a piece of advice on the different challenges I faced. Their love and comfort was present every moment. They taught me the path of success under God’s will, being patient and working hard to succeed while serving others. My siblings Daniel and Martha even though far away were always cheering for me, their love and joyfulness was unconditionally present every day during my studies abroad, the distance was no barrier for them to show me their love and support every moment. My family is the greatest team I have ever been part of. xii I have received the support of many friends, colleagues and Professors in Guatemala. I owe special thanks to Dr. Héctor Monzón Despang for his support and confidence. I had the opportunity to attend civil engineering courses taught by Prof. Efraín Aguilera and Prof. Juan José Barrios, young professors that inspired me and supported me to enter into my graduate studies. A great deal of support was given by Ing. Juan Mini and his family, his advice and guidance have been of great use for my life. I got the opportunity to bond with many people in Vancouver. I owe special thanks to my close friends Yann Dumont and Susana Casado and their lovely daughters Anais and Emma who became a second family for me in Canada. Hugón Juarez and Elsa Pelcastre supported me during my stay in Vancouver, I have learned a lot from their example of caring for others without expecting any reward other than the satisfaction of helping. I thank my friends and research colleagues Sekú Catacolí, José Centeno, Freddy Pina, Juan Carlos Carvajal and Bishnu Pandey for sharing their experience with me, because always provided time for discussions about research and life at UBC. The technical support and advice from Felix Yao at the Earthquake Engineering Research Facility is gratefully appreciated. I also thank to Edwin Guerra and Nicole Jansen for their support and friendship. The care and love in Christ shared by people of the congregation of the Kerrisdale Presbyterian Church was of great comfort and rest for my soul in this journey. Last but not least, I thank God for the immense blessings brought to my life, because each day alive is a gift he gives me and all the successes I achieve are for his glory. xiii Dedication To my family 1 1. Introduction 1.1 Problem statement The seismic input for nonlinear time history analysis (NLTHA) of modern high-rise buildings includes definition of site seismic hazard, selection and scaling of ground motion records representative of the seismic hazard at the site. Although NLTHA provides for our best estimates of building nonlinear dynamic response to ground motion it does not considers the influence the ground motion directionality has on the building seismic response. This influence is deemed to be significant when a high-rise building experience very strong shaking. No buildings will be exempt from this influence unless they are provided with a radial symmetric distribution of stiffness and strength. 1.2 Motivation This study was developed to provide the practicing structural engineering involved in seismic design of high-rise buildings insight on how the directionality of horizontal ground motion components influences calculation of seismic response parameters. In specific when a nonlinear time history analysis is implemented to evaluate the building’s performance and the seismic input corresponds to a strong level of shaking. 1.3 Goal and objectives The goal of this study was to contribute to the state of knowledge regarding the influence of the ground motion directionality on seismic response of a high-rise building. The knowledge gained will serve to motivate future studies on this topic and make practicing engineers aware of the importance of this ground motion parameter in seismic design. The objectives of this study were 1) evaluate high-rise building nonlinear dynamic response under many ground motion scenarios considering ground motion directionality, 2) evaluate the influence of ground motion directionality on several seismic response parameters, 3) assess the relation between ground motion directionality and the critical angle of incidence for several seismic response quantities. 2 1.4 Scope of study • The seismic input in Vancouver city, British Columbia was defined through a probabilistic seismic hazard assessment (PSHA). The same way as provided in the Vancouver Uniform Hazard Spectrum of the National Building Code of Canada 2005. • The ground motions selected and scaled were representative of crustal earthquakes only, subcrustal and subduction earthquake motions were not included. • The case study was developed only for one reinforced concrete high-rise building of downtown Vancouver design according to modern seismic design philosophy. • The influence of ground motion directionality was evaluated using a suite of 3 pairs of horizontal ground motion. In order to ensure enough excursions into nonlinear range additional analyses were conducted at several seismic hazard levels using only one pair of ground motion. 1.5 Organization of this thesis The thesis is organized in seven chapters and three appendices. Forthcoming Chapter 2 presents an overview of previous research and building code provisions regarding the ground motion directionality. Chapter 3 deals with the analysis methodology implemented to conduct this study, includes method for definition of nonlinear building model, nonlinear static analysis, definition of seismic input, nonlinear time history analysis, assessment of post-processed data results. Chapter 4 describes the case study that was used to which the methodology was applied, it includes seismic hazard in Vancouver city, selected and scaled ground motions, building model assumptions and nonlinear time history analysis implementation. Chapter 5 presents the results of the analyses in great detail for one ground motion. Chapter 6 is devoted to the interpretation and discussion of results. Chapter 7 presents the conclusions, recommendations for practicing engineer and future research work. The appendices include complementary analysis results for the dynamic response of the building to the 3 ground motions considered for the study. 3 1.6 Definitions A careful examination of the definitions of terms below is recommended for a thorough understanding of the methodology and results presented in this study. Ground displacement vector: During an earthquake the ground displacement vector is defined as the difference between final and initial position vectors of the ground at instant ti+Δt and ti, respectively. In this study criterion for selection of ti+Δt and ti was the span of time over which the ground motion orbit exhibited almost rectilinear pattern with a significant amount of displacement. In Figure 1.1 a horizontal ground motion orbit is presented along with a displacement vector, with hidden line the initial and final position vectors are shown. Figure 1.1 Horizontal ground motion orbit and displacement vector Directionality of ground motion: It is a characteristic of the ground displacement vector. In this study directionality of ground motion was taken as the direction of the ground displacement vector, and measured counterclockwise with respect to X-axis. In Figure 1.2 is shown the directionality and magnitude of the ground displacement vector presented in Figure 1.1. -30 -15 0 15 30 -30 -15 0 15 30 C om po ne nt 2 o f g ro un d d is pl ac em en t ( cm ) Component 1 of ground displacement (cm) t1 t1+Δt 4 The directionality of a ground displacement vector represents the angle that the ground motion would have to be rotated clockwise in order to be aligned parallel to the structural X-axis. Figure 1.2 Ground displacement vector Ground motion coordinate system: The horizontal ground motions are recorded by strong motion instruments at two orthogonal components typically oriented along North-South and East-West which define a ground motion coordinate system. Horizontal Ground Motion Component Angle of incidence: For dynamic analysis of a building the horizontal ground motion component angle of incidence was regarded as the angular deflection the recorded ground motion coordinate system had with respect to the building structural axes X-positive and Y-positive, which usually are orthogonal. If the recorded ground motion components were applied along X-positive and Y-positive axes the angle of incidence would be zero. The angle of incidence was measured clockwise as shown in Figure 1.3. Figure 1.3 Ground Motion component angle of incidence α = 50º X-axis Y-axis α = 30º Structural X-axis (East direction) α = 30 º St ru ct ur al Y -a xi s( N or th d ire ct io n) 5 From the definitions above it follows that ground motion directionality and ground motion component angle of incidence are related, however the former represents a variable characteristic of the ground motion while the latter is a constant characteristic given by the ground motion component and structural axes. 6 2. Background 2.1 Overview In this chapter a literature survey is presented on the effect of ground motion directionality on seismic response of buildings. This survey draws from up to date building code seismic design provisions, seismic design recommendations and research. The building code provisions are presented in the first part of the chapter (section 2.2). In the second part follows an overview of research conducted on critical angle of incidence (section 2.3-2.4). A chronological development of these studies is presented, starting from early response spectrum analysis with modal combinations rules to multi-component response history analysis. The research overview is limited to some of the fundamental studies published up to date on this topic. For ease of reference each section is designated with the name the document was published. 2.2 Building codes seismic design provisions The issue of ground motion directionality is not dealt in detail within the building codes. Their provisions for seismic design of buildings recommend the use of a specific combination of bi-directional loading that does not incorporate the effect of ground motion directionality. Although most of the building codes recognize the nature of load directionality, none of them provides guidance on how to consider it when nonlinear analysis (static or dynamic) is implemented. An overview of seismic design provisions related to ground motion directionality and bidirectional effects of earthquake excitations given in building codes from several countries follows. The building code provisions of interest are directly quoted here because of their prescriptive nature and no attempt was made to rephrase the code provisions statements. 2.2.1 National Building Code of Canada (2005) According to the seismic design provisions for the Seismic Force Resisting System (SFRS) in section 4.1.8.8 of National Building Code of Canada (NBCC, 2005) direction of earthquake forces shall be assumed to act in any horizontal direction, except that the following shall be considered to provide adequate design force levels in the structure: a) Where components of the SFRS are oriented along a set of orthogonal axes, independent analyses about each of the principal axes of the structure shall be performed, 7 b) Where the components of the SFRS are not oriented along a set of orthogonal axes and IeFaSa(0.2) is less than 0.35, independent analyses about any two orthogonal axes is permitted, or c) Where the components of the SFRS are not oriented along a set of orthogonal axes and IeFaSa(0.2) is equal to or greater than 0.35, analysis of the structure independently in any two orthogonal directions for 100% of the prescribed earthquake loads applied in one direction plus 30% of the prescribed earthquake loads in the perpendicular direction, with the combination requiring the greater element strength being used in the design. The Ie factor accounts for a building occupancy importance, Fa accounts for site amplification due to ground conditions and Sa is the design spectral acceleration at the period of the building in the site. 2.2.2 International Building Code (2006) Section 1613 of the International Building Code (2006) requires every structure to be designed and constructed for seismic demands in accordance with ASCE 7. Section 12.5 of ASCE 7-05 defines the direction of seismic loading as follows: 12.5.1 Direction of Loading Criteria. The directions of application of seismic forces used in the design shall be those which produce the most critical load effects. It is permitted to satisfy this requirement using the procedures of Section 12.5.2 for Seismic Design Category B, Section 12.5.3 for Seismic Design Category C, and Section 12.5.4 for Seismic Design Catergories D, E and F. 12.5.2 Seismic Design Category B. For structures assigned to Seismic Design Category B, the design seismic forces are permitted to be applied independently in each of two orthogonal directions and orthogonal interaction effects are permitted to be neglected. 12.5.3 Seismic Design Category C. Loading applied to structures assigned to Seismic Design Category C shall, as a minimum, conform to the requirements of section 12.5.2 for Seismic Design Category B and the requirements of this section. Structures that have horizontal structural irregularity Type 5 in Table 12.3-1 shall use one of the following procedures: a. Orthogonal Combination Procedures. The structure shall be analyzed using the equivalent lateral force analysis procedure of Section 12.8, the modal response spectrum analysis procedure of Section 12.9, or the linear response history procedure of Section 16.1, as permitted under Section 12.6, with the loading applied independently in any two orthogonal directions and the most critical load effect due to direction of application of seismic forces on the structure is permitted to be assumed to be satisfied if components and their foundations are designed for the following combination of prescribed load: 100 percent of 8 the forces for one direction plus 30 percent of the forces for the perpendicular direction; the combination requiring the maximum component strength shall be used. b. Simultaneous Application of Orthogonal Ground Motion. The structure shall be analyzed using the linear response history procedure of Section 16.1 or the nonlinear response history procedure of Section 16.2, as permitted by Section 12.6, with orthogonal pairs of ground motion acceleration histories applied simultaneously. 12.5.4 Seismic Design Categories D through F. Structures assigned to Seismic Design Category D, E, or F shall, as a minimum, conform to the requirements of Section 12.5.3. In addition, any column or wall that forms part of two or more intersecting seismic force-resisting systems and is subjected to axial load due to seismic forces acting along either principal plan axis equaling or exceeding 20 percent of the axial design strength of the axial design strength of the column or wall shall be designed for the most critical load effect due to application of seismic forces in any direction. Either of the procedures of Section 12.5.3 a or b are permitted to be used to satisfy this requirement. Except as required by Section 12.7.3, 2-D analyses are permitted for structures with flexible diaphragms. 2.2.3 EuroCode 8 (prEN 1998-1:2003) The provisions of the English version of the Eurocode 8: Design of structures for earthquake resistance, regarding bi-directional loading of structures require the following: 3.2.3 Alternative representations of the seismic action 3.2.3.1 Time-history representation 3.2.3.1.1.1 General (1)P The seismic motion may also be represented in terms of ground acceleration time-histories and related quantities (velocity and displacement). (2)P When a spatial model is required, the seismic motion shall consist of three simultaneously acting accelerograms. The same accelerogram may not be used simultaneously along both horizontal directions. Simplifications are possible in accordance with the relevant Parts of EN 1998. 4.2.1.3 Bi-directional resistance and stiffness. (1)P Horizontal seismic motion is a bi-directional phenomenon and thus the building structure shall be able to resist horizontal actions in any direction. (2) To satisfy (1)P, the structural elements should be arranged in an orthogonal in-plan structural pattern, ensuring similar resistance and stiffness characteristics in both main directions. 9 (3) The choice of the stiffness characteristics of the structure, while attempting to minimize the effects of the seismic action (taking into account its specific features at the site) should also limit the development of excessive displacements that might lead to either instabilities due to second order effects or excessive damages. 4.3.3 Methods of Analysis 4.3.3.1 General … (11)P Whenever a spatial model is used, the design seismic action shall be applied along all relevant horizontal directions (with regard to the structural layout of the building) and their orthogonal horizontal directions. For buildings with resisting elements in two perpendicular directions these two directions shall be considered as the relevant directions. 4.3.3.2 Nonlinear methods … (7)P The seismic action shall be applied in both positive and negative directions and the maximum seismic effects as a result of this shall be used. 4.3.3.5 Combination of the effects of the components of the seismic action 4.3.3.5.1 Horizontal Components of the seismic action … (7)P When using nonlinear time-history analysis and employing a spatial model of the structure, simultaneously acting accelerograms shall be taken as acting in both horizontal directions. 2.2.4 Chile Official Normative (1996) The provisions of the Building Code of Chile (1996) for analysis of structures subjected to earthquake loading states: 5.8 Seismic Actions on the Structure. 5.8.1 The structure shall be analyzed, as a minimum, for independent seismic actions along each of the two orthogonal or approximately orthogonal horizontal directions. 10 2.2.5 Mexico Federal District Code (2004) The provision of the Mexico Federal District Code (2004) for static analysis of structures subjected to earthquake loading state: 8.7 Bi-directional Effects. The effects of both horizontal components of ground movement will be combined considering, in each direction of analysis of the structure, the 100 percent of the effects of the component that acts in that direction and the 30 percent of the effects of the orthogonal action, with the signs that provide the most unfavorable condition. And dynamic analysis of structures subjected to earthquake loading considering bidirectional effects in accordance to Section 9.4 shall follow the same provisions given in Section 8.7. 2.2.6 Recommendations for the seismic design of high-rise buildings (2008) The response of the building is obtained for ‘n’ pairs of selected ground motions in applying the nonlinear time history analysis (NLTHA). These ground motions are scaled to the target spectra. This modified suite of ground motions is used to produce a set of 2n dynamic analyses, out of this total ‘n’ analyses correspond to the components of the ground motion applied along the structural axes of the structure x and y (orthogonal axes) and ‘n’ complementary analyses with the components of the ground motion rotated 90° with respect to the structural axes of the building. The procedure described above is recommended for the collapse-level assessment of the building (Willford et. al., 2008). 2.3 Response spectrum analysis This method has been widely used for seismic design of structures. It has been extensively used because of its significant computational advantages over the time history analysis. The method involves the calculation of maximum values of response quantities in different modes. The method has been implemented to evaluate the critical angle of incidence in different studies as follows. 11 2.3.1 Characteristics of 3-dimensional earthquake ground motions Author: Penzien, Joseph ; Watabe, Makoto Date: 1975 Journal: Earthquake Engineering and Structural Dynamics Volume/Issue: 3/4 Pages: 365-373 Publisher: John Wiley and Sons Ltd. A characterization of tridimensional earthquake ground motions was presented by Penzien and Watabe (1975). They predicted an increased need to conduct multi-component seismic response analysis would be required with the future progress of earthquake engineering. The ground motion was defined to possess an orthogonal set of principal axes along which the components are uncorrelated. Applying a transformation of the covariance matrix of the ground motion components the principal variances of ground motion and their corresponding principal directions can be obtained. Examination of recorded accelerograms revealed that the major principal axis points in the direction of the epicenter while the minor principal axis oriented along the vertical direction. 2.3.2 Modal combination rules for multi-component earthquake excitation Author: Smeby, Wiggo ; Der Kiureghian, Armen Date: 1985 Journal: Earthquake Engineering and Structural Dynamics Volume/Issue: 13/1 Pages: 1-12 Publisher: John Wiley and Sons Ltd. A modal combination rule for multi-component seismic input was presented by Smeby et. al. (1985). This application of the response spectrum method is a good approximation for dynamic analysis of linear structures. The method accounts for the correlation between the modal responses, also it considers the correlation between the ground motion components. This method is defined for ground motions that follow the Penzien-Watabe model. The method allows estimates of the root-mean-square, mean and standard deviation of the peak value for a response quantity. If the direction of the ground motion principal axes is unkown two alternatives are (i) to provide a conservative design using the direction most critical for each member in the structure, (ii) considers the direction of the ground motion principal axes as a random variable performing an integration over the range of possible values. 12 For practical purposes, the input response spectra should be specified for the components of ground motions along the principal directions. If these components have similar intensity the rule unfolds into two calculations, first the Complete Quadratic Combination (CQC) rule for each ground motion component modal response is applied and subsequently a Square-Root-of-Sum-of-Squares (SRSS) rule to combine the responses each of the three ground motion components. 2.3.3 A clarification of the orthogonal effects in a three-dimensional seismic analysis Author: Wilson, Edward L. ; Suharwardy,Iqbal ; Habibullah,Ashraf Date: 1995 Journal: Earthquake Spectra Volume/Issue: 11/4 Pages: 659-666 Publisher: Earthquake Engineering Research Institute The work done by the authors shows that percentage rules used for response spectra analyses that apply a 100% of the prescribed forces in one direction of the structure concurrent with 30 or 40 % of the forces in the perpendicular direction produces estimates that are dependent on the user’s selection of the reference system. In their work they assumed that ground motion components are uncorrelated along the principal directions, and follows the Penzien-Watabe model. The analytic formulation presented is a SRSS for spectral forces. The comparison is performed with a SRSS combination of two 100% percent of the spectral demands, which proves to produce design forces not dependent on the user’s selection of the reference system. It is shown that the maximum force is obtained with the SRSS of individual responses to both spectra in direction θ=0° and θ=90° oriented along the structural axes. Hence the result is a structural design which has equal resistance to seismic demands from all directions. They developed an example which depicts how the percentage rules fail to accurately estimate the demands for a non-rectangular structure with one axis of symmetry. The 100% plus 30/40% produces design forces that are not always rational. 13 2.3.4 A replacement for the 30%, 40%, and SRSS rules for multicomponent seismic analysis Author: Menum, Charles ; Der Kireghian, Armen Date: 1998 Journal: Earthquake Spectra Volume/Issue: 14/1 Pages: 153-164 Publisher: Earthquake Engineering Research Institute A method for Multi-component Seismic Analysis using a Complete Quadratic Combination (CQC3), was presented by Menun et. al. (1998). This method is based on the modal combination rules presented by the second author (1985). A minor change in the sign convention measuring the relative rotation of the principal to structural axes was implemented with respect to Smeby’s convention. The authors develop a numerical example to compare the response quantity estimates between the 30%, 40%, SRSS rules and the CQC3 method, using a curved bridge. A single dynamic analysis was performed for the altenative methods and a parametric analysis for the CQC3 method was conducted. It is shown that for other methods the response quantities estimates are sensitive to direction of the structural axes. In most practical situations the 40% rule yields more conservative estimates than the SRSS and CQC3. The SRSS underestimates response quantities compared to the CQC3, because it disregards the correlation among the ground motion components. If the spectral ratio among response spectra of both horizontal components is unity, the CQC3 reduces to a SRSS formulation. The other case where the SRSS yields the same estimate as the CQC3, is when the structural axes are oriented along the principal axes of the earthquake. Therefore the SRSS can be used to replicate the maximum estimates from the CQC3 but requires modifications in the model and more dynamic analyses. The critical orientation for the response quantity can be easily predicted using the CQC3 method. On the other hand, percentage rules and SRSS allow to compute the critical orientation following a trial and error approach. It was concluded that the CQC3 method is straightforward in assessing the elastic dynamic response of a structure. 14 2.3.5 Critical response of structures to multi-component earthquake excitation Author: Lopez, Oscar A.; Chopra, Anil K.; Hernandez, Julio J. Date: 2000 Journal: Earthquake Engineering and Structural Dynamics Volume/Issue: 29/12 Pages: 1759-1778 Publisher: John Wiley and Sons Ltd. In this study the critical response of structures to multi-component seismic motion due to three uncorrelated components defined along principal axis is investigated. An explicit formulation to evaluate the critical response due to the two horizontal components and the vertical component was developed. This was presented as an improved modal combination rule, which accounts for the correlation of the responses through the modal correlation coefficient. The spectrum intensity ratio γ is defined as the ratio between two principal components of ground motion, characterized by design spectra A(Tn) and γA(Tn). The ratio γ is between 0 and 1. When the spectrum intensity ratio is zero i.e, one component of horizontal motion, it is observed that the ratio between the critical response and the SRSS response rcr/rSRSS has the largest range of values, with a lower bound equal to unit and the upper bound equal to√2. For typical values of γ=0.75 the largest value of critical response relative to the SRSS response is 1.13. As an example a parametric study to evaluate the response of one-storey symmetrical-plan building and one-story unsymmetrical-plan building was presented. 2.4 Time history analysis The computational advantages of the response spectrum analysis are offset by ignoring the effect the duration of strong motion has over the response of a structure, since only the peak response is obtained. The time history analysis is more accurate than the Response Spectrum to obtain the seismic response of a structure. It can be performed as an elastic analysis at moderate levels of shaking, but when large seismic demands are present a nonlinear analysis should be employed. 15 2.4.1 Critical orientation of three correlated seismic components Author: Athanatopoulou, A.M. Date: 2005 Journal: Engineering Structures Volume/Issue: 27/2 Pages: 301-312 Publisher: Elsevier Ltd. This study presents an analytical formulation to estimate the critical angle of seismic incidence and the corresponding maximum value of a response parameter for structures excited under three seismic correlated components: two horizontal components applied along arbitrary directions and a vertical component of ground motion. The formulation considers only linear seismic response. It requires to perform two specific time history analysis for the horizontal components and one for the vertical component. The analyses results presented show that the critical value of a response quantity can be up to 80% larger than the predicted response applying the horizontal ground motion components along structural axes. It is concluded that for the same ground motion record exists noticeable variability of the critical angle of incidence for different response quantities. Different earthquake records have different critical angles for the same response quantity. For a given ground motion record the maximum relative variation of a response quantity can be up to 200% for angles of seismic incidence in the range 0-180°. 2.4.2 Variation of response with incident angle under two horizontal correlated seismic components Author: Athanatopoulou, A. M. ; Tsourekas, A.; Papamano, G. Date: 2005 Journal: Earthquake Resistant Structures Volume: 81 Pages: 183-192 Publisher: Transactions of the Wessex Institute A parametric study is presented by the authors to evaluate the variation of seismic response with the angle of incidence under bi-directional horizontal ground motion components. The study included a 16 variety of RC building models with the following characteristics: symmetric, one and two way un- symmetric, regular plan and elevation, irregular plan and elevation for single, five and ten stories. The study implemented the methodology developed by Athanatopoulou (2004) and concluded that the response quantity can be up to 76% higher than the estimated when seismic components are applied along structural axes. They did not find any dependence of the response variation on the geometric characteristics of the building. Besides they confirmed the conclusions gathered in the previous study. 2.4.3 Incremental dynamic analysis Author: Vamvatsikos, Dimitrios; Cornell Allin C. Date: 2001 Journal: Earthquake Engineering and Structural Dynamics Volume/Issue: 31/3 Pages: 491-514 Publisher: John Wiley and Sons Ltd. This method is described by the authors to provide more thoroughly estimates of structural response to seismic excitation. The method is based on maximum estimates of a response parameter to ground motion records scaled to increasing levels of intensity, therefore it is a suitable tool to assess structural performance, ranging from elastic response through inelastic response and global collapse capacity. The Incremental Dynamic Analysis (IDA) curve can be implemented to both single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF), which becomes feasible on the grounds of increased computer processing power. The method can be extended to obtain IDA curves from several records (multi-record IDA), which can later be summarized to evaluate the variability in the structural response that can be expected due to ground motion randomness. An example of IDA curve obtained for a 44 storey building using a single record is shown in Figure 1. The right side of the figure show the peak insterstorey drift ratio over building’s height under increasing levels of shaking characterized by the first mode spectral acceleration from the uniform hazard spectrum in Vancouver city, these are snapshots of the Incremental Dynamic Analysis conducted for the case study of this research. 17 Figure 2.1 IDA curve from a single-record of a T1=4.27s, 44 storey RC building 2.4.4 Influence of angle of incidence on seismic demands for inelastic single-storey structures subjected to bi-directional ground motions Author: Rigato, Antonio B.; Medina, Ricardo A. Date: 2007 Journal: Engineering Structures Volume/Issue: 29/10 Pages: 2593-2601 Publisher: Elsevier Ltd. This work evaluates the influence the angle of incidence of the ground motion has on several engineering demand parameters (EDP) for a single-storey structure, subjected to bi-directional ground motions. 15 35 55 75 95 115 135 155 -0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 H ei gh t a bo ve fo un da tio n le ve l ( m ) - G ro un d le ve l i s 1 5m Interstory Drift 50% in 50 yr 10% in 50 yr 5% in 50 yr 2% in 50 yr 18 A set of 39 ground motion pairs were used to perform the nonlinear time histories analyses. The ground motions were scaled at the fundamental period of the structure to match the design spectrum for a coastal site in California, NEHRP site class D. The study included two plan layouts, a torsionally unbalanced structure (TUB) and a torsionally balanced structure (TB). The evaluation was performed for various degrees of inelasticity, which were attained varying the plastic moment capacity in the columns in accordance to force reduction factors Rd of 1, 2, 4 and 6. The influence of the ground motion directionality was studied over ductility demands, column drifts and slab rotations. It was concluded that maximum values of the EDPs examined have the potential to occur when horizontal ground motion components are applied at a different angle than the structural axes. In general peak inelastic deformations are underestimated when ground motion components are applied along structural axes. The critical angle is dependent on the fundamental period, model and degree of inelasticity, therefore is difficult to determine its value a priori as opposed to an elastic structure. The ratios of peak deformation responses based on all angles of incidence to the peak deformation when the ground motions are applied along the building structural axes on average is in the range 1.1 to 1.6. However for an individual ground motion the ratio of deformation response can be as large as 5.0. The results suggested the angle of incidence of the ground motion has a significant influence over the performance estimates and further studies are necessary to provide a straightforward method to estimate the critical angle and the respective maximum demands. 2.4.5 Incremental dynamic analysis with two components of motion for a 3D steel structure Author: Vamvatsikos, D. Date: 2006 Proceedings: 8th National Conference on Earthquake Engineering, San Francisco. The IDA framework developed for planar structures and uniaxial loading in this paper is implemented with an application extended to 3D structures. The process involves a set of nonlinear time history analysis of a structure performed under a suite of ground motion records, having each component equally scaled to several levels of intensity and collecting the response parameters. The method application is illustrated through the analysis of a 20 story steel frame subjected to biaxial seismic loading. A suite of twenty-six (two components each) ground motions is used. A proper selection of intensity measures and engineering demand parameters is given by spectral acceleration coordinates of the record components and maximum interstory drifts respectively. 19 A novel definition of IDA surfaces is introduced and used to define the limit-states. This IDA application to 3D structures is deemed as a powerful tool for analysis capable of thoroughly assessing the seismic performance of spatial structures. 2.4.6 Multicomponent incremental dynamic analysis considering variable incident angle Author: Lagaros, Nikos D. Date: 2010 Journal: Structure and Infrastructure Engineering Volume/Issue: 6/2 Pages: 77-94 Publisher: Taylor and Francis The author presents a novel approach to Multicomponent Incremental Dynamic Analysis (MIDA) considering variable incident angle ground motions. The proposed methodology is a MIDA implemented using N-incident angle record-pairs generated through the Latin Hypercube Sampling (LHS) technique. It is noted in this study the scarcity of available literature where the critical incident angle is examined when the time history is employed. Here the angle of incidence of the ground motion is defined as the direction where the principal axes of the ground motion are oriented with respect to the structural axes and is measured counterclockwise. The first part of the study, evaluates the influence of the incident angle in the framework of MIDA through a parametric study. The ground motions employed were recorded from Loma Prieta, the Imperial Valley and Northridge earthquakes. Two RC structures are evaluated, having a symmetrical and non- symmetrical plan layout. The ground motions are applied in a range of directions from 0° to 360°, with angle step of 5° and scaled to Sa(T1) of 0.05g, 0.30g and 0.50g. It is observed that seismic response varies significantly with respect to the incident angle and the COV increases with the intensity level. Another remark is that critical seismic response is encountered for different incident angles when a different ground motion record is considered. It is concluded from this study case that it is not possible to predict the critical incident angle where the response measured by interstory drift is maximum for a given intensity level. The second part of the study, includes six implementations of MIDA which are compared based on limit state fragility curves developed. These implementations are with bi-directional horizontal ground 20 motions are applied as follows: (i) along the structural axes and their complementary direction, (ii) along the principal axes and their complementary direction, (iii) along a random direction and its complementary direction, where the angle of incidence is fixed for all the ground motions. An additional set of ground motions generated with the LHS is included in the parametric study as follows: (iv) applying the methodology over a sample of 15 pairs –LHS 15–, (v) applying the methodology over a sample of 30 pairs –LHS 30–, (vi) applying the methodology over a sample of 100 pairs –LHS 100–. A suite of 15 bi-directional horizontal records is considered enough to predict the seismic demand with acceptable accuracy, therefore 15 ground motions are employed in this study. For the proposed methodology is proven that when incident angle is taken into account with LHS technique, the results do not vary significantly on the number of pairs sampled, having LHS-30 implementation almost identical to LHS-100. It is concluded that the incident angle must be considered in the MIDA, this is based on the grounds that the response varies with respect to the record, intensity level and incident angle. 2.5 Summary The current building code provisions recognize the nature of ground motion directionality in determining the seismic demands. However they do not provide guidance on how to properly account for ground motion directionality when nonlinear time history analysis is conducted. The recommendations for seismic design of the Council of Tall Buildings requires the ground motions for seismic input at 0° and then rotate them 90° to partially cope with the effect of ground motion directionality when nonlinear time history analysis is used for high-rise building. Most of the studies developed on the critical angle of incidence for response spectrum analysis and time history analysis are constrained to elastic dynamic analysis. Meanwhile the limited studies that have been conducted on the effect of ground motion directionality on nonlinear dynamic response are for low-rise buildings. The study conducted by Rigato (2007) showed that for an individual ground motion the ratio of peak deformation response of the model over all angles of incidence to the peak deformation when the ground motion was applied along the structural axes could be as large as 5.0. 21 3. Analysis methodology 3.1 Overview The methodology that follows was developed to assess the effect of ground motion directionality on the nonlinear seismic response of a reinforced concrete (RC) high-rise building. The methodology is comprised of several methods to define (i) mathematical model of the building for nonlinear analysis, (ii) multi-directional nonlinear static analysis, (iii) seismic input for dynamic analysis considering ground motion directionality, (iv) multi-directional Incremental Dynamic Analysis (MIDA), (v) assessment of ground motion directionality on building response and critical angle of incidence of ground motion. The structure of this methodology is illustrated in Figure 3.1. Figure 3.1 Structure of methodology. (i) Mathematical Model (ii) Multi-directional nonlinear static analysis (iii) Seismic Input (iv) Multi-directional Incremental Dynamic Analysis (v) Assessment of ground motion directionality effect on building response and critical angle of incidence Repeat step (ii) to (v) for each ground motion 22 In order to consider bi-directional loading a full 3D building model was created. This choice would provide for a more accurate analysis if the building structural configuration presents eccentricities or irregularities. The model included the following elements: shear walls, columns and flat plates. For simplicity of the evaluation and ease of modeling the model did not accounted for soil-structure interaction. A multi-directional nonlinear static analysis (multi-directional pushover) was conducted to investigate building’s model stiffness and strength variability along structural axes with different orientations. The response curve related base shear with roof displacement along the loading direction. A family of response curves was obtained for several angles of loading, which provided insight on how the expected nonlinear response could be related for different angles of incidence of dynamic loading. The seismic input was defined for dynamic analysis of the high-rise building. The steps included definition of site seismic hazard through a Probabilistic Seismic Hazard Analysis (PSHA), selection and scaling of recorded ground motion compatible with site seismic design hazard level (2% in 50 years) followed by creation of ground motion scenarios which accounted for different ground motion directionality. A Multi-directional Incremental Dynamic Analysis (MIDA) was implemented to consider the effect of ground motion directionality over dynamic response for several ground shaking levels. This analysis included the nonlinear time history analysis under seismic input and post-processing of results. The nonlinear response history analyses were performed under bi-directional horizontal ground motions and considered different ground motion angles of incidence. The response of the building was evaluated considering overall response quantities such as interstorey drift, storey shear, overturning moment and floor displacement. A sensitivity analysis was performed for different response quantities with respect to the angle of incidence. The assessment of the effect ground motion directionality on building seismic response and critical angle of incidence aimed to quantitatively evaluate the relationship between the ground motion directionality, the multi-directional incremental dynamic analyses and the multi-directional nonlinear static response of the building. The case study in Chapter 4 elaborates on the application of this methodology to a RC high-rise building located in downtown Vancouver, British Columbia, Canada. The RC building is 44 storey which resembles the typical structural configuration of high-rise buildings in Vancouver, where Seismic Force 23 Resisting Sytem (SFRS) is provided by a central RC core shear wall and the gravity load carrying system is provided by columns and flat plate slabs. The results of building analysis are presented in Chapter 5. 3.2 Mathematical model The simulation of the dynamic response of the building was based on a fully tridimensional nonlinear model using the CANNY program (Li, 2010). The model included elements to represent the behavior of shear walls, coupling beams, columns and flat plate slabs. The tridimensional model enabled to capture the torsional response that could occur in the building under ground shaking. In the following sections a description of the modeling assumptions for the different components of the high-rise building. 3.2.1 Columns and shear walls The CANNY program includes a force-based fiber model to account for interaction of bending and axial force on reinforced concrete elements. This model assumes distributed inelasticity. The fiber model was used to model the flexural behavior of columns and shear walls elements to represent multi-axial interacting loads. The respective material stress-strain relationships of concrete and reinforcing steel bars used for the case study are described in Chapter 4. The fiber model used to model a column element in CANNY has two fiber slices at the end of the element. The extreme fiber slices in are used to define the moment-curvature and axial stress-strain relations in that section. The assumption of flexibility distribution along the element is multi-linear if the inflexion-point lies within the element however if the inflexion-point lies out of the element deformable span a linear distribution is used. The multi-spring (MS) model that is used as a basis for the fiber model, this MS model requires the definition of the element section dimensions and steel reinforcement arrangement. For the shear walls the inelastic response was considered only for flexure and axial response, the shear response was modeled as elastic only, based on the gross section properties. Flexural response is captured only in the plane, shear response was be modeled in and out of plane. For further details about the fiber model for columns and shear walls in CANNY refer to the user’s manual of the program (Li, 2010) 24 3.2.2 Coupling beams The one-component element model defined by Giberson (1967) was used to idealize coupling beams between cantilevered walls. This model is comprised of a perfectly elastic element and one rigid inelastic rotational spring at each end of the elastic element. All the inelastic deformations of the beam are lumped in the two rotational springs. The one-component has its rotational springs connected in series with the elastic beam. This one-component model as provided for flexural response in CANNY is shown in Figure 3.2. In CANNY program the uniaxial hysteresis model is used to characterize the moment- rotation relation at the end of the beams using the beam stiffness and capacity (Li, 2010). Figure 3.2 Uniaxial spring model for flexural response. Moment capacity, shear capacity, section secant stiffness and effective inertia were defined with the following models (Paulay and Priestley, 1992). The moment capacity was estimated neglecting the contribution from the slab reinforcement, as follows: M ൌ Aୱୢcosα f୷ zୠ (Eqn 3.1) The shear capacity of the beam was defined as: V ൌ M౨ାMౢ ୪ (Eqn 3.2) where ln , Ast, fy, zb and α are shown in Figure 3.3. Figure 3.3 Coupling beam forces. MbMaA B θbθa Td Td Cd + Cc Mr Asd AsdMl Th zb Cd 2α ln 25 The secant modulus of elasticity of concrete was defined as: E ൌ 4700ඥfୡ′ [MPa] (Eqn 3.3) The effective stiffness for diagonally reinforced coupling beams was defined as: Iୣ ൌ .ସI ଵାଷቀ୦ ୪ൗ ቁ మ (Eqn 3.4) The effective stiffness for conventional reinforced coupling beams was defined as: Iୣ ൌ .ଶI ଵାଷቀ୦ ୪ൗ ቁ మ (Eqn 3.5) Where f’c is the concrete compressive strength, h is the beam depth, ln is the clear span length and Ig the gross moment of inertia. For flexural response the degrading bilinear model BL2 from CANNY was adopted to consider the degrading response of coupling beams. This model is shown in Figure 3.4. In this BL2 model K is defined as the product of Ie and E, as given by equations above. The fy represents the moment capacity of the beam as given by equations above. The post-yield stiffness was set with a ratio β = 0.0015. The Ku is the unloading degraded stiffness and controlled by parameter γ = 0.05. Figure 3.4 Degrading bilinear model BL2 in CANNY program. The shear response was modeled to be elastic, hence one shear spring was placed for each beam based on coupling beam cross section properties. Ku 1 K 1 Rotation Ku=f(K,γ)β=0.01 γ=0.05 Moment 1βK 26 3.2.3 Flat slabs For analysis, the floor slabs are normally considered to be fully rigid within their own planes. Then no relative movement occurs between the vertical elements supports at each storey level. The program CANNY has a feature called supernode, which allows to couple the degrees of freedom of many nodes using a one, two and three dimensional element. Those elements coupled with the supernode will follow the same response history in acceleration, velocity and displacement. The supernode used as a two dimensional element is ideal to model the rigid diaphragm that is commonly used in multi-story buildings (Li, 2010). The flat slab system in the study building consists of solid post-tensioned concrete slabs, supported directly on columns, where the framing is provided by the column and the slab band working as a shallow beam. In this manner the storey height is optimized and provides economic advantage. Slabs without drop panels are commonly called flat plates. The frame provided by the slab and column usually has low lateral stiffness, usually shear walls are provided to increase the lateral stiffness. Because of the monolithic condition of concrete buildings, lateral forces are resisted by shallow beam-column frames and shear walls. The behavior of the lateral force resisting system is similar to a core frame system as depicted in Figure 3.5. The lateral forces acting on the upper portion of the structure are primarily resisted by frame action and those in the lower portion by the shear wall (Schueller, 1977). The frame action of a flat plate structure can be relied on provided the column-slab joint is properly designed to transfer the shear forces. Figure 3.5 Interaction frame-shear wall under lateral forces. a b c =+ 27 3.2.4 P-delta effects The P-delta effect is caused by gravity loads acting on the deformed shape of the structure. This effect should be considered in the nonlinear time history analysis. The effect is moderately pronounced if the effective stiffness is positive at large deformations, however if the effective stiffness is negative large displacements could take place during dynamic response and potential for collapse exists. The influence of P-Delta on the collapse capacity of a moment resisting frame system is larger than on a shear wall structure given a plastic flexural mechanism is developed on the wall instead of a shear failure (ATC 72, 2010). The P-Delta effect was included in the model for nonlinear time history analysis as it is included in the program CANNY. The effect is included as an additional lateral force on the member at a given storey. 3.2.5 Gravity load effects The building model included definition of the weight due to structural and non-structural elements and occupancy loads. The gravity load estimates correspond to the expected loads in the scenario of an earthquake, which is different from the factored loads assumed for design check. The dead load includes self-weight of the structure, architectural finishes (floor finishes, ceiling finishes, partition walls, façade cladding), storage tanks, mechanical distribution systems and so on. To evaluate the overall response of high-rise buildings, the live load should be reduced from the nominal (full) design value to account for (a) low probability of nominal design load acting simultaneously throughout the building, (b) the low probability of the nominal live load acting simultaneously with the earthquake. A reduction factor factor of 0.4 and 0.5 is applied to account for these conditions, leading to a total reduction factor of 0.2 (ATC 72, 2010), where traditionally a factor of 0.25 has been used. Finally the gravity load combination used for nonlinear response of tall buildings of residential occupancy is: 1.0 ܦܮ 0.2 ܦܮ (Eqn 3.6) The gravity loading in the program CANNY was managed through definition of load cases using the following elements: Element self-weight: this instructed the program to calculate the dead-load due to self-weight of structural elements included in the model and integrate it into the static load. Initial load force: This allowed the input of initial forces on vertical structural elements. This input accounted for dead load of structural and non-structural elements not included in the model. In addition reduced nominal live load was included. 28 3.2.6 Mass source for dynamic response The mass used for dynamic analysis was obtained from load combination given in equation 3.6. This load combination provided for weight values assigned to the nodes in the model. The distribution of loads to each node was based on tributary areas, where nodes are placed at the columns locations. The program CANNY considers different sources of mass for dynamic analysis, the options used for the model were: Element self-weight: This instructed the program to calculate the self-weight of structural elements included in the model. Collect weight: This instruction enables the program to use the calculated structural self-weight and incorporate it into the mass matrix for dynamic analysis. Node weight: This input accounts for mass of structural and non-structural elements not included in the model. In addition a reduced live load was included. The program integrates this input into the mass matrix for dynamic analysis. 3.2.7 Damping and energy dissipation If the dynamic analysis is intended to calculate the nonlinear response of the structure the damping matrix must be defined completely. The classical damping matrix is an appropriate idealization when damping sources are similarly distributed throughout the building height. A way to construct the classical damping matrix is through Rayleigh damping as follows (Chopra, 2007). ൌ ܽ ܽଵ (Eqn 3.7) The damping ratio for the nth mode of this system is: ζ ൌ బଶ ଵ ω భ ଶ ω (Eqn 3.8) If the modal damping ζ is assumed to be equal for mode ith and mode jth coefficients a0 and a1 are determined as follows: ܽ ൌ ζ ଶ ωωೕωାωೕ (Eqn 3.9) ܽଵ ൌ ζ ଶ ωାωೕ (Eqn 3.10) 29 For the building model using CANNY program the classical damping matrix was defined proportional to the mass and the tangent stiffness matrix, while modal damping ratios were defined using Rayleigh damping as described above. Modal damping ratios were set for the first and second translational modes of vibration of the building to provide reasonable damping ratios at higher modes. Typical estimates of modal damping ratios are available in extensive studies for tall buildings and are in the range of 2% to 3%, which is significantly lower compared to the conventional 5% commonly used in building codes (Wilford, 2008). The additional energy dissipation from hysteretic damping was captured through the fiber model of columns, core shear walls and coupling beams after yielding and cracking of steel and concrete happens. 3.3 Multi-directional nonlinear static analysis The pushover analysis was conducted to gain insight on the nonlinear response of the building model under different loading directions. The pushover load pattern was defined as an inverted triangle load pattern (first mode loading). The pushover analysis was performed through a parametric study to account for load directionality. The loading was applied along the X-axis and Y-axis shown in Figure 3.6. The ratio of loading along X-axis and Y-axis was in the range 0 to 1 to create scenarios of loading in the range 0º to 90º. The pushover was conducted to reach a minimum roof drift of 2% which will ensure significant inelastic response was attained. The response of the building was obtained along the direction of loading. The variability in stiffness and strength of the building across a variety of load intensity and directions was captured through this analysis. 3.4 Seismic input This section describes the methodology adopted to define the seismic ground motion input for time- history analyses. The methodology described here can be used to obtain the seismic input in any location of British Columbia, Canada. The definition of seismic input for the seismic analysis of the model comprises: site seismic hazard, selection, scaling and rotation of ground motions. The site response and soil-structure interaction were not included in the definition of seismic input. 3.4.1 Seismic hazard analysis The basic elements used in this study to define the seismic input were hazard curves and Uniform Hazard Spectrum (UHS) associated with specific hazard level. 30 The calculation of ground motion hazard was obtained with the state-of-the art probabilistic seismic hazard analysis (PSHA) at a ground reference condition. The PSHA results were used as a reference to select and scale records from different earthquakes to be compatible with the UHS. The seismic hazard analysis was done following closely the methodology adopted by the Geological Survey of Canada (GSC) in the Open File 4459 (Adams and Halchuk, 2003). The software EZ-FRISK v 7.26 (Risk Engineering Inc.) was used for the hazard calculation, applying a probabilistic seismic hazard analysis method. In this study the spectral accelerations were obtained for a soil site class C soil (firm soil or soft rock). According to the National Building Code of Canada 2005 the ground motion amplification factors for firm soil are: Fa=Fv=1.0 (Finn et. al 2003). In the NBCC 2005 the site class C soil is characterized by having a time averaged shear wave velocity between 360 – 750 m/s in the uppermost 30 meters. The hazard calculated for the analysis included ground motions at several return periods, data presented corresponds to return periods of 72 years, 475 years, 975 years, 2475 years and 4975 years at mean confidence levels. The results of this seismic hazard assessment are presented in chapter 4. In the Pacific Northwest, the ground motions are attributed to five types of earthquakes: (1) crustal earthquakes occurring in the continental crust, (2) subcrustal earthquakes occurring within the subducting Juan de Fuca plate, (3) megathrust earthquakes occur at the interface of Juan de Fuca plate and North America plate, where Juan de Fuca plate is subducting beneath North America plate, (4) crustal earthquakes occurring in the oceanic crust and (5) earthquakes occurring in the transition zone below the locked zone (Atkinson, 2005). In the seismic hazard analysis of this study in British Columbia, the ground motion was attributed mainly to three types of events: (i) crustal continental earthquakes (shallow), (ii) subcrustal earthquakes occurring within the Juan de Fuca slab subducting beneath North America plate and (iii) subduction earthquakes at the interface of Juan de Fuca plate and North America plate descending beneath the North American plate. 3.4.2 Site location The specific location was needed to be provided as part of the input information for the PSHA. In the framework of EZFRISK program it can be a single location or a gridded zone supplied as latitude and longitude. Chapter 4 defines the location utilized for the case study. 31 3.4.3 Treatment of uncertainty The probabilistic hazard calculations inherently present two categories of uncertainty, aleatory and epistemic. Aleatory uncertainty arises from physical variability expected in events, this is the inherent randomness in natural hazard events. It is characterized by probability distributions. The aleatory variability cannot be reduced to zero by collection of additional data. The PSHA deal with aleatory uncertainty when the earthquake source characterization is performed, including spatial uncertainty, size uncertainty, time distribution uncertainty. The epistemic uncertainty, is due to lack knowledge about the phenomenon. It can be reduced by collection of additional – new – data regarding the event. The results presented here did not account for the epistemic uncertainty as is usually considered through a logic tree approach. This produced seismic hazard estimates that do not account for different expert opinions making the estimates moderately reliable, however the comparison of hazard estimates obtained here showed to be within a 10% of the hazard estimates in the Open File 4459 with for a probability of exceedance of 2% in 50 years. 3.4.4 Magnitude recurrence relations The earthquake catalogue used in this hazard analysis is the same as the one used for the 4th generation seismic hazard maps for Canada (Adams and Halchuk, 2003). This earthquake catalogue includes current information up to the year 1991 in western Canada. The earthquake catalogue is used to define the magnitude-recurrence parameters. The magnitude-recurrence parameters are provided in the appendix of the GSC Open File 4459 (OF4459), they are also available in the built-in database of the EZFRISK program for the Canada seismic sources. 3.4.5 Upper and lower bound for magnitude The lower bound for magnitudes in the hazard calculations was M4.75, to be consistent with the 4th generation GSC model (Adams and Halchuk, 2003). This lower threshold is generally accepted as the lowest earthquake magnitude that produce ground motions with duration and intensity capable of causing damage to engineered structures. The maximum magnitude for each source is defined by the GSC. All these parameters are provided also in the built-in database of EZ-FRISK for Canada seismic sources. 3.4.6 Seismic sources in southwestern BC In Canada, the definition of seismic sources is not unique, for this reason the GSC provides two probabilistic seismic source models for the west of Canada, the Historical model (H model) and Regional model (R model). These source models are provided with the respective coordinates that define the boundary of each zone and respective magnitude-recurrence parameters. The source-to-site distance is 32 internally calculated by the program and performed for all the sources. Using a distance filter, sources within a radius of 500km of the site are included. The seismic sources are defined as areal sources, while the Queen Charlotte and Cascadia Subduction are treated as fault sources. Refer to Figures in page 111 of Appendix C3 of the OF4459 (Adams and Halchuk, 2003) for a map with geographic location of each seismic source zone for the two probabilistic seismic source models defined by the GSC for western Canada, Historical (H model) and Regional (R model). The hatched area represents subcrustal sources and without hatch are the crustal. It is important to note that subcrustal sources are located beneath some of the crustal sources. 3.4.7 Ground motion prediction equations The ground motion prediction equations are also known as attenuation relationships or attenuation equations. Estimates of the ground motion are obtained by attenuation equations that predict the ground motion hazard parameter based on characteristics of the earthquakes such as magnitude and source-to- site distance. The current attenuation equations used to predict the ground motion parameter in BC are as follows: for crustal earthquakes the equation proposed by Boore et al. (1997) and adjusted by the GSC was used – BJFWall – and for the subcrustal earthquakes the equations proposed by Youngs et al. (1997) was used. The range of parameters where these equations are valid is presented in table 3.1. Table 3.1 Ground motion prediction equations used in analysis. Equation Moment Magnitude range Epicentral Distance Range (km) Upper bound for spectral periods (sec) Distance Youngs et al. (1997) 5.0 - 8.2 8.5 – 550 3 Rrup Boore et al. (1997) 5.2 - 7.4 0 – 118 2 Rjb The Youngs et al. (1997) equation predicts the horizontal component of ground motion parameter as the geometric mean of two perpendicular horizontal components. The Boore et al. (1993) relationship was provided with a set of equations that predict either the larger of the two horizontal components or a randomly-oriented horizontal component of ground motion, the latter was derived from the geometric mean of two horizontal components. For this study it was deemed that the ground motion parameter 33 predicted by these two equations should be considered as a geometric mean. This latter choice was the basis to scale the records selected to the target ground shaking level. 3.4.8 Selection of ground motion records This study used a suite of ground motions recorded from different earthquakes to consider the ground motion randomness. The ground motions were selected from a online strong motion database. The motions selected correspond to earthquakes having a magnitude and distance similar to the mean values presented in the M-R deaggregation at 2% in 50 years for the fundamental period of the building model. In addition the site condition and spectral shape were part of the selection criteria. The suite of ground motions selected was meant to be representative only of the crustal earthquake type in the Pacific Northwest, specifically in Vancouver, British Columbia. 3.4.9 Scaling of horizontal bi-directional ground motion records There are two techniques customarily employed to scale ground motions records to a target spectrum, the amplitude scaling and the spectral matching. The former modifies the amplitude of the ground motion to obtain an elastic response spectrum with spectral ordinates that match the target spectrum at a specific range. The latter modifies the frequency content of the ground motion to obtain an elastic response spectrum that closely matches the target spectrum. The spectral matching technique was deemed to produce ground motions that will provide similar estimates of building response to successive rotated ground motions. Since one of the objectives of this study is to demonstrate the variability in building response when the ground motion angle of incidence is considered to its greatest extent, the spectral matching is not utilized to obtain motions compatible with the target spectrum. Instead of performing spectral matching, the technique of amplitude scaling was adopted. The target spectrum was the Uniform Hazard Spectrum obtained from the PSHA. The ground motion prediction equations used for the seismic hazard analysis in British Columbia provide the ground motion intensity as a geometrical mean (geomean) value of two horizontal ground motion components. The geometric mean spectrum was scaled to the target spectrum. The target period was the fundamental period of the building model. However the compatibility of the geometric mean spectrum and the target spectrum was considered as well within a period band that ranged from the period corresponding to the 3rd translational mode up to 1.5 times the fundamental period of the building model. In this period range the geometric mean spectrum was not allowed to have amplitude below 70% of the target spectrum. This was done to ensure enough participation of the higher 34 modes of the building model in the seismic response. Many ground motion candidates were considered until the criteria described above was satisfied. Each ordinate of the geometric mean spectrum is defined as the square root of the product of the spectral ordinates of the two ground motion components (X and Y), as expressed in equation 3.11. ܵܽ ݃݉ ሾܶሿ ൌ ඥܵܽݔሾܶሿ כ ܵܽݕሾܶሿ (Eqn 3.11) where Sa gm [T] represents the geometric mean ordinate at period T, Sax [T] represents the spectral ordinate from ground motion component X at period T and Say [T] the corresponding spectral ordinate to ground motion component Y at period T. As stated by Baker (2006), if a three-dimensional model of a structure is to be analyzed, the most straightforward method is to use Sa gm as the intensity measure for both the ground motion hazard and the structural response. Since the geometric mean response spectrum was scaled to match Uniform Hazard Spectrum the ground motion was consistently scaled with the same factor for component X and Y. An advantage of this procedure was that the relative ratio between both horizontal components of motion was preserved. The scaling factor, α, used to modify the ground motions was computed using equation below. ןൌ ௌ ሾ்ଵሿ ሺுௌሻ ௌ ሾ்ଵሿ (Eqn 3.12) The amplitude scaled spectral ordinates ܵܽ തതതത݃݉ሾܶሿ were computed using equation below. ܵܽ തതതത݃݉ሾܶሿ ൌ ߙ כ ܵܽ ݃݉ሾܶሿ (Eqn 3.13) 3.4.10 Ground motion component angle of incidence The ground motion components as recorded were scaled to the target spectrum using the method described above. The ground motions for angles of incidence other than zero were obtained through a lineal combination of the two horizontal recorded components of ground motion. This produced ground motion components that would have different angles of incidence, at the end it is the same motion but is referenced to a different coordinate system as shown in Figure 3.6. The angle of incidence was measured clockwise from the X-positive axis. The step angle was set equal to 10° to provide a reasonable resolution to capture the variability in the response of the building model to different angles of incidence. The angle was varied from 0° to 360°, however 360° was taken the same as 0° and not included in the analysis. Additional motions at 45°, 135°, 35 225° and 315° were added to capture the response of the building to a symmetric diagonal angle of incidence. The variation of the angle of incidence produced 40 pairs of horizontal motion. The equations used for the lineal combination are. a(t)x’ = a(t)x cos(α) + a(t)y sin(φ) (Eqn. 3.14) a(t)y’= - a(t)x sin(α) + a(t)y cos(φ) (Eqn. 3.15) Figure 3.6 Transformation of ground motion to different coordinate system. 3.5 Multi-directional incremental dynamic analysis (MIDA) The seismic input provided several sets of 40 pairs of horizontal ground motions for dynamic analyses. Each set of motions was obtained from scaling and rotating recorded ground motions considered as representative of the seismic design hazard level (2% in 50 years) at the site. The MIDA was obtained through nonlinear time history analysis of the model subject to the bi- directional ground motion described above. Ground Motion Component 1 Structural X-axisφ Ground Motion Component 2 & Structural Y-axis 36 3.6 Assessment of ground motion directionality effect on calculated building nonlinear dynamic response and critical angle of incidence The results were evaluated to assess the influence that ground motion directionality had on the response of the building. For each set of ground motions the evaluation was conducted by comparing envelopes of largest response obtained with the 40 pairs of motion having different angle of incidence. The comparisons were developed for several response quantities, including overturning moments, storey shears, floor displacements and interstorey drifts. To gain insight on the extent of ground motion directionality influence the comparison considered the overall response of the building and response of some structural elements as well. 37 4. Case study 4.1 Description of case study In this case study the methodology described in chapter 3 was followed to define the computer building model, perform the multi-directional nonlinear static analysis, define the seismic input and perform the multi-directional Incremental Dynamic Analysis. The high-rise building model considered for this case study is roughly based on the construction blueprints of an existing building in downtown Vancouver, in British Columbia in Canada. This model does not represent the actual building exactly, but resembles the general structural features commonly provided in Vancouver RC high-rise buildings. Modeling seismic response of high-rise buildings is a challenging task. In this case study the soil- structure-interaction was not included to streamline the modeling process. The soil conditions at the site were considered to correspond to site class C (NBCC, 2005). Although the modeling techniques employed pose limitations to predict response under large demands at the onset of structural degradation, the results provided here give insight into different scenarios of the expected response for typical RC high-rise buildings in Vancouver city when subjected to strong ground motion. 4.2 Building description The building used as a reference of this case study was a reinforced concrete (RC) 44 storey. It was designed in accordance to the National Building Code of Canada 1995. The design of reinforced concrete elements and components was carried out using the CSA A24 (1994). The building tower is for residential occupancy. It is part of a complex that comprises a residential tower and a hotel tower. The residential tower plan layout 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 plan layout distribution in the lower and upper levels is shown in Figure 4.1 and Figure 4.2. The residential tower section along East-West and South-North is presented in Figure 4.3 and Figure 4.4 respectively. The residential tower extends 130m above the podium level. The podium structure includes two storey above ground level and 5 underground parking storey. The total height of the podium is 24m from the slab on grade at underground parking level 5 to the floor slab of the level 3. The underground structure height is 15m. 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 38 shear walls, and the gravity load resisting system is provided by reinforced concrete columns and shear walls. The slab is part of the SFRS and the gravity load carrying system. Figure 4.1 Plan layout of the residential tower and podium. Figure 4.2 Plan layout of the residential tower upper levels. 39 Figure 4.3 Building section A-A (East West). 40 Figure 4.4 Building section B-B (North South). 41 4.3 Structural system The building seismic resistant design was provided in accordance to the National Building Code of Canada 1995. The seismic zone of the site was categorized as 4. The force reduction factor was R=3.5, and the importance factor was 1.0. The main seismic force resisting system (SFRS) of the building is a non-symmetrical reinforced concrete core with cantilever and coupled shear walls around the utilities shaft. The core plan layout has external dimensions of 8.38m along east-west and 8.69m along north-south. The thickness of the walls in the core is 660 mm (26 in) for east and west wall, the thickness of the middle wall is 300mm (12 in). The height- to width ratio of the seismic force resisting system is 16.5 along east-west direction and 16.0 along north- south. Secondary system to resist lateral forces is provided by the slab-columns frames. The columns are reinforced concrete with different shapes circular, square and rectangular. The slab is a post-tensioned concrete flat plate. The gravity load resisting system in the tower comprises slabs, columns and the core walls. Some structural features in the upper storey are shown in Figure 4.5. Additionally along the perimeter of the podium in the underground parking levels, thick continuous walls are provided to 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 transfer beams indicated at floor 3 in the construction blueprints were not included in the model and all the vertical load carrying elements in the tower were modeled as continuous down to the foundation. No offsets in the gravity columns were included, although the building blueprints presented some offsets in vertical elements. Figure 4.5 3D view of structural elements in upper levels of tower. Columns Post-tensioned concrete slab 180mm thick (flat plate) Shear Walls 42 4.4 Computer building model Since the building has a non-symmetrical plan distribution of stiffness and mass a 3D model was deemed the best option to properly account for the inherent torsional response due to eccentricities. The CANNY included all the structural elements but foundation mat raft and isolated footings. The model included 3746 elements and 1478 nodes. 4.4.1 Columns and shear walls As described in Chapter 3 columns and shear walls were modeled with the fiber model. This required introducing the element section dimensions and reinforcement arrangement into CANNY program. The section dimensions and reinforcement distribution was introduced in CANNY as presented in building construction blueprints. An example of shear wall and column sections introduced in CANNY program is shown in Figure 4.6. The material properties and stress-strain relationships of the concrete and reinforcement steel are presented in section 4.7 and 4.8 respectively. The column sections in CANNY were modeled using the rectangular and circular column section option. The shear walls sections were modeled using rectangular column and wall sections. The shear walls sections at the boundary elements were modeled as column sections because are zones with high density of reinforcement, the low reinforced sections were modeled as wall sections as shown in Figure 4.6. The transverse reinforcement was introduced in the section properties to allow the program CANNY model the confinement effect on concrete due to the combined action of longitudinal and transverse reinforcement. Confinement is very important in the seismic response of columns (Paulay et. al., 1992). Figure 4.6 Typical shear wall and columns sections of the building modeled in CANNY. Wall Panel Column in Boundary zone Column in Boundary zone Longitudinal reinforcement Transverse reinforcement Longitudinal reinforcement Transverse reinforcement 43 4.4.2 Coupling beams The RC coupling beams in the construction blueprints of the building were diagonally reinforced, and framed into shear walls along east-west. The reinforcement detail given to these coupling beams was considered to provide high levels of ductility during seismic response. The coupling beams dimensions were typically 660mm wide by 510mm height. The moment capacity, effective section inertia and modulus of elasticity was calculated evaluating equations 3.1 through 3.5 as presented in Chapter 3. The one-component element as described in section 3.2.2 was used to model flexural and shear response of the coupling beams. 4.4.3 Slabs In the building construction blueprints the slabs were post-tensioned and were 178mm (7 in) thick flat plates. At the ground level a transfer slab was constructed, with slab-bands. The slabs bands at ground level were 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. The entire slabs were not modeled directly in CANNY. Instead the supernode element was formed at each floor level to account for the rigid diaphragm action of the flat slab. In order to include the frame action, slab bands acting as equivalent beams were included using the slab CANNY element. These wide shallow elastic beams were 178mm (7 in) thick and 1000mm (40 in) wide, typical reinforcement as specified in blueprints was provided. The prestress in the concrete was not included in the model, it is expected that lower cracking would occur in the slab as a consequence of the prestress. The columns in the residential tower had an elongated cross section, therefore punching shear stresses are unlikely to impose severe demands on the column-slab joint. Under these circumstances no joint shear degradation was expected and no further consideration was given to the capacity of the joint in evaluating the building response (PEER, 2010). 4.4.4 Frame reference system to define the model The definition of columns sections, shear wall sections and coupling beams was used to build the 3D model in CANNY. To this purpose a frame reference system was defined in CANNY. This reference system was defined to allow the exact location of structural elements within the building model and was based on building construction blueprints. The X, Y and Z location was required for definition of the frame reference system. 44 The CANNY program created nodes at specific locations within this frame reference system. The column sections, shear wall sections and coupling beam elements were used to model columns, shear walls and coupling beams along line elements connecting two nodes. This is illustrated in Figure 4.7 for the building model in CANNY. Figure 4.7 View of frame reference system for building model in CANNY. 4.4.5 Degrees of freedom The dynamic analysis of the model required the definition of degrees of freedom (DOF) of each node. The horizontal translational DOF of the nodes (X and Y) were prescribed to zero from the foundation level up to the upper Lower Lobby (LL), this level of fixity is indicated in Figure 4.3 and 4.4. The rotational DOF around Z-axis was prescribed to zero for all the levels underground. At the base (foundation level) of the columns and shear walls all the DOF were prescribed to zero. Given soil- structure interaction was not considered all these DOF constraints were incorporated to consider fixity due to the surrounding basement walls and soil around them. The nodes found in the same floor of the model were prescribed to have the same translational DOF this was done using the Master node feature in CANNY. 4.4.6 Mathematical model calibration The case study building was subjected to an ambient vibration test to determine its natural modes of vibration on 2006. The test was conducted by a research group from UBC and results are presented by Turek et. al. (2007). Prior to perform dynamic analysis it is recommended to correlate the mathematical model of the structure to the modal model derived from ambient vibration measurements of the structure. This approach aims to correct the mathematical model of the structure to have a more realistic representation of the dynamic properties of the building and reliably perform structural analysis (He et. al., 2001). 45 Since the building modal model was known the building model in CANNY was calibrated to exhibit vibration modes close to the ones derived from AV test measurements. The calibration process included variation of parameters in the mathematical model that defined building mass and concrete modulus of elasticity to yield natural periods and modal shapes similar to those measured. The foundation and soil influence on dynamic properties of the building model was regarded to provide a fix base and lateral restraint to building displacement in the underground storey. 4.4.7 Material properties The concrete and reinforcing steel material properties were established by nominal values specified in construction blueprints for initial analysis in the model calibration process. In the blueprints the concrete was specified to be 50 MPa (7.25 ksi) compressive strength and steel reinforcing bars 400 MPa (58 ksi) yielding strength. The concrete modulus of elasticity was calculated using equation 3.3 because construction records with cylinder tests were not available for this study. The 50 MPa concrete is a High-Strength-Concrete. Some studies have shown large variability in the concrete properties, reference values for High-Strength-Concrete (HSC) such as unit weight, compressive strength and modulus of elasticity are presented in Figure 4.8 (Al-Omaishi et. al, 2009). The building model was updated or calibrated to the modal model from ambient vibration (AV) tests prior to performing dynamic analysis. Considering that actual values of material properties are very different from nominal specified values, these material properties were modified in the building model. On the grounds of the variability shown in Al Omaishis research for concrete modulus of elasticity and unit weight, these parameters were modified until a reasonable correlation between building model modes and modal model from AV test was achieved. Differences in periods were within 5% for all translational modes but in the range of 20-35% torsional modes exhibited a larger difference. The original and modified modulus and unit weight are presented in table 4.1, the calibration of the model required increasing the modulus by 3% and the unit weight of slabs reduced by 9%. The concrete properties compressive strength and modulus of elasticity were regarded as constant for the entire building. Table 4.1 Concrete material properties. Material Ec (MPa) ε c f’c (MPa) Density (kN/m3) columns and walls Density (kN/m3) slabs Concrete (nominal properties) 33,234 0.002 50 25 25 Concrete (modified properties) 34,470 0.002 50 25 23 46 The steel reinforcing bar properties were kept as nominal specified values in construction blueprints and presented in table 4.2. Table 4.2 Steel material properties. Material Es (MPa) ε y fy (MPa) Steel rebar 200,000 0.002 400 4.4.8 Material stress-strain models The steel reinforcing bars hysteresis model were modeled in CANNY using the SS3 model. This is a trilinear model that includes stiffness degradation before onset of yielding to account for rebar slippage at the joints. The hysteresis model is shown in Figure 4.8. The onset of yielding in the skeleton curve was controlled by parameters ν and κ. The parameters that defined the unloading branch were φ = 0 (without stiffness degradation prior to yielding), θ = 0.75 (set the onset of yielding under a reverse load) and γ = 0.2 for post-yield stiffness degradation. Figure 4.8 Hysteresis model SS3 for steel rebars (Li, 2010). The concrete model hysteresis used in CANNY was CS3, it is shown in Figure 4.9. It is a trilinear model with stiffness degradation after the cracking point. The skeleton curve loading in compression was defined with ν=0.5, f’c=50 MPa, tangent stiffness K=34,470 MPa and peak strain εc=0.002. The ultimate stress was defined with parameter λ=0 and λ=0.2 for unconfined and confined concrete, respectively. The ultimate strain was defined by μ=5, five times the peak strain. The unloading stiffness degradation βK 1 1 Strain Stress fy νfy K θfy εy κεy Ku 1 Ku=f(Kuy,γ,φ) fy=400 MPa K=200,000 MPa εy=0.002β=0.01 ν=0.5 κ=1.8 γ=0.2 θ=0.75 φ=0 κεy 47 was set as φ = 0 (without stiffness degradation prior to peak stress), and γ = 0.2 for post-peak stiffness degradation. The tension relation was set to a maximum strain parameter τ=3. Figure 4.9 Hysteresis model CS3 for concrete (Li, 2010). 4.4.9 Dynamic characteristics of calibrated model As presented in table 4.3 the model natural periods are close to the vibration test. In addition the modal shapes were compared to ensure the patterns of deformation were properly correlated. The first two translational modes presented moderate coupling in the EW-NS directions. The animated mode shapes are shown in Figure 4.10. Table 4.3 Comparison of building model dynamic characteristics and modal model. Mode Description Period (s) from CANNY Model Period (s) from Ambient Vibration Test 1 Translational E-W 3.74 3.78 2 Translational N-S 3.30 3.45 3 Torsional 1.14 1.54 4 Translational E-W 0.84 0.81 5 Translational N-S 0.74 0.71 6 Torsional 0.41 0.49 7 Translational E-W 0.38 0.35 8 Translational N-S and Torsional 0.31 0.30 9 Torsional and Translational 0.29 0.28 10 Torsional 0.24 0.21 1 εc Stress f’c νfc K ft μεc Ku 1 Ku=f(Kcu,γ,φ) ft=5 MPa f’c=50 MPa K=34,470 MPa εc=0.002λ=0.2 (confined) λ=0 (unconfined) γ=0.4 φ=0.5 τεt εt Strain μ=5 ν=0.5 τ=3 λfc 48 Figure 4.10 Mode shapes of building. First mode Period: 3.74s Translation E-W Pattern: Cantilever Second mode Period: 3.30s Translation N-S Pattern: Cantilever Third mode Period: 1.14s Torsional Pattern: Cantilever Fourth mode Period: 0.84s Translation E-W Pattern: C shape Fifth mode Period: 0.74s Translation N-S Pattern: C shape Sixth mode Period: 0.41s Torsional 49 Figure 4.10 Mode shapes of building (continued). Seventh mode Period: 0.38s Translational E-W Pattern: S shape Eighth mode Period: 0.31s Torsional Ninth mode Period: 0.29s Translational N-s and Torsional Tenth mode Period: 0.24s Torsional 50 4.4.10 Gravity load criteria The building model in CANNY was calibrated to dynamic characteristics from the ambient vibration test considering only dead load as the source of mass because the building was under construction during the measurements. However in predicting the response of the building to ground shaking, expected occupancy loads should additionally be considered which contribute to the source of mass for dynamic response. The dead load was calculated from the dimensions of structural elements and non-structural elements. Dead loads caused by non-structural elements were estimated with nominal loads and tributary area, the same applies to 20% of the nominal live load. Based on NBCC 2005 residential occupancy loads are as follows: Table 4.4 Nominal values for superimposed dead and live load. Type of load Æ Dead load Live load Use of area N/m2 psf N/m2 psf Underground parking 1436 30 2394 50 Main floor (lower lobby) 2729 57 5985 125 Second floor (upper lobby) 2729 57 3591 75 Tower (residential) 2729 57 1915 40 The distributed loads were lumped as weights in each node of the model. The weights were allocated to each node based on tributary areas. This distribution of tributary areas is shown for the typical tower plan layout in Figure 4.11. Figure 4.11 Distribution of tributary areas. 51 4.4.11 Dynamic characteristics of the building for analysis The dynamic properties of the building obtained during the ambient vibration test did not include the modal damping ratios. In selecting modal damping values for dynamic response of the building model, recommended values for modal damping ratios were taken from literature, the ATC-72 (2010) suggests a 3% and the Council of Tall Buildings for Urban Habitat (Wilford et. al., 2008) recommends also a 3% for RC buildings for response in the first mode. These moderate values reflect how damping ratios are lower in high-rise buildings, because non- structural components are usually provided with connections that reduce interaction with structural elements and the soil-structure damping mechanism is less significant (Wilford et. al., 2008). The energy dissipated when the structure is undergoing inelastic response was captured through the hysteretic response of the structural elements included in the model. The energy dissipated by elements not included in the model was estimated through modal damping and the classical damping matrix as presented in Chapter 3, using equations 3.7 through 3.10. The resulting modal damping ratios are shown in table 4.5. Table 4.5 Periods of vibration and Rayleigh modal damping ratios. Mode Natural Frequency (rad/s) Natural Period (s) Modal Damping (%) 1 0.234 4.27 3.00% 2 0.267 3.75 2.78% 3 0.741 1.35 2.52% 4 1.04 0.96 3.00% 5 1.18 0.85 3.25% 6 2.08 0.48 5.17% 7 2.27 0.44 5.60% 8 2.86 0.35 6.92% 9 2.94 0.34 7.11% 10 3.57 0.28 8.56% 52 4.5 Seismic input The seismic input was developed as presented in Chapter 3. All the motions required for dynamic analysis were selected, scaled and rotated as presented in the following sections. The location of the site was downtown Vancouver city and defined by latitude 49.25 North and -123.12 West, values are given in decimal degrees. 4.5.1 Seismic hazard analysis The seismic hazard in Vancouver city was obtained through a probabilistic seismic hazard analysis (PSHA). It was conducted including all the seismic sources defined in the Regional model as provided in the Open File 4459 of Geological Survey of Canada (Adams et. al, 2004). The ground motion hazard was defined through a UHS (Uniform Hazard Spectrum) at a site with soil condition site class C (NBCC, 2005). The dynamic response of the building was estimated for several hazard levels, with probabilities of exceedance of 1%, 2%, 5%, 10% and 50% in 50 years, which correspond to different return periods of the ground motion intensity. The respective return periods are 72 years, 475 years, 975 years, 2475 years and 4975 years. 4.5.2 Uniform hazard spectra The Uniform Hazard Spectrum (UHS) was obtained for a 5% damping ratio in the period range of 0.03sec to 4sec. The UHS for different return periods is shown in Figure 4.12. These elastic spectra were used to define the ground motion intensity levels for building analysis. Figure 4.12 Uniform hazard spectra for Vancouver city (5% damping ratio). 0.00 0.20 0.40 0.60 0.80 1.00 0 1 2 3 4 5 6 Sp ec tr al a cc el er at io n (g ) Period (s) 50% in 50 yr 10% in 50 yr 5% in 50 yr 2% in 50 yr 53 4.5.3 Contribution to hazard level in UHS The relative contribution from crustal and subcrustal earthquakes to the Vancouver city Uniform Hazard Spectra for a hazard level of 2% in 50 year is shown in Figure 4.15. The seismic hazard in Vancouver city for long period spectral accelerations is largely contributed by crustal earthquakes and by subcrustal earthquakes in the short period. For the high-rise building seismic response is influenced by long period demands in the fundamental mode and short period demands on higher modes. The relative contribution shown in Figure 4.13 gives insight on how the selection of motions that pose a higher hazard for Vancouver city should be. The crustal earthquakes are more relevant for the long period demands and the subcrustal earthquakes for the short period demands. Figure 4.13 Crustal and subcrustal relative contribution to the uniform hazard spectrum in Vancouver city for a hazard level of 2% in 50 year. 54 4.5.4 Seismic hazard deaggregation A magnitude-distance deaggregation at T=4.27s for hazard level 2% in 50 year in Vancouver city is presented in Figure 4.14. It provides a probability distribution of the earthquake magnitudes and distances that most significantly contribute to the seismic hazard for spectral acceleration. The M-R pairs were integrated by crustal and subcrustal earthquakes. Two clusters of M-R can be differentiated in Figure 4.14, large magnitude at short distance pairs and moderate magnitude at long distance pairs. Figure 4.14 Magnitude-distance deaggregation of hazard level 2% in 50 year at Vancouver. Period : 4.27s Amplitude : 0.086g Mean Magnitude Mw : 6.99 Mean Distance : 46 km 55 4.5.5 Selection of ground motions for time history analysis A total of three ground motions from different earthquakes were chosen based on their potential to represent the characteristics of the seismic hazard in Vancouver city. From the M-R deaggregation a range of magnitudes around the mean Mw was defined as 6.4