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Seismic Performance of concrete buildings reinforced with superelastic shape memory alloy rebars Hossain, Afrin 2013

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  SEISMIC PERFORMANCE OF CONCRETE BUILDINGS REINFORCED WITH SUPERELASTIC SHAPE MEMORY ALLOY REBARS  by Afrin Hossain  B.Sc., Bangladesh University of Engineering and Technology, 2009, October  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA  (Okanagan)  July 2013  ? Afrin Hossain, 2013  ii  Abstract  In this study, superelastic shape memory alloy (SMA) rebar was used as reinforcement in the plastic hinge regions of reinforced concrete beams. Twenty different reinforced concrete (RC) moment resisting frames of three different heights (3, 6, and 8-storys) were used in this study where SMA is gradually introduced from level 1 to the top most floor. The frames were designed according to the recent code of (CSA A23.3-04 2004) and assumed to be located in the high seismic zone of Western Canada. Nonlinear static pushover analysis and incremental dynamic analysis, (   ) considering 20 earthquake records were performed to determine the best distribution of SMA rebars and investigate the seismic performance factors (    ). The best distribution of SMA rebars was determined based on the results of the seismic performance of steel and SMA frames obtained from the nonlinear incremental dynamic analysis (   ) in terms of the maximum inter-story drift ratio (        ), maximum residual inter-story drift ratio (         ), roof drift ratio (      ) and residual roof drift ratio (       ). The seismic performance of the SMA RC frames was evaluated where the acceptability of a trial value of the response modification coefficient,   factor was assessed and appropriate values of system overstrength factor,    and the deflection amplification factor,    were determined. The recent FEMA P695 (2009) methodology, was followed for this purpose. The obtained results on      of all individual frames from nonlinear static pushover analysis (   ) and incremental non-linear dynamic analysis, (   ) represent that the proposed seismic factors were within the range of permissible limit and when subjected to maximum considered earthquake, (   ) a sufficient  iii  margin could be provided against collapse. Steel-SMA-RC frames experienced 4%-17% lower probability of collapse compared to the steel-RC frames. Keywords: Shape memory alloy; incremental dynamic analysis; seismic performance factors; limit states.    iv  Table of Contents Abstract .............................................................................................................................. ii Table of Contents .............................................................................................................. iv List of Tables ................................................................................................................... viii List of Figures ......................................................................................................................x List of Symbols ................................................................................................................ xiii List of Abbreviations ........................................................................................................xiv Acknowledgements ............................................................................................................ xv Dedication .........................................................................................................................xvi  Introduction .....................................................................................................1 Chapter  1:1.1 General ..................................................................................................................1 1.2 Thesis Organization ...............................................................................................3  Shape Memory Alloys: Characteristics and Use ............................................5 Chapter  2:2.1 General ..................................................................................................................5 2.2 Shape Memory Alloys ...........................................................................................5 2.2.1 NiTi Shape Memory Alloy .................................................................................7 2.3 Microstructure of NiTi SMAs ................................................................................8 2.3.1 Shape Memory Effect.........................................................................................9 2.3.2 Superelasticity or Pseudo Elasticity .................................................................. 12 2.3.3 Characteristics under Repetitive or Cyclic Loading .......................................... 14 2.4 Mechanical Properties of SMA ............................................................................ 16 2.4.1 Behavior under Tension and Compression........................................................ 16 2.4.2 Behavior under Torsion and Shear ................................................................... 17  v  2.5 Constitutive Material Model of SMA ................................................................... 18 2.5.1 Micromechanically Based Modeling ................................................................ 19 2.5.2 Thermodynamics Based Modeling ................................................................... 19 2.5.3 Phenomenological Modeling ............................................................................ 19 2.6 Seismic Applications of SMA in RC Buildings .................................................... 21 2.6.1 SMA Rebars .................................................................................................... 21 2.6.2 SMA Joint Connectors ..................................................................................... 22 2.6.3 SMA Bracing Systems ..................................................................................... 24 2.6.4 SMA-based Isolation Devices .......................................................................... 24 2.6.5 SMA Dampers ................................................................................................. 25 2.7 Constraints of Using SMAs .................................................................................. 26 2.8 Objectives of the Study ........................................................................................ 26  Development of Structural Systems and Nonlinear Models ........................ 29 Chapter  3:3.1 General ................................................................................................................ 29 3.2 Methodology ........................................................................................................ 29 3.3 Design of the Building Frames ............................................................................. 32 3.4 Development of Nonlinear Models ....................................................................... 36 3.5 Eigen Value Analysis ........................................................................................... 38 3.6 Nonlinear Static Pushover Analysis...................................................................... 39 3.7 Nonlinear Incremental Dynamic Analysis ............................................................ 40 3.7.1 Ground Motions Considered ............................................................................ 42  Pushover Analyses and Limit States ............................................................. 44 Chapter  4:4.1 General ................................................................................................................ 44  vi  4.2 Overstrength and Ductility ................................................................................... 45 4.3 Inter-storey Drift Ratio ......................................................................................... 51 4.4 Failure Mechanism and Limit States .................................................................... 53 4.4.1 Local Failure Criteria ....................................................................................... 54 4.4.2 Damage Schemes ............................................................................................. 55 4.5 Conclusion ........................................................................................................... 62  Incremental Dynamic Analyses ..................................................................... 63 Chapter  5:5.1 General ................................................................................................................ 63 5.2 Previous Studies................................................................................................... 63 5.3 Collapse Limits .................................................................................................... 65 5.4 Statistical Assessment .......................................................................................... 65 5.5 Results ................................................................................................................. 65 5.5.1 Maximum Inter-storey Drift Ratio .................................................................... 66 5.5.2 Roof Drift Ratio ............................................................................................... 71 5.5.3 Maximum Residual Inter-Storey Drift Ratio ..................................................... 76 5.5.4 Residual Roof Drift Ratio ................................................................................. 82 5.5.5 Case Study: El Centro Earthquake .................................................................... 87 5.6 Best Distribution of SMA Rebars ......................................................................... 90 5.7 Conclusion ........................................................................................................... 93  Seismic Performance Factors ........................................................................ 95 Chapter  6:6.1 Introduction ......................................................................................................... 95 6.2 Overview of Methodology ................................................................................... 97 6.3 Performance Group .............................................................................................. 99  vii  6.4 Median Collapse Intensity and Collapse Margin Ratio ....................................... 100 6.5 Adjusted Collapse Margin Ratio ........................................................................ 104 6.5.1 Evaluation of Collapse Margin Ratio and Acceptance Criteria ....................... 105 6.6 Evaluation of the Deflection Amplification Factor ............................................. 107 6.7 Conclusion ......................................................................................................... 108  Conclusion.................................................................................................... 109 Chapter  7:7.1 Summary ........................................................................................................... 109 7.2 Limitations......................................................................................................... 109 7.3 Conclusions ....................................................................................................... 110 7.3.1 Nonlinear Static Pushover Analyses ............................................................... 110 7.3.2 Nonlinear Dynamic Incremental Analyses ...................................................... 111 7.3.3 Seismic Performance Factors ......................................................................... 112 7.4 Recommendations for Future Research .............................................................. 113 Bibliography or References ............................................................................................. 114 Appendices ....................................................................................................................... 128 Appendix A Limit States Results ................................................................................... 128 A.1 3-storey .......................................................................................................... 128 A.2 6-storey .......................................................................................................... 130 A.3 8-storey .......................................................................................................... 134   viii  List of Tables  Table 2.1       Typical properties of NiTi compared with structural steel (Boyd and Logoudas (2008) and Penar (2005)) ................................................................. 8 Table 2.2       Mechanical properties of Ni-Ti alloy (Alam 2009) ......................................... 17 Table 3.1       Considered frames.......................................................................................... 30 Table 3.2       Column size and reinforcement arrangements ................................................ 34 Table 3.3       Beam reinforcement details ............................................................................ 34 Table 3.4       Material properties used in the finite element analysis .................................... 35 Table 3.5       Fundamental period and its corresponding spectral acceleration ..................... 39 Table 3.6       Ensemble of ground motion records ............................................................... 43 Table 4.1       Summary of pushover analyses results ........................................................... 47 Table 4.2       Comparison of overstrength and ductility results with Moni (2011) ................ 48 Table 4.3       Inter-storey drift ratio (%) for 3-storey ........................................................... 52 Table 4.4       Inter-storey drift ratio (%) for 6-storey ........................................................... 52 Table 4.5       Inter-storey drift ratio (%) for 8-storey ........................................................... 53 Table 4.6       Limit states results for 3-storey ...................................................................... 54 Table 4.7       Limit states results for 6-storey ...................................................................... 54 Table 4.8       Limit states results for 8-storey ...................................................................... 54 Table 5.1       Summary of maximum inter-storey drift ratio,         at design level intensity ......................................................................................................... 71 Table 5.2       Percentage difference of maximum inter-storey drift ratio,         (w.r.to steel-RC frames) ................................................................................. 71 Table 5.3       Summary of roof drift ratio,        at design level intensity ......................... 76  ix  Table 5.4       Percentage difference of roof drift ratio,        (w.r.to steel-RC frames) ..... 76 Table 5.5       Summary of maximum residual inter-storey drift ratio,          at design level intensity ..................................................................................... 81 Table 5.6       Percentage difference of maximum residual inter-storey drift ratio,          (w.r.to steel-RC frames) .................................................... 82 Table 5.7       Summary of residual roof drift ratio,         at design level intensity ......... 87 Table 5.8       Percentage difference of residual roof drift ratio,         (w.r.to steel-RC frames) .................................................................................................... 87 Table 5.9       Performance (%) comparison of different steel-SMA-RC frames w.r.to steel frames .................................................................................................... 92 Table 5.10     Rank for different Steel-SMA-RC frames....................................................... 93 Table 6.1       Performance group summary ........................................................................ 100 Table 6.2       Summary of collapse results ......................................................................... 101 Table 6.3       Summary of final collapse margins and comparison to acceptance criteria ... 107    x  List of Figures Figure 2.1       2D representation of the microstructure of different phases of SMAs .............. 9 Figure 2.2       Schematic depiction of the shape memory effect ........................................... 11 Figure 2.3       Schematic depiction of superelasticity........................................................... 13 Figure 2.4       Typical stress-strain curve of austenitic SMA under cyclic forces (Dolce and Cardone (2001) and Vivet et al. (2001)) ................................................. 15 Figure 2.5       Typical stress-strain curve of martensite SMA under cyclic forces (Dolce and Cardone (2001) and Liu et al. (1999)) .................................................... 15 Figure 2.6       Typical stress-strain curve of SMAs under tension/compression (Alam 2009) ............................................................................................................ 17 Figure 2.7       Typical stress-strain curve of SMAs under torsion/shear (Alam 2009) .......... 18 Figure 2.8       1D- superelastic model of SMA incorporated in FE pack .............................. 20 Figure 3.1       Flow chart of analytical research ................................................................... 29 Figure 3.2       Configuration of a typical 6 storey RC building ............................................ 33 Figure 3.3       Longitudinal section of beam reinforcement ................................................. 34 Figure 3.4       Stress-strain curve for concrete ..................................................................... 37 Figure 3.5       Uniaxial bilinear stress-strain curve for steel ................................................. 37 Figure 3.6       Stress-strain curve for SMA .......................................................................... 38 Figure 3.7       Variation of spectral acceleration with period of structure ............................. 42 Figure 4.1       Pushover response curves ............................................................................. 45 Figure 4.2       Inter-storey drift distributions for (a) 3-storey, (b) 6-storey, and (c) 8-storey frames ................................................................................................ 52 Figure 4.3       Sequences of local damages in individual members for 3-storey ................... 57  xi  Figure 4.4       Sequences of local damages in individual members for 6-storey ................... 59 Figure 4.5       Sequences of local damages in individual members for 8-storey ................... 61 Figure 5.1           curves of first mode spectral acceleration         , plotted against maximum inter-storey drift ratio,         for 3 storey .............................. 67 Figure 5.2           curves of first mode spectral acceleration         , plotted against maximum inter-storey drift ratio,         for 6 storey .............................. 69 Figure 5.3           curves of first mode spectral acceleration         , plotted against maximum inter-storey drift ratio,         for 8 storey .............................. 70 Figure 5.4           curves of first mode spectral acceleration         , plotted against roof drift ....................................................................................................... 72 Figure 5.5           curves of first mode spectral acceleration         , plotted against roof drift ratio,        for 6 storey .............................................................. 74 Figure 5.6           curves of first mode spectral acceleration         , plotted against roof drift ratio,        for 8 storey .............................................................. 75 Figure 5.7           curves of first mode spectral acceleration         , plotted against maximum residual inter-storey drift ratio,          for 3 storey .............. 78 Figure 5.8           curves of first mode spectral acceleration         , plotted against maximum residual inter-storey drift ratio,          for 6 storey .............. 79 Figure 5.9           curves of first mode spectral acceleration         , plotted against maximum residual inter-storey drift ratio,          for 8 storey .............. 81 Figure 5.10         curves of first mode spectral acceleration         , plotted against residual roof drift ratio,         for 3 storey .............................................. 83  xii  Figure 5.11         curves of first mode spectral acceleration         , plotted against residual roof drift ratio,         for 6 storey .............................................. 85 Figure 5.12         curves of first mode spectral acceleration         , plotted against residual roof drift ratio,         for 8 storey .............................................. 86 Figure 5.13     Storey drift time histories of second (maximum) and top floor due to ground motion record 11 scaled for a    equal to 0.7g .................................. 89 Figure 5.14     Storey drift time histories of second (maximum) and top floor due to ground motion record 11 scaled for a    equal to 1.3g .................................. 90 Figure 6.1       Illustration of seismic performance factors (FEMA P695 (2009)).................. 96 Figure 6.2       Process for quantitatively establishing and documenting seismic performance factors (    ) (adapted from FEMA P695 (2009)) ................... 98 Figure 6.3       Incremental dynamic analysis to collapse, showing the maximum considered earthquake ground motion intensity,     median collapse capacity intensity,      and collapse margin ratio,     for 3-storey frames ........................................................................................................ 104 Figure A.1      Limit states results for 3-storey ................................................................... 129 Figure A.2      Limit states results for 6-storey ................................................................... 133 Figure A.3      Limit states results for 8-storey ................................................................... 138   xiii  List of Symbols       Adjusted collapse margin ratio for ground motion record set   (     )    Calculated adjusted collapse margin ratio for ground motion record set       ?? ?? ?? ?? ?    Average value of calculated collapse margin ratio         Adjusted collapse margin ratio at 10% collapse probability         Adjusted collapse margin ratio at 20% collapse probability    Austenitic finish temperature    Austenitic start temperature     Deflection amplification factor          Maximum inter-storey drift ratio           Maximum residual inter-storey drift ratio    Martensitic finish temperature     Martensitic start temperature n Number of storeys in a frame where SMA was substitute for steel in the plastic hinge region of all the beams   factor Response modification coefficient        Roof drift ratio         Residual roof drift ratio    (     ) 5% damped spectral acceleration at the fundamental mode period of the structure / Spectral acceleration for DE     Spectral acceleration at which the         is too high (collapse occurred) for ground motion record set    ?   5%-damped median spectral acceleration of the collapse level ground motions      Spectral acceleration for MCE      Seismic performance factors     First modal time period   Ductility    System overstrength factor  xiv  List of Abbreviations     Collapse margin ratio    Damage measure    Degraded performance     Incremental dynamic analysis    Intensity measure     Improved performance IO Immediate occupancy IP Improved performance LS Life safety MCE Maximum considered earthquake ground motion PGA Peak Ground Acceleration      Pushover analysis SAW Simple additive weighting method SE Superelasticity or Pseudoelasticity    Scale factor  SME Shape memory effect      Seismic performance factors   xv  Acknowledgements In the name of Allah, the Most Gracious and the Most Merciful All praises go to the almighty Allah for allowing me to bring this effort to fruition. At first, I express my sincere appreciation and deepest gratitude to my esteemed supervisors, Dr. Alam and Dr. Rteil for giving me the opportunity to work in the very interesting area of shape memory alloys. In particular, I would thank them for their systematic guidance, constant persuasion, and continued encouragement throughout my M.A.Sc. Program. I would like to express my indebtedness for their belief in my potential. Their superior technical expertise, impromptu answers and solution to any thesis related problems have made a great contribution to the success of this research.  I offer my enduring gratitude to the faculty, staff and my fellow students at UBC who helped me with the necessary advice and information during the course of this work. I would also like to acknowledge Natural Sciences and Engineering Research Council of Canada (NSERC)?s support in this research work.  Special thanks are owed to my parents, sister, and family, whom I believe to be the cardinal source of inspiration for all my achievements. I am also thankful to those who indirectly contributed to this work.     xvi  Dedication   Dedicated to my Parents & Sister      1   Introduction Chapter  1:1.1 General For safety reasons concrete buildings reinforced with conventional steel are mostly designed such that the seismic performance depends on the dissipated energy through the yielding of steel reinforcing bars. While this plastic deformation allows to dissipate the seismic energy, thus, preventing the collapse of a structure, it comes at a cost of causing a permanent residual deformation in the structure which jeopardizes its serviceability. After the 1985 Michoac?n earthquake (Mexico), and the 1994 Northridge earthquake (United States) many structures needed to be rebuilt because of the excessive permanent deformation as repair was not technically and economically feasible. Again in the 1995 Hyogo-Ken Nanbu earthquake (Kobe, Japan), more than 100 reinforced concrete (RC) bridge piers experienced more than 1?(1.75%) of permanent deformation, which compels the government to demolish and rebuild the structures as it was hard to straighten them (Ramirez and Miranda 2012). Beside this, 240,000 building structures partially collapsed with an approximate economic loss of $50 to $100 billion (U.S.) (Comartin et al. 1995; Eguchi et al. 1998).  Another major earthquake in Chile in 2010 with an 8.8 degree magnitude on the Richter scale and subsequent tsunami caused damages in 80 buildings among 3000 (>10 storey) resulting in a $30 billion estimated economic loss (Wen et al. 2011). More recently in 2011, an earthquake with a 9 degree magnitude on the Richter scale, (T?hoku earthquake) shook Japan triggering powerful tsunami waves resulting in the collapse of 129,225 buildings, with a further 254,204 buildings 'half collapsed', and another 691,766 buildings partially damaged (National Police Agency of Japan 2012).   2  To avoid this type of damage, academic and structural engineering community preferred a performance-based seismic design methodology, where the seismic performance of a structure would remain within a range of specific limits even after the ground motion excitation ensuring occupants safety (large deformation) and structural functionality (re-centering) (Jason McCormick et al. 2008). This performance-based seismic design is focused on reducing the residual lateral deformations by using re-centering devices, i.e. post tensioned re-centering device (Priestley et al. 1999; Valente et al. 1999), passive energy dissipating devices i.e. tuned mass and tuned liquid dampers (Clark et al. 1995; Symans et al. 2008), and / or smart materials like shape memory alloys (SMAs) (Alam et al. 2009).  SMA is a unique material that has the ability of reverting back to its original shape after undergoing large deformation. If SMAs can be incorporated in a RC structure as reinforcing bars, then the structure will be able to dissipate seismic energy by undergoing large deformation while sustaining minimum residual deformations (Alam et al. 2008; Moni 2011; Saiidi and Wang 2006; Youssef et al. 2008). However, due to its high cost, SMA use is very limited (Alam et al. 2008; Moni 2011). Hence, with the call for robust, resilient, more sustainable and serviceable structures, this study aims at determining the best distribution of the SMA rebars in RC frames.       3  1.2 Thesis Organization This thesis was segmented into six chapters in addition to this one.  In Chapter 2 a state-of-art literature review relevant to the scope of this research is summarized. SMA basic concepts, and its properties are presented along with the seismic applications of SMAs in RC building structures.  Other additional background literature is presented in the following chapters.  Chapter 3 discusses the development of the numerical models used in this study along with their design detailing. It also presents the different techniques used to perform the analysis to meet the set objectives. Chapter 4 investigates the system overstrength,   , ductility (period based ductility,   ), inter-storey drift ratio (    ), failure mechanism, and limit states of different steel-RC and steel-SMA-RC building with three different heights (3, 6, and 8-storeys) using nonlinear static pushover analyses (   ).  In Chapter 5 the best distribution of the SMA rebars was determined by assessing the seismic vulnerability for 20 different Steel-RC and Steel-SMA-RC frames in terms of their maximum inter-storey drift ratio (        ), roof drift ratio (      ), maximum residual inter-storey drift ratio (         ), and residual roof drift ratio (       ). These parameters were determined using nonlinear incremental dynamic analyses (   ), considering twenty far-field earthquake records using finite element software.  In Chapter 6 seismic performance factors,      of different steel-RC (3 frames) and steel-SMA-RC (17 frames) frames considering different building heights (3, 6, and 8-storeys) and degree of replacement of steel by SMA (from level 1 to top level) were determined. The  4  values had sufficient margin of safety against collapse following the methodology depicted in FEMA P695 (2009). Finally, in Chapter 7 major findings from this study are summarized along with the limitations and future recommendations.     5   Shape Memory Alloys: Characteristics and Use Chapter  2:2.1 General Smart structures are structures which can automatically change their structural characteristics through the integration of smart materials in response to external disturbance and / or unexpected severe loading thus improving their structural safety and serviceability resulting in extended lifetime (Alam 2009; S. Otani et al. 2000). Self-recovery, healing, actuating, sensing are some synergetic advantages of using smart materials (Hu 2008).  One example of smart material is shape memory alloys (SMAs).  SMAs are unique materials, since they can undergo excessive deformations but can revert to their parent shape by applying heat or by removing the load (i.e. earthquake) (Alam 2009; Moni 2011). The following sections of this chapter will present a review on the basic concepts of the SMA, and their microstructure, mechanical properties, and the constitutive material modeling of NiTi shape memory alloys along with the seismic applications of SMAs in RC building structures. 2.2 Shape Memory Alloys   Shape memory alloys (SMAs) are metals which "remember" their parent shape, for this they are called shape memory alloys. SMAs are usually produced by vacuum melting technique or in inert gas environment to avoid contamination due to oxidization. Then hot working or cold working is done to achieve the desired shape, i.e. wire, ribbon, tube, sheet, or bar. The fabrication process is completed by the shape memory treatment, in which the alloys undergo appropriate thermomechanical processing resulting in distinct characteristics which help the alloy to remember its original shape (Alam 2009).   Different types of SMAs  6  with various compositions have been developed, such as: NiTi, Cu-Al-Ni, Cu-Zn-Al, and Fe-Mn-Si (Borden 1991); Ag-CD, Au-CD, Cu-Zn, Fe-Mn, Mn-Cu, Fe-Pd, Cu-Zn-Al-Mn-Zr, Cu-Al-Be, Ti-Ni-Cu, Ti-Ni-Hf, and Ni-Ti-Fe (Otuska and Wayman 1999).  NiTi SMA has been found to be the most useful one and is the most used in structural engineering (Alam 2009; Elfeki 2009; Hu 2008; Moni 2011; Ozbulut 2010).  The unique behavior of the shape memory effect, unlike traditional materials used in the construction of structures, were first observed by Chang and Read in 1932 in gold-cadmium (AuCd) alloy (Chang and Read 1951). However, it was not until the 1960s that researches have started to advance the field of SMAs after the invention of Nitinol in 1962 by Buchler and Wiley (1961). This alloy has found applications only in the past 15-20 years due to its high cost, lack of understanding of its complex behavior, and difficulty found in modeling its unusual behavior (Veronica 2008).  Due to its exceptionally good biocompatibility, the potential use of SMA was first exploited in the field of biomedical applications and then gradually in aerospace engineering (Schetky 1999). Over the last several years with the advance in technology, higher quality and reliability, capability for energy dissipation, high / low cycle fatigue resistance, the re-centering property of SMAs under cyclic loading, excellent corrosion resistance coupled with a significant reduction in price have made it exceptionally attractive to use in various structures to enhance their seismic performance in the form of bolted connections, bracing systems, dampers, prestressing strands and finally, as reinforcing bars (Alam 2009; Veronica 2008). The SMAs can be available in wires, bars, tubes and plates. Though the use of SMA in retrofitting of existing structures is not widespread, recently SMA devices have been used to retrofit several historic structures in Italy, such as: St. Giorgio Church bell tower in Rio,  7  St. Feliciano Cathedral in Folingno, and St. Frances Basilica in Assisi which was seriously damaged by the earthquake of Oct. 15th, 1996 (Castellano et al. 2001). These retrofits were positively verified when another earthquake occurred (on June 18th, 2000, with the same epicenter and a comparable Richter Magnitude) where no substantial damage was found (Indirli et al. 2001). Some of the applications of SMAs in new RC buildings are discussed in the subsequent sections.    2.2.1 NiTi Shape Memory Alloy  The shape memory characteristic of equi-atomic alloy of Ni and Ti, known as 'Nitinol' (acronym for Nickel Titanium Naval Ordinance Laboratory) was first observed in 1962 at the US Naval Ordinance Laboratory (Jackson et al. 1972). Among all the alloys, several distinct and advantageous properties of NiTi were observed such as: large recoverable strain (up to 8%), superelasticity, remarkably good corrosion resistance and excellent fatigue strength, biocompatibility, and stable hysteretic behavior. These properties made NiTi-SMA most applicable in the medical, aerospace, and structural engineering sectors. Throughout this thesis SMA will refer to NiTi-SMAs unless otherwise specified.  A comparative analysis showing the typical properties of NiTi along with structural steel is shown in Table 2.1.        8  Table 2.1       Typical properties of NiTi compared with structural steel (Boyd and Logoudas (2008) and Penar (2005))     Different Properties Properties NiTi Alloy Structural Steel Austenite Martensite Physical Properties Melting Point 1240-1310?C 1500?C Density 6.45 g/cm3 7.85 g/cm3 Thermal Conductivity 0.28 W/cm ?C 0.14 W/cm ?C 0.65 W/cm ?C Thermal Expansion 11.3   10-8 / ?C 6.6  10-8 / ?C 11.7  10-8 / ?C Magnetive No Yes Electrical Resistivity 80 to 100      72      Mechanical Properties Recovered Elongation up to 8% 0.2% Young?s Modulus 30-83 GPa 21-41 GPa 200 GPa Yield Strength 195-690 MPa 70-140 MPa 248-517 MPa Ultimate Tensile Strength 895-1900 MPa 448-827 MPa Elongation at failure 5-50% (typically  25%)   20% Possion?s Ratio 0.33 0.27-0.3 Hot workability Quite good Good Cold Wokabilty Difficult due to rapid work hardening Good Mechinability Difficult, abrasive techniques preferred Good Hardness 30-60 Rc Varies Weldability Quite good Very good Biocompatibility Excellent Fair Torqueablity Excellent Poor Chamical Properties Corrosion performance Excellent  Fair 2.3 Microstructure of NiTi SMAs NiTi SMA microstructure has two stable phases ? the high temperature and low stress phase known as austenite which is highly symmetric having a body-centered cubic atomic structure (B2) (Figure 2.1 (a)) and the low temperature and high stress phase known as martensite which is stable having rhombic geometry (B19). Martensite phase can be in any of two forms depending on the crystal orientation direction: twinned (twin variants) and detwinned (single favored variant) as presented in Figure 2.1 (b) and (c) respectively.     9      Austenite Martensite  (a) (b) (c)  Figure 2.1       2D representation of the microstructure of different phases of SMAs The unique properties exhibited by SMAs are the result of the transformation that occurs between these two phases (austenite and martensite) upon heating and cooling or load removal. The resulting effects associated with this phase transformation cause the material to return to its original phase due to the application of heat known as shape memory effect (SME) or after the removal of stress, known as superelasticity or pseudoelasticity (SE).  2.3.1 Shape Memory Effect Shape memory effect (SME) is the ability of SMA material to revert back to its initial shape even after being deformed following a thermal cycling. Four characteristic temperatures are defined to show the transformation temperature cycle. Martensitic start temperature (  ) at which the alloy starts to transform from austenitic to martensitic phase; martensitic finish temperature (  ) at which the alloy completely transformed into martensitic phase; austenitic start temperature (  ) at which the reverse transformation initiates from martensitic to austenitic  phase; and finally the austenitic finish temperature (  ) at which the martensitic alloy fully turned into austenitic alloy completing the reverse phase transformation.   Ti Ni Twinned Detwinned  10  The transformation occurs when the SMA is in its twinned martensitic phase at a temperature below    (at point A in Figure 2.2 (a)). When a stress above a critical level is applied, the twinned martensitic SMA starts to reorient into detwinned martensitic phase accommodating the increased strain (at point B in Figure 2.2 (a)) and the deformed detwinned structure is retained even after the removal of the stress (at point C in Figure 2.2 (a)). When the material is heated to a temperature above   , it transforms into austenitic phase with a complete shape recovery (at point D in Figure 2.2 (a)). As the SMA cools back to a temperature below   , the austenitic SMA would revert back to the low symmetric initial twinned martensitic phase without any permanent deformation (at point A in Figure 2.2 (a)). A stress-strain curve along with a stress-temperature diagram depicting the shape memory effect (SME) phenomenon of SMAs is shown in Figure 2.2.          11   (a) Stress-strain curve   (b) Temperature-stress diagram Figure 2.2       Schematic depiction of the shape memory effect   B C A Recovery upon heating Detwinned Martensite Austenite Detwinned Martensite Twinned MartensiteA  <    Strain Stress D    COOLING       + ?  1. Twinned Martensite (unstressed)  2. Detwinned Martensite (stressed-deformed) 4. Austenite (unstressed-undeformed)  3. Detwinned Martensite (unstressed-deformed)       Temperature Stress ?  ?             HEATING  12  2.3.2 Superelasticity or Pseudo Elasticity Superelasticity phenomenon of SMAs is shown in Figure 2.3. Only above the austenitic finishing temperature superelasticity of SMA occurs (i.e. austenitic SMA). During loading and unloading of an austenitic SMA, six distinctive characteristics are observed in the stress-strain diagram (Figure 2.3 (a)); (a) the alloy is in its austenitic phase and shape when it is not subjected to any load or at low strains (<1%) (segment AB in Figure 2.3 (a)), (b) after the stress is induced, the alloy starts to accommodate the strain by transforming into detwinned martensitic phase at the martensitic transformation stress,     (segment BC in Figure 2.3 (a)) with a long and constant stress plateau at intermediate and large strains, (c) the alloy completely transforms into the detwinned  martensitic phase when the stress reaches above      and the elastic response is observed at large strains (segment CD in Figure 2.3 (a)). Irreversible plastic deformation may occur beyond this stress level if the stress continues to increase, (d) the detwinned martensite starts to revert back to its original phase with the decreasing stress,      showing elastic strain recovery (segment DE in Figure 2.3 (a)), (e) as the stress is removed gradually, reverse transformation of unstable martensite occurs followed by a constant stress path with instinctive strain recovery (segment EF in Figure 2.3 (a)) (Wang et al. 2006), and (f) when the stress is very low then the alloy reverts back to its austenitic phase followed by elastic unloading (segment FA in Figure 2.3 (a)).  As a result, one hysteresis loop is formed during the above mentioned loading-unloading process by a closed stress-strain curve. The area formed by this loop is equal to the energy dissipation capacity of the SMA material (Pieczyska et al. 2005; De Silva 2000).  It should be noted that, a partial shape recovery occurs in between the temperatures    and   . Whereas,  13  if the temperature is greater than a critical temperature (  ) martensitic transformation cannot be exhibited even after the removal of the load.  (a) Stress-strain curve   (b) Temperature-stress diagram Figure 2.3       Schematic depiction of superelasticity         + ?  Detwinned Martensite (stressed-deformed)    Austenite (unstressed-undeformed)  Temperature Stress ?  ?             Partial Recovery No transformation Full Recovery                 Austenite Detwinned Martensite Strain Stress A B C E D F Martensite ? Austenite (Reverse Transformation)  Austenite ? Martensite  (Forward Transformation)   >     14  2.3.3 Characteristics under Repetitive or Cyclic Loading It is very important to investigate the effect of SMA under cyclic loading as an earthquake induces such type of loads on a structure. Many studies have been reported investigating the cyclic properties of SMA under different loading conditions namely tension, compression, shear, and torsion forces.   Some researchers have conducted tests to investigate the effect of cycling loading on NiTi SMA wires with a diameter of 1-2 mm (DesRoches et al. 2004; Dolce and Cardone 2001; Gall et al. 2001; Gong et al. 2002; Kawaguchi et al. 1991; Malecot et al. 2006; Miyazaki et al. 1986; Strnadel et al. 1995a; b; Tamai and Kitagawa 2002; Wolons et al. 1998). The outcome of these studies showed an increase in the residual strain, a decrease in the loading plateau, a decrease in the forward phase transformation stress level with the number of loading cycles and decreased hysteresis loop area i.e. the dissipated energy. To improve the results, many researchers tried to train the NiTi SMA alloy resulting in a stabilized hysteresis, enhanced re-centering ability, and decreased residual strain (MANSIDE 1998; C. McCormick and DesRoches 2006; Miyazaki et al. 1986; Strnadel et al. 1995a; Z. G. Wang et al. 2003).   The characteristic stress-strain behavior of an austenitic SMA under cyclic axial, shear, and torsion forces is shown in Figure 2.4 and the characteristic stress-strain behavior of martensitic SMA under cyclic axial and torsion forces is shown in Figure 2.5.  When austenitic SMA is subjected to a repetitive cyclic loading, a certain amount of energy is dissipated either without (under axial force) or with (under shear and torsion forces) very low permanent deformation. On the other hand, the martensitic SMA shows more permanent deformation than the austenitic SMA but it shows a greater and full hysteresis loop around  15  the origin thus dissipating higher amount of energy compared to the austenitic SMA. Due to these disadvantages in both of the SMAs, martensite-austenite co-existence was found to be better compared to an indivdual phase under cyclic loading (Ma and Song 2005).  According to Orgeas et al. (1997) the austenitic SMA shows some residual deformation under shear stress even after the removal of the stress, which could be due to the presence of partially stabilzed martensite. Frequency of the applied load plays an important role in the austenitic SMA behavior under torsion stress unlike the martesnitic SMA behavior. Both the SMAs produce stable hysteresis loops under torsion  (Dolce and Cardone 2001).   (a) Axial  (b) Shear (c) Torsion Figure 2.4       Typical stress-strain curve of austenitic SMA under cyclic forces (Dolce and Cardone (2001) and Vivet et al. (2001))    (a) Axial (b) Torsion Figure 2.5       Typical stress-strain curve of martensite SMA under cyclic forces (Dolce and Cardone (2001) and Liu et al. (1999))  16  2.4 Mechanical Properties of SMA Compared to the regular structural steel, SMA exhibits distinct mechanical properties which interests a large number of researchers to investigate its proper application. The mechanical properties of SMA depend on its chemical composition, atomic arrangement, and temperature. According to Strnadel et al. (1995) mechanical properties may be affected substantially due to a small percentage of changes in the constitutive metals. Some researchers conducted experiments on different SMA specimen in the form of bars and wires considering different diameters and loading conditions.   2.4.1 Behavior under Tension and Compression  A typical stress-strain curve of the SMA under tension and compression (Figure 2.6) has four distinct segments. In the first segment, a linear elastic response is observed which could be applicable for both austenitic and martensitic SMAs with a modulus of elasticity,   . The second phase starts when the stress and strain increase beyond    and    respectively forming stress-induced martensitic specimen having modulus of elasticity,    , which is 10% to 15% of   . The third phase starts when the strain increases above     indicating an elastic deformation of martensite with slip and dislocation motion. The modulus of elasticity of this stage is denoted by    , which is 50% to 60% of   . The final stage occurs when the strain increases beyond      indicating the plastic deformation of martensitic phase. The modulus of elasticity corresponding to this phase is denoted by   , which is 3% to 8% of    and it continues until failure. Typical values of   ,   ,   ,   ,    , and    are presented in Table 2.2.  17   Figure 2.6       Typical stress-strain curve of SMAs under tension/compression (Alam 2009) Table 2.2       Mechanical properties of Ni-Ti alloy (Alam 2009) Studies Test Types Property Unit Austenite Martensite Manach and Favier 1997 Tension Young?s modulus,    GPa 30-98 21-52 Otsuka and Wayman 1999 Yield strength,    MPa 100-800 50-300 Mazzolani and Mandara 2002 Ultimate strength,    MPa 800-1900 800-2000 Rejzner et al. 2002 Elongation at failure,    % 5-50 20-60 Zak et al. 2003 Recovered pseudoelastic strain,     %  8 - DesRoches et al. 2004 Maximum recovery stress,     MPa 600-800 - Orgeas and Favier 1995 Compression  Young?s modulus,    GPa 56-69 20-80 Liu et al. 1998 Yield strength,    MPa 550-800 125-190 Lim and McDowell 1999 Ultimate strength,    MPa 1500 1800-2120 Fukuta and Iiba 2002 Elongation at failure,    % - 17-24  Recovered pseudoelastic strain,     % 3-6 -  Maximum recovery stress,     MPa 650-820 - Manach and Favier 1997 Liu and Favier 2000 Delgadillo-Holtfort et al. 2004  Shear  Shear modulus    GPa 18-25 25-40 Yield strength,    MPa 186 42-100 Ultimate strength,    MPa >515 <515 Elongation at failure,    %  40  40 Recovered pseudoelastic strain,     % 3-6 - Maximum recovery stress,     MPa 300 - Melton 2000 Torsion Shear modulus    GPa 6-28 4-9  Yield strength,    MPa 220-350 88 Lim and McDowell 1999 Ultimate strength,    MPa >500 210-380 Dolce and Cardone 2001 Elongation at failure,    % 10 - >24 - McNaney et al. 2003 Recovered pseudoelastic strain,     % - 1-6  Maximum recovery stress,     MPa - 270-500 2.4.2 Behavior under Torsion and Shear A typical stress-strain curve of SMA material under torsion and shear forces is presented in Figure 2.7. It also has four segments and similar pattern to the stress-strain curve of SMAs under tensile and compressive forces. Here the stress ( ), the strain ( ), and the modulus of  18  elasticity ( ) under tension and compression are replaced with shear stress ( ), shear strain ( ), and shear modulus of elasticity ( ). Typical values of   ,   ,   ,   ,    , and    are presented in Table 2.2.  Figure 2.7       Typical stress-strain curve of SMAs under torsion/shear (Alam 2009) 2.5 Constitutive Material Model of SMA The unique properties of SMA material require new constitutive relationship which has been the focus of research studies by many researchers during the last decade. Due to the novelty and the complexity of generating the SMA behavior, most of the researchers ended up with simple, one-dimensional constitutive equations considering simple stress in their initial experimental investigations. However, recently with the additional knowledge achieved on SMA production technology combined with the increasing sophistication in the micro and macro-scale phenomena lead researchers to perform more complex experimental investigations considering combined effect such as tension-torsion. The reported material models can be divided into three groups: micromechanically based approach, concept based approach or thermodynamics, and phenomenological models (Veronica 2008).   19  2.5.1  Micromechanically Based Modeling The origin of the micromechanically based approach is in the models for single-crystal shape memory alloys investigating the kinematics of martensitic phase transformations which can be either homogenous plane invariant strain or heterogeneous invariant slip. These types of models are based on the micro-scale level information, such as nucleation or interface motion (Goo and Lexcellent 1997; Hall and Govindjee 2002; Levitas et al. 1998; Patoor et al. 1998; Sun and Hwang 1993).  2.5.2 Thermodynamics Based Modeling Thermodynamics based modeling is based on the fundamental concept of equilibrium thermodynamics. The effect of interfacial energy on the phase boundaries and the phase equilibrium minimizing the total free energy were considered when developing these models (Boyd and Logoudas 1996; Brinson 1993; Brinson et al. 1996; Liang and Rogers 1990, 1992; Raniecki and Lexcellent 1998). 2.5.3 Phenomenological Modeling Only macroscopic behavior is considered in one dimensional phenomenological or macro-level model which made it best suited in most of the civil engineering applications. The advantage of this model is that the classical experimental test results can be used to identify the material parameters. The equations used to describe the material structures are well compatible to be implemented into numerical methods like finite element (FE) programs to analyze a structure. A number of uniaxial constitutive models representing the complex behavior of SMA with path dependency, phase transformations, and hysteresis have been proposed in the literature (Auricchio and Lubliner 1997; Auricchio and Sacco 1997a; b,  20  1999, 2001; Auricchio et al. 2003, 2008; Qidwai and Lagoudas 2000). Superelastic behavior of SMAs has been integrated in a number of FE packages of current commercial general purpose FE software, e.g., ANSYS (2005) v.14.5 , SeismoStruct (2010) v.6 and ABAQUS (2003) v.6.11.  One dimensional superelastic model of SMA incorporated in FE packages is shown in Figure 2.8 where SMA experiences stress induced austenite-martensite transformation subjecting to multiple stress cycles at a constant temperature. Six parameters are used to define the material model:    (austenite to martensite starting stress);     (austenite to martensite finishing stress);     (martensite to austenite finishing stress);     (martensite to austenite finishing stress);    (superelastic plateau strain length or maximum residual strain); and    and    (moduli of elasticity).   Figure 2.8       1D- superelastic model of SMA incorporated in FE pack Comparing all three groups of models, the phenomenological models were found to be less complicated and less computationally demanding (Moni 2011) since the other models were based on the exact examination of the physics of the materials (Brocca et al. 2002). Therefore, phenomenological models were found to be best suited for civil engineering  21  applications to be easily implemented into FE programs involving SMA wires and bars (Alam 2009; Moni 2011; Veronica 2008).   2.6 Seismic Applications of SMA in RC Buildings 2.6.1 SMA Rebars During a strong earthquake, buildings reinforced with steel are yielded to dissipate energy and experience permanent deformation, whereas if the buildings were reinforced with SMA rebars they would have the ability to undergo large deformations and come back to the original shape after a strong ground motion.  Saiidi and Wang (2006) investigated the seismic performance of two RC columns applying SMA in the plastic hinge region and regular steel bars in other regions using shake table test. The SMA reinforced concrete column was found to have a superior performance for its recoverable post-yield deformations compared to steel-RC column. Alam et al. (2009) analyzed two 8-storey concrete frames where one was reinforced with regular steel and the other was reinforced with SMA in the plastic hinge region of the beams and steel in the other parts. They considered 10 accelerograms and reported the results in terms of maximum inter-storey drift, top-storey drift, inter-storey residual drift, and residual top-storey drift. They found that most of the post-yield deformations are recovered by SMA frames even after a strong ground motion and the frames reinforced with steel were experiencing 13 times higher residual drift compared to the frames reinforced with SMA.   Alam et al. (2012) analytically investigated the effect of SMA rebars in concrete building considering three different heights (3, 6, and 8-storeys) and three different reinforcement detailing: (i) beams reinforced with steel reinforcement only (steel), (ii) SMA  22  rebar applied in the plastic hinge region of the beams and regular steel rebars in the rest of the regions of the beams (steel-SMA), and (iii) beams reinforced with SMA rebars throughout (SMA). For all three cases, the columns were reinforced with regular steel. They reported the structures' overstrength and ductility; seismic demand and capacity (in terms of base shear and drift) after performing nonlinear static pushover analysis and nonlinear dynamic time history analysis. Their results showed that low and medium rise buildings having SMA had 15%-20% less seismic demand compared to the buildings having regular steel.         Abdulridha et al. (2013) experimentally investigated the performance of beams reinforced with SMA bars in the plastic hinge region and steel elsewhere, and beams reinforced with conventional deformed steel reinforcement under monotonic, cyclic, and reverse cyclic loading. They found that beams with SMA rebars had superior capacity for their higher (93%) recoverable inelastic deformations compared to the beams with conventional deformed steel bars.  Shrestha et al. (2013) investigated the feasibility of Cu-Al-Mn superelastic alloy bars as reinforcement elements in concrete beams both analytically and experimentally. They found that the superelastic alloy bars are superior over the conventional steel bars for 89% higher crack recovery capabilities.  2.6.2 SMA Joint Connectors According to Alam et al. (2008), beam-column and column-foundation joints are the weakest link in a frame system, therefore, the seismic performance of a building can be enhanced by reinforcing the connections using SMA rebars.  They conducted experimental  23  investigation applying SMA in the beam-column joints. Based on their results, they established guidelines to predict the seismic behavior of a concrete beam-column element reinforced with superelastic SMAs. They also examined the disparities in the moment-curvature relationship between regular steel and SMA sections and modeled the behavior of SMA using finite element (FE) program.  Hu and Leon (2010) used SMA in the partial restraint (PR) connection between steel beams and concrete-filled tube columns in composite moment frames (C-MF) for the re-centering capacity of SMA. Their analytical investigation demonstrated that PR connections with SMA have less residual drift and better distribution of the demand over the height of the frames compared to the traditional welded connections. To improve the ductility of FRP RC elements Nehdi et al. (2010), and Billah and Alam (2012) proposed SMA-FRP RC hybrid beam-column joint and column, where SMA rebars were applied in the plastic hinge regions of the beam and column, respectively and FRP bars were applied in the other regions. They reported the results in terms of load-storey drift, moment-rotation, and energy dissipation capacity and compared the results with the similar conventional RC beam-column elements. They found that the SMA-FRP beam-column elements have adequate energy dissipation under seismic loading. Their findings proved that it could be a great benefit to use this type of hybrid beam-column elements in highly corrosive exposure requiring less or no maintenance / repairing.     24  2.6.3 SMA Bracing Systems SMA can be used as active or passive bracing members. To achieve an active control system, heating is required through the SMA tendons unlike a passive control system which is achieved by the superelasticity of the SMA tendons.  Youssef et al. (2010) analytically investigated the seismic performance of buckling restrained braces (BRBs) and shape memory alloy braces (SMABs) in a 3-storey RC building. The SMAB system was found to have a substantial reduction in seismic residual deformations.  The NiTi based bracing system was found to be more effective compared to the conventional steel bracing system in minimizing the inter-storey drift, the column rotations, and the residual inter-storey drift (Bernardini et al. 1999; McCormick and DesRoches 2003; and Zhu and Zhang 2007). Some researchers proposed diagonal SMA bracing system in a frame structure (Han et al. 2003; Lafortune et al. 2007; McCormick, DesRoches, et al. 2007; Saadat et al. 2001). Some more studies were performed by Eaton (1999); Saadat et al. (2002); and Salichs et al. (2001) showed that the addition of SMA cross bracing enhanced the dynamic damping capacity and re-centering capability of a structural system.   2.6.4 SMA-based Isolation Devices Isolators connect the superstructure with the ground thus limiting the transfer of the seismic energy from the ground to the structure.  A single degree of freedom system (SDOF) was analytically isolated using SMA springs by Khan and Lagoudas (2002) with the help of a shake table whereas a multi-storey shear frame was isolated using SMA tendon by Corbi  25  (2003) from ground excitation. In both of these cases good results were obtained in supressing the plastic deformations.   SMA wires incorporated with an elastomeric bearing was proposed by Choi et al. (2005) as an isolation device to improve the seismic performance of conventional lead-rubber bearings in terms of instability and residual deformation.  Dezfuli and Alam (2013) used SMA-based natural rubber bearings (SMA-NRBs) for base isolation and found that it increased the dissipated energy by 74% and reduced the residual deformation by 15%.  2.6.5 SMA Dampers The austenitic SMA has great potential to be used as a passive damping system due to its high damping capacity ( Humbeeck and Kustov 2005; Humbeeck 1999; Saadat et al. 2002; Tamai and Kitagawa 2002) or as a vibration control system for building structures (Higashino and Aizawa 1996; Krumme et al. 1995; Salichs et al. 2001; Suduo and Xiongyan 2007; Zuo et al. 2008).  Han et al.(2005) proposed superelastic NiTi based damper which can work under tension, compression and torsion simultaneously. Pan and Cho (2007) used NiTi alloy as shape memory based passive damper exploiting its pseudoelasticity and found that it has a good energy dissipation ability compared to typical materials damper by converting the mechanical energy to thermal energy.   26  2.7 Constraints of Using SMAs  SMA has great potential to be used in civil engineering structures but the great obstacle in its application is its high cost compared to the conventional reinforcing material. Though in the last decade, the cost of SMA has decreased 10 times than before (from more than US$1000/kg to below US$150/kg) (Alam et al. 2007a), it is still more expensive compared to the price of conventional reinforcing materials.  Improper bridging between the material scientists group who develop SMAs and the civil engineering group who implements SMAs is another reason for not finding its path to wide spread application (Tyber et al. 2007). Difficulty in large scale implementation (Weinert and Petzoldt 2004) and lack of knowledge for large scale performance  (McCormick, Tyber, et al. 2007) are also hindering its application. Despite of these challenges, researchers are trying to come up with new SMA materials, such as: Fe-Mn-Si-Cr (Janke et al. 2005) to reduce the cost and conduct further investigations to enable its real world implementation for making smart structures.   2.8 Objectives of the Study Previously, Moni (2011) presented an analytical study of the behavior of reinforced concrete moment resisting frames of varying heights (3, 6, and 8-storey).  For each frame height, three different reinforcing configurations were considered: all steel reinforcement, all steel reinforcement except in the plastic hinge region where SMA reinforcement was used, and steel reinforcement in the columns and SMA reinforcement in all the beam members. The use of SMA reinforcement in two of the configurations were intended to take advantage of the unique flag shape hysteresis and re-centering capability of superelastic  27  SMAs. Both nonlinear pushover analysis and nonlinear incremental dynamic analysis were performed to analyze the behavior of the structures with respect to inter-storey drift, roof drift, and base shear.  From these, both the overstrength and ductility of the systems were evaluated.  The results suggest the effectiveness of the SMA-RC frame over the other two systems in terms of base shear capacity/demand ratio. This research aims to further extend the study of Moni (2011) by investigating the performance of frames where SMA being utilized in the plastic hinge region of the beams in less number of floors in order to minimize cost. The main objectives of this research include: 1. Determining the limit state of different concrete frames reinforced with steel and SMA rebars (steel-SMA-RC frames) and compare those with the limit states of concrete frames reinforced with steel rebars only (steel-RC frames) where the designs are based on the current seismic standards (CSA A23.3-04 2004). 2. Comparing the seismic performance of different steel-RC frames with steel-SMA-RC frames designed based on the current seismic standards (CSA A23.3-04 2004) in terms of maximum inter-storey drift ratio (        ), maximum residual inter-storey drift ratio (         ), roof drift ratio (      ) and residual roof drift ratio (       ) in order to determine the best distribution of SMA rebars in RC structures. 3. Determining the seismic performance factors, (    ) following the methodology presented in FEMA P695 (2009), including determining the response modification coefficient,   factor, the system overstrength factor,   , and the deflection amplification factor,    of different steel-SMA-RC frames and compare these with  28  steel-RC frames designed based on the current seismic standards (CSA A23.3-04 2004).  29    Development of Structural Systems and Nonlinear Models Chapter  3:3.1 General The methodology followed to accomplish the objectives of this study is described in this chapter. In particular, the development of the structural models, their design, and the development of nonlinear models which will be used in the subsequent chapters are discussed. Different analyses, i.e. eigen value analysis, nonlinear static pushover analysis, and nonlinear incremental dynamic analysis are also described in this chapter.  3.2 Methodology The procedure to accomplish the listed objectives in Chapter 2 is shown in Figure 3.1.  Figure 3.1       Flow chart of analytical research Three frames were reinforced with either steel (will refer to them as steel-RC frames) or with SMA in the plastic hinge section and steel elsewhere (will be called steel-SMA-RC Design of the frames  (Chapter 3) Analytical models (Chapter 3) Limit States (Chapter 4) Seismic Performance and  Optimization (Chapter 5)       ( ,   , and   ) (Chapter 6)             and     Results and Discussions  30  frames). All the columns were reinforced with regular steel in all the frames. Those ductile moment resisting reinforced concrete frames were designed according to NBCC (2005) and CSA A23.3-04 (2004) standards. In order to determine the best distribution of SMA rebars, two different parameters have been investigated: building height (3, 6, and 8-storeys representing low, mid, and high rise buildings) and replacement of steel by SMA in the plastic hinge region of the beams starting from the first storey to the top storey. For this purpose, two different reinforcement detailings were used in the frames: (i) steel reinforcement along the full length of the beams at all levels (ii) replacement of steel by SMA rebars used in the plastic hinge region of the beams in the first storey only then gradually increasing the use of SMA in the upper levels up to the top storey while steel rebars were used in other regions of the beams. Frame ID was denoted by Bh_n, where h is the building height (3, 6, and 8) and n is the number of storeys in a frame where SMA was substitute for steel in the plastic hinge region of all the beams starting from level 1. So, for 3, 6, and 8 storey buildings 4, 7, and 8 frames were considered respectively which are shown in Table 3.1.  Table 3.1       Considered frames Building ID 3-storey 6-storey 8-storey Frame ID B3_0 B6_0 B8_0 B3_1 B6_1 B8_1 B3_2 B6_2 B8_2 B3_3 B6_3 B8_3  B6_4 B8_4  B6_5 B8_5  B6_6 B8_6   B8_7   B8_8  31  In order to better utilize the effect of SMA, steel was replaced one storey at a time and the performance was analyzed. In order to do so, analytical 2D models of the steel-RC and the steel-SMA-RC frames were generated using the software SeismoStruct (2010). Therefore, a total of 20 frames were analyzed.  A nonlinear push over analysis (   ) was performed on these frames. Limit states of different steel-RC and steel-SMA-RC frames were defined investigating their global and local damages based on the results from nonlinear static pushover analysis (   ).  Seismic performance of steel-RC and steel-SMA-RC frames were evaluated in terms of maximum inter-storey drift ratio (        ), maximum residual inter-storey drift ratio (         ), roof drift ratio (      ) and residual roof drift ratio (       ) from the results of incremental dynamic analyses (   ). Based on these results, the best locations of SMA rebars were determined using simple additive weighting method (SAW). An appropriate value of system overstrength factor,    using the results from nonlinear static pushover analysis was determined and the acceptability of a trial value of the response modification coefficient,   factor and the deflection amplification factor,    (derived from an acceptable    value considering effective damping of the structure) were checked using incremental nonlinear dynamic analysis, (   ) of 20 earthquake records to provide a sufficient margin against collapse under the maximum considered earthquake, (   )  for different steel-RC and newly developed steel-SMA-RC frames.  The evaluation procedure, developed by FEMA P695 (2009) was followed for this purpose.     32  3.3 Design of the Building Frames  Each frame has equidistant 5 bays of 5 m length in both directions. Three metres storey height was considered for all the frames. Three steel-RC frames (B3_0, B6_0, B8_0) have been analyzed based on NBCC (2005) and designed according to CSA A23.3-04 (2004) following the equivalent static force procedure as moderately ductile moment resisting frames based on the previous work of Moni (2011). A typical plan for different building heights was similar and is shown in Figure 3.2 (a) and the elevation of 3, 6, and 8-storey frames are shown in Figure 3.2 (b), (c) and (d) respectively. For different steel-SMA-RC frames, same section sizes and reinforcement detailing have been used as the steel-RC frames. The slab effect was taken into account by considering T-beam sections in the frames. Section sizes used for the beam-column members along with the reinforcement detailing according to the CSA A23.3-04 (2004) standards are tabulated in Table 3.2 and Table 3.3 respectively. A typical beam reinforcement detailing is presented in Figure 3.3.         33      (a) Plan (b) Elevation of 3-storey     (c) Elevation of 6-storey (d) Elevation of 8-storey Figure 3.2       Configuration of a typical 6 storey RC building    34  Table 3.2       Column size and reinforcement arrangements  Building ID Floor level Column ID (Figure 3.2 (a)) Size (mm x mm) Section ID Main Reinforcement (M) 3-Storey  Up to roof C1 C2 375 x 375 300 x 300 8-15 M 4-20 M 6-Storey  Up to 3rd floor  3rd floor to roof  C1 C2 C1 C2 450 x 450 300 x 300 450 x 450 300 x 300 8-25 M 6-20 M 8-20 M 4-20 M 8-Storey  Up to 3rd floor  3rd floor to roof C1 C2 C1 C2 500 x 500 300 x 300 500 x 500 300 x 300 8-25 M 6-25 M 6-25 M 6-20 M Table 3.3       Beam reinforcement details   Building ID  Beam ID  (Figure 3.2 (b), (c), (d)) Size        (mm x mm) Section ID (Figure 3.3) Section 1-1 Section 2-2 Section 3-3 Main Reinforcement Main Reinforcement Main Reinforcement Top (M) Bottom (M) Top (M) Bottom (M) Top (M) Bottom (M) 3-Storey B1 300 x 450 2-20 2-20 2-20 2-20 2-20 2-20 6-Storey  B1 B2 300 x 500 300 x 500 3-25 2-20 4-25 2-20 3-25 2-20 4-25 3-20 5-25 2-20 4-20 3-20 8-Storey  B1 B2 300 x 500 300 x 500 3-25 3-20 4-25 3-20 3-25 3-20 4-25 3-20 5-25 3-20 4-20 3-20 Note: B1 beams were used for the first three storeys otherwise B2 beams were used.   Figure 3.3       Longitudinal section of beam reinforcement   35  The frames are assumed to be sited in Vancouver, a high seismic region in Western Canada.  The material properties used for designing the frames are shown in Table 3.4.  Table 3.4       Material properties used in the finite element analysis  Material Mechanical property Unit Value Concrete Compressive strength  Tensile strength  Compressive strain at peak stress  Compressive strain at crushing  MPa MPa % % 35 3.5 0.2 0.0035 Steel Modulus of elasticity  Yield strength  Strain hardening parameter   MPa MPa % 200,000 400 0.5 SMAs Modulus of elasticity  Austenite to martensite starting stress  Austenite to martensite finishing stress  Martensite to austenite starting stress  Martensite to austenite finishing stress  Super elastic plateau strain length MPa MPa MPa MPa MPa % 60,000 400 500 300 100 6  As SMA rebars have been applied in the plastic hinge region of the beams, the length of the plastic hinge region,   , had to be calculated using Equation 3.1 which was proposed by Paulay and Priestley (1992) and suggested by Alam et al. (2008) and Wang (2004) for SMA-RC elements. The rest of the beam was reinforced using steel rebars. Mechanical couplers / anchorages were assumed to be used for coupling steel and SMA rebars together (Alam et al. 2010), therefore ensuring continuous reinforcement.                                            (  ? )                   3.1 where   ?  = Element length from the face of beam-column joint to beam mid-span         = Diameter of steel bar    = Yield stress of steel bar  36  In this study, the length of the plastic hinge region of a beam reinforced with 25M bar was calculated as 450 mm. 3.4 Development of Nonlinear Models In order to perform the numerical analysis on the frames, nonlinear models of the steel-RC and steel-SMA-RC frames were developed using nonlinear finite element (FE) software SeismoStruct (2012). To account for the distribution of material nonlinearity along the length and cross-sectional area of a member, the fibre modeling approach was used. The elements used for modeling the beam-column joints were 3D beam-column inelastic displacement based frame elements. The sectional stress-strain state of those elements was obtained considering the integral nonlinear uniaxial material response of each fibre subdividing the sections according to the fibre modeling approach.  All the material models used in this study were built-in in SeismoStruct (2012). For concrete, constitutive relationship of uniaxial nonlinear constant confinement model by Mander and Priestley (1988) and cyclic rules proposed by Martinez-Rueda and Elnashai (1997) were used (Figure 3.4). To present steel, uniaxial bilinear stress-strain model with kinematic strain hardening was used (Figure 3.5). SMA has been represented by the constitutive relationship of uniaxial model for superelastic shape-memory alloy proposed by Auricchio and Sacco (1997a) (Figure 3.6). Both the beams and the columns were subdivided into 4 elements longitudinally and again each element was subdivided into 200 by 200 fibre elements in the transverse direction. Two of the longitudinal elements of the beams represent the plastic hinge regions at the beam-column joint. In different steel-SMA-RC frames the  37  beam-column joints have been modeled according to Alam et al. (2008) to take into account the SMA slippage inside the couplers (Alam et al. 2007b).  Figure 3.4       Stress-strain curve for concrete   Figure 3.5       Uniaxial bilinear stress-strain curve for steel  38    Figure 3.6       Stress-strain curve for SMA 3.5 Eigen Value Analysis Eigen value analysis was performed using SeismoStruct (2012) to obtain the modal time period for each frame. It was observed that the first modal time period,   , governs compared to the time periods corresponding to other modes. Then,    was used to determine the corresponding spectral acceleration,    , from the city of Vancouver hazard spectra (design spectra in Figure 3.7). In Table 3.5, the values of the time period obtained from eigen value analysis and the associated spectral acceleration for the different frames along with the time period obtained from NBCC (2005) code are shown.  It is observed that as the steel is being replaced by SMA, the fundamental period of the frames was increased. This is because of the lower modulus of elasticity of the SMA rebars, which decreased the stiffness and increased the time period of a structure, thus decreasing the corresponding spectral acceleration. It is to be noted that the time period for 3-storey frames are lower than the NBCC (2005) value. It might be because of the frame being slightly overdesigned.   39  Table 3.5       Fundamental period and its corresponding spectral acceleration Building ID Fundamental period, T1 (s), and spectral acceleration,       (sec) from Code NBCC (2005)  Eigen Value Analysis Frames     (sec.)    (  ) (g) 3-Storey 0.39 B3_0 0.332 0.810 B3_1 0.334 0.810 B3_2 0.336 0.800 B3_3 0.336 0.800 6-Storey 0.66 B6_0 0.696 0.528 B6_1 0.696 0.528 B6_2 0.701 0.516 B6_3 0.705 0.516 B6_4 0.706 0.516 B6_5 0.706 0.516 B6_6 0.706 0.516 8-Storey 0.81 B8_0 0.889 0.404 B8_1 0.895 0.404 B8_2 0.900 0.392 B8_3 0.900 0.392 B8_4 0.910 0.379 B8_5 0.910 0.379 B8_6 0.910 0.379 B8_7 0.910 0.379 B8_8 0.910 0.379 3.6 Nonlinear Static Pushover Analysis The pushover analysis was performed using SeismoStruct (2012). A lateral triangular load (  ) was applied, where the vertex and the apex of the load were at the roof and base levels of the frames respectively. According to Khoshnoudian et al. (2011) uniform load pattern provides higher capacity curve compared to the dynamic capacity therefore it is not reliable. Whereas triangular load pattern is reliable as it underestimates the capacity of the buildings. For this reason, inverted triangular load pattern was used in this study. The applied incremental load was kept proportional in such a way (       ), where the load factor,  , was increased monotonically by the program until a user defined limit or a numerical failure (depending on the convergence conditions at the previous step) was achieved. Response control strategy was followed in this study for the incrementation of the loading factor. It refers to direct incrementation of the global displacement (0.9 m was considered in this  40  study) of the top node and the calculation of the loading factor that corresponds to this target displacement.  3.7 Nonlinear Incremental Dynamic Analysis Nonlinear incremental dynamic analysis (   ) is a parametric analysis method used to capture the structural performance more thoroughly under seismic loads (Vamvatsikos and Cornell 2002). It was performed to evaluate the nonlinear time-history response for each of the frames for a set of predefined earthquake ground motions. Each ground motion was scaled up and down to arrest a large variety of ground motions using a scale factor (  ). For this, a monotonic scalable non-negative ground motion intensity measure,    [    ) was required. Some quantities such as: moment magnitude, duration and modified mercalli intensity proposed to characterize the intensity of a ground motion are non-scalable which cannot be used as intensity measure for     whereas, some other quantities such as: peak ground acceleration (PGA), peak ground velocity, the 5% damped spectral acceleration at the fundamental modal period of a structural system,    (     ) are scalable which can be used as intensity measure for    . According to Shome et al. (1998),    (     ) is more closely representative to the response of moderate to long period structures compared to the previously most widely used   , PGA. Moreover, Vamvatsikos and Cornell (2005) proved that  PGA is deficient compared to    (     ) in expressing the limit states capacities as it increases the variability between the     curves resulting dispersion in capacities. For this reason in this study, 5% damped spectral acceleration at the structure's first modal period,    (     ) was considered as   . In other words, it is the input of    .  41  The scaling was performed by multiplying the    of a natural accelerogram by a     denoted by       is a non-negative scalar where a simple transformation is used scaling up or down the    of a natural accelerogram by a scalar   [    )                    . To get a scaled accelerogram (   ), the    of a un-scaled (natural) accelerogram (   ) was multiplied by  . Therefore, for natural accelerogram,     1, whereas   < 1 and  > 1 indicates scaled-down and scaled-up accelerograms respectively. In this study, to cover a wide range of accelerograms ranging from milder to severe compared to the natural accelerograms    [       ] was considered.  The     continued by increasing the intensity of each of the 20 ground motions until the median collapse was reached. Each point along the     curve was obtained performing one single nonlinear dynamic time history analysis for one frame subjected to one ground motion record scaled to one intensity level. This procedure was repeated to get the full range of     curves.  Different responses were arrested considering damage measure (  ).    is a structural state variable indicating the damage state of a structure in terms of its response due to the application of seismic load, scaled accelerogram (  ), which is also a non-negative scalar     [    ]. In other words, it is the output of    . In this study, maximum inter-storey drift ratio (        ), maximum residual inter-storey drift ratio (         ), roof drift ratio (      ) and residual roof drift ratio (       ) were considered as   .     42  3.7.1 Ground Motions Considered Twenty far-field (10 km or more from the fault site) earthquake ground motion records available in PEER database (PEER 2006) were used for     to assess the frames. A summary having the key information of these ground motions are shown in Table 3.6.  Acceleration response spectrum (5% damped) of the selected ground motion set is presented in Figure 3.7 (a). Two steps scaling process was performed. In the first step, the earthquake records were matched with Vancouver hazard spectrum using the software SeismoMatch (2010) based on the wavelet algorithm proposed by Abrahamson (1992) and Hancock et al. (2006) to represent the local condition (Figure 3.7 (b)). Then in the second step, the intensity of those matched ground motions was scaled up until median collapse was reached as a part of the nonlinear         (a) Un-scaled (b) Scaled  Figure 3.7       Variation of spectral acceleration with period of structure 43  Table 3.6       Ensemble of ground motion records EQ No. PEER EQ ID Earthquake Recording Station Epicentral Distance (km) PGAmax           (g) PGVmax      (cm/s.) PGV/PGA      (sec) T1 (sec) Magnitude Year Name Name 1 120111 6.7 1994 Northridge Beverly Hills - Mulhol 13.3 0.42 58.95 0.145 0.853 2 120911 7.3 1992 Landers Yermo Fire Station 86 0.24 52 0.214 1.365 3 120121 6.7 1994 Northridge Canyon Country-WLC 26.5 0.41 42.97 0.107 0.569 4 120921 7.3 1992 Landers Coolwater 82.1 0.28 26 0.092 0.525 5 120411 7.1 1999 Duzce, Turkey Bolu 41.3 0.73 56.44 0.079 0.543 6 121011 6.9 1989 Loma Prieta Capitola 9.8 0.53 35 0.068 0.731 7 120521 7.1 1999 Hector Mine Hector 26.5 0.27 28.56 0.11 0.188 8 121021 6.9 1989 Loma Prieta Gilroy Array #3 31.4 0.56 36 0.066 0.577 9 120611 6.5 1979 Imperial Valley Delta 33.7 0.24 26 0.111 0.396 10 121111 7.4 1990 Manjil, Iran Abbar 40.4 0.51 43 0.084 0.341 11 120621 6.5 1979 Imperial Valley El Centro Array #11 29.4 0.36 34.44 0.096 0.244 12 121211 6.5 1987 Superstition Hills El Centro Imp. Co. 35.8 0.36 46 0.132 0.65 13 120711 6.9 1995 Kobe, Japan Nishi-Akashi 8.7 0.51 37.28 0.075 0.482 14 121221 6.5 1987 Superstition Hills Poe Road (temp) 11.2 0.45 36 0.082 0.46 15 120721 6.9 1995 Kobe, Japan Shin-Osaka 46 0.24 38 0.158 0.694 16 121321 7.0 1992 Cape Mendocino Rio Dell Overpass 22.7 0.39 44 0.116 1.412 17 120811 7.5 1999 Kocaeli, Turkey Duzce 98.2 0.31 59 0.192 3.724 18 121411 7.6 1999 Chi-Chi, Taiwan CHY101 32 0.35 71 0.204 2.979 19 120821 7.5 1999 Kocaeli, Turkey Arcelik 53.7 0.22 17.69 0.082 0.165 20 121421 7.6 1999 Chi-Chi, Taiwan TCU045 77.5 0.47 37 0.079 0.64  44   Pushover Analyses and Limit States  Chapter  4:4.1 General The analytical study reported by Moni (2011) focused on the effectiveness of using SMA reinforcement in reinforced concrete moment frame systems.  The moment frames considered varied in height (3, 6, and 8-storeys) and had one of three different reinforcing layouts: all steel reinforcement, SMA reinforcement within the plastic hinge regions and steel reinforcement elsewhere, and steel reinforcement in the columns and SMA reinforcement in the beam members. Both inelastic pushover analysis and nonlinear incremental dynamic analysis were conducted to evaluate the inter-storey drift, roof drift, and base shear. The capacity-demand ratios, overstrength factors, and ductility associated with each of the frames also were used to further evaluate the behavior of the reinforced concrete and SMA reinforced concrete moment frames. The results suggest the effectiveness of the SMA-RC frame over the other two systems in terms of base shear capacity/demand ratio. Therefore, in this study, steel at the plastic hinge regions of beam sections at less number of floors in a building are gradually replaced by SMA rebars in order to minimize cost and their responses were observed to find the best seismic performance. Nonlinear static pushover analysis, (   ), was performed on 20 different frames of steel-RC and steel-SMA-RC in order to investigate their system overstrength (  ), ductility ( ), inter-storey drift ratio, failure mechanism, and limit states using SeismoStruct (2012). The analysis were performed in 2D-interface.     45  4.2 Overstrength and Ductility  Pushover response curves for 20 different frames of 3, 6, and 8-storeys along with the idealized 3-storey response are shown in Figure 4.1 (a), (b), (c), and (d) respectively and the pushover analysis results are summarized in Table 4.1.    (a) 3-storey (b) 6-storey   (c) 8-storey (d) Idealized 3-storey response Figure 4.1       Pushover response curves  For 3-storey frames as presented in Table 4.1 and Figure 4.1 (a) it can be observed that, the actual lateral capacities ( ) for B3_0, B3_1, B3_2, and B3_3 were 1.47, 1.41, 1.41 and 1.40 times greater than the design base shear,   , respectively. The maximum capacities  46  reached at 180 mm (2%), 190 mm (2.11%), 210 mm (2.22%), and 220 mm (2.33%) of the lateral roof displacement, respectively for B3_0, B3_1, B3_2, and B3_3. It should be noted that the roof displacement corresponds to the global displacement of the entire structure.  47  Table 4.1       Summary of pushover analyses results Frame ID Design base shear,    1 (kN) Base shear capacity,       (kN) Ultimate roof drift displacement,     (mm) Effective yield roof drift displacement,         (mm) Overstrength factor,    2 Ductility,  3 % difference of     w.r.to steel frames  % difference of    w.r.to steel frames  B3_0 394.00 579.74 180 70 1.47 2.57 0.00 0.00 B3_1 388.77 547.88 190 80 1.41 2.38 4.22 7.39 B3_2 388.04 545.18 210 100 1.40 2.10 4.52 18.29 B3_3 387.89 544.59 220 110 1.40 2.00 4.58 22.18 B6_0 610.00 939.36 207 75 1.54 2.76 0.00 0.00 B6_1 604.65 919.02 210 80 1.52 2.63 1.30 4.71 B6_2 604.12 917.62 225 90 1.52 2.50 1.36 9.42 B6_3 603.07 913.33 252 105 1.51 2.40 1.65 13.04 B6_4 602.72 912.59 270 115 1.51 2.35 1.68 14.86 B6_5 601.65 909.91 280 120 1.51 2.33 1.79 15.58 B6_6 600.58 908.90 290 125 1.51 2.32 1.73 15.94 B8_0 682.00 1067.99 380 160 1.57 2.38 0.00 0.00 B8_1 674.79 1039.69 390 170 1.54 2.29 1.61 3.78 B8_2 673.78 1033.92 400 180 1.53 2.22 2.01 6.72 B8_3 672.51 1022.94 420 190 1.52 2.21 2.87 7.14 B8_4 671.09 1020.83 420 190 1.52 2.21 2.86 7.14 B8_5 670.44 1019.83 420 195 1.52 2.15 2.86 9.66 B8_6 670.04 1019.28 440 205 1.52 2.15 2.86 9.66 B8_7 669.98 1015.96 440 210 1.52 2.10 3.17 11.76 B8_8 669.77 1015.54 450 215 1.52 2.09 3.18 12.18  1Base shear capacity,   was calculated using NBCC (2005) 2 Overstrength factor,    =       ?  3 Ductility,              ?   48  For 6-storey frames, the lateral capacities for B6_0, B6_1, B6_2, B6_3, B6_4, B6_5, and B6_6 were 1.54, 1.52, 1.52, 1.51, 1.51, 1.51, and 1.51 times greater than    and these values were achieved at 207 mm (1.15%), 210 mm (1.17%), 225 mm (1.25%), 252 mm (1.40%), 270 mm (1.50%), 280 mm (1.56%), and 290 mm (1.61%) of lateral roof displacement, respectively which is shown in Table 4.1 and Figure 4.1 (b). For 8-storey frames, the lateral capacities for B8_0, B8_1, B8_2, B8_3, B8_4, B8_5, B8_6, B8_7 and B8_8 were 1.57, 1.54, 1.53, 1.52, 1.52, 1.52, 1.52, 1.52, and 1.52 times greater than    and the corresponding roof displacements were 380 mm (1.58%), 390 mm (1.63%), 400 mm (1.67%), 420 mm (1.75%), 420 mm (1.75%), 420 mm (1.75%), 440 mm (1.83%), 440 mm (1.83%), and 450 mm (1.88%), respectively as presented in Table 4.1 and Figure 4.1 (c). The overstrength results of frames B3_0, B3_3, B6_0, B6_6, B8_0, and B8_8 were compared with the results obtained by Moni (2011) where good alignment was observed (Table 4.2). Table 4.2       Comparison of overstrength and ductility results with Moni (2011) Frame ID Overstrength Ductility This study Moni (2011) This study Moni (2011) B3_0 1.47 1.45 2.57 2.34 B3_3 1.40 1.45 2.00 1.92 B6_0 1.54 1.52 2.76 1.85 B6_6 1.51 1.53 2.32 1.61 B8_0 1.57 1.55 2.38 2.12 B8_8 1.52 1.61 2.09 1.96  From the results, it can be observed that the initial stiffness for all steel-RC and steel-SMA-RC fames were similar before the concrete experienced cracking. However, with the increased use of SMA in different storeys, the stiffness of the steel-SMA-RC frames start to  49  decrease. This could be attributed to the lower modulus of elasticity of the SMA. SMA rebars became active in resisting the tensile forces immediately after the concrete cracking was observed in beams resulting in 3%-15% reduced stiffness in the  steel-SMA-RC frames compared to the steel-RC frames for 3, 6, and 8 storey frames which was also observed in the experimental results of Youssef et al. (2008) and numerical investigations of Alam et al. (2007, 2009) and Roh and Reinhorn (2010). The maximum load capacity was reached at a higher displacement experiencing lower post crack stiffness for steel-SMA-RC frames compared to the Steel-RC frames. It can be also observed that as the steel was replaced by SMA, the maximum load capacity was decreasing by 3%-6%. System overstrength factor, (  ), for different steel-RC and steel-SMA-RC frames were calculated using the ratio of       ? , where the design base shear was obtained using NBCC (2005) (Table 4.1). Redistribution of internal forces and consideration of confinement or use of conservative models to predict the capacities might be the sources of this overstrength (Humar and Rahgozar 1996). Percentage of system overstrength factor, (  ), with respect to steel-RC frames were also calculated and are tabulated in Table 4.1. The results show that the steel-SMA-RC frames experienced 1%-5% reduced overstrength compared to the steel-RC frames. SMA frames experienced higher flexibility and reduced stiffness due to its lower modulus of elasticity resulting in reduced capacity and extended displacement to achieve that capacity.  To get the value of the effective yield displacement, force-displacement curve obtained from the pushover analysis was converted into an equivalent bilinear elastic perfectly plastic envelope (Figure 4.1 (d)) (FEMA 356 2000). In order to do so, a horizontal line was drawn through the maximum base shear, then an inclined line was drawn from the origin to connect  50  to the horizontal line in such a way that the area enclosed by the two lines and the actual area (indicated as 'a' Figure 4.1 (d)) is equal to the area under the actual capacity envelope (indicated as 'b' in Figure 4.1 (d)). The point of intersection of these two lines is identified as the point of yield. The calculated effective yield displacements are shown in Table 4.1. The whole procedure is depicted in Figure 4.1 (d). For the 3-storey building, the effective yield roof displacements for B3_1, B3_2, and B3_3 were 14%, 43%, and 57% higher than the effective yield roof displacement for B3_0 frame, respectively. For the 6-storey building, the effective yield roof displacements for B6_1, B6_2, B6_3, B6_4, B6_5, and B6_6 are 7%, 20%, 40%, 53%, 60%, and 67% higher than the effective yield roof displacement for B6_0 frame, respectively. For the 8-storey building, the effective yield roof displacements for B8_1, B8_2, B8_3, B8_4, B8_5, B8_6, B8_7 and B8_8 are 6%, 12%, 19%, 19% 22%, 28%, 31%, and 34% higher than the effective yield roof displacement for B8_0 frame, respectively. These results indicate that the replacement of steel with SMA rebars would increase the displacement at yield load.  Ductility ( ) can be defined as the ratio between the ultimate roof drift displacement,    at maximum base shear,      and the effective yield roof displacement,        (           ? ) (FEMA 356 2000) or the ratio between the roof displacement,    at the post peak value of              and the effective yield roof displacement,        (              ? ) (FEMA P695 2009).  In this study, the FEMA 356 (2000) definition was followed and the calculated values of   are tabulated in Table 4.1.  The percentage difference of   for different steel-SMA-RC frames were also calculated and compared to the steel-RC frames. Observing the results in Table 4.1, as the steel is being  51  replaced by SMA rebars, the ductility decreased by 3%-23% for steel-SMA-RC frames. This could be due to the lower modulus of elasticity for the SMA, which results in a more flexible frame with higher drift. In spite of experiencing higher drift, the ductility for the steel-SMA-RC frames is lower than the steel-RC frames due to the yield loads being experienced at higher displacement in the steel-SMA-RC frames.  The ductility results of frames B3_0, B3_3, B6_0, B6_6, B8_0, and B8_8 were compared with the results obtained by Moni (2011) (Table 4.2). The observed disparities might be the reason of calculating the areas shown in Figure 4.1 (d). 4.3 Inter-storey Drift Ratio Ratio of the difference in lateral displacement between two floors to the height between those two floors is called the inter-storey drift ratio (    ). The distribution of the inter-storey drift ratio at the maximum base shear (                      ) over different floor levels from the results of the pushover analyses for different 3, 6, and 8-storey frames are presented in Figure 4.2 (a), (b), and (c) and  Tables 4.3, 4.4, and 4.5 respectively. As expected, all the steel-SMA-RC frames exhibited higher inter-storey drift ratio compared to the steel-RC frames due to the lower modulus of elasticity of SMA and higher flexibility of frames. The maximum inter-storey drift ratio for all the frames of steel-RC and steel-SMA-RC occurred in level 2 for 3-storey frames whereas the maximum inter-storey drift ratio was in level 5 for both 6 and 8-storey frames which is in alignment with the finding of Alam et al. (2012).     52    (a) 3-storey (b) 6-storey  (d) 8-storey Figure 4.2       Inter-storey drift distributions for (a) 3-storey, (b) 6-storey, and (c) 8-storey frames Table 4.3       Inter-storey drift ratio (%) for 3-storey Storey Level Frame ID B3_0 B3_1 B3_2 B3_3 0 0.00 0.00 0.00 0.00 1 2.30 2.03 2.23 2.32 2 3.19 3.07 3.28 3.41 3 0.81 0.90 1.40 1.47 Table 4.4       Inter-storey drift ratio (%) for 6-storey Storey Level Frame ID B6_0 B6_1 B6_2 B6_3 B6_4 B6_5 B6_6 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1 0.46 0.66 0.73 0.73 0.74 0.73 0.73 2 0.86 1.24 1.41 1.44 1.46 1.45 1.44 3 0.90 1.03 1.44 1.57 1.59 1.58 1.58 4 1.54 1.62 1.63 1.73 2.16 2.17 2.17 5 2.55 2.65 2.58 2.34 3.33 3.24 3.22 6 0.59 0.60 0.61 0.58 0.62 1.03 1.05   53  Table 4.5       Inter-storey drift ratio (%) for 8-storey Storey Level Frame ID B8_0 B8_1 B8_2 B8_3 B8_4 B8_5 B8_6 B8_7 B8_8 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1 0.78 1.03 0.98 0.92 1.02 0.99 0.99 0.99 0.91 2 1.74 2.17 2.07 1.96 2.14 2.09 2.10 2.09 1.93 3 2.31 2.66 2.62 2.48 2.70 2.64 2.66 2.66 2.47 4 2.41 2.68 2.57 2.52 2.74 2.69 2.73 2.73 2.56 5 3.10 3.13 3.25 3.44 3.67 3.70 3.64 3.77 3.77 6 2.07 2.32 2.17 2.06 2.29 2.24 2.29 2.31 2.15 7 0.93 0.95 0.93 0.91 0.93 0.92 1.25 1.32 1.31 8 0.37 0.37 0.36 0.34 0.36 0.36 0.43 0.65 0.67 4.4 Failure Mechanism and Limit States Limit states results for the different frames investigated in this study (3, 6 and 8-storeys) are summarized in Table 4.6, 4.7, and 4.8 respectively. Percentage of damage was calculated in terms of number of crushed beams and columns with respect to total number of beams and columns (Elfeki 2009).  From these tables it is clear that the frames with SMA at all levels result in less damage for all three frame heights, whereas frames with steel at all levels result in higher damage for all frame heights. For this reason, in order to investigate the failure mechanism and limit states in details, only frames with steel in all levels and SMA in all levels, namely B3_0 and B3_3 (for 3-storey), B6_0 and B6_6 (for 6-storey), and B8_0 and B8_8 (for 8-storey) were presented and discussed in the subsequent section. However, the limit states results of all other frames are presented in Appendix A. The sequence of yielding and crushing was noted in the structure (Appendix A, Figure A.1, A.2, and A.3). The      values shown in Table 4.6, 4.7, and 4.8 also indicate the sequence shown in Appendix A.   54  Table 4.6       Limit states results for 3-storey Frame ID 1st yielding in column (Figure A-1) 1st concrete crushing (Figure A-1) Collapse (Figure A-1) % Damage at      (%) Point at      (%) Point at      (%) Point B3_0 0.79 1 2.85 12 4.20 16 34.85 B3_1 0.84 1 2.89 12 4.25 15 31.82 B3_2 0.88 1 2.94 12 4.36 15 30.30 B3-3 0.92 1 3.24 13 4.52 16 28.79 Table 4.7       Limit states results for 6-storey Frame ID 1st yielding in column (Figure A-1) 1st concrete crushing (Figure A-1) Collapse (Figure A-1) % Damage at      (%) Point at      (%) Point at      (%) Point B6_0 0.82 3 1.97 15 4.32 21 18.94 B6_1 0.84 6 2.02 21 4.48 23 18.18 B6_2 0.88 6 2.08 21 4.60 23 18.18 B6_3 0.92 6 2.1 20 4.65 22 18.18 B6_4 0.93 9 2.13 22 4.66 25 17.42 B6_5 0.94 10 2.14 26 4.72 28 16.67 B6_6 0.94 10 2.15 27 4.75 30 15.91 Table 4.8       Limit states results for 8-storey Frame ID 1st yielding in column (Figure A-1) 1st concrete crushing (Figure A-1) Collapse (Figure A-1) % Damage at      (%) Point at      (%) Point at      (%) Point B8_0 0.92 10 2.36 19 4.49 22 12.50 B8_1 0.92 5 2.44 25 4.51 28 12.50 B8_2 0.92 5 2.51 27 4.56 31 11.93 B8_3 0.98 5 2.59 22 4.61 26 11.93 B8_4 1.03 4 2.64 23 4.62 27 11.93 B8_5 1.03 5 2.69 20 4.68 23 11.93 B8_6 1.04 1 2.72 19 4.78 22 11.93 B8_7 1.06 1 2.73 19 4.85 22 11.36 B8_8 1.06 1 2.76 15 4.90 17 10.23 4.4.1 Local Failure Criteria Local failure of the elements can be defined when the tensile strain of the longitudinal reinforcing bars reach the yield strain, and the compressive strain of concrete sections reach the crushing strain. The yield strain in the case of steel was 0.002 and for SMA rebars was 0.0067. In this study, the concrete crushing strain was assumed to be 0.0035.  55  4.4.2 Damage Schemes In this study three damage states were defined in the pushover curves: ? Minor damage: Defined when the strain in the reinforcing rebar is below the first yield. ? Major damage: Defined when the strain in the reinforcing bars are higher than the first yield but lower than the strain due to the first observed concrete crushing in any member of the frame. ? Extensive damage: Defined when the strain in five columns at the same level reach the crushing state at which point it is defined as the collapse state (Elfeki 2009).  From Figure 4.2 it was observed that the         has occurred at level 2, 5, and 5 for 3, 6, and 8-storey frames, respectively. This observation was used to define the global damage levels. Therefore, pushover curves were developed between base shear and      at those critical levels for the frames B3_0, B3_3, B6_0, B6_6, B8_0, and B8_8 which are presented in Figures 4.3, 4.4, and 4.5 respectively. These figures also show the limit states along with their corresponding local damage sequence. The local damages of all the frames were tracked down to investigate the global limit states. Both steel and SMA exhibted same stiffness before concrete cracking. Once the concrete is cracked, the SMA becomes effective and starts resisting the force and thus resulting overall reduction in the frame stiffness due to the lower stiffness of SMA (Figures 4.3, 4.4, and 4.5).  For 3-storey, frames B3_0 and B3_3 remained elastic without cracking up to      of 0.15% and 0.1%, respectively and these two points are shown as elastic stages in Figure 4.3 (c) and (d). From Figure 4.3 (a) and (b) it can be observed that, for frame B3_0, the first  56  yielding of steel rebar took place in the lower end of the three columns in the first floor with the sequence presented by point 1 at a corresponding      of 0.79%. For frame B3_3, the first yielding of steel rebar initiated in the lower end of two columns in the same floor with the sequence presented by point 1 at a corresponding drift of 0.92% which is higher than frame B3_0 and it can be considered as the minor damage states for the frames. With further progress in the pushover loading, the first concrete crushing was observed at lower end of one of the columns in the first storey which is denoted by point 12 and point 13 for frames B3_0 and B3_3, respectively at corresponding      of 2.85% and 3.15% which can be segmented as major damage states. Finally, for both the frames at point 16, five columns from all six columns in the first storey reached the crushing state at corresponding      of 4.20% and 4.52% which were presented as extensive damage states. The 4% collapse drift suggested by FEMA 356 (2000) was found to be conservative for both the frames. Percentage of crushed beams and columns was 21% higher in B3_0 frame compared to the B3_3 frame.        57           (a) B3_0 (b) B3_3   (c) B3_0 (d) B3_3 Figure 4.3       Sequences of local damages in individual members for 3-storey  Steel Yielding SMA Yielding Core Concrete Crushing  58  In Figure 4.4 (b) and (c), it is shown that the 6-storey frames B6_0 and B6_6, remained elastic without cracking up to      of 0.10% and 0.07% respectively and these two points are shown as elastic stages. From Figure 4.4 (a) and (b) it can be observed that in the case of B6_0 frame, the first yielding of steel rebar took place in the lower end of one of the columns in the first storey with the sequence presented by point 3 at a corresponding      of 0.82% whereas in the case of frame B6_6, the first yielding of steel rebar also initiated in the same column with the sequence presented by point 10 at a corresponding drift of 0.94% which is higher than the B6_0 frame and it can be considered as the minor damage state for the frames. In both frames, the plastic hinges were formed in some of the beams in this minor damage state. With further progress in pushover loading, first concrete crushing was observed at the lower end of one of the columns in the first storey which is denoted by points 15 and 27 for B6_0 and B6_6 frames, respectively at corresponding      of 1.90% and 2.15% which can be segmented as major damage state. Finally, for B6_0 and B6_6 frames, at points 21 and 30, five columns in the first storey reached the crushing state at corresponding      of 4.32% and 4.63% which were presented as extensive damage states and the 4% collapse drift suggested by FEMA 356 (2000) was found to be conservative for both frames. Percentage of crushed beams and columns was 19% higher in B6_0 frame compared to the B6_6 frame.       59           (a) B6_0 (b) B6_6   (c) B6_0 (d) B6_6 Figure 4.4       Sequences of local damages in individual members for 6-storey  Steel Yielding SMA Yielding Core Concrete Crushing  60  From Figure 4.5 (b) and (c), it can be observed that the 8-storey frames B8_0 and B8_8, remained elastic without cracking up to      of 0.09% and 0.07%, respectively and these two points are shown as elastic stages of the frames. From  Figure 4.5 (a), (b) it can be observed that in the case of frame B8_0, the first yielding of the steel rebar took place in the lower end of one of the columns in the first storey with the sequence presented by point 10 at corresponding      of 0.92% whereas in the case of frame B8_8, the first yielding of steel rebar also initiated in the same column with the sequence presented by point 1 at a corresponding drift of 1.06% which is higher than frame B8_0 and it can be considered as the minor damage state for the frames. In the case of frame B8_0 in this minor damage state, plastic hinges were formed in most of the beams whereas in frame B8_8 no hinge was formed in the beams. With further progress in pushover loading, the first concrete crushing was observed at lower end of one of the columns in the first storey which is denoted by points 19 and 15 for B8_0 and B8_8, respectively at corresponding      of 2.20% and 2.76% which can be segmented as major damage states. Finally, for frames B8_0 and B8_8 at points 22 and 17, five columns from all six columns in the first storey reached the crushing state at corresponding      of 4.49% and 4.90% which were presented as extensive damage states. The 4% collapse drift suggested by FEMA 356 (2000) was found to be conservative for both frames. Percentage of crushed beams and columns was 22% higher in frame B8_0 compared to frame B8_8. No crushing was observed at higher storey columns for all the different storey buildings for all frames.  61           (a) B8_0 (b) B8_8   (c) B8_0 (d) B8_8 Figure 4.5       Sequences of local damages in individual members for 8-storey   Steel Yielding SMA Yielding Core Concrete Crushing  62  4.5 Conclusion This chapter investigated the system overstrength and ductility (period based ductility,   ), inter-storey drift ratio, failure mechanism, and limit states of different steel-RC and steel-SMA-RC frames of three different heights (3, 6, and 8-storeys) using nonlinear static pushover  analyses (   ). The major findings of this chapter are:  ? Frames with SMA bars have 3%-6% less load carrying capacity and 3%-23% less ductility compare to the frames with steel.  ? Largest number of beam failure was observed in the first and second floors in all 3, 6 and 8-storey frames. ? Replacement of steel with SMA rebars provided yield at higher drift. ? The frame deformations show that the collapse for      varied from 4.20% to 4.90% which is conservative compared to the 4% value proposed by FEMA 356 (2000). ? By investigating the damage scheme, it was observed that using SMA bars led to a better collapse mechanism during the pushover analyses. Percentage of crushed beams and columns in frames with SMA bars was 19%-20% lower compared to the frames with steel bars.  ? Frames with SMA bars experience 7%-26% higher      compared to the frames with steel with the replacement of steel by SMA.    63   Incremental Dynamic Analyses  Chapter  5:5.1 General Previously, determining the seismic performance and the economic losses of a building after an earthquake were limited to calculating the peak inter storey drift ratios (     ), member ductility and peak floor accelerations (    ). Inelastic behavior due to yielding of reinforcing steel bars causes permanent deformation. Therefore, it is essential to calculate the residual drift of a structure in order to evaluate its performance and potential damage after an earthquake event (Christopoulos et al. 2003; Pampanin et al. 2003; Wu et al. 2004). Residual drift also helps to determine the technical and economical feasibility in repairing a damaged structure (Ramirez and Miranda 2012).  In this chapter, the seismic performance of the different steel-RC and steel-SMA-RC frames analyzed in this study (Chapter 3) were assessed in terms of their maximum inter-storey drift ratio (        ), maximum residual inter-storey drift ratio (         ), roof drift ratio (      ) and residual roof drift ratio (       ) by performing nonlinear incremental dynamic analyses (   ) using 20 earthquake records. 5.2 Previous Studies Very few studies were reported which measure the residual drift of a structure after a ground motion excitation. Residual drift can be reduced or completely removed by building smart structures using self-centering components exhibiting flag shaped hysteretic response in structures or by  increasing the post yield stiffness of traditional framed and braced systems (Pettinga et al. 2007). Ruiz-Garcia and Miranda (2006) considered 12 one-bay 2-D generic frame models. Incremental dynamic analysis were performed by applying 40 ground  64  motions scaled to different intensities to the frames. They reported the distribution of the residual drift along the building height. They analyzed the effect of the number of storeys, the fundamental period of vibration, the building frame mechanisms, the element hysteretic behavior, and the system over strength on central tendency. Erochko et al. (2011) presented an expression to predict the amplitude of the residual drift. In this study they considered special moment resisting frames (SMRFs) and buckling restrained braced frames (BRBs) strictly designed as per ASCE 7-05 (ASCE 2005). Ramirez and Miranda (2012) used 4 and 12-storey ductile and non-ductile buildings to determine their residual drift. The results were used to propose an advanced performance-based loss estimation methodology by considering the economic losses due to the demolishing of RC buildings after the ground motion excitation. Tremblay et al. (2008) studied the behavior of 5 different building heights (2, 4, 8, 12, and 16-storey) with self-centering energy dissipative (SCED) bracing members and buckling restrained braces (BRB) located in Los Angeles. They reported the peak storey drift angle, the residual storey drift angle, the damage concentration factor, and the normalized peak absolute floor acceleration ratio. They used push over analysis and nonlinear dynamic analysis with three groups of 20 earthquake records representing three different seismic hazard levels (50% in 50 year, 10% in 50 year and 2% in 50 year). They found that the SCED could be a potential alternative to BRB systems for improved seismic performance.      65  5.3 Collapse Limits For    , the spectral acceleration at which a sudden increase was observed in the           was defined as the collapse acceleration, i.e. the frame reached its collapse state FEMA P695 (2009). 5.4 Statistical Assessment     curves help to determine the natural random variability within the earthquake records which need a statistical assessment. Engineering design is based on the mean, median, or 84% of a damage measure (        ,          ,               ). Therefore, the 16%, median, and 84%     curves for the considered ground motion records are considered as statistical assessment which is more reliable and convenient to use. To construct percentile     curve (for instance, 16%     curve), 16 percentile value of    (maximum inter-storey drift ratio (        ), maximum residual inter-storey drift ratio (         ), roof drift ratio (      ), or residual roof drift ratio (       ))  for all 20 ground motion records at each increment of 5% damped spectral acceleration of the scaled earthquake records at the fundamental mode period of the structure,    (     ) was calculated. Then, the calculated 16% values of    were plotted against the    (     ). 5.5 Results The     results for different steel-RC and steel-SMA-RC frames were reported in terms of maximum inter-storey drift ratio (        ), maximum residual inter-storey drift ratio (         ), roof drift ratio (      ) and residual roof drift ratio (       ) along with their 16th, 50th and 84th percentile     curves.  66  5.5.1 Maximum Inter-storey Drift Ratio The maximum inter-storey drift ratio,          of 3, 6, and 8-storey buildings for different frame types (steel-RC and steel-SMA-RC) are plotted against the 5% damped spectral acceleration of the scaled earthquake records at the fundamental mode period of the structure,    (     ) along with their 16th, 50th and 84th percentile     curves which are shown in Figures 5.1, 5.2, and 5.3 respectively.      curves seem to show weaving / twisting patterns displaying successive segments of softening and hardening. It might be because of the frame experiencing the acceleration rate of          accumulation corresponding to one    (     ) is less powerful than the deceleration rate corresponding to the next    (     ). This results in pulling back the      curves to relatively lower value of          (Vamvatsikos and Cornell 2002). All the percentile results from the     curves at design level (Vancouver spectrum) ground motion intensity are summarized in Table 5.1. At all 3 percentile values, frames B3_0, B6_0, and B8_0 experienced the lowest            compared to the steel-SMA-RC frames whereas frames B3_3, B6_6, and B8_8 experienced the highest            compared to the steel-RC frames.  According to FEMA 356 (FEMA 356 2000) guidelines the permissible limits for drift ratio (transient) of concrete buildings are 1%, 2% and 4% at design level spectral acceleration for immediate occupancy (IO), life safety (LS) and collapse prevention (CP), respectively. For all the 3-storey frames the results are within the limiting value of life safety (2%) at design level spectral acceleration. However, in the case of 6-storey frames, all the frames are safe at 16th percentile and median value, whereas only the 16th percentile values were categorized as life safety for the 8-storey frames.   67  The percentage difference of          for all the steel-SMA-RC frames calculated with respect to the steel-RC frames are presented in Table 5.2. The values show that the maximum increase in the            were in the B3_3, B6_6, and B8_8 frames at all 3 percentile values.   It can be concluded that the          is increasing with the replacement of steel by SMA rebars due to its lower modulus of elasticity, which is about one third of the steel modulus of elasticity and also with the increment of the height of the frames due to the increased time period.    (a) B3_0          (b) B3_1   (c) B3_2      (d) B3_3  Figure 5.1           curves of first mode spectral acceleration    (     ), plotted against maximum inter-storey drift ratio,         for 3 storey   68    (a) B6_0          (b) B6_1   (c) B6_2            (d) B6_3   (e) B6_4             (f) B6_5   69   (g) B6_6  Figure 5.2           curves of first mode spectral acceleration    (     ), plotted against maximum inter-storey drift ratio,         for 6 storey    (a) B8_0 (b) B8_1   (c) B8_2 (d) B8_3   70    (e) B8_4 (f) B8_5   (g) B8_6 (h) B8_7  (i) B8_8  Figure 5.3           curves of first mode spectral acceleration    (     ), plotted against maximum inter-storey drift ratio,         for 8 storey     71  Table 5.1       Summary of maximum inter-storey drift ratio,         at design level intensity Frame ID Building ID Frame ID Building ID Frame ID Building ID 3-storey 6-storey 8-storey 16th P Median 84th P 16th P Median 84th P 16th P Median 84th P B3_0 0.70 0.90 1.10 B6_0 1.00 1.75 2.00 B8_0 1.70 1.95 2.70 B3_1 0.80 1.00 1.20 B6_1 1.00 1.80 2.00 B8_1 1.80 2.00 2.80 B3_2 0.90 1.05 1.30 B6_2 1.00 1.80 2.00 B8_2 1.90 2.00 2.80 B3_3 0.95 1.10 1.35 B6_3 1.00 1.85 2.05 B8_3 2.00 2.10 2.90     B6_4 1.00 1.95 2.10 B8_4 2.00 2.10 3.00     B6_5 1.00 1.95 2.20 B8_5 2.00 2.15 3.10     B6_6 1.10 2.00 2.30 B8_6 2.00 2.15 3.10         B8_7 2.00 2.15 3.10                B8_8 2.00 2.20 3.10 Table 5.2       Percentage difference of maximum inter-storey drift ratio,         (w.r.to steel-RC frames) Frame ID Building ID Frame ID Building ID Frame ID Building ID 3-storey 6-storey 8-storey 16th P Median 84th P 16th P Median 84th P 16th P Median 84th P B3_0 NA NA NA B6_0 NA NA NA B8_0 NA NA NA B3_1 14.29 11.11 9.09 B6_1 0.00 2.86 0.00 B8_1 5.88 2.56 3.70 B3_2 28.57 16.67 18.18 B6_2 0.00 2.86 0.00 B8_2 11.76 2.56 3.70 B3_3 35.71 22.22 22.73 B6_3 0.00 5.71 2.50 B8_3 17.65 7.69 7.41     B6_4 0.00 11.43 5.00 B8_4 17.65 7.69 11.11     B6_5 0.00 11.43 10.00 B8_5 17.65 10.26 14.81     B6_6 10.00 14.29 15.00 B8_6 17.65 10.26 14.81         B8_7 17.65 10.26 14.81                B8_8 17.65 12.82 14.81 5.5.2 Roof Drift Ratio The roof drift is considered an important parameter to assess the seismic vulnerability of a building. Figures 5.4, 5.5, and 5.6 present the results of     for the 20 far-field ground motion records performed on the 20 frames as     curves where roof drift ratio,        for different frame types (steel-RC and steel-SMA-RC) are plotted against the 5% damped spectral acceleration of the scaled earthquake records at the fundamental mode period of the structure,    (     ) along with their 16th, 50th and 84th percentile     curves.  All the percentile results from the     curves at design level (Vancouver spectrum) ground motion intensity are summarized in Table 5.3. At all 3 percentile values, frames B3_0, B6_0, and B8_0 experienced the lowest        compared to the steel-SMA-RC  72  frames whereas, frames B3_3, B6_6, and B8_8 experienced the highest        compared to the steel-RC frames. The percentage difference of        for all the steel-SMA-RC frames calculated with respect to the steel-RC frames are presented in Table 5.4. The values show that the maximum increase in the        were in frames B3_3, B6_6, and B8_8 at all 3 percentile values.  It can be concluded that the roof drift ratio is increasing with the replacement of steel by SMA rebars due to its lower modulus of elasticity and also with the increment of the height of the frames due to the increased time period but all the results are within the limiting value of life safety (2%).   (a) B3_0          (b) B3_1   (c) B3_2      (d) B3_3  Figure 5.4           curves of first mode spectral acceleration    (     ), plotted against roof drift   73    (a) B6_0          (b) B6_1   (c) B6_2            (d) B6_3   (e) B6_4             (f) B6_5   74   (g) B6_6  Figure 5.5           curves of first mode spectral acceleration    (     ), plotted against roof drift ratio,       for 6 storey    (a) B8_0 (b) B8_1   (c) B8_2 (d) B8_3   75    (e) B8_4 (f) B8_5   (g) B8_6 (h) B8_7  (i) B8_8  Figure 5.6           curves of first mode spectral acceleration    (     ), plotted against roof drift ratio,       for 8 storey     76  Table 5.3       Summary of roof drift ratio,       at design level intensity Frame ID Building ID Frame ID Building ID Frame ID Building ID 3-storey 6-storey 8-storey 16th P Median 84th P 16th P Median 84th P 16th P Median 84th P B3_0 0.30 0.40 0.70 B6_0 0.35 0.60 0.90 B8_0 0.90 0.90 1.40 B3_1 0.30 0.50 0.70 B6_1 0.35 0.65 0.90 B8_1 0.90 1.00 1.40 B3_2 0.40 0.60 0.80 B6_2 0.40 0.70 0.90 B8_2 0.95 1.10 1.40 B3_3 0.50 0.60 0.90 B6_3 0.50 0.75 0.95 B8_3 0.95 1.10 1.40     B6_4 0.60 0.80 0.98 B8_4 1.00 1.10 1.40     B6_5 0.60 0.80 1.00 B8_5 1.00 1.12 1.50     B6_6 0.70 0.90 1.00 B8_6 1.00 1.15 1.50         B8_7 1.00 1.15 1.50                 B8_8 1.00 1.15 1.50 Table 5.4       Percentage difference of roof drift ratio,       (w.r.to steel-RC frames) Frame ID Building ID Frame ID Building ID Frame ID Building ID 3-storey 6-storey 8-storey 16th P Median 84th P 16th P Median 84th P 16th P Median 84th P B3_0 NA NA NA B6_0 NA NA NA B8_0 NA NA NA B3_1 0.00 25.00 0.00 B6_1 0.00 2.86 0.00 B8_1 5.88 2.56 3.70 B3_2 33.33 50.00 14.29 B6_2 0.00 2.86 0.00 B8_2 11.76 2.56 3.70 B3_3 66.67 50.00 28.57 B6_3 0.00 5.71 2.50 B8_3 17.65 7.69 7.41     B6_4 0.00 11.43 5.00 B8_4 17.65 7.69 11.11     B6_5 0.00 11.43 10.00 B8_5 17.65 10.26 14.81     B6_6 10.00 14.29 15.00 B8_6 17.65 10.26 14.81         B8_7 17.65 10.26 14.81                 B8_8 17.65 12.82 14.81 5.5.3 Maximum Residual Inter-Storey Drift Ratio Residual drift ratio also plays an important role in assessing the seismic vulnerability along with the drift ratio. Maximum residual inter-storey drift ratio,          of 3, 6, and 8 storey buildings for different frame types (steel-RC and steel-SMA-RC) are plotted against the 5% damped spectral acceleration of the scaled earthquake records at the fundamental mode period of the structure,    (     )  in the form of    curves as shown in Figures 5.7, 5.8, and 5.9 respectively. The 16th, 50th and 84th percentile values of           for the selected ground motion records are also calculated to construct the percentile     curves which are also shown in the same figures (Figures 5.7, 5.8, and 5.9).   77  All the percentile results from the     curves at design level (Vancouver spectrum) ground motion intensity are summarized in Table 5.5. It is clear that, at all 3 percentile values, frames B3_0, B6_0, and B8_0 experienced the highest              compared to the steel-SMA-RC frames whereas, frames B3_3, B6_6, and B8_8 experienced the lowest           compared to the steel-RC frames. According to FEMA-356 (FEMA 356 2000) guidelines the permissible limits for drift ratio (residual / permanent) of concrete building are negligible, 1% and 4% for immediate occupancy (IO), life safety (LS) and collapse prevention (CP), respectively. It can be concluded that the           is decreasing with the replacement of SMA for its self-centering property. All values, except the median value of frames B8_0 and B8_1, and the 84th percentile value of frames B8_0, B8_1, B8_2, B8_3 and B8_4 are less than 0.5% residual drift at which people starts feeling nausea and dizziness. The greater values are reduced or completely removed by applying SMA due to its re-centering capability after removal of load at all storey levels for all 3, 6, and 8-storey frames.  The percentage difference of            for all the steel-SMA-RC frames calculated with respect to the steel-RC frames are presented in Table 5.6. The values show that the maximum reduction in the            were in frames B3_3, B6_6, and B8_8 at all 3 percentile values.   78    (a) B3_0          (b) B3_1   (c) B3_2      (d) B3_3  Figure 5.7           curves of first mode spectral acceleration    (     ), plotted against maximum residual inter-storey drift ratio,          for 3 storey    (a) B6_0          (b) B6_1   79    (c) B6_2            (d) B6_3   (e) B6_4             (f) B6_5  (g) B6_6  Figure 5.8           curves of first mode spectral acceleration    (     ), plotted against maximum residual inter-storey drift ratio,          for 6 storey   80    (a) B8_0 (b) B8_1   (c) B8_2 (d) B8_3   (e) B8_4 (f) B8_5   81    (g) B8_6 (h) B8_7  (i) B8_8  Figure 5.9           curves of first mode spectral acceleration    (     ), plotted against maximum residual inter-storey drift ratio,          for 8 storey  Table 5.5       Summary of maximum residual inter-storey drift ratio,          at design level intensity Frame ID Building ID Frame ID Building ID Frame ID Building ID 3-storey 6-storey 8-storey 16th P Median 84th P 16th P Median 84th P 16th P Median 84th P B3_0 0.15 0.25 0.40 B6_0 0.20 0.30 0.40 B8_0 0.20 0.60 0.70 B3_1 0.10 0.20 0.35 B6_1 0.15 0.29 0.35 B8_1 0.20 0.50 0.65 B3_2 0.10 0.20 0.22 B6_2 0.15 0.25 0.30 B8_2 0.20 0.40 0.65 B3_3 0.00 0.00 0.10 B6_3 0.10 0.22 0.30 B8_3 0.20 0.40 0.60     B6_4 0.05 0.10 0.25 B8_4 0.15 0.30 0.55     B6_5 0.05 0.06 0.25 B8_5 0.10 0.15 0.22     B6_6 0.00 0.01 0.10 B8_6 0.10 0.10 0.15         B8_7 0.08 0.08 0.15                 B8_8 0.00 0.01 0.05  82  Table 5.6       Percentage difference of maximum residual inter-storey drift ratio,          (w.r.to steel-RC frames) Frame ID Building ID Frame ID Building ID Frame ID Building ID 3-storey 6-storey 8-storey 16th P Median 84th P 16th P Median 84th P 16th P Median 84th P B3_0 NA NA NA B6_0 NA NA NA B8_0 NA NA NA B3_1 -33.33 -20.00 -12.50 B6_1 -25.00 -3.33 -12.50 B8_1 0.00 -16.67 -7.14 B3_2 -33.33 -20.00 -45.00 B6_2 -25.00 -16.67 -25.00 B8_2 0.00 -33.33 -7.14 B3_3 -100.00 -100.00 -75.00 B6_3 -50.00 -26.67 -25.00 B8_3 0.00 -33.33 -14.29     B6_4 -75.00 -66.67 -37.50 B8_4 -25.00 -50.00 -21.43     B6_5 -75.00 -80.00 -37.50 B8_5 -50.00 -75.00 -68.57     B6_6 -100.00 -96.67 -75.00 B8_6 -50.00 -83.33 -78.57         B8_7 -60.00 -86.67 -78.57                 B8_8 -100.00 -98.33 -92.86 5.5.4 Residual Roof Drift Ratio Considering residual roof drift ratio like residual inter-storey drift ratio along with inter-storey drift ratio and roof drift ratio is a new trend to assess the seismic performance of a building especially after an earthquake event. Figures 5.10, 5.11, and 5.12 present the results of     for 20 far-field ground motion records performed for different frame types (steel-RC and steel-SMA-RC) against the 5% damped spectral acceleration of the scaled earthquake records at the fundamental mode period of the structure,    (     ). The 16th, 50th and 84th percentile values of demand parameters for the selected ground motion records are also calculated to construct the percentile     curves which are shown in the same figures (Figures 5.10, 5.11, and 5.12).  All the percentile results from the     curves at design level (Vancouver spectrum) ground motion intensity are summarized in Table 5.7. It can be concluded that the residual roof drift ratio,         was completely removed with the replacement of steel by SMA due to its re-centering capability after removal of the load in all storey levels for all the 3, 6 and 8-storey frames at all probability levels.  83  The percentage difference of         for all the steel-SMA-RC frames calculated with respect to the steel-RC frames are presented in Table 5.8. The values show that the maximum reduction in the         were in frames B3_3, B6_6, and B8_8 at all 3 percentile values.   (a) B3_0          (b) B3_1   (c) B3_2      (d) B3_3  Figure 5.10         curves of first mode spectral acceleration    (     ), plotted against residual roof drift ratio,         for 3 storey   84    (a) B6_0          (b) B6_1   (c) B6_2            (d) B6_3   (e) B6_4             (f) B6_5   85   (g) B6_6  Figure 5.11         curves of first mode spectral acceleration    (     ), plotted against residual roof drift ratio,         for 6 storey    (a) B8_0 (b) B8_1   (c) B8_2 (d) B8_3   86    (e) B8_4 (f) B8_5   (g) B8_6 (h) B8_7  (i) B8_8  Figure 5.12         curves of first mode spectral acceleration    (     ), plotted against residual roof drift ratio,         for 8 storey     87  Table 5.7       Summary of residual roof drift ratio,         at design level intensity Frame ID Building ID Frame ID Building ID Frame ID Building ID 3-storey 6-storey 8-storey 16th P Median 84th P 16th P Median 84th P 16th P Median 84th P B3_0 0.10 0.20 0.40 B6_0 0.00 0.12 0.25 B8_0 0.00 0.25 0.30 B3_1 0.00 0.10 0.40 B6_1 0.00 0.08 0.12 B8_1 0.00 0.20 0.20 B3_2 0.00 0.00 0.20 B6_2 0.00 0.06 0.19 B8_2 0.00 0.05 0.10 B3_3 0.00 0.00 0.00 B6_3 0.00 0.00 0.05 B8_3 0.00 0.05 0.10     B6_4 0.00 0.00 0.02 B8_4 0.00 0.00 0.10     B6_5 0.00 0.00 0.01 B8_5 0.00 0.00 0.00     B6_6 0.00 0.00 0.00 B8_6 0.00 0.00 0.00         B8_7 0.00 0.00 0.00                 B8_8 0.00 0.00 0.00 Table 5.8       Percentage difference of residual roof drift ratio,         (w.r.to steel-RC frames) Frame ID Building ID Frame ID Building ID Frame ID Building ID 3-storey 6-storey 8-storey 16th P Median 84th P 16th P Median 84th P 16th P Median 84th P B3_0 NA NA NA B6_0 NA NA NA B8_0 NA NA NA B3_1 -100.00 -50.00 0.00 B6_1 0.00 -33.33 -52.00 B8_1 0.00 -20.00 -33.33 B3_2 -100.00 -100.00 -50.00 B6_2 0.00 -50.00 -24.00 B8_2 0.00 -80.00 -66.67 B3_3 -100.00 -100.00 -100.00 B6_3 0.00 -100.00 -80.00 B8_3 0.00 -80.00 -66.67     B6_4 0.00 -100.00 -92.00 B8_4 0.00 -100.00 -66.67     B6_5 0.00 -100.00 -96.00 B8_5 0.00 -100.00 -100.00     B6_6 0.00 -100.00 -100.00 B8_6 0.00 -100.00 -100.00         B8_7 0.00 -100.00 -100.00                 B8_8 0.00 -100.00 -100.00 5.5.5 Case Study: El Centro Earthquake In this section, a case study is presented considering frames B3_0 and B3_3 under the El Centro Array #11 earthquake (record no. 11 in Table 3.6) which was chosen randomly. The ground motion was scaled to   [       ] but as both frames were collapsed at    (     ) of 2.5g, the results for 0.7g and 1.3g are shown in Figures 5.13 and 5.14 showing the displacement-time histories exhibited by the second and top floors of frames B3_0 and B3_3.  When the El Centro Array #11 earthquake was scaled to 0.7g, frames B3_0 and B3_3 experienced          of 0.34% and 0.55%, and 0.25% and 0.31% in second and top floors, respectively. The maximum inter-storey drift ratio,          for frame B3_3 was 62% (second floor) and 24% (top floor) higher than B3_0 due to the lower stiffness of SMA  88  frames. The characteristics difference between these two frames was found in the maximum residual inter-storey drift ratio,          , which is shown in Figures 5.13. Frame B3_0 experienced           of 0.08% and 0.05% in the second and top floors respectively, whereas frame B3_3 experienced no      in either of the floors.  The seismic response of both frames was also investigated when the El Centro Array #11 earthquake was scaled to 1.3g (Figures 5.14). At this higher ground motion intensity, frames B3_0 and B3_3 experienced          of 0.7% and 1.9%, and 0.9% and 1.54% in the second and top floors, respectively. In this case, the          for frame B3_0 was 35-60% that of frame B3_3. This is because of the lower stiffness of SMA RC frames. Frame B3_0 experienced           of 0.7% and 0.2% in the second and top floors, respectively, whereas frame B3_3 experienced no       in any floor even though the ground motion intensity was high because of the shape recovery ability of SMA.  In both cases, though experiencing higher     , frame B3_3 performed better in reducing / completely removing the residual drift under both excitations by utilizing the re-centering capability of SMA rebars.   89              (a)          (b)             (c)          (d) Figure 5.13     Storey drift time histories of second (maximum) and top floor due to ground motion record 11 scaled for a    equal to 0.7g         90              (a)            (b)               (c)          (d) Figure 5.14     Storey drift time histories of second (maximum) and top floor due to ground motion record 11 scaled for a    equal to 1.3g  5.6 Best Distribution of SMA Rebars  In order to determine the best distribution of the SMA rebars from all the steel-SMA-RC frames without increasing the residual deformations, the simple additive weighting method (SAW) (Yoon and Hwang 1947) was used.  In this method, a score value is achieved by multiplying comparable normalized attributes by the importance weight assigned to each attribute, then these products were summed up over all the attributes to get the total score for an alternative which can be expressed as  (  )       ?                               5.1  91  where,   (  ) is the value function of alternative   , and    and     are the weight and comparable normalized attribute.  Analyzing the     results, it was observed that as the steel rebars were replaced with SMA rebars gradually from the lower to the top most level, the maximum inter-storey drift ratio (        ) and roof drift ratio (      ) were increasing, which indicates a degraded performance, (  ) of the steel-SMA-RC frames compared to the steel frames. Whereas, the maximum residual inter-storey drift ratio (         ) and residual roof drift ratio (       ) were reduced which indicated an improved performance, (  ) of the steel-SMA-RC frames compared to the steel frames. The degraded performance, (  ) of 50 percentile value of          and        in comparison to the steel frames are denoted as        and          and similarly the improved performance, (  ) of 50 percentile value of          and         in comparison to the steel frames are denoted as         and           for different steel-SMA-RC frames for all 3, 6, and 8-storey buildings which are shown in Table 5.6. It can be observed that the percentage improvement in terms of residual drift is quite significant with the replacement of SMA. The values in Table 5.6 were normalized following the linear normalization where greater    values indicate more preference and smaller    values indicate less performance. The normalized value of     can be obtained as            ?                                5.2 where,      are different attribute values and      is the maximum value of the     attribute.    attributes were transformed to    attributes by taking the inverse ratings (i.e.,     ? ).  The  92  attribute is more favorable as     approaches 1. In this case,   denotes       ,         ,        , and         . Table 5.9       Performance (%) comparison of different steel-SMA-RC frames w.r.to steel frames Building ID Frames        (%)         (%)          (%)           (%) 3-storey B3_0 0 0 0 0 B3_1 11 20 14 50 B3_2 17 20 29 100 B3_3 22 100 29 100 6-storey B6_0 0 0 0 0 B6_1 3 3 8 17 B6_2 3 17 17 50 B6_3 6 27 25 100 B6_4 11 67 33 100 B6_5 11 80 33 100 B6_6 14 97 50 100 8-storey B8_0 0 0 0 0 B8_1 3 17 11 20 B8_2 3 33 22 44 B8_3 8 33 22 52 B8_4 8 50 22 100 B8_5 10 75 24 100 B8_6 10 83 28 100 B8_7 10 87 28 100 B8_8 13 98 28 100  The importance weights,    , assigned to       ,         ,         and          were assumed as 0.25 for all the parameters (i.e. all have the same importance). Depending on the total score (Equation 4.1), ranking for different frames were done and tabulated in Table 5.7. It can be concluded that replacement of steel by SMA in all storey levels gives the best result for all the 3, 6, and 8-storey frames. Frame B3_1 performed better compared to frame B3_2; similarly frame B6_1 compared to frames B6_2, B6_3, and B6_4; and frame B8_1 compared to frames B8_2, B8_3, and B8_4. This might be because of the partial recovered residual drift as the steel in the beams of only few levels were replaced in those frames.  93  Table 5.10     Rank for different Steel-SMA-RC frames Frames Building ID Frames Building ID Frames Building ID 3-storey 6-storey 8-storey Score Rank Score Rank Score Rank B3_1 0.68 2 B6_1 0.55 3 B8_1 0.59 5 B3_2 0.59 3 B6_2 0.54 5 B8_2 0.57 7 B3_3 0.75 1 B6_3 0.53 6 B8_3 0.42 8    B6_4 0.55 4 B8_4 0.59 6    B6_5 0.58 2 B8_5 0.62 4    B6_6 0.59 1 B8_6 0.62 3       B8_7 0.63 2             B8_8 0.65 1 5.7  Conclusion  This study attempts to optimize the use of SMA as rebar due to its high cost in steel-SMA-RC frames of different heights (3, 6, and 8-storeys) by investigating the results of     in terms of maximum inter-storey drift ratio (        ), roof drift ratio (      ), maximum residual inter-storey drift ratio (         ), and residual roof drift ratio (       ). The general conclusions drawn from this study are summarized below: ? The maximum inter-storey drift ratio, (        ) increased with the replacement of steel by SMA rebars and also with the increment of the height of the frames. For all the frames of 3-storey buildings, all the results are within the limiting value of life safety (2%) at design level earthquake. However, for 6 and 8-storey frames all the results of the 16th percentile values of the maximum inter-storey drift ratio are within the range of limiting value of life safety. ? The roof drift ratio, (      ) also increased with the replacement of steel by SMA rebars and also with the increment of the height of the frames but all the results are within the limiting value of life safety (2%). ? The maximum residual inter-storey drift ratio, (         ) decreased with the replacement of steel by SMA rebars for its self-centering property. The greater  94  values of           in the steel frames are reduced or completely removed by applying SMA in all levels at all the three probability levels for all 3, 6 and 8-storey frames.  ? The residual roof drift ratio,         is completely removed with the replacement of steel by SMA in all levels for all the 3, 6 and 8-storey frames at all probability levels.  ? The replacement of steel by SMA in all storey levels gives the best result for all the 3, 6, and 8-storey frames which can reduce the repair cost and make the structure serviceable even after an earthquake event.     95   Seismic Performance Factors   Chapter  6:6.1 Introduction Response modification coefficient,   factor, system overstrength factor,    and the deflection amplification factor,    are collectively known as the seismic performance factors (    ) (FEMA P695 2009). The Response modification coefficient,   factor, can be defined as the ratio of the force level (  ) that the seismic-force-resisting system would experience if it remained entirely linearly elastic under a design earthquake ground motion to the lateral force at the base of the system (i.e., base shear,   ) (Figure 6.1).          6.1 The system overstrength factor,    can be defined as the ratio between the actual maximum strength of the fully-yielded system      and the seismic base shear required for design,   (Figure 6.1).            6.2 The deflection amplification factor,    can be defined as fractional part of   factor.           6.3 where,   is the assumed roof drift of the yielded system corresponding to design earthquake ground motions and    is the assumed roof drift of a system that remains entirely linearly elastic for design earthquake ground motion  96    Figure 6.1       Illustration of seismic performance factors (FEMA P695 (2009)) Compared to regular steel, SMA has different mechanical properties which might result in a changed structural response under seismic loads if SMAs are used as reinforcement in RC buildings. This requires calculating the changed seismic performance factors (    ). Previously some studies were done in this field. Alam et al. (2012) investigated the response modification factors (     ) of RC buildings reinforced with SMA rebars where the factors were based on Canadian standards. Ghassemieh and Kargarmoakhar (2013) evaluated the seismic response of steel braced frames employing SMA braces. The overall behavior  of the structural systems examining the response factors were studied using static pushover analysis, incremental non-linear dynamic analysis and linear dynamic analysis in terms of overstrength, ductility and response modification factors. They considered the effect of building height, number of spans and two different types of bracings: diagonal X and Chevron.  To date, no studies were reported that follow the more recent FEMA P695 (2009) guidelines.   97  The main objective of this chapter is to determine an appropriate value of the system overstrength factor,   , utilizing the results from the non-linear static pushover analysis. A trial value of the response modification coefficient,   factor, and the deflection amplification factor,   (derived from an acceptable    value considering effective damping of the structure) will be checked utilizing the incremental non-linear dynamic analysis results to provide sufficient margin against collapse under the maximum considered earthquake for the different frames evaluated in this study.  6.2 Overview of Methodology Seismic performance factors (    ) for different steel-SMA-RC frames were determined by iteration.  The initial trial value for response modification coefficient,   factor for the iteration was assumed to be 3.5 unlike the system overstrength factor,    which was determined from the measured lateral overstrength of the frames. The deflection amplification factor,    was determined from the acceptable value of   factor later on. The whole process is summarized in Figure 6.2.  98   Figure 6.2       Process for quantitatively establishing and documenting seismic performance factors (    ) (adapted from FEMA P695 (2009)) At first, structural systems of the new proposed frame of steel-SMA-RC were designed using a trial value of   factor which was considered the same as the value used in steel frames (in Chapter 3).  Non-linear structural model was developed using SeismoStruct (2012) (in Chapter 3). Non-linear static pushover analysis results were used to determine the system overstrength and period based ductility (Chapter 4) and non-linear incremental dynamic results (Chapter To evaluate performance Starting with an initial assumed trial value of   factor Design of building frames   To simulate structural collapse Development of nonlinear models  From non-linear static analyses   Calculation of system overstrength and ductility From non-linear dynamic analyses   Calculation of median collapse spectrum    Calculation of collapse margin ratio Calculation of adjusted collapse margin ratio  Calculation of acceptable values of adjusted collapse margin ratio Checking the acceptability of adjusted collapse margin ratio AACMR  OK? Calculation of deflection amplification factor  Yes Redefine system with new   factor   To PEER review Documentation of results No   99  5) were used to determine the median collapse spectrum in this chapter to evaluate the      . The adequacy of the trial value of  the   factor was validated by checking the limiting value of the adjusted collapse margin ratio (discussed in the subsequent section) suggested by FEMA P695 (2009).  If the   factor does not satisfy the acceptable value of the adjusted collapse margin ratio, then a new trial value of   factor was assumed and the whole procedure was repeated until a satisfactory trial value of   factor was obtained. Then, the deflection amplification factor,    will be determined using the acceptable value of   factor.  6.3 Performance Group Performance group was identified based on the basic structural configuration (varying building height-3, 6, and 8-storeys), seismic design category (design earthquake, (DE) considered for all the frames was the city of Vancouver hazard spectra which fall in the maximum seismic design category SDC Dmax) and period domain (short and long period systems). The systems were designed based on the design earthquake (DE) but the collapse assessment was done based on the maximum considered earthquake (MCE) ground motion.   Eigen value analysis were performed using SeismoStruct (2012) to obtain the first modal time period    for each frames of steel-RC and steel-SMA-RC.T hen that time period was used to determine the corresponding spectral acceleration,    of that frame from Vancouver hazard spectra (design spectra). Table 6.1 presents the performance group summary along with the values of the time period with their corresponding spectral acceleration,   and     for DE and MCE, respectively of different frames. It can be observed that as the steel is being replaced by SMA, the fundamental period of the frames increased. This is because of the lower modulus of elasticity of the SMA rebars, which decreased the stiffness and  100  increased the time period of a structure, thus decreasing the corresponding spectral acceleration.  Table 6.1       Performance group summary  Gr. No. Frame ID Basic Config. Design Load Level Key Archetype Design Parameter No. of Storey Seismic      (sec.) Period Domain    (T1) (g)      (g) PG-1 B3_0 3 SDC Dmax 3.5 0.332 Short 0.820 0.820 PG-2 B3_1 3 SDC Dmax 3.5 0.334 Short 0.820 0.820 B3_2 0.336 0.800 0.800 B3_3 0.336 0.800 0.800 PG-3 B6_0 6 SDC Dmax 3.5 0.696 Long 0.528 0.528 PG-4 B6_1 6 SDC Dmax 3.5 0.696 Long 0.528 0.528 B6_2 0.701 0.516 0.516 B6_3 0.705 0.516 0.516 B6_4 0.706 0.516 0.516 B6_5 0.706 0.516 0.516 B6_6 0.706  0.516 0.516 PG-5 B8_0 8 SDC Dmax 3.5 0.889 Long 0.404 0.404 PG-6 B8_1 8 SDC Dmax 3.5 0.895 Long 0.404 0.404 B8_2 0.900 0.392 0.392 B8_3 0.900 0.392 0.392 B8_4 0.910 0.379 0.379 B8_5 0.910 0.379 0.379 B8_6 0.910 0.379 0.379 B8_7 0.910 0.379 0.379 B8_8 0.910 0.379 0.379 6.4 Median Collapse Intensity and Collapse Margin Ratio Collapse level ground motion is defined as the median collapse when a seismic-force-resisting system experiences an intensity causing some sort of life-threatening collapse in one half of the earthquake records considered. Collapse margin ratio,    , which is the primary parameter to characterize collapse assessment can be defined as the ratio between the 5%-damped median spectral acceleration of the collapse level ground motions,  ?   (or corresponding displacement,     ) and the 5%-damped spectral acceleration of the maximum considered earthquake (MCE),     (or corresponding displacement,     ) measured at the fundamental period of the system. MCE ground motion would cause less  101  probability of collapse as MCE ground motions are less than the collapse level ground motions. The probability of exceedance of Vancouver hazard spectrum is 2% in 50 years NBCC (2005). For this reason     was kept equal to DE spectral acceleration,    instead of multiplying with 1.5. Summary of collapse results are presented in Table 6.2.        ?               6.4 Table 6.2       Summary of collapse results Gr. No. Frame ID Basic Config. Design Load Level  IDA results No. of Storey Seismic  ?   (g)     (g)        PG-1 B3_0 3 SDC Dmax 2.24 0.820 2.73 PG-2 B3_1 3 SDC Dmax 2.34 0.820 2.85 B3_2 2.34 0.800 2.93 B3_3 2.35 0.800 2.94 PG-3 B6_0 6 SDC Dmax 1.77 0.528 3.35 PG-4 B6_1 6 SDC Dmax 1.80 0.528 3.41 B6_2 1.80 0.516 3.49 B6_3 1.81 0.516 3.51 B6_4 1.82 0.516 3.53 B6_5 1.82 0.516 3.53 B6_6 1.82 0.516 3.53 PG-5 B8_0 8 SDC Dmax 2.56 0.404 6.34 PG-6 B8_1 8 SDC Dmax 2.92 0.404 7.23 B8_2 2.92 0.392 7.45 B8_3 2.93 0.392 7.47 B8_4 2.93 0.379 7.73 B8_5 2.93 0.379 7.73 B8_6 3.00 0.379 7.92 B8_7 3.00 0.379 7.92 B8_8 3.00 0.379 7.92  A typical single seismic-force-resisting system subjected to full set of ground motions with varying intensities is shown in Figure 6.3 (a) and (c) in which spectral intensity of the ground motion and the maximum storey drift ratio are plotted in vertical and horizontal axes, respectively. Structural collapse was predicted based on the lateral dynamic instability or  102  excessive lateral displacements from the dynamic analysis results. To calculate the value of median collapse accelerations, fragility curves were developed relating the ground motion intensity to the probability of collapse through a cumulative distribution function (   ) using the collapse data obtained from     results (Ibarra et al. 2002). A sample fragility curve is shown in Figure 6.3 (b) and (d) by fitting a lognormal distribution to the collapse data.   Porter et al. (2007) proposed 4 different methods to develop fragility curves: (i) all frames collapsed at collapse level spectral acceleration, (ii) some frames collapsed and the corresponding collapse spectral acceleration is known, (iii) no specimens are collapsed, collapse level spectral acceleration is known, (iv) derived fragility functions. The first method was adopted in this study which considered the collapse accelerations for all the ground motions to develop fragility curves. In this method    (   ) or          (   ) represents the fragility function for damage state    (here maximum inter storey drift ratio,          was considered), given in Equation 6.5. The value of the spectral acceleration at which the          is too high (i.e. causing collapse) is idealized by a lognormal distribution. This distribution was used as it fits a variety of building collapse by     (Cornell et al. 2005).              (   )    (      (    ?  ? ) )     6.5    103  where,     is the standard normal cumulative distribution function (e.g., NORM. S. DIST. in Microsoft Excel),   ?   is the median value of the distribution which can be calculated using Equation 6.6, and    is the logarithmic standard deviation which can be calculated using Equation 6.7.  ?      (  ?         ) 6.6    ?     ?(  (    ?  ? ))      6.7 where,   is the number of ground motions considered to collapse (here,   = 20)   is the index of ground motions     is the value of the spectral acceleration at which the          is too high (collapse occurred) for ground motion record set  .  For example, in Figure 6.3 the spectral intensities,  ?   of 2.24g and 2.35g are called the median collapse accelerations since half of the record set causes collapse to frames B3_0 and B3_3 respectively.       104    (a) B3_0 (b) B3_0   (c) B3_3 (d) B3_3 Figure 6.3       Incremental dynamic analysis to collapse, showing the maximum considered earthquake ground motion intensity,     median collapse capacity intensity,  ?    and collapse margin ratio,     for 3-storey frames 6.5 Adjusted Collapse Margin Ratio The collapse margin ratio,   , is converted to an adjusted collapse margin ratio,      for each building,  , in order to account for the unique characteristics of extreme ground motion (spectral shape). It was calculated using spectral shape factor,    , which depends on the system ductility (ductility,  ) and period of vibration (fundamental period,  ) and is calculated from the tabulated values in FEMA P695 (2009). The system having longer period of vibration and larger ductility is benefited by larger adjustment. The calculated  105  values of adjusted collapse margin ratio, (     )    for all 20 frames are shown in Table 6.3.                   6.8 6.5.1 Evaluation of Collapse Margin Ratio and Acceptance Criteria  The adjusted collapse margin ratio is compared to the acceptable criteria which represent the collapse uncertainty. If the value is large enough, then the structure is safe with a less probability (10% for average and 20% for individual according to FEMA P695 2009) of collapse at MCE level ground motions and the assumed value of   factor is acceptable. If not, then a new value of    factor needs to be assumed for the next trial.  Acceptable criteria for the probability collapse of a structural system at the maximum considered earthquake, MCE ground motions is limited to 20% or less, which was set based on judgement and total system collapse uncertainty,     . The total system collapse uncertainty is a function of record-to-record (   ) uncertainty, design requirements-related (  ) uncertainty, test data related (  ) uncertainty, and modeling (   ) uncertainty and was assumed to be 0.5 for medium quality ratings for this study.  According to FEMA P695 (2009) acceptable performance is identified by the following two basic criteria of collapse prevention: ? The average probability of collapse across a performance group for MCE ground motions is approximately 10%, or less. ? The probability of collapse of an individual archetype,   within a performance group for MCE ground motions is approximately 20%, or less.  106  Therefore, in this study acceptable performance was achieved when the following two criteria were satisfied: ? Average value of calculated collapse margin ratio,      ?? ?? ?? ??    for the corresponding performance group exceeds        :      ?? ?? ?? ??   >         6.9 ? Collapse margin ratio for an individual frame,   with a performance group,       exceeds        :       >         6.10 The acceptable value of collapse margin ratio at 10% and 20% collapse probability,        and        , based on          and were 1.9 and 1.52 respectively from the tabulated values in FEMA P695 (2009). From of results shown in Table 6.3, it can be concluded that all the values of the calculated collapse margin ratio are within the range of acceptable criteria. Not only the mean average value      ?? ?? ?? ??    of performance group for steel-SMA-RC frames are within the        , but also each archetype in the performance groups satisfies the criteria for both         and        . This indicates that the probability of collapse for all the frames is low at MCE ground motion, especially for 8-storey buildings as the calculated value of       for all the frames are much higher than the required value mentioned in the code. Steel-SMA-RC frames experienced 4%-17% lower probability of collapse compared to the steel-RC frames. All those results indicate that it is safe to use SMA rebars in the buildings.    107  Table 6.3       Summary of final collapse margins and comparison to acceptance criteria Group No. Frame ID Basic Config. Computed overstrength and collapse margin parameters Acceptance Check No. of Storey                    (calculated)       (assume)     (code) Pass/Fail PG-1 B3_0 3 2.73 2.67 1.16 3.17 0.5 1.90 Pass PG-2 B3_1 3 2.85 2.50 1.16 3.31 0.5 1.52 Pass B3_2 2.93 2.43 1.15 3.36 0.5 1.52 Pass B3_3 2.94 2.42 1.15 3.38 0.5 1.52 Pass Mean of performance group 2.91 2.45 1.15 3.34 0.5 1.90 Pass PG-3 B6_0 6 3.35 2.76 1.20 4.02 0.5 1.90 Pass PG-4 B6_1 6 3.41 2.63 1.19 4.06 0.5 1.52 Pass B6_2 3.49 2.50 1.18 4.12 0.5 1.52 Pass B6_3 3.51 2.40 1.17 4.10 0.5 1.52 Pass B6_4 3.53 2.35 1.17 4.13 0.5 1.52 Pass B6_5 3.53 2.33 1.17 4.13 0.5 1.52 Pass B6_6 3.53 2.32 1.17 4.13 0.5 1.52 Pass Mean of performance group 3.50 2.42 1.18 4.13 0.5 1.90 Pass PG-5 B8_0 8 6.34 2.38 1.20 7.60 0.5 1.90 Pass PG-6 B8_1 8 7.23 2.29 1.19 8.60 0.5 1.52 Pass B8_2 7.45 2.22 1.19 8.86 0.5 1.52 Pass B8_3 7.47 2.21 1.18 8.82 0.5 1.52 Pass B8_4 7.73 2.21 1.18 9.12 0.5 1.52 Pass B8_5 7.73 2.10 1.18 9.26 0.5 1.52 Pass B8_6 7.92 2.00 1.17 9.26 0.5 1.52 Pass B8_7 7.92 1.91 1.16 9.18 0.5 1.52 Pass B8_8 7.92 1.88 1.16 9.18 0.5 1.52 Pass Mean of performance group 7.67 2.10 1.18 9.05 0.5 1.90 Pass 6.6 Evaluation of the Deflection Amplification Factor The deflection modification factor,    is calculated from the reduced value of acceptable   factor by damping factor,    corresponding to the system archetype damping.          6.11  For this study, the inherent damping was assumed to be 5% of the critical damping value which gives a corresponding value of damping factor,     . This resulted with values of    being the same as those of the   factor.  108  6.7 Conclusion  A total of 20 different steel-RC and steel-SMA-RC frames were analyzed based on the methodology presented in FEMA P695 (2009) using non-linear static pushover analysis and nonlinear incremental dynamic analysis. For this study, two parameters were considered: different building heights (3, 6, and 8-storeys) and gradual replacement of steel by SMA starting from level 1 to all level in the plastic hinge region of the beams only. This gave a total of 20 different steel-RC and steel-SMA-RC frames to be considered. For all the frames the columns were reinforced with the regular reinforcement.  It can be concluded that all 3 steel-RC and 17 steel-SMA-RC frames of three different storeys met the FEMA P695 (2009) acceptance criteria. The proposed seismic performance factor,     (     ) was acceptable as it provided a satisfactory margin of safety against collapse when subjected to the maximum considered earthquake ground motions.  Steel-SMA-RC frames experienced 4%-17% lower probability of collapse compared to the steel-RC frames.    109   Conclusion Chapter  7:7.1 Summary This study investigates the potentiality of applying SMA as rebars in the RC buildings along with the determination of best distribution of SMA due to its high cost. This study provides the state-of-art on the outstanding characteristics, mechanical properties, constitutive modeling of SMA along with their different applications in RC buildings. Furthermore, this study determines the seismic vulnerability of steel-RC and steel-SMA-RC frames in terms of both inter-storey drift and residual drift and checks the acceptability of proposed seismic performance factors following FEMA P695 (2009) methodology.  7.2 Limitations Despite the extensive work presented, the study did not address all possible scenarios or variables. Therefore, the limitations of the current study are ? The ductility was calculated using equivalent bilinear model instead of following  FEMA P695 (2009). This simplified response idealization is well representative for single degree of freedom system (SDOF) which can dissipate energy in a stable manner. Whereas for multiple degree of freedom systems (MDOF) which exhibit significant strength degradation, the definition of the effective yield displacement is more complicated and this simple equivalent bilinear model may not be very reliable in calculating ductility (Annan et al. 2009).  ? For developing fragility curves only one component of 20 ground motions records were considered but according to FEMA P695 (2009) two components of these 20 ground motions should have been considered.   110  ? To develop performance group having all possible frames varying maximum and minimum seismic load, high and low gravity loads were not considered for assessing the acceptability of seismic performance factors as mentioned in FEMA P695 (2009).   ? Damping value was assumed to be 5% of the critical value for all frames. 7.3 Conclusions In this study two parameters were varied: building height (3, 6, and 8-storey) and replacement of steel by SMA starting from level 1 to all level in the plastic hinge region of the beams only. For all the frames the columns were reinforced with the regular reinforcement. The frames were designed according to the CSA standards (A23.3-04 2004). The frames were assumed to be located in the high seismic zone in Vancouver. In this study the best distribution of SMA as rebar was determined due to its high cost.  7.3.1 Nonlinear Static Pushover Analyses ? Frames with SMA bars have 3%-6% less load carrying capacity and 12%-22% less ductility compare to the frames with steel.  ? The frame deformations show that the collapse for      varied from 4.20% to 4.90% which is conservative compared to the 4% value proposed by FEMA 356 (2000). ? By investigating the damage scheme, it was observed that using SMA bars led to a better collapse mechanism during the pushover analyses. Percentage of crushed beams and columns in frames with SMA bars was 19%-20% lower compared to the frames with steel bars.   111  ? Frames with SMA bars experience 7%-26% higher      compared to the frames with steel with the replacement of steel by SMA. 7.3.2 Nonlinear Dynamic Incremental Analyses The recently popular and demanding nonlinear dynamic analyses known as incremental dynamic analyses (   ) were used in this study to arrest a wide range of earthquake records varying milder and severe ones using finite element software considering twenty far-field earthquake records matched with the Vancouver spectrum. The results were reported in terms of maximum inter-storey drift ratio (        ), roof drift ratio (      ), maximum residual inter-storey drift ratio (         ), and residual roof drift ratio (       ) plotting different     curves. A statistical assessment developing 16th, 50th (median), and 84th percentile     curves was also conducted for different steel-RC and steel-SMA-RC frames. And finally, depending on all these results the best distribution of SMA rebars was determined using simple additive weighting (SAW) method.  The general conclusions drawn from this study are summarized below: ? The maximum inter-storey drift ratio, (        ) increased with the replacement of steel by SMA rebars and also with the increment of the height of the frames.  ? The roof drift ratio, (      ) also increased with the replacement of steel by SMA rebars and also with the increment of the height of the frames but all the results are within the limiting value of life safety (2%). ? The maximum residual inter-storey drift ratio, (         ) decreased with the replacement of steel by SMA rebars for its self-centering property. The greater  112  values are reduced or completely removed by applying SMA in all level at all three probability levels for all 3, 6 and 8-storey frames.  ? The residual roof drift ratio,         is completely removed with the replacement of steel by SMA in all levels for all 3, 6 and 8-storey frames for all probability levels.  ? Replacement of steel by SMA in all storey levels gives the best result for all 3, 6, and 8-storey frames which can reduce the repair cost and make the structure serviceable even after an earthquake.  7.3.3 Seismic Performance Factors  Seismic performance factors, (    ) including the acceptability of response modification coefficient,   factor and determining appropriate values of system overstrength factor,    and the deflection amplification factor,    of total 20 different steel-RC and steel-SMA-RC frame archetypes were conducted based on the methodology in FEMA P695 (2009) using the results from nonlinear static pushover analyses (   ) and nonlinear incremental dynamic analyses (   ). The general conclusions drawn from this study are summarized below: ? The proposed seismic performance factor,     (     ) was acceptable for all 20 frames of steel-RC and steel-SMA-RC frames as it provided a satisfactory margin against collapse when subjected to the maximum considered earthquake ground motions. ? Steel-SMA-RC frames experienced 4%-17% lower probability of collapse compared to the steel-RC frames.  113  7.4 Recommendations for Future Research This study only focused on the application of SMA in the plastic hinge region of the beams but further study can be conducted applying SMA rebars in the columns or combination of beams and columns after identifying the locations of collapse beforehand using nonlinear static pushover analysis and nonlinear dynamic time history analysis. Then the best location can be chosen based on better seismic performance.   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Journal of Intelligent Material Systems and Structures, 19(6), 631?639.     128  Appendices Appendix A  Limit States Results A.1 3-storey          (a) B3_0 (b) B3_1   (c) B3_0 (d) B3_1       Core Concrete Crushing Steel Yielding SMA Yielding  129           (e) B3_2 (f) B3_3   (g) B3_2 (h) B3_3 Figure A.1      Limit states results for 3-storey Core Concrete Crushing Steel Yielding SMA Yielding  130  A.2 6-storey          (a) B6_0 (b) B6_1   (c) B6_0 (d) B6_1 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 3 15 21 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 6 21 23 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage Core Concrete Crushing Steel Yielding SMA Yielding  131           (e) B6_2 (f) B6_3   (g) B6_2 (h) B6_3 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 6 21 23 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 6 20 22 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage Core Concrete Crushing Steel Yielding SMA Yielding  132           (i) B6_4 (j) B6_5   (k) B6_4 (l) B6_5 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 9 22 25 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 10 26 28 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage Core Concrete Crushing Steel Yielding SMA Yielding  133          (m) B6_6  (n) B6_6 Figure A.2      Limit states results for 6-storey     0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 10 27 30 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage Core Concrete Crushing Steel Yielding SMA Yielding  134  A.3 8-storey          (a) B8_0 (b) B8_1   (c) B8_0 (d) B8_1 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 10 19 22 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%)   5 25 28 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage Core Concrete Crushing Steel Yielding SMA Yielding  135           (e) B8_2 (f) B8_3   (g) B8_2 (h) B8_3 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 5 27 31 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 5 22 26 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage Core Concrete Crushing Steel Yielding SMA Yielding  136           (i) B8_4 (j) B8_5   (k) B8_4 (l) B8_5 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 4 23 27 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 5 20 23 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage Core Concrete Crushing Steel Yielding SMA Yielding  137           (m) B8_6 (n) B8_7   (o) B8_6 (p) B8_7 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 1 19 22 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage 0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 1 19 22 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage Core Concrete Crushing Steel Yielding SMA Yielding  138          (q) B8_8  (r) B8_8 Figure A.3      Limit states results for 8-storey  0200400600800100012000 1 2 3 4 5 6 7 8Base Shear (kN) ISDR_F5 (%) 1 15 17 Minor Damage Major Damage Extensive Damage Collapse  Limit Elastic  Stage Core Concrete Crushing Steel Yielding SMA Yielding 

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