Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Rodriguez Barriga, Jhordy Ismael
Master of Applied Science - MASc
University of British Columbia
Seismic performance assessment is of critical importance for structures in regions where earthquakes are prevalent. Identifying the inelastic capacity of structures undergoing seismic loads is one of the main concerns in seismic structural design. Such assessment in terms of determining the inelastic capacity of structures is often performed in laboratories at universities, government, and industrial facilities through quasi-static cyclic loading test programs. In these programs, selecting an appropriate loading protocol is crucial for achieving an accurate assessment of the inelastic capacity. Appropriate loading protocols need to be representative of the seismic demands to which a structural component may be subjected during seismic events. A standard loading protocol available in the literature will not necessarily yield a meaningful response. For the quasi-static cyclic test program, a loading protocol should be developed specifically to the site, structural component, and system of the structure. In this study, an Incremental Dynamic Analysis-based approach is introduced to develop component-specific quasi-static cyclic loading protocols, which is illustrated for a single-column reinforced concrete bridge bent in Vancouver, British Columbia. The sensitivity of the development procedure to different design parameters (e.g., mechanical, and geometric properties) was evaluated through a factorial analysis to propose a single cyclic loading protocol for several bridge pier designs. Here, target displacement ductility demand levels of two, four, and eight, and crustal, subcrustal, and subduction earthquakes were considered. The number of inelastic cycles and cumulative ductility damage were the primary target demand parameters in the loading protocol development. Conventional loading protocols were found unrealistically more damaging than those proposed for crustal and subcrustal earthquakes. The proposed loading protocols for subduction earthquakes were consistent with those developed by other researchers for the same but employed the classical constant ductility design approach to achieve the target displacement ductility levels, which is unrealistic.
QUASI-STATIC CYCLIC LOADING PROTOCOL DEVELOPMENT FOR THE SEISMIC PERFORMANCE ASSESSMENT OF BRIDGE PIERS by Jhordy Ismael Rodríguez Barriga B.Sc., Monterrey Institute of Technology and Higher Education, Mexico, 2018 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) March 2021 © Jhordy Ismael Rodríguez Barriga, 2021 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis/dissertation entitled: QUASI-STATIC CYCLIC LOADING PROTOCOL DEVELOPMENT FOR THE SEISMIC ASSESSMENT OF BRIDGE PIERS submitted by Jhordy Ismael Rodríguez Barriga in partial fulfillment of the requirements of the degree of Master of Applied Science. Dr. Shahria Alam Supervisor Dr. Ahmad Rteil Supervisory Committee Member Dr. Hewage Kasun Supervisory Committee Member Dr. Kenneth Chau University Examiner iii ABSTRACT Seismic performance assessment is of critical importance for structures in regions where earthquakes are prevalent. Identifying the inelastic capacity of structures undergoing seismic loads is one of the main concerns in seismic structural design. Such assessment in terms of determining the inelastic capacity of structures is often performed in laboratories at universities, government, and industrial facilities through quasi-static cyclic loading test programs. In these programs, selecting an appropriate loading protocol is crucial for achieving an accurate assessment of the inelastic capacity. Appropriate loading protocols need to be representative of the seismic demands to which a structural component may be subjected during seismic events. A standard loading protocol available in the literature will not necessarily yield a meaningful response. For the quasi-static cyclic test program, a loading protocol should be developed specifically to the site, structural component, and system of the structure. In this study, an Incremental Dynamic Analysis-based approach is introduced to develop component-specific quasi-static cyclic loading protocols, which is illustrated for a single-column reinforced concrete bridge bent in Vancouver, British Columbia. The sensitivity of the development procedure to different design parameters (e.g., mechanical, and geometric properties) was evaluated through a factorial analysis to propose a single cyclic loading protocol for several bridge pier designs. Here, target displacement ductility demand levels of two, four, and eight, and crustal, subcrustal, and subduction earthquakes were considered. The number of inelastic cycles and cumulative ductility damage were the primary target demand parameters in the loading protocol development. Conventional loading protocols were found unrealistically more damaging than those proposed for crustal and subcrustal earthquakes. The proposed loading protocols for subduction earthquakes were consistent with those developed by other researchers for the same but employed the classical constant ductility design approach to achieve the target displacement ductility levels, which is unrealistic. iv LAY SUMMARY One of the biggest concerns of quasi-static cyclic tests is the ability to accurately represent the inelastic demands from dynamic tests. Moreover, it is common for experimentalists to select cyclic loading protocols without regard to their development background and applicability. This work presents a relatively simple approach that leads to the development of suitable quasi-static cyclic loading protocols for bridge pier columns, and that can also be implemented for different structural systems. Several bridge pier column designs were evaluated to propose a single protocol along with the recommendations of its selection and applicability. The implementation of component-specific cyclic loading protocols might result in improved structural design and retrofit solutions. v PREFACE The findings presented in this thesis are based on original work carried out at the University of British Columbia (Okanagan) by the author. The production of the entire work, including literature review, collection and analysis of data, numerical analysis, and thesis writing was the sole responsibility of the author. Ph.D. candidate Saif Aldabagh provided guidance and discussion time throughout the entire work and contributed to the preparation of journal and conference manuscripts. Dr. Shahria Alam provided guidance and supervision throughout the entire process. Portions of Chapters 2, 3, 4 and 5 were submitted to peer-reviewed conferences and journals for publication, as listed below: ▪ Rodríguez, J., Aldabagh, S., and Alam, M.S. (2021). “Quasi-static Loading Protocols for Bridge Piers”. Journal of Bridge Engineering (To be submitted). ▪ Rodríguez, J., Aldabagh, S., and Alam, M.S. (2021). “Incremental Dynamic Analysis-Based Procedure for the Development of Loading Protocols”. Journal of Bridge Engineering (Under review). ▪ Rodríguez, J., Aldabagh, S. and Alam, M.S. (2020). “Development of Loading Protocols for Bridge Piers Considering Subduction Earthquakes”. Canadian Society of Civil Engineers Annual Conference 2020. (Accepted) vi TABLE OF CONTENTS ABSTRACT .................................................................................................................................. iii LAY SUMMARY ......................................................................................................................... iv PREFACE .......................................................................................................................................v TABLE OF CONTENTS ............................................................................................................ vi LIST OF TABLES ....................................................................................................................... ix LIST OF FIGURES ..................................................................................................................... xi LIST OF ABBREVIATIONS .....................................................................................................xv ACKNOWLEDGEMENTS ..................................................................................................... xvii DEDICATION.......................................................................................................................... xviii CHAPTER 1: INTRODUCTION .................................................................................................1 1.1 Background ................................................................................................................... 1 1.2 Research Objectives ...................................................................................................... 3 1.3 Thesis Layout ................................................................................................................ 4 CHAPTER 2: LITERATURE REVIEW ....................................................................................5 2.1 Seismic Design Provisions in the Canadian Highway Bridge Design Code (CHBDC) 5 2.3 Seismicity of Southwestern British Columbia .............................................................. 9 2.4 Methodologies for Predicting the Seismic Response of RC Bridge Piers .................. 11 2.5 Cumulative Damage.................................................................................................... 14 2.6 Experimental Seismic Assessment of RC Bridge Piers .............................................. 16 CHAPTER 3: PROPOSED PROCEDURE METHODOLOGY ............................................39 vii 3.1 General Methodology and Implications...................................................................... 39 3.2 Proposed Procedure .................................................................................................... 40 CHAPTER 4: CASE STUDY .....................................................................................................45 4.1 Selection of Ground Motion Records ......................................................................... 45 4.2 Design and Geometry of the Example Prototype Bridge Pier .................................... 48 4.3 Numerical Model of the Bridge Pier ........................................................................... 51 4.4 Cumulative Damage Model ........................................................................................ 54 4.5 Target Response - Incremental Dynamic Analysis ..................................................... 55 4.6 Cycle Counting ........................................................................................................... 58 4.7 Construction of Loading Protocols ............................................................................. 60 4.8 Proposed Testing Protocols......................................................................................... 63 CHAPTER 5: SENSITIVITY ANALYSIS ................................................................................67 5.1 Preliminary Screening of Bridge Engineering Design Parameters ............................. 67 3.1 Determination of Inelastic Cycles and Total Damage Demands ................................ 71 3.1 Proposed Testing Protocols......................................................................................... 75 3.1 Comparison of the Proposed Loading Protocols ........................................................ 79 CHAPTER 6: CONCLUSIONS .................................................................................................84 6.1 Summary and Conclusions ............................................................................................. 84 6.2 Limitations ...................................................................................................................... 85 6.3 Research Contribution .................................................................................................... 86 6.4 Future Research .............................................................................................................. 86 BIBLIOGRAPHY ........................................................................................................................87 viii APPENDICES ..............................................................................................................................93 Appendix A: RC bridge pier displacement-based fibre model example for Opensees ........ 93 Appendix B: Rain-flow cycle counting algorithm MATLAB code ..................................... 95 ix LIST OF TABLES Table 1. Performance levels in CHBDC (CSA S6 2019) ............................................................... 7 Table 2. Typical bridge classes listed in the BC bridge inventory (Siddiquee and Alam, 2017) ... 8 Table 3. Bridge superstructure system listed in the BC bridge inventory (Siddiquee and Alam, 2017) ............................................................................................................................................... 8 Table 4. Bridge pier type classification (Siddiquee and Alam, 2017) ............................................ 9 Table 5. Sequence of loading of FEMA protocol ......................................................................... 20 Table 6. Sequence of loading of the ECCS protocol .................................................................... 22 Table 7. Sequence of loading of the ATC-24 protocol ................................................................. 24 Table 8. Sequence of loading of the SAC protocol ...................................................................... 25 Table 9. Sequence of loading of the JISF loading protocol .......................................................... 26 Table 10. Sequence of loading of the AISC loading protocols ..................................................... 27 Table 11. Sequence of loading of the FCC loading protocol ........................................................ 29 Table 12. Sequence of loading of the ASTM 564 loading protocol ............................................. 30 Table 13. Sequence of loading of the ISO loading protocol ......................................................... 31 Table 14. Sequence of loading of the CUREE-Caltech loading protocol..................................... 32 Table 15. Sequence of loading of the New Zealand protocol ....................................................... 34 Table 16. Sequence of loading of the Modified New Zealand loading protocol .......................... 35 Table 17. Sequence of loading of the ACI 374.2R loading protocol ............................................ 36 Table 18. Ground motions suites .................................................................................................. 46 Table 19. Natural frequency and elastic period of the bridge pier ................................................ 49 Table 20. Moment-Curvature analysis results .............................................................................. 50 Table 21. Material properties ........................................................................................................ 53 x Table 22. Scaling factor extracted from the IDA analysis. ........................................................... 57 Table 23. Target values for inelastic cycles and CDD for the prototype bridge pier ................... 61 Table 24. Case study loading protocol for crustal and subcrustal earthquakes. ........................... 65 Table 25. Case study loading protocol for subduction earthquakes ............................................. 66 Table 26. Level of factors ............................................................................................................. 68 Table 27. Results from the preliminary factorial analysis ............................................................ 70 Table 28. Final experiment design ................................................................................................ 71 Table 29. Structural properties of all the factor combinations ...................................................... 72 Table 30. Average values for the number of inelastic cycles ....................................................... 74 Table 31. Proposed inelastic sequence of loading for crustal earthquakes. .................................. 76 Table 32. Proposed inelastic sequence of loading for subcrustal earthquakes. ............................ 77 Table 33. Proposed inelastic sequence of loading for subduction earthquakes. ........................... 78 Table 34. Comparison of conventional and proposed protocols................................................... 83 xi LIST OF FIGURES Figure 1. Seismotectonic of the Pacific Northwest ....................................................................... 10 Figure 2. Uniform radial discretization of a bridge circular column ............................................ 13 Figure 3. Bridge pier column discretization scheme .................................................................... 14 Figure 4. Shake table test illustration ............................................................................................ 16 Figure 5. Quasi-static uniaxial cyclic test illustration ................................................................... 17 Figure 6. FEMA loading protocol ................................................................................................. 20 Figure 7. ECCS loading protocol .................................................................................................. 22 Figure 8. ATC-24 loading protocol .............................................................................................. 23 Figure 9. SAC loading protocol .................................................................................................... 24 Figure 10. JISF loading protocol .................................................................................................. 25 Figure 11. AISC loading protocols: a) beam-to-column connections b) link-to column connections ................................................................................................................................... 27 Figure 12. FCC loading protocols ................................................................................................. 28 Figure 13. ASTM E564 loading protocol: a) static load and b) cyclic loading ............................ 30 Figure 14. ISO loading protocol ................................................................................................... 31 Figure 15. CUREE-Caltech loading protocol ............................................................................... 32 Figure 16. New Zealand protocol ................................................................................................. 33 Figure 17. Modified New Zealand loading protocol .................................................................... 35 Figure 18. ACI 374.2R loading protocol ...................................................................................... 36 Figure 19. Duscika and Bazaez loading protocol ......................................................................... 38 Figure 20. Flow chart of the methodology for the development of loading protocols ................. 42 Figure 21. Mean response spectrums matched to the NBCC spectrum for 2% in 50 years ......... 47 Figure 22. Bridge bent: (a) configuration and geometry, (b) cross-section .................................. 49 xii Figure 23. Bridge pier analysis: a) moment-curvature and b) pushover curves ........................... 51 Figure 24. Numerical model of the case study prototype bridge pier. .......................................... 52 Figure 25. Comparison of the experimental and numerical time-history displacement responses: a) EQ1 and b) EQ3 ........................................................................................................................ 54 Figure 26. IDA-curve example for the three target ductility scaling factors. ............................... 56 Figure 27. Illustrative procedure for the simplified rain-flow counting: (a) inelastic ductility response history; (b) pre-peak inelastic response history; (c) ordered cycles including mean effect; (d) final ordered cycles. ..................................................................................................... 59 Figure 28. Comparison of target demand parameters: (a) inelastic cycles; (b) CDD ................... 62 Figure 29. Grouping of pre-peak inelastic cycles (N) for (a) crustal and subcrustal earthquakes and (b) subduction earthquakes at target displacement ductility level of 4. ................................. 63 Figure 30. Crustal and subcrustal case study loading protocol ..................................................... 65 Figure 31. Subduction case study loading protocol ...................................................................... 66 Figure 32. Numerical model (discretization of the bridge columns) ............................................ 69 Figure 33. Proposed testing protocols for crustal earthquakes: target ductility of a) 2, b) 4 and c) 8..................................................................................................................................................... 76 Figure 34. Proposed testing protocol for subcrustal earthquakes: target ductility of a) 2, b) 4 and c) 8 ................................................................................................................................................ 77 Figure 35. Proposed testing protocol for subduction earthquakes: target ductility of a) 2, b) 4 and c) 8 ................................................................................................................................................ 78 Figure 36. Comparison of protocols considering a) crustal earthquakes: b) subcrustal earthquakes; and c) subduction earthquakes. ................................................................................ 82 xiii LIST OF SYMBOLS 𝐴𝑔 = gross cross-sectional area 𝑎𝑖 = amplitude of the FEMA loading protocol cycles 𝑎𝑛 = largest amplitude of the FEMA loading protocol cycles C = type of component parameter c = failure mode parameter D = diameter of the column 𝐷𝑝𝑎 = cumulative dissipated energy 𝑑𝑏 = longitudinal rebar diameter 𝑑𝐸 = dissipated hysteretic energy 𝑑𝑖 = standardized residuals 𝑒𝑖 = residual value F = Fisher–Snedecor test F0 = F-test statistics 𝑓𝑦 = steel yield strength 𝑓′𝑐 = concrete compressive strength GPa = giga-pascals units H/D = aspect ratio 𝐼𝐸 = importance factor 𝐾𝑆𝐸 = rotational spring stiffness kN = kilo-newtons unit L = height of the pier 𝑙𝑠𝑝 = effective strain penetration length m = metres unit mm = millimetres unit M = earthquake moment magnitude MPa = mega-pascals units 𝑀𝑁 = analytical nominal moment 𝑀𝑝 = plastic moment 𝑀𝑦′ = first yield moments xiv N = number of cycles P = axial load P/Po = axial load ratio 𝑄𝑦 = specimen yield strength r = concrete cover R = modification factor 𝑅𝑝 = closest distance to fault rupture t0 = t-test statistics 𝑉𝑖 = specimen theoretical strength α = parameter describing the rate of amplitude 𝛽 = hysteretic energy constant 𝛿 = displacement 𝛿𝑖 = plastic displacement 𝛿𝑚𝑎𝑥 = maximum displacement 𝛿0 = threshold for damaging cycles 𝛿𝑢 = ultimate displacement 𝛿𝑦 = yield displacement 𝛿𝑦′ = measured first yield displacement 𝜃 = drift angle 𝜃𝑝 = beam rotation 𝜇𝑖 = displacement ductility 𝜌𝑙 = longitudinal reinforcement ratio 𝜌𝑠 = transverse reinforcement ratio 𝜑𝑦 = curvature at first yield ∆𝛿𝑝𝑖 = range of plastic displacement ∆m = deformation amplitude ∆max = maximum deformation ∆0 = minimum deformation amplitude xv LIST OF ABBREVIATIONS AASHTO American Association of State Highway and Transportation Officials ACI American Concrete Institute ANSI American National Standards Institute AISC American Institute of Steel Construction ASTM American Society for Testing and Materials ATC Applied Technology Council BC British Columbia BMS Bridge Management System CDE Cumulative Displacement Energy CDD Cumulative Displacement Ductility CHBDC Canadian Highway Bridge Design Code COSMOS Consortium of Organizations for Strong Motion Observation Systems CUREE Consortium of Universities for Research in Earthquake Engineering CREW Cascadia Region Earthquake Workgroup CSA Canadian Standards Association DBD Displacement-Based Design ECCS European Convention for Constructional Steelwork FEMA Federal Emergency Management Agency IDA Incremental Dynamic Analysis ISO International Organization for Standardization ISPA Inelastic Static Pushover Analysis JISF Japan Iron and Steel Federation K-NET Kyoshin Network MD Maximum Displacement MNZ Modified New Zealand MOT Ministry of Transportation NBCC National Building Code of Canada PBD Performance-Based Design PEER Pacific Earthquake Engineering Research xvi RC Reinforced Concrete SDOF Single Degree of Freedom System SEA Structural Engineers Association of California SHM Structural Health Monitoring TD Total Damage THA Time-History Analysis UHS Uniform Hazard Spectrum xvii ACKNOWLEDGEMENTS I would like to express my sincere appreciation to my supervisor Dr. Shahria M. Alam, for the continuous support of my masters and research. His motivation and immense knowledge encouraged me to keep going and improve my work every time. As well, I would like to thank the School of Engineering at the University of British Columbia for providing an adequate environment for me to successfully complete my work. Furthermore, the financial support of Mitacs is greatly appreciated, without this aid the project would not have been possible. I sincerely appreciate my colleagues who helped me during my thesis: Saif Aldabagh, who put enormous dedication to introduce me to bridge engineering and shared his knowledge to guide me in all the time of my research and writing of this thesis, Faroque Hossain, who spent a lot of time studying and sharing his experience with me, Anas Issa, who shared his experience and time to support me when I needed guidance, and Sherif M. Osman, who provided unconditional support in everything I needed in my masters. As well, I would like to thank all my research teammates who spent time with me when I needed their help and all my friends who were with me during my master's study. xviii DEDICATION I would like to dedicate my work to all my family in Ecuador, especially to my father Edwin, my mother Tina, my sister Denise, my brother Ronny, my nephew Emilio, and my niece Carol for their unconditional support. Also, I deeply appreciate my wife Cassy for her love and for always being there with me in the easiest and the hardest moments of this journey. I want all of them to know that all my achievements are theirs too. Thank you very much, always. 1 CHAPTER 1: INTRODUCTION 1.1 Background In earthquake engineering, understanding the behaviour of structural components has always been one of the major challenges. In the conventional force-based design approach, critical components of a structure are designed to sustain certain levels of damages but without a global collapse. To prevent collapse, it is critical to understand the post-yield characteristics of structural systems to ensure they exhibit appropriate levels of ductility and energy absorption. We know that the strength and deformation capacities of all structural elements are limited, and damage control depends on the ability to accurately determine these capacities. Currently, the approach to obtain such values in structural engineering is to implement modification coefficients (e.g., CSA S6 2019); however, for some cases, applying these factors might be sufficient but for many other cases may not. So, experimentalists and researchers must resort to testing and to analytical modelling to evaluate the performance of structures. Here, cumulative damage needs a lot of attention because the strength and deformation capacities of a structure highly depend on it. This concept implies that every component has a “memory” of a previous damaging event and in turn indicates that performance is related to it (Krawinkler et al., 2001). A reasonable way to assess the consequences of history is to replicate the deformation (also load) histories a structural component will undergo in a seismic event. While improving the design of an existing structural component/system or understanding the behaviour of a newly proposed component/system, it is a common practice to experimentally determine their inelastic behaviour by performing quasi-static cyclic loading tests. Here, the load is often applied in a slow, controlled, and predetermined manner, which allows relating specific structural measurements (e.g., displacement) to visual damages. Results from such tests have been utilized to establish current seismic design procedures for various structural components and systems. Additionally, design codes, including modern bridge design codes (e.g., AASHTO 2017, CSA S6 2019), have been gradually moving towards the performance-based design (PBD) approach in which structures are designed to achieve predefined minimum performance levels. Extensive experimental work involving quasi-static cyclic tests is often required to establish performance levels in terms of engineering parameters such as material strains and drift ratios. For instance, in the Canadian Highway Bridge Design Code (CHBDC) (CSA S6 2019), material strains identifying the onset of 2 different damage states in reinforced concrete (RC) members were established based on the experimental programs by Kowalsky (2000) and Goodnight et al. (2012). Bridge piers are essential components that support the superstructure and transfer the loads to the foundation, therefore, their vulnerability in earthquakes is a big concern for civil engineers. How to correctly describe and predict the nonlinear hysteretic behaviour is the focus issue for their seismic analysis because of its direct relationship with performance and deformation capacity. Unfortunately, for quasi-static cycle tests of bridge piers, there are currently a few guidelines available to develop experimental procedures for seismic performance assessments. The very first efforts for developing such guidelines focused on the structural components of steel, concrete, and wood frames (ATC 1992; ACI 1999; Krawinkler et al., 2001). Intending to bring consistency to research work involving quasi-static tests, further efforts have led to establishing general testing programs for different materials and structural systems (e.g., FEMA 2007, and ACI 2013). However, loading protocols obtained from these procedures may not always be adequate, for example, different RC bridge piers could have substantially different structural characteristics, such as strength and deformation capacities. According to El-Bahy et al. (1999), Moyer and Kowalsky (2003), and Tsuno and Park (2004), damage initiations appear to be noticeably dependent on the history of the inelastic loading protocol used in the quasi-static cyclic tests. It has also been found that the failure mode of RC bridge piers is highly influenced by the type of load path applied (Kunnath et al. 1997; McDaniel et al. 2006). For example, studies have shown that there is a noticeable difference in terms of damage of RC columns under long-duration protocols compared to conventional protocols (Pujol et al. 2006 and Ou et al. 2014). This indicates that before any quasi-static cyclic tests, it is of critical importance to select loading protocols with amplitudes and the number of inelastic cycles that are representative of the seismic demands. Clearly, representative and properly developed structural component-specific loading protocols will help better utilize the results whether for establishing damage states or general seismic performance evaluation. The unavailability of a universal loading protocol applicable to all structural systems stems from the fact that the performance of structural systems is largely dependent on the history of inelastic loading, known as the cumulative damage process (Krawinkler et al. 2001). Consensus-based, yet representative and properly developed loading protocols for quasi-static cyclic test programs of bridge piers are highly required to produce suitable data for the seismic assessment of RC bridge piers. Such programs will strongly influence 3 the future update of limit states and acceptance criteria for the PBD of different bridge codes. The correct assessment of different structural components plays a fundamental role (FEMA 2007), therefore appropriate cyclic loading histories that allow one to obtain consistent, reliable, and comparable results are in great demand. 1.2 Research Objectives This thesis presents a numerical procedure leading to the development of component-specific loading protocols for bridge piers. The proposed approach will be illustrated using a prototype RC bridge pier susceptible to crustal, subcrustal and subduction earthquakes. Incremental Dynamic Analysis (IDA) using fibre element-based modelling was performed to determine the displacement time history responses of the prototype bridge pier at target ductility demand levels of two, four and eight. The recommendations of Krawinkler et al. (2001), Mergos and Beyer (2014), and Bazaez and Dusicka (2016) to develop loading protocols were partly considered. In previous studies in which the loading protocols were developed, the target ductility demand levels are often achieved using the classical constant ductility design approach. As an alternative approach, the target ductility demand levels were achieved utilizing the well-established IDA (Vamvatsikos and Cornell 2002). Finally, the proposed procedure will be implemented in a sensitivity analysis to propose generalized loading protocols for cyclic tests of RC bridge piers. Thus, the primary objectives of this work were: • Develop an Incremental Dynamic Analysis (IDA)-based simplified procedure to develop suitable component-specific quasi-static loading protocols. • Evaluate the sensitivity of the loading protocols to several bridge design parameters and propose generalized loading protocols applicable to bridge piers with a wide range of geometrical and reinforcement details. • Compare and validate the proposed loading protocols with those available in the literature and standard codes. 4 1.3 Thesis Layout Throughout this thesis, a literature review of different aspects related to the seismic assessment of bridge piers and quasi-static loading protocols is presented in Chapter 2. Chapter 3 describes all the main considerations taken to propose a component-specific development procedure of cyclic loading protocols. To illustrate the procedure, a case study is given in Chapter 4. Chapter 5 presents a sensitivity analysis to validate the procedure and evaluate its applicability. Conclusion and recommendations are summarized in Chapter 6. 5 CHAPTER 2: LITERATURE REVIEW This chapter presents a comprehensive review of different aspects of bridge engineering that are related to the seismic evaluation of bridge components. An overview of the seismic design provision of the Canadian Highway Bridge Design Code (CSA S6 2019) is presented. A bridge inventory of the province of British Columbia, Canada, will be reviewed along with the seismology of the region. Moreover, methodologies for predicting the seismic response of bridges will be presented, emphasizing the experimental testing techniques, especially for the quasi-static cyclic test. Due to the lack of standards and/or guidelines for the development of quasi-static loading protocols, the most used protocols in experimental programs will be presented along with their correspondent development background, implications, and applicability. 2.1 Seismic Design Provisions in the Canadian Highway Bridge Design Code (CHBDC) 2.1.1 General Seismic Design In the conventional force-based design approach, critical components of a structure are designed to sustain certain levels of damages but without a global collapse. For example, a ductile column subjected to reversed-cyclic inelastic deformations is expected to resist the load without significant loss of strength and develop the appropriate level of ductility while remaining stable. To achieve this goal, it is critical to identified plastic hinge regions and to understand the post-yield characteristics of structural systems to ensure they exhibit appropriate levels of ductility and energy absorption. The approach of modern seismic design codes (e.g., AASHTO 2017, CSA S6 2019) is to provide ductile substructure elements (e.g., columns) and eliminate brittle failure modes. One way to address this approach is by implementing the capacity design methodology. Here the main idea is to avoid failure modes such as shear failure of joints or columns and the yielding in foundations and/or the superstructure elements by providing adequate detailing of reinforcement (flexural and transverse) to allow the development of high strains. Moreover, design codes implement quantitative limitations over design parameters such as column aspect ratio, axial load ratio, longitudinal reinforcement ratio, volumetric transverse reinforcement ratio, and material properties, to ensure the desired behaviour. This way, essential structural components will behave in the elastic range and remain protected without any loss of strength. 6 2.1.2 Force-Based Design (FBD) In this methodology, the seismic demand that is imposed into the structure is translated as the application of a lateral force. The stiffness of the system is the main variable implemented to distribute the later force among the structural members according to their proportion. Moreover, ductility is introduced into the system by the implementation of a force modification factor (R) that depends on the type of structural system. This factor is utilized to divide the forces from an elastic analysis, which in turn include a modification factor (IE), and obtain the seismic design forces in each orthogonal axis of the structure. For capacity-protected elements, the design forces are obtained considering both factors (R and IE) equal to 1. This way, the lateral load resisting substructure elements must be designed and detailed such that ductile responses can be achieved. According to CHBDC, the FBD approach for ductility substructure elements is permitted only in special cases, which in turn depends on whether the bridge is classified as regular or irregular, the importance of the bridge, and the seismic performance category. 2.1.3 Displacement-Based Design (DBD) In DBD, the seismic demand is characterized as imposed structural displacement. Parameters such as the cross-section curvature and node displacements are established as the fundamental column properties. In this methodology, predetermined regions of the structure are designed to withstand most of the damage. In other words, the seismic damage is localized in specific locations of the structure. Even though the CHBDC allows the implementation of this approach, the code rather remains silent on the details and suggests the performance-based design (PBD) as the preferred method used for the design. 2.1.4 Performance-Based Design (PBD) A relatively new methodology that modern bridge design codes have adopted is the PBD approach in which structures are designed to achieve predefined minimum performance levels. The CHBDC specifies criteria to meet specific structural, functional, and service performance levels depending on various hazard levels, as shown in Table 1. It is worth mentioning that extensive experimental work involving quasi-static cyclic tests is often required to establish performance levels in terms of engineering parameters such as material strains and drift ratios. For instance, the CHBDC, material strains identifying the onset of different damage states in RC 7 members were established based on the experimental programs by Kowalsky (2000) and Goodnight et al. (2012). Table 1. Performance levels in CHBDC (CSA S6 2019) Service Damage Immediate Minimal Damage Limited Repairable Damage Service Disruption Extensive Damage Life Safety Probable Replacement 2.2 Bridges of the province of British Columbia Relatively recent seismic events that impacted highway bridges around the world have raised a big concern in Canada for evaluating the vulnerability of these structures. For that reason, Siddiquee and Alam (2017) developed an inventory for highway bridges in British Columbia (BC) that are under the responsibility of the BC Ministry of Transportation and Infrastructure (BC MOT). The objective of the study was to develop an advanced Bridge Management System (BMS) for the efficient assessment of the existing structural health condition of Canadian bridges, predicting their future deterioration, and prioritizing their maintenance and retrofitting works. The structures were identified in terms of their location, structural and geometric parameters such as construction materials, bridge length, number of spans, deck width, skew angle, bridge pier, and foundation type, structural health condition rating and construction period. In total, 2555 highway bridges in BC were examined and classified into 13 typical bridge classes, which are about 96% of the total highway bridges in the region. All the bridge classes along with their numbers and correspondent percentage are shown in Table 1, where RC bridges represent 46.5% of the total inventory, whereas steel and timber bridges represent 29.3% and 20.1%, respectively. Additionally, the most common bridge superstructure systems are provided in Table 2, where most of the bridges are constructed using I girder (rectangular) and box girder sections, and some others using suspension and cable-stayed bridge systems. 8 Table 2. Typical bridge classes listed in the BC bridge inventory (Siddiquee and Alam, 2017) SL. Type Abbreviation # % 1 Single Span Concrete Girder Bridge SSG Concrete 218 8.5 2 Single Span Concrete Box Girder Bridge SSBG Concrete 347 13.6 3 Single Span Steel Girder Bridge SSG Steel 233 9.1 4 Single Span Steel Truss Bridge SST Steel 138 5.4 5 Single Span Timber Girder Bridge SSG Timber 301 11.8 6 Multi-span Continuous Concrete Girder Bridge MSCG Concrete 147 5.8 7 Multi-span Continuous Concrete Box Girder Bridge MSCBG Concrete 160 6.3 8 Multi-span Simply Supported Concrete Girder Bridge MSSG Concrete 252 9.9 9 Multi-span Simply Supported Concrete Box Girder Bridge MSSS Concrete 62 3.4 10 Multi-span Continuous Steel Girder Bridge MSCG Steel 165 6.5 11 Multi-span Simply supported Steel Girder Bridge MSSG Steel 153 6.0 12 Multi-span Timber Girder Bridge MSG Timber 211 8.3 13 Multi-span Steel Truss Bridge MST Steel 80 3.1 14 Others 88 3.4 Total 2555 100 Table 3. Bridge superstructure system listed in the BC bridge inventory (Siddiquee and Alam, 2017) SL. No. Construction Type SL. No. Construction System 1 Rectangular Girder 7 Suspension 2 Box Girder 8 Cable-Stayed 3 I Girder 9 Balanced Cantilever 4 Truss-Through 10 Tunnel/Culvert 5 Truss-Deck 11 Miscellaneous 6 Structural Arch To complement the characterization of a bridge type, the bridges were also broken down into pier and foundation types. Eight different foundation types were identified including concrete piles, steel H piles, steel pipe piles, steel pipe piles filled with concrete, timber or log piles, single column footing, combined footing, and wall footing. Likewise, 10 typical pier types were identified based on their construction material type, the number of columns in each pier, and other geometrical parameters, as shown in Table 3. Among the pier types, RC multi-column bents are 9 the most common for highway bridges in BC. Multi-column bent, pile bent, wall pier, and timber pier are the governing pier types among BC highway bridges. The preferred option for the footing of bridges with a single column, flared column, and multi-column bent pier types was always a single column footing. For bridges with piles bent with or without concrete diaphragm, the steel pipe pile is the most common among the others. Depending on the soil conditions, when using wall piers, steel H pile and steel pipe pile were identified as the common option. Table 4. Bridge pier type classification (Siddiquee and Alam, 2017) SL. Pier Type Number of Bridges % of Bridges 1 Single Column 11 0.85 2 Flared Column 77 5.96 3 Multi-Column Bent 283 21.89 4 Pile Bent 178 13.77 5 Pile Bent with Concrete Diaphragm Steel Bent 96 7.42 6 Steel Bend 36 2.78 7 Solid Wall Pier of Uniform Cross Section 119 9.20 8 Solid Wall Pier Varying Cross Section 85 6.57 9 Hammerhead or Tee Type Wall Pier 35 2.71 10 Timber Pier 272 21.04 11 Others/Information not available 103 7.97 Total 1295 100 Finally, it is worth mentioning that the information gather by the authors is of great importance because it will also allow experimentalists as well as authorities to have effective infrastructure management, proper rehabilitation solutions, and efficient design of a Structural Health Monitoring (SHM) and Control System for enhancing structural resilience of highway bridges in BC. 2.3 Seismicity of Southwestern British Columbia The seism tectonic and seismicity relationship in the Cascadia subduction zone has been well established. This region stretches 1,300 km from the Brooks Peninsula, Vancouver Island in Canada to Cape Mendocino, California in the United States. Here, the Explorer, Juan de Fuca, and Goda plates move to the east and slide below the much larger mostly continental North American 10 Plate, as shown in Figure 1. Moreover, three different earthquake types have been identified in the Cascadia subduction zone: shallow (also called crustal), deep (also called subcrustal), and subduction earthquakes (CREW 2009). Deep events start below the interface between the Juan de Fuca and Gorda plates and the overlying North American plate (CREW 2008). Shallow earthquakes are hosted by the zone lying within the crust of the overlying North America plate (CREW 2009). Subduction earthquakes occur on the interface between the subducting plate and the North American plate and represent a big concern for the Cascadia subduction zone because of its great extent and potential to break over an enormous area (CREW 2013). Each type exhibits specific characteristics and implications that need to be considered for structural design and/or post-earthquake evaluation in this region. Figure 1. Seismotectonic of the Pacific Northwest 2.3.1 Crustal Ground Motions Crustal earthquakes occur within the continental crust of the overlying North American plate. Generally, they occur at depth of less than 35 km and are expected to be less than M7.5 (CREW 2009). Their frequency is relatively high because of the large number of shallow faults. However, damaging events of this type will occur only every few decades and their duration is not expected to last more than a few seconds to a minute. Aftershocks are common and need to be 11 considered, even though tsunamis are unlikely (CREW 2009). Examples of crustal earthquakes are the 1946 Vancouver Island (M7.3) and the 1993 Scotts Mills Oregon (M5.6) earthquakes. 2.3.2 Subcrustal Ground Motions Subcrustal earthquakes (also called intraplate earthquakes) take place within the oceanic plate as it subducts underneath the North American plate. This type of earthquake occurs at depths of about 45 to 80 km and usually, they are less than M7.5. Moreover, their frequency is relatively low and changes depending on the area; damaging earthquakes of this type typically occur every 10 to 30 years (CREW 2008). Since the epicentre of these events is also deep, the seismic wave energy radiates over a much larger area when compared to Crustal events. After subcrustal events occur, few, if any, aftershock will occur. Likewise, no tsunami is expected (CREW 2008). Some examples of these seismic events are the 2001 Nisqually (M6.8) and the 1965 Seattle (M6.5) earthquakes. 2.3.3 Subduction Ground Motions Subduction earthquakes occur offshore Cascadia affecting long areas in southwestern Canada and the northwestern United States. The events can be as large as M9 and are expected to happen roughly every 500 years (CREW 2013). Moreover, the shacking might be felt for several minutes and several aftershocks will occur, some being as large as M7. A destructive tsunami is also expected to hit the coast. Clearly, the destructive potential of these events is high and strong attention should be given to these cases to avoid many injuries, fatalities, and infrastructure loss. 2.4 Methodologies for Predicting the Seismic Response of RC Bridge Piers 2.4.1 Eigenvalue Analysis One of the first steps before performing a nonlinear time-history analysis is performing an eigenvalue analysis to determine the periods of vibration as well as the mode shapes of the structure (damping is neglected). Both parameters are a function of the structural properties and boundary conditions. They are essential because they characterize the basic dynamic behaviour of the structure and are an indicator of how the structure will respond under seismic excitation; therefore, a good knowledge of natural frequencies and mode shapes is required for all types of dynamic analysis. 12 2.4.2 Moment-Curvature Analysis The moment-curvature (𝑀− 𝜑) analysis is an important framework when carrying out inelastic analysis of RC bridge piers. It can be understood as the measure of strength, stiffness, ductility, and energy dissipation capacity of the structure. Appropriate material stress-strain relationships are required to compute a reliable moment-curvature analysis, which can be characterized by parameters such as the yield and idealized yield moments, the nominal flexural capacity, and the ultimate flexural resistance. Moreover, the assumption of “plain-sections-remain-plain” is also required. The inclusion of axial load is another parameter that should be considered since it can be varied to study the flexural behaviour of the column while satisfying the strain compatibility assumption and equilibrium of forces. 2.4.3 Pushover Analysis (Curvature Integration) By performing a pushover analysis, a simplified nonlinear analysis technique is implemented to estimate the dynamic demands (e.g., deformations) imposed into the structure by seismic excitation. Basically, in this technique, the curvature response of the section is numerically integrated over the length of the structure. Therefore, it is crucial to determine the curvature response distribution by utilizing the correspondent moment-curvature relationship. Generally, a lumped-plasticity model is used to determine the ultimate displacement capacity, where the curvature is idealized with a linearly varying curvature up to the yield curvature and a trapezoidal lumped curvature over the plastic hinge length. Several plastic hinge models have been proposed in the literature, with the one from Priestley et al. (1996) being the most known models. 2.4.4 Fibre-Element Analysis Since the PBD restricts local seismic demand parameters to specific limits states for certain hazard levels, more precise analytical models are needed to estimate demands in critical regions of a bridge pier. Fibre-based distributed plasticity is an analytical technique that estimates the plasticity spread over the height of the structure by defining integration points. The cross-section of the structure is subdivided into fibres that are then assigned uni-axial stress-strain relationships of the correspondent material. An example of cross-sectional discretization is shown in Figure 2. Each fibre section is an integration point and, in the case of RC, strain compatibility between the materials. Additionally, it is worth mentioning that the influence of actions such as confinement in concrete and strain hardening can be accounted for with fibre-based analysis; however, for 13 simulation of effects such as bar buckling or bar fracture, more precise models are needed (e.g., finite element analysis). Figure 2. Uniform radial discretization of a bridge circular column 2.4.5 Nonlinear Beam-Column Elements (Distributed Plasticity) As previously discussed, distributed plasticity models allow yielding to occur anywhere in the column height. To formulate these scenarios, two methods can be followed: displacement-based and force-based formulations. In force-based beam-column elements, equilibrium is satisfied in both elastic and inelastic ranges and the section forces are interpolated in relationship with the correspondent element forces (Scott and Fenves 2006). Generally, in force-based formulations, only one element per structural member is implemented with the related section fibre-discretization, as shown in Figure 2. On the other hand, in displacement-based formulations, the section forces are characterized by the section deformation and forces. Usually, several displacement-based elements are required along the length of, for example, a bridge column, to match the force-displacement behaviour. In both cases, the finer the discretization of fibres, the more precise the results, improving the accuracy of the response and decreasing the susceptibility to converge in the analysis. For bridge piers, the general practice is to discretize the structure with a minimum of 5 elements of equal length and 200 to 400 section fibres. Examples of software available for these types of analysis are OpenSees (McKenna et al. 2011) and SiesmoStruct (SeismoSoft 2021). 14 Figure 3. Bridge pier column discretization scheme 2.5 Cumulative Damage In bridge engineering, basic seismic capacity parameters for a bridge component are strength, stiffness, inelastic deformation capacity, and cumulative capacity parameters such as damage and energy dissipation (Krawlinkler 1983). The main connection with cumulative damage concepts lies in the fact that all these parameters are expected to decrease as the number of damaging cycles from seismic excitation increases, depending on the properties of the structure. Likewise, the effect of the number of inelastic cycles on structural elements is a concern because it is associated with fatigue, which is a result of the gradual loss of element resistance due to the inelastic deformation. (Krawinkler et al., 1983). Therefore, cumulative damage is an essential concept in the framework of the seismic assessment of bridge piers, where the general concept has been commonly applied as per the Coffin-Mason model and the Miner’s rule as a baseline (Krawinkler et al., 1983). The theory assumes that a stress cycle that alternates above the fatigue limit (endurance limit) inflicts a measurable permanent total damage. Thus, if inelastic cycles are predominant for example, due to their relatively low number, low-cycle fatigue should be considered, and the damage potential could be quantified utilizing a cumulative damage model. However, it is important to carefully consider select the appropriate approach to account for Confined concrete Steelreinforcement Unconfinedconcrete Discretized section LpL -LpLocation of applied axial loadNodeColum Height (L)Displacement-based elementStrain penetration sectionZero-length element 15 cumulative damage in structural systems that are susceptible to strength and stiffness degradation (as is the case for bridge piers) because a cycle that occurs after the peak excursion would present significant degradation of these parameters, consequently, adding little hysteretic energy dissipation and cumulative damage even if its amplitude is large (Park, 1989). The most common approaches for considering cumulative damage in bridge components has been those related to deformation and/or ductility like the following general damage model: 𝑪𝑫𝑬 = 𝑪 ∑(∆𝜹𝒑𝒊)𝒄𝑵𝒊=𝟏 (1) where ∆𝛿𝑝𝑖 is the range of each plastic deformation, and C and c are parameters that depend on the type of component and failure mode. The parameter c is usually greater than 1.0, indicating that large inelastic cycles cause more damage than low ones. Similarly, Park (1989) developed a cumulative damage index referred to as the cumulative displacement ductility factor CDD which account for cumulative damage by accumulating the ratio of plastic displacement (𝛿𝑖) to the correspondent yield displacement (𝛿𝑦) of the structure, as shown in the equation below: 𝑪𝑫𝑫 = ∑𝜹𝒊𝜹𝒚𝑵𝒊=𝟏= ∑ 𝝁𝒊𝑵𝒊=𝟏 (2) Other cumulative damage models have been developed from considerations of energy-dissipation demand and capacity, like the model developed by Park and Ang (1985) where the damage is considered as the maximum deformation and the cumulative dissipated energy: 𝑫𝒑𝒂 =𝜹𝒎𝒂𝒙𝜹𝒖+𝜷𝑸𝒚𝜹𝒖∫ 𝒅𝑬 (3) where 𝛿𝑚𝑎𝑥 is the maximum response deformation, 𝛿𝑢 is the ultimate deformation capacity under monotonic load, 𝑄𝑦 is the specimen yield strength, 𝛽 is a constant that depends on the duration effect of the hysteretic energy and 𝑑𝐸 is the dissipated hysteretic energy. Other approaches also include hybrid formulations combining some aspects of structural parameters and more complex theories based on concepts derived from fatigue models. (Kunnath et al. 1997). Finally, it is important to mention that the ability to predict damage in a structural member does not necessarily reflect the success of a damage model, rather its correct calibration is what determines if it is effective or not. 16 2.6 Experimental Seismic Assessment of RC Bridge Piers An experimental study of the seismic behaviour of a structural element may involve monotonic tests, force or displacement is imposed with no alternations; shaking table tests, in which the structure is subjected to simulated ground motion; and quasi-static tests, in which equivalent static displacements or forces are imposed on a structural element to induce nonlinear behaviour. Results from these types of tests have been implemented to determine limit states for design codes and develop numerical models to study the seismic behaviour of a structure. 2.6.1 Monotonic Load Test This type of test corresponds to a uni-axial test with no alternating cycles that are normally carried out in civil engineering laboratories as a preliminary portion of a complete experiment that includes cyclic and/or shake table testing (e.g., Kunnath et al. 1997 and Kwan and Billington 2003). The objective of the test generally lies in the determination of strength and deformational characteristics of structural components (e.g., bridge piers). Parameters such as ultimate displacement (𝛿𝑢) and yield displacement (𝛿𝑦) are mainly obtained from this type of test, which will then be useful to characterize certain parameters for further tests. 2.6.2 Shake Table Test The shaking table test is a well-known technique to assess the seismic performance of a variety of structures, such as shear walls (e.g., Martinelli and Filippou, 2009), steel frames (e.g., Saranik et al. 2012) and bridge piers (e.g., Schoettler et al., 2015). The test can be implemented for the evaluation of the elastic and/or inelastic dynamic response of usually, scaled models of a structure. The test setup consists of hydraulic actuators that can apply the displacement component of ground motion records in both orthogonal directions as shown in Figure 4. Figure 4. Shake table test illustration -0.4-0.3-0.2-0.100.10.20.30.40 20 40 60 80Acceleration (g)Time (seconds) 17 2.6.3 Quasi-static Uniaxial Cyclic Test The main idea of quasi-static tests is to replace the inertial forces generated by an earthquake on a structure with equivalent static loads. Hydraulic actuators induce predetermined statics forces (loading protocol) or displacements between a strong wall or frame and a test specimen, as shown in Figure 5. Basic information, such as strength, stiffness, and ductility, is obtained to predict the behaviour of structures during earthquake loading (e.g., Dazio et al., 2009 and Mehary et al., 2017). The main advantages of quasi-static tests are that large-scale specimens can be tested, and a test is not affected if it needs to be stopped for reasons such as assessing the condition of the specimen. The main weakness and concern of this test reside in the uncertainty about how well it is correlated to a dynamic test. The specimen should be able to dissipate energy and its response should be comparable to that from a dynamic test. Therefore, the choice of an adequate testing protocol is a critical step in any quasi-static test program. Generally, the methodology of testing is based on the imposition of a reverse cyclic displacement protocol where the failure (end of the test) is conventionally pre-defined. Figure 5. Quasi-static uniaxial cyclic test illustration The very first efforts for developing guidelines for experimental studies involving cyclic test focused on the structural components of steel, concrete, and wood frames (e.g., ECCS 1986, ATC 1992; JSIF 1996; SAC, 1997; ACI 1999;). Likewise, many loading protocols have been proposed in the literature for different testing programs (e.g., ATC 1992, Krawinkler et al. 2001, Dusicka and Bazaez 2016). Further efforts for developing consensus-based guidelines for experimental studies involving cyclic tests, general testing programs for different materials and structural systems were adopted as standards in many countries around the world (e.g., FEMA 18 2007, ACI 2013, AISC 2016) have been established. These protocols have been widely implemented for several structural systems such as steel frames, steel connections, beam-column connections, reinforced concrete columns, etc. have been generally tested regarding such recommendations. It should be noted that no single protocol can be characterized as complete or better than the others. The protocols recommend somewhat different loading histories, but in most cases, they differ more in detail than in concept. These recommendations all share the following same basic characteristics: • Testing loading protocols are displacement or force-controlled. • Specimens are subjected to symmetrical reversed cyclic displacement history. • The loading history is expressed in terms of a reference parameter. • Loading cycles are repeated in the post-elastic range. • Failure is conventionally pre-defined. These generally accepted recommendations have served as a common ground for the construction of different loading protocols, the comparison of different testing programs, and the derivation of detailed rules (e.g., performance limit states) for national building codes. Moreover, performance indexes such as strength and/or stiffness degradation and hysteretic energy-dissipation can be obtained from cyclic tests carried out regarding these recommendations. Several loading protocols have been proposed for the cyclic test of structural components and systems. It would be unrealistic to review every single protocol implemented in experimental studies, the next section will be limited to present formal loading protocols that have been proposed and widely utilized. The subsequent section will discuss loading protocols that have been suggested for structural and non-structural components and, since there are many combinations of both materials and structural configurations, loading protocols will be classified by types of structural materials: steel structures, wood structures, and concrete structures. It is worth mentioning that most of the standard loading protocols have been developed to represent the demands of short period structural components and/or systems. Finally, loading protocols specifically developed for bridge piers will also be included. 19 2.6.4 Standard Loading Protocols for Structural and Non-structural Components 2.6.4.1 FEMA A-461 Loading Protocol (2007) The U.S. Federal Emergency Management Agency (FEMA) originally developed this protocol for testing drift-sensitive non-structural components; however, the protocol is also applicable to drift-sensitive structural components. The objective of this loading protocol is the determination of performance characteristics of components and its behaviour is controlled either by the application of seismic forces (not shown in this section) or seismic-induced displacements. The protocol can be used to determine fragility data for several structural and non-structural components. Moreover, constitutive force-deformation properties and hysteretic data for structural components can be derived for structural analysis and assessments. Shear walls, beam-column assemblies, and frame assemblies are examples of structural components that may be tested following this protocol. Examples of non-structural components for which this protocol is suitable include cladding panels, glazing panels, and drywall partitions. The protocols utilized two different reference values, a target maximum deformation amplitude (∆m), and a target minimum deformation amplitude (∆0). Likewise, a predetermined number of increments higher than 10 is recommended to determine the loading history. The amplitude 𝑎𝑖 of the stepwise increasing deformation cycles is given by the following equation: 𝒂𝒊+𝟏𝒂𝒏= 𝟏. 𝟒 (𝒂𝒊𝒂𝒏) (4) where 𝑎𝑖 is equal to ∆0 and 𝑎𝑖+1 is equal to ∆m. Two cycles are executed at each amplitude, and if the last damage state has not occurred at the target maximum displacement, the protocol continues further using cycles of amplitude equal to 0.3∆m. The loading protocol is shown in Figure 6 and Table 5 considering the largest amplitude (an) to be exactly equal to ∆m. 20 Figure 6. FEMA loading protocol Table 5. Sequence of loading of FEMA protocol Cycle Group Number of Cycles 𝑎𝑖𝑎𝑛⁄ 1 2 0.018 2 2 0.025 3 2 0.035 4 2 0.048 5 2 0.068 6 2 0.095 7 2 0.133 8 2 0.186 9 2 0.26 10 2 0.364 11 2 0.51 12 2 0.714 13 2 1 -1-0.75-0.5-0.2500.250.50.7510 4 8 12 16 20 24 28Relative Amplitude (ai/∆m)Number of Cycles 21 2.6.4.2 Mergos and Beyer (2014) This research work presented the development of cyclic loading protocols for European regions of low to moderate seismicity. Sixty ground motion records were utilized to capture the response of a wide variety of Single-Degree-of-Freedom (SDOF) systems that reflect the fundamental properties of a large portion of structural systems including: elastic systems, systems for which lateral resistance is provided by timber walls, RC frames, RC walls, unreinforced masonry shear or rocking walls. The structural fundamental period ranges from 0.1 to 0.5 seconds, which will be taken also for those structures that exceed 0.5 seconds. The ground motions are representative of a seismic hazard level of 2475 return period in a European moderate seismicity region. The proposed loading protocols are characterized by cycle amplitudes that are normalized by the maximum displacement (∆max) of the structure and expressed as analytical functions of each load step, which allows describing different loading protocols for different structural systems by only two parameters. Each loading protocol consists of n load steps with n1 cycles of the same amplitude per step, which is taken separately as 1, 2, and 3. Therefore, the protocols comprise a total of n1.n cycles and are represented by the following equation: 𝒇(𝒙) =𝟏𝒆 − 𝟏[𝜹𝟎(𝒆 − 𝟏) + (𝟏 − 𝜹𝟎)(𝒆𝒙𝒑(𝒕𝜶))] (5) where 𝛿0 is the threshold for damaging cycles, assumed as 0.05, t = x/n, with x being the current load step, n is the number of load steps and α is a parameter describing the rate of amplitude increase and that depends on the structural system considered (Mergos and Beyer, 2014). The motivation of this study lies in the fact that existing loading protocols for quasi-static cyclic testing of structures are based on ground motion records of high seismicity; therefore, they overestimate the imposed cumulative damage demands of low to moderate seismicity regions. 2.6.5 Standard Loading Protocols for Steel Structures For steel components and systems, different loading protocols have been extensively used in experimental programs, such as the ECCS, ATC-24, SAC, JISF, and AISC protocols. The ATC-24 and SAC protocols were developed at Stanford University by Helmut Krawinkler, a researcher that has been highly engaged in these topics. The ECCS and the last two protocols are part of the European Convention for Constructional Steelwork guideline, the Japan Iron and Steel Federation, and the seismic provisions of the American Institute of Steel Construction, respectively. 22 2.6.5.1 ECCS Loading Protocol (1986) One of the first efforts to define standard loading histories may be found in the testing procedure for assessing the behaviour of structural steel elements as part of the ECCS guidelines. The recommendation is to implement three specimens for three separate tests: two monotonic tests (not shown here) and one cyclic test. The monotonic tests are aimed to provide information about the reference parameters for the loading protocols obtained when the specimen is subjected to reverse bending, namely the yield displacement (δy) and the yield force (𝑓𝑦). The cyclic test is presented in Figure 7 and shown in Table 6. Figure 7. ECCS loading protocol Table 6. Sequence of loading of the ECCS protocol Cycle Group Number of Cycles % of Yield Displacement 1 1 25 2 1 50 3 1 75 4 3 100 5 3 200 > 5 3 2 + 2(cycle group) 23 2.6.5.2 ATC-24 Loading Protocol (1992) This protocol was developed in 1992 specifically for components of steel structures (ATC 1992). It represents one of the first formal protocols developed in the U.S. for seismic performance evaluation of components using a cyclic loading history. It is based on the yield deformation (δy) as the reference for increasing the amplitude of cycles. The recommendation to determine such value during a preliminary monotonic test or by extrapolating the deformation of the test specimen at 75% of its yield strength (𝑄𝑦) measured either during the initial stage of the protocols. The ATC-24 loading protocol is illustrated in Figure 8 and presented in Table 7. The relative amplitudes and progression of the cycles were derived from statistical studies of time history responses of Single-Degree-of-Freedom (SDOF) systems, and therefore represent global deformation demands and not local deformation histories that would be experienced, for example, by links in eccentrically braced frames. The parameters analyzed in the development procedure of the protocols were: the number of inelastic cycles, the plastic deformation ranges, and the cumulative plastic deformation range (Hadidi-Tamjed 1987, Nassad and Krawinkler 1991). The ATC-24 document presents another loading protocol for multi-specimen cumulative damage testing protocol that is based on constant amplitude testing. Figure 8. ATC-24 loading protocol -5-4-3-2-10123450 4 8 12 16 20Amplitude 𝛿/𝛿yNumber of Cycles 24 Table 7. Sequence of loading of the ATC-24 protocol Cycle Group Number of Cycles Control Variable Amplitude 1 3 𝑄𝑦 0.5 𝑄𝑦 2 3 0.75 𝑄𝑦 3 3 δy 1 δy 4 3 2 δy 5 3 3 δy 6 2 4 δy 7 2 5 δy 8 and higher 2 Increment of δy until failure 2.6.5.3 SAC Loading Protocol (1997) SAC is a joint venture of SEA, ATC and CUREE. In 1997, as part of a research project for the cyclic testing of steel moment-resisting connections, Krawinkler developed two loading protocols: a standard loading protocol and a near-fault effect loading protocol. The standard protocol is based on inter-storey drift angle (ratio of inter-storey displacement and the storey height) time history response and was developed by subjecting several hypothetical steel moment-resisting frame structures to a wide range of seismic input (SAC, 1997). This protocol (shown in Figure 9) does not require previous tests to obtain the parameters that characterize the protocol. The sequence of loading is presented in Table 8. Figure 9. SAC loading protocol 25 Table 8. Sequence of loading of the SAC protocol Cycle Group Number of Cycles Drift Angle Amplitude (rad) 1 6 0.00375 2 6 0.005 3 6 0.0075 4 4 0.01 5 2 0.015 6 2 0.02 7 2 0.03 8 and grater 2 Increment of 0.01 until failure 2.6.5.2 JISF Loading Protocol (2002) This protocol is recommended by the Building Research Institute and the Japan Iron and Steel Federation and constitutes a very well accepted protocol in Japan for steel structures. The protocol was obtained through the response analyses of steel frame specifically. The reference parameter that characterized the loading protocol is the elastic beam rotation (θp) when the moment of the beam’s cantilever end equals its full plastic moment (Mp). The cyclic loading starts from ±2 θp, with intervals of 2 θp. For each loading step, two cycles are applied, as shown in Figure 10 and Table 9. Figure 10. JISF loading protocol 26 Table 9. Sequence of loading of the JISF loading protocol Cycle Group Number of Cycles Elastic Beam Rotation (θp) 1 2 2 2 2 4 3 2 6 4 2 8 5 and grater 2 Increment of 2 until failure 2.6.5.2 AISC-341 Loading Protocol (2016) The ANSI/AISC 341-16 (an American National Standard) “Seismic Provisions for Structural Steel Buildings” provide recommendations for testing procedures that are contained in Appendix S “Qualifying Cyclic Tests of Beam-to-Column and Link-to Column Connections”. The recommendations are based on the results of the 1997 SAC project previously discussed and were adopted to update the seismic design provisions after significant damages to frame buildings that came because of the 1994 Northridge earthquake (AISC 2016). The main objective of the testing protocols is towards a qualification-oriented procedure for beam-to-column and link-to-column connections. For beam-to-column connections, the inter-story drift angle θ is directly assumed as the control parameter. As it is a qualifying test, there are specified requirements for strength and inter-story drift angle θ. The two loading protocols are shown in Figure 11 and broken down in Table 10. Cycles greater than the cycle group for beam-to-column moment connections and link-to-column connections will carry on with cycle increments of 0.01 and 0.02, respectively. 27 Figure 11. AISC loading protocols: a) beam-to-column connections b) link-to column connections Table 10. Sequence of loading of the AISC loading protocols Beam-to-Column Moment Connections Link-to-Column Connections Cycle Group Number of cycles Drift Angle Amplitude (rad) Cycle Group Number of cycles Drift Angle Amplitude (rad) 1 6 0.00375 1 6 0.00375 2 6 0.005 2 6 0.005 3 6 0.0075 3 6 0.0075 4 4 0.01 4 6 0.01 5 2 0.015 5 4 0.015 6 2 0.02 6 4 0.02 7 2 0.03 7 2 0.03 8 2 0.04 8 1 0.04 8 or grater 2 Increments of 0.01 until failure 9 1 0.05 - - - 10 1 0.07 - - - 11 1 0.09 - - - 11 or grater 1 Increments of 0.02 until failure -0.04-0.03-0.02-0.0100.010.020.030.040 5 10 15 20 25 30Drift Angle (rad)Number of Cycles-0.09-0.06-0.0300.030.060.090 5 10 15 20 25 30 35 40Drift Angle (rad)Number of Cyclesa) b) 28 2.6.6 Standard Loading Protocols for Wood Structures Several damaging seismic events, like the above-mentioned 1994 Northridge earthquake, increased the interest in the seismic response of wood components and structural systems in the United States. Due to the different characteristics of wood structures, such as the non-isotropic properties of the material, there has been an increment in the number of experimental studies of full-scale cyclic tests on wood structures. Therefore, several loading protocols have been proposed for these cases, and the most common ones will be presented in the following sections. 2.6.6.1 FCC Loading Protocol (1993) This loading protocol was developed by the Forint Canada Corporation (Karacabeyli, 1998) and consists of a progression of 3 cycles arranged in different groups of equal magnitude, as shown in Figure 11. The reference parameter is taken as the nominal yield slip, defined as half the ultimate load displacement, and obtained from a preliminary monotonic test. The protocol starts at a cycle amplitude of 50% of the reference displacement followed by 100% and then back to 50% of the reference displacement. A similar trend is followed until the end of the loading protocols as shown in Figure 12 and Table 11. Figure 12. FCC loading protocols 29 Table 11. Sequence of loading of the FCC loading protocol Cycle Group Number of Cycles Amplitude (% of Yield Slip) 1 3 50 2 3 100 3 3 50 4 3 150 5 3 100 6 3 200 7 3 150 8 3 250 9 3 300 10 3 350 10 and grater 3 Increment of 50% until failure 2.6.6.2 ASTM E564 Loading Protocol (1995) The American Society of Testing and Materials (ASTM) developed a standard loading protocol to evaluate entire wood frame shear walls. The standard provides two load sequences: a static load sequence (not shown here) with only positive cycles and a cyclic load sequence. The static loading consists of a pre-load of 10% of the expected ultimate load (Fu) and it is meant to “seat” the connections. After that, the protocol starts with cycles of one-third of the expected ultimate load followed by increases of the same amplitude until the ultimate load is reached. Five cycles are completed at each one-third load increment and after each increment group, the specimen is unloaded before the application of the next higher load increment. Moreover, the duration of each cycle increment should be enough to permit measuring and recording of load and deflection. The cyclic loading protocol is shown in Figure 13 and Table 12. 30 Figure 13. ASTM E564 loading protocol: a) static load and b) cyclic loading Table 12. Sequence of loading of the ASTM 564 loading protocol Static Loading Cyclic Loading Cycle Group Number of half-cycles Amplitude (% of Fu) Cycle Group Number of cycles Amplitude (% of Fu) 1 1 10 1 1 10 2 1 33 2 5 33 3 1 66 3 5 66 4 1 100 4 5 100 2.6.6.3 ISO Loading Protocol (1997) The ISO loading protocol was developed by the working group 7 of the ISO Technical Committee on Timber Structures. Even though it was originally developed for wood connections, it has been also considered suitable for wood frame shear wall cyclic testing. The loading is based on the displacement at ultimate load, which is taken as the average of previous values from a monotonic test. The loading protocol is presented in Figure 14 and Table 13. 31 Figure 14. ISO loading protocol Table 13. Sequence of loading of the ISO loading protocol Cycle Group Number of Cycles Amplitude (% of Maximum Displacement) 1 3 2.5 2 3 5 3 3 7.5 4 3 10 5 3 12.5 6 3 25 7 3 50 8 3 75 9 3 100 > 9 3 Increment of 50% until failure 2.6.6.1 CUREE-Caltech Loading Protocol (2001) Krawinkler et al. (2001) developed new cyclic loading protocols with the objective of testing wood frame components. Two loading protocols were developed based on a statistical analysis of several nonlinear dynamic analyses of SDOF systems: a standard loading protocol and a near-fault loading protocol (not presented here). The standard protocol is formed by initiation cycles, primary cycles, and trailing cycles. The reference parameter is the displacement taken as a fraction of the ultimate displacement obtained from a preliminary monotonic test. The protocol 32 begins with a sequence of 6 initiation cycles that represent the cumulative damage that comes from small earthquakes that could have occurred during the life of the specimen prior to the main seismic event. Then, these cycles are followed by the primary and trailing cycles, which are applied in an interspersal way. The amplitude of the trailing cycles is 75% of the amplitude of the previous group of primary cycles. The CUREE protocol is shown in Figure 15 and Table 14. Figure 15. CUREE-Caltech loading protocol Table 14. Sequence of loading of the CUREE-Caltech loading protocol Primary Cycles Trailing Cycles Cycle Group Number of half-cycles Amplitude (% displacement) Cycle Group Number of cycles Amplitude (% displacement) 1 6 5 1 6 --- 2 1 7.5 2 1 5.625 3 1 10 3 1 7.5 4 1 20 4 1 15 5 1 30 5 1 22.5 6 1 40 6 1 30 7 1 70 7 1 52.5 8 1 100 8 1 75 9 1 150 9 1 112.5 > 9 1 Increments of 50% until failure > 9 1 Increments of 37.5% until failure 33 2.6.5 Standard Loading Protocols for Concrete Structures To date, several cyclic testing of RC structures has been conducted by many researchers implementing a variety of loading protocols. However, only two loading protocols have been highlighted among those cases: The New Zealand and ACI 374.2R protocols. In addition, another protocol derived from New Zealand, known as the Modified New Zealand protocol, will also be reviewed. 2.6.5.1 New Zealand Protocol (1991) The New Zealand protocol was developed as part of a research project collaboration between the United States, New Zealand, Japan, and China. The protocol is based on a yield displacement (δy), which is obtained by extrapolating the displacement of the test specimen at 75% of the theoretical strength (Vi) measured during the third cycle of the loading sequence (Cheung et al., 1991). One of the advantages of this loading protocol is that it does not require the execution of a preliminary monotonic test; however, it does require carrying out a preliminary analysis to determine the strength of the specimen. The cyclic loading of the protocols is shown in Figure 16 and Table 15. Interestingly, the first three cycles of the protocol are load controlled, where the first two impose a lateral force of 50% of the theoretical strength (Vi) of the specimen and the third cycle is used to determine the yield displacement (δy) of the specimen as described above. The subsequent group of cycles is displacement controlled with the increasing stepwise ductility factors (μ) shown in Table 15 and calculated as δ δy⁄ . Figure 16. New Zealand protocol 34 Table 15. Sequence of loading of the New Zealand protocol Cycle Group Number of Cycles Control Variable Amplitude 1 2 𝑉𝑖 0.5 𝑉𝑖 2 1 δy 1 δy 3 2 2 δy 4 2 4 δy 5 2 6 δy 6 2 8 δy 7 2 10 δy 8 and grater 2 Increment of 2δy until failure 2.6.5.2 Modified New Zealand Protocol (2002) The Modified New Zealand protocol was developed at the University of California San Diego to evaluate the seismic performance of hollow rectangular RC piers with highly confined boundary elements (Hines et al., 2002). However, it has been also widely implemented in RC bridge piers of circular and rectangular column cross-section. The protocols require the same reference parameters as the New Zealand protocol to be characterized, i.e., the theoretical strength (Vi) of the specimen and yield displacement (δy). The difference lies in the number of cycles from the force and displacement-controlled section. The force-based portion has two more cycles at 0.25Vi, 0.75Vi amplitudes. Likewise, the number of cycles in the displacement-controlled portion of the protocols is increased by 1 cycle per cycle group, i.e., now three cycles are applied at each amplitude. Additionally, one extra group of 3 cycles at 1.5 δy is included. The entire loading protocol is shown in Figure 17 and Table 16. 35 Figure 17. Modified New Zealand loading protocol Table 16. Sequence of loading of the Modified New Zealand loading protocol Cycle Group Number of Cycles Control Variable Amplitude 1 1 𝑉𝑖 0.25 𝑉𝑖 2 1 0.5 𝑉𝑖 3 1 0.75 𝑉𝑖 5 3 δy 1 δy 6 3 1.5 δy 7 3 2 δy 8 3 4 δy 9 3 6 δy 10 3 8 δy 11 and grater 3 Increment of 2δy until failure 2.6.5.3 ACI 374.2R (2013) This loading protocol is suggested by the American Concrete Institute (ACI) with the objective of complementing those loading protocols with a specific focus like the ATC-24 protocol for steel, the Seible and Hosee (2000) protocol for bridges, and the FEMA 461 protocol for structural and non-structural components. Moreover, the ACI report highlights that the objective of the protocols also is aimed to increase the effectiveness of the research effort that is affected by the lack of consistency when selecting loading histories, measurement of data, and the presentation 36 of test observations and results. For all that, a consensus-based loading protocol is provided for RC structural components, as shown in Figure 18. The protocol is characterized by the normalized yield displacement (δ/δy) and the cycle groups are arranged in groups of two following the amplitudes presented in Table 17. Figure 18. ACI 374.2R loading protocol Table 17. Sequence of loading of the ACI 374.2R loading protocol Cycle Group Number of Cycles δ/δy 1 2 0.5 2 2 1 3 2 2 4 2 3 5 2 4 6 2 5 7 2 6 8 2 7 9 2 8 > 9 2 Increment of 2 δ/δy until failure 37 2.6.7 Loading Protocols and Performance Evaluation Guidelines Specifically Developed for Bridge Piers 2.6.7.1 Hose and Seible (1999) In the year 1999, Hose and Seible published a document containing guidelines for the performance evaluation of bridge piers called “Performance Evaluation Database for Concrete Bridge Components and Systems under Simulated Seismic Loads” (Hose and Seible, 1999). The research was part of a comprehensive program instigated by the Pacific Earthquake Engineering Research (PEER) and was conducted at the University of California San Diego to develop a parametric basis for performance-based bridge engineering in the form of a bridge performance catalogue. Even though this work did not provide a loading protocol as such, it was focused on determining the applicability of different approaches (e.g., multi-step design) for the capacity evaluation of bridge components, sub-assemblages, and systems to accurately characterize and predict structural behaviour. The database presented in this paper is formed by a five-level performance evaluation approach divided into three different stages that translates performance descriptions into engineering limit states. The three stages are: “Bridge Damage Assessment”, where classifications of structural damage are related to socio-economic descriptions and the seismic performance levels are defined in terms of repair effort; “Bridge Performance Assessment”, which qualitatively and quantitatively relates the structural damage to the capacity of the structure with parameters that range from cracking to strength degradation; and “Bridge Performance/Design Parameters”, where a framework of evaluation parameters is given for the five qualitative performance levels of the previous stage. This parameterization helps to establish a consistent performance-based design methodology of structures that are susceptible to seismic hazards, such as bridges. For more details and case studies, the complete report is available online in the PEER database. 2.6.7.2 Duscika and Bazaez (2016) In 2016, Duscika and Bazaez developed a loading protocol specifically for subduction megathrust earthquakes for circular bridge piers. The authors followed a methodology like the one presented by Krawinkler et al. (2001). Results from nonlinear time-history analyses were used considering several subduction ground motions imposed on structures with a wide range of 38 structural periods and target ductility values. The target demand responses that were transformed into loading protocols were obtained employing a constant ductility design. In other words, each ground motion and structural period served as the basis to design the structure to produce a nonlinear response (i.e., the normalized yield displacement) that will be further transformed into a stepwise increasing sequence of cycles. Different protocols were developed for short-, medium-, and long-period responses and were represented by the following expression: 𝒇(𝑵) = (𝒂 ∙ 𝒆𝒃𝑵 + 𝒄 ∙ 𝒆𝒅𝑵)𝜹𝒚 (6) where 𝑁 is the number of inelastic cycles, 𝛿𝑦 is the yield displacement and a, b, c, and d are coefficients that characterized the progression of cycles in the loading protocol and are dependent on the period of the structure and the target ductility demand selected. As an example, a loading protocol calculated for a bridge column with 1 second fundamental period is shown in Figure 19. Figure 19. Duscika and Bazaez loading protocol 39 CHAPTER 3: PROPOSED PROCEDURE METHODOLOGY In this chapter, a numerical procedure leading to the development of quasi-static loading protocols for RC bridge piers will be presented. The general methodology in the development of loading protocols is discussed first. Then the initial considerations for the proposed procedure will be presented to establish the implications and the scope of the study. Later in the thesis, this methodology will be implemented in a case study. The task at hand is to present a relatively simple approach to develop suitable loading protocols for RC bridge piers. The need of sharing experimental data around the world makes it necessary to establish loading protocols that can be employed universally without subjective opinions. 3.1 General Methodology and Implications The importance of cyclic testing lies in the fact that it allows the acquisition of knowledge on the structure’s cyclic behaviour that is useful, for example, for defining numerical models or improving structural detailing. Thus, the development of loading protocols for seismic performance testing requires the execution of time history analysis to capture the demand characteristics peculiar to any structural system of interest (e.g., bridge piers). However, to develop component-specific loading protocols, assumptions and judgment need to be used to transform random structural responses to a simple yet statistically representative series of symmetric cycles. Before the development and/or selection of a loading protocol, one must first define the objective of the testing program because it will directly provide the main arguments for adopting one loading protocol over another. The number, ranges, and amplitudes of all damaging cycles that a structure may experience are considered in a loading protocol and can be obtained from representative analytical models of the structural system subjected to ground motions of various characteristics. Results are statistically processed, which also involves the transition, rearrangement, and shift of cycles from the structural response where and when necessary. The following three major steps are normally employed: 1) Modelling and analysis of structural models. This step typically starts by determining a seismic hazard from the seismic de-aggregation of the zone considered. This parameter is used to establish a representative protocol, which in turn indicates that a single protocol that accounts for all cases is not suitable. Then, the desired structure is subjected to the considered ground motions. The structure can be designed as per the correspondent design code, it can be a specimen taken from previous experimental studies or also an existing structure that 40 needs performance evaluation. 2) Identification of structural response. A critical response parameter needs to be identified such that the resulting loading protocol is representative of the most damaging demand. Then, the control parameter that will characterize the loading protocol is to be selected, which mainly depends on whether the member is displacement- or force-sensitive. Moreover, the loading protocol is typically constructed considering a performance assessment at various performance levels (emphasis on performance-based design). Likewise, it is recommended to consider the situation of the structure before the performance assessment as it can represent an important source of damage. 3) Cycle processing and development of loading protocol. After the response of the structure is obtained and processed, the random nature of it must be transformed into a series of uniform cycles of different amplitudes. The most widely used methodologies to work with the time-history response include the rain flow counting and the level crossing counting. These methods can be easily found in fatigue analysis problems, but they can be also extended as cumulative damage is a common factor between both cases. Seismic demand parameters are normally characterized by the total number of damaging cycles, plastic deformation range, and peak deformation, and represent the basic information that controls the pattern of the upcoming loading protocol. These parameters are statistically processed and arranged with an appropriate sequence subjected to both the rain flow-counting original pattern and engineering judgement. 3.2 Proposed Procedure As shown before, loading protocols must be an accurate representation of the demands certain structure is likely to be subjected to and should be developed considering cumulative damage implications. Such a process requires statistical values from structural responses of several ground motion records to decide on the appropriate number and sequence of cycles that will form the loading protocol. The proposed methodology allows one to obtain the information required to construct the loading protocols. It is organized and described as follows: a) The objective of the loading protocol is established along with the implications and limitations of the testing program that might apply (e.g., maximum capacity of hydraulic jacks). b) The appropriate seismic hazard level(s) along with the number and type of earthquakes to analyze the structure are defined per the location of the structure. 41 c) An appropriate cumulative damage model is selected based on the structural component. Sensitivity to stiffness and strength degradations should be considered. d) The reference response parameters that will be used to construct the loading protocols are established (i.e., parameters related to structural performance and cumulative damage) e) Demand levels of interest (target demand levels) are selected to construct the loading protocols for each selected value (e.g., ductility demand levels). f) A suitable numerical model capable of representing the peculiar hysteretic characteristics of the structure is created (e.g., a displacement-based fibre model for a bridge pier). g) The model is subjected to the ground motion records utilizing an Incremental Dynamic Analysis (IDA). This approach is considered such that each ground motion is scaled up until the maximum demand level in the column exceeds the target (the scaling factor step is arbitrarily selected and depends on the desired level of precision). The maximum demand for each scaling step is monitored and stored along with the correspondent factor (i.e., creation of IDA curves). The specific scaling factors that will result in demand equal to the target defined are calculated by linearly interpolating the values from the IDA curves. A new set of ground motions records is then created by scaling the original ground motions with the calculated factors for all the target demand levels. h) A Time-History Analysis (THA) is performed with the new set of ground motions to analyze the performance of the structure that corresponds to the target desired. Each nonlinear time-history response is normalized by the reference response parameter. This variable must agree with the target demand parameters. i) The correspondent parameters are analyzed from all the normalized structural responses obtained in the previous step. The number of cycles is extracted utilizing a cycle counting method (e.g., the rain-flow cycle counting algorithm) and the cumulative damage is calculated for each cycle counted. j) Statistical values are obtained of the reference parameters for all ground motion records and all ductility demand levels. The values are used to construct the loading protocols with engineering judgement and following the results from the rain-flow cycle counting, the target demands, and the cumulative damage calculations. As it has been discussed in this chapter, the nonlinear time-history response of the structure is the main source of information that needs to be processed and analyzed to propose loading 42 protocols for quasi-static cyclic testing. Therefore, the development procedure implemented to develop the loading protocols can be consistently expressed in five main steps: 1) Selection and matching of ground motion records. 2) Design and numerical modelling of a representative bridge pier. 3) Iterative determination of targeted seismic demands (IDA). 4) Statistical post-processing of results. 5) Construction of loading protocols. The procedure is illustrated in Figure. 20: Figure 20. Flow chart of the methodology for the development of loading protocols 3.2.1 Recommendations for Bridge Piers The procedure presented for the development of quasi-static loading protocols is surrounded by many considerations related to the objective of the test, the type of structural component, the seismic hazard considered, among others. For that reason, the following recommendations and observations are presented specifically for RC bridge piers of a single column. The first thing to discuss is the determination of the objective of the quasi-static test. For RC bridge piers, typically experimentalists want to evaluate the performance of the structure by subjecting it to a loading protocol that allows the observation of stiffness and strength degradation. 43 For that reason, the applicability of the loading protocols to limit states other than capacity is not addressed. Another important point is the selection of the seismic hazard considered. As discussed, they must be representative of the zone where the structure is located and therefore target design spectrum can be defined considering approaches such as the Uniform Hazard Spectrum (UHS), Conditional Mean Spectrum (Baker 2011), or Conditional Spectrum (Jayaram et al. 2011) which have different characteristics and depending on the case account for variables including source mechanisms, paths, and site effects. Moreover, ground motions can come from different sources (e.g., crustal, subcrustal and subduction earthquakes) with different characteristics such as frequency content, duration and return periods. Sets of 11 records are recommended for each type (NBCC 2019). Emphasis is usually placed on performance assessment of components at ground motion levels associated with long return periods, i.e., 475 and 2475 years return periods. The variability between records can also be tackled by considering an appropriate percentile value of the concerned response, normally the 84th percentile (Krawlinkler et al. 2001). Additionally, it is worth mentioning that the proposed procedure involved the scaling of records (IDA approach); therefore, spectral matching of selected ground motions to the design spectrum of the correspondent location can be an option to bring some consistency to the analysis since now all the records will have a similar spectrum. The definition of a target demand is also an important step before constructing the loading protocols. Common target demands have been ductility displacement, strain, axial deformation, rotation, etc., for structures such as RC bridge columns, steel braces and connections. With this, also comes the determination of an appropriate reference response parameter, which for RC bridge piers is normally taken as the displacement of the tip of the column specimen. The nonlinear time-history response of the structure is normalized with yield displacement (𝛿𝑦) of the structure. Thus, the target demands can be taken as ductility values of 2, 4 and 8 to account for low, moderate, and high demands, respectively. Each target demand level is represented by a separate loading protocol (lower demand level might be contained in higher demand levels). Since the loading protocol will be a representation of the response of the structure, the selection of a structural system for predicting the response is also important. For RC bridge piers, fibre modelling has been a widely implemented technique capable of representing the hysteretic characteristics peculiar of RC bridge piers. The nonlinearity of the system is captured by material and geometric nonlinearity coming from the constitutive laws of concrete and steel reinforcement, and the P-delta effect. Once the 44 response has been obtained, an important consideration for RC bridge piers is the sensitivity to stiffness and strength degradation. Here the concept of cumulative damage becomes crucial since it is implemented as a reference to ensure the test subjects the specimen to comparable damage that would be expected from dynamic analysis. Moreover, a direct relationship can be established between the number of inelastic cycles and the cumulative damage experienced by the structure (Bazaez and Dusicka 2016). Approaches such as distinguishing between inelastic cycles before (pre-peak) and after (post-peak) the maximum (or minimum) peak in the normalized nonlinear time-history response can also be examined to construct the loading protocol. However, one must be careful considering effects such as mean and sequence effect of the cycles. For RC bridge piers, usually, the mean and sequence effects are ignored, meaning that the inelastic cycles are centred to the origin and ordered in decreasing magnitude, respectively. Finally, it is worth mentioning that the implications and the procedure presented in this chapter were defined considering the cyclic loading protocol development procedures by Krawinkler et al. (2001), Mergos and Beyer (2014), and Bazaez and Dusicka (2016). Likewise, it is worth noting that many of the previous studies on loading protocol development have used results from time-history analysis (THA) of Single-Degree-of-Freedom Systems (SDOF) and a constant ductility design to reach targeted ductility demands through an iterative process (i.e., varying the structure’s stiffness to reach predetermine responses) (Bazaez and Dusicka 2016). This approach is not considered in this work, rather an IDA procedure is proposed, keeping constant the structure’s properties, and instead varying the intensity of the seismic input to achieve certain target ductility demand. 45 CHAPTER 4: CASE STUDY As presented in previous chapters, in the literature several loading protocols have been proposed for quasi-static cyclic testing; however, these procedures may not always be adequate, for example, different reinforced concrete (RC) bridge piers could have substantially different structural characteristics, such as strength and deformation capacities. The unavailability of a universal loading protocol applicable to all structural systems stems from the fact that the performance of structural systems is largely dependent on the history of inelastic loading, known as the cumulative damage process (Krawinkler et al. 2001). In this chapter, the procedure presented in Chapter 3 for the development of quasi-static loading protocols of RC bridge piers will be implemented in a case study of an archetype bridge pier located in Vancouver city, British Columbia, Canada. 4.1 Selection of Ground Motion Records The selection of the appropriate ground motions is one of the most important steps for the development of representative loading protocols. Ground motions should reflect the seismic hazard for the region where the tested component would be constructed (Krawinkler et al. 2001). For this study, the region stretching from northern Vancouver Island in Canada to Northern California in the United States, i.e., Cascadia subduction zone, was selected. This region is considered a high seismic zone. The most recent major seismic event in this region was the 7.7 magnitude Haida Gwaii earthquake in 2012. Three different earthquake types have been identified in the Cascadia subduction zone: shallow (also called crustal), subcrustal, and deep (also called subduction) earthquakes (CREW 2009). Thus, in this study, three ground motion sets representative of crustal, subcrustal, and subduction earthquakes were considered. Each set contains 11 ground motion records. The ground motion records, for crustal, subcrustal, and subduction earthquakes, were obtained from the Pacific Earthquake Engineering Research Centre (PEER) (PEER 2011), Consortium of Organizations for Strong Motion Observation Systems (COSMOS), and Kyoshin Network (K-Net) databases, respectively. The selection process was carried out discarding near-fault and soft soil ground motions. For each selected ground motion recording station, the two orthogonal horizontal ground motion components were utilized, whereas the vertical ground motion component was not considered. Details of selected ground motion sets are given in Table 18. 46 Table 18. Ground motions suites Type # Earthquake Event Year Record Station 𝑴∗𝒂 𝑹𝒑∗𝒃 (km) PGA (g) Crustal 1 San Fernando 1971 Old Ridge Route 6.61 23 0.275 2 San Fernando 1971 CIT Athenaeum 6.61 25 0.159 3 Loma Prieta 1989 Coyote Lake Dam 6.93 20 0.109 4 Loma Prieta 1989 Gilroy Array #6 6.93 18 0.152 5 Northridge 1994 Burbank - Howard Rd. 6.69 17 0.126 6 Northridge 1994 Castaic - Old Ridge Route 6.69 21 0.568 7 Northridge 1994 Brentwood VA Hospital 6.69 23 0.185 8 Northridge 1994 Sunland - Mt Gleason Ave 6.69 13 0.133 9 Loma Prieta 1989 Hollister & Pine 6.93 51 0.179 10 Loma Prieta 1989 Palo Alto - SLAC Lab 6.93 54 0.277 11 Kern County 1952 Taft Lincoln School 7.40 46 0.111 Subcrustal 1 Nisqually 2001 0720a 6.8 70 0.187 2 Nisqually 2001 1416a 6.8 64 0.387 3 Nisqually 2001 1416c 6.8 64 0.356 4 Nisqually 2001 1421a 6.8 59 0.322 5 Nisqually 2001 1421c 6.8 59 0.489 6 El Salvador 2001 4359a 7.6 76 0.478 7 El Salvador 2001 4359c 7.6 76 0.495 8 El Salvador 2001 6987c 7.6 70 0.228 9 El Salvador 2001 7133c 7.6 82 0.278 10 El Salvador 2001 7147c 7.6 93 0.303 11 El Salvador 2001 7157c 7.6 83 0.257 Subduction 1 Tohoku, Japan 2011 AKT017 9 152 0.812 2 Tohoku, Japan 2011 AKT017 9 152 1.246 3 Tohoku, Japan 2011 AKTH191 9 155 1.021 4 Tohoku, Japan 2011 AKTH191 9 155 1.138 5 Tohoku, Japan 2011 AOM008 9 166 0.830 6 Tohoku, Japan 2011 AOM008 9 166 0.871 7 Tohoku, Japan 2011 AOM028 9 167 0.642 8 Tohoku, Japan 2011 IBR008 9 161 1.185 9 Tohoku, Japan 2011 IBR005 9 162 1.072 10 Tohoku, Japan 2011 YMT002 9 161 0.800 11 Tohoku, Japan 2011 YMT009 9 156 0.753 a Moment magnitude b Closest distance to fault rupture 47 All the selected ground motion records were spectrally matched to the hazard level of a site in Vancouver (i.e., where the prototype example bridge to be constructed) for 2% in 50 years probability of exceedance (2,475-year return period). The target site-specific design response spectrum was extracted from the National Building Code of Canada (NBCC 2019), which corresponds to a Uniform Hazard Spectrum (UHS). Here, it is worth mentioning that experimentalists employing the proposed procedure are not obligated to use the UHS for generating the ground motions. In fact, they are even encouraged to explore other types of target spectrums such as Conditional Mean Spectrum (Baker 2011), Conditional Spectrum (Jayaram et al. 2011), among others. Spectrally matching the selected records is an important step in the loading protocol development to ensure that the selected records are representative of the seismic hazard to which the tested specimen may be susceptible. Also, spectral matching is useful in bringing consistency to the scaling factors when achieving the target ductility levels as part of the IDA, as will be illustrated in subsequent sections of this manuscript. The matching process was carried out using SeismoMatch (Seismosoft 2020) software. The software adjusts any ground motion accelerogram to match a target response spectrum using a wavelet algorithm proposed by Hancock et al. (2006), which often results in minor intervention in the frequency content of the records if applied over small ranges of periods. The matching was performed within the period range of interest, which was taken as 0.2T to 1.5T following the ASCE 7-10 (AISC 2016) specifications, where T is the fundamental period of the structure. T of the prototype example bridge pier in this study (introduced in the next section) was 0.7 seconds. The mean response spectrum of the matched ground motion records and the target response spectrum are shown in Figure 21. Figure 21. Mean response spectrums matched to the NBCC spectrum for 2% in 50 years 48 4.2 Design and Geometry of the Example Prototype Bridge Pier One of the recommended steps in the loading protocol development is to select a prototype building/bridge representative of a significant portion of the overall building/bridge inventory in a region. This would save effort and time for experimentalists intending to develop loading protocols for such structures having similar characteristics to the prototype. In this study, a loading protocol is developed for a bridge pier as a demonstration. Siddiquee and Alam (2017) developed an inventory for highway bridges in British Columbia (BC) that are under the responsibility of the BC Ministry of Transportation and Infrastructure (BC MoT). In brief, in the bridge inventory, 2555 highway bridges in BC were examined and classified into 11 classes depending on their bent configurations. 11 bridges with bents having a single column, which are the scope of this study, were identified. One bridge with the most representative structural and geometric characteristics was selected. The selected bridge was a three-span continuous concrete-box girder bridge and classified as a major route bridge. The three spans have an equal length of 30.3 m and are composed of multiple precast segments, which are post-tensioned together in the longitudinal direction. The bridge piers are monolithically connected to the deck. The bridge middle pier was seismically designed following the force-based design approach in CHBDC (CSA S6 2019). For bridge design purposes, the bridge was assumed to be located in Vancouver, BC. A response modification factor (R) of 4 and an importance factor (𝐼𝐸) of 1.5 were considered. The effective height of the pier (L) was 9.14m and the diameter of the column (D) was 1.83m, corresponding to an aspect ratio (𝐷 𝐿⁄ ) of 5, ensuring a flexure dominated behaviour. To meet the design requirements, the pier was reinforced with 40 longitudinal reinforcing bars of 32 mm diameter (longitudinal reinforcement ratio of 1.22%) and 20-mm-diameter steel spirals at 80 mm pitch (transverse reinforcement ratio of 0.91%). The transverse reinforcement ratio was determined per Clause 4.7.5.2.5 in CHBDC (CSA S6 2019) to ensure adequate confinement in the plastic hinge region. The bridge bent configuration and geometry, and column cross-section are shown in Figure 22. 49 Figure 22. Bridge bent: (a) configuration and geometry, (b) cross-section 4.2.1 Analysis of the Structure The weight of the superstructure from the two adjacent half spans was represented by a constant mass of 10912 kN lumped at the centre of mass of the bridge deck, which corresponds to an axial load (P) ratio [𝑃 (𝑓𝑐′ ∗ 𝐴𝑔)⁄ ] of 10%, where 𝑓𝑐′ and 𝐴𝑔 are the concrete compressive strength and gross cross-sectional area, respectively. With this information, the natural frequency of the bridge pier was calculated through an eigenvalue analysis resulting in an elastic period of 0.78 seconds for the first vibration mode and 0.126 seconds for the second one. The two vibration mode shapes are shown in Table 19. Table 19. Natural frequency and elastic period of the bridge pier Mode Natural Frequency (rad/seconds) Elastic Period (seconds) 1 8.06 0.78 Using a moment-curvature analysis, the yield, maximum and ultimate moment capacities of the bridge column were calculated. The correspondent values are shown in Table 20. An inelastic static pushover analysis (ISPA) in the transverse direction was carried out to compute the displacement ductility capacity of the pier. This step was critical to ensure that the pier exhibit ductility capacity greater than 8 which is the maximum target ductility demand considered. Displacement ductility is the ratio of the displacement at ultimate (𝛿𝑢) to the displacement at yield a) b) 50 (𝛿𝑦). 𝛿𝑢 was established when the load-carrying capacity is reduced by 20%, whereas 𝛿𝑦 was computed as follows (Elnashai and Di Sarno 2008): 𝜹𝒚 = 𝜹𝒚′ (𝑴𝑵𝑴𝒚′) (7) where, 𝛿𝑦′ is the measured first yield displacement, and 𝑀𝑁 and 𝑀𝑦′ are the analytical nominal and first yield moments, respectively. The moment-curvate and pushover curves are shown in Figure 23. Based on the static pushover analysis, the 𝛿𝑦 and 𝛿𝑢 of the selected bridge pier were 55 and 640 mm, respectively, which corresponded to a displacement ductility capacity of 11.6. In addition, the modified compression field theory (Vecchio and Collins 1986), which predicts the experimentally determined shear failure within 1% error (Bentz et al. 2006), was used to compute the shear capacity of the pier. The shear capacity of the pier was 7130 kN which is noticeably greater than the shear force demand corresponding to the pier’s overstrength moment (i.e., 2600 kN), consequently indicating that the pier is capacity-protected against shear failure. Table 20. Moment-Curvature analysis results Yielding Moment (My) (kN-m) Maximum Moment (M max) (kN-m) Ultimate Moment (Mult) (kN-m) 14620 𝑥 103 19440 𝑥 103 18486 𝑥 103 0400080001200016000200000 5 10 15 20 25Moment (kN-m)Curvature (rad/km)a) 51 Figure 23. Bridge pier analysis: a) moment-curvature and b) pushover curves 4.3 Numerical Model of the Bridge Pier Accurate quantification of the seismic bridge response in terms of overall structural displacement requires generating numerical models capable of capturing material and bridge component nonlinearities. Fibre element-based nonlinear analysis program SeismoStruct (Seismosoft 2020) software was employed to create the numerical model of the bridge pier. In this study, the time-history analysis was performed to illustrate the proposed procedure of the loading protocol development, and therefore, for simplicity, the earthquake ground motions were applied to the bridge in the transverse direction only. For this reason, the response of the overall bridge in the transverse direction was represented by a single-degree-of-freedom (SDOF) model of the middle bridge pier (Priestley et al. 1996). In the SDOF model, the mass from the two adjacent half spans was lumped at an effective height of 9.14 m, which represents the distance from the ground level to the centroid of the superstructure, as illustrated in Fig. 4 (b). The bridge pier was modelled using a three-dimensional inelastic force-based element with five Gauss–Lobatto quadrature integration points. Material nonlinearities were considered through the concrete and reinforcement steel constitutive laws which were defined as per the Mander et al. (1988) and Menegotto and Pinto (1973) models, respectively. It should be noted that the Menegotto and Pinto (1973) steel model considers the Bauschinger effect, which captures the stiffness degradation under cyclic loading. Details of both concrete and steel reinforcement constitutive models employed in this study are given in Table 21. A rigid link was used to connect the top of the column to the centroid 050010001500200025000 100 200 300 400 500 600Base Shear (kN)Displacement (mm)Pushover Curveb) 52 of the superstructure. To simulate bar-slip rotations, an elastic rotational spring was added at the bottom end of the column. The stiffness of the rotational spring (𝐾𝑆𝐸) was computed using the following relationship (Elwood and Eberhard, 2009): 𝑲𝑺𝑬 = 𝟐𝑴𝒚𝝋𝒚𝒍𝒔𝒑 (8) where 𝑀𝑦 and 𝜑𝑦 are the moment and curvature at first yield, respectively, and 𝑙𝑠𝑝 is the effective strain penetration depth. A proposed expression by Elwood and Eberhard (2009) was used to compute 𝑙𝑠𝑝 which is as follows: 𝒍𝒔𝒑 = 𝟏𝟑. 𝟐 𝒇𝒚𝒇𝒄′𝒅𝒃 (9) where 𝑓𝑦 is the steel reinforcement yield strength in MPa, 𝑓𝑐′ is the concrete compressive strength in MPa, and 𝑑𝑏 is the longitudinal rebar diameter in mm. To account for damping, 5% Rayleigh damping was used in the model. The damping coefficients were evaluated such that the 5% damping ratio occurs in the first two modes as computed by the elastic modal analysis. Figure 24. Numerical model of the case study prototype bridge pier. Integration PointLumped MassInelastic Force-Based Element Column SectionFixed SupportGlobal AxesReinforcing SteelConfined Concrete Unconfined Concrete Rigid LinkRotational Spring 53 Table 21. Material properties Material Property Value Concrete Compressive strength (MPa) 41.5 Corresponding strain 0.0028 Tensile strength (MPa) 0 Elastic modulus (GPa) 23.6 Steel Elastic modulus (GPa) 200 Yield stress (MPa) 519 Ultimate strain 0.15 Strain hardening parameter 0.008 Transition curve initial shape parameter 20 Transition curve shape calibrating coefficient (A1) 18.5 Transition curve shape calibrating coefficient (A2) 0.15 4.1.1 Validation of the Numerical Model Before utilizing the numerical model, it is critical to validate it using available experimental data. For this study, a single-column bridge bent utilized in a shake-table tests program at the University of California, Berkley, was considered (Schoettler et al. 2015). The column was designed based on Caltrans Bridge Design specifications (Caltrans 2004) and Seismic Design Criteria (Caltrans 2006a). The specimen was subjected to uniaxial shake table excitation to evaluate the nonlinear response (seismic performance) under such specifications. For more information, the reader can refer to the complete research report available on the PEER website (PEER 2011). For the validation, the numerical model described in the previous section was employed to replicate test ‘EQ1’ (elastic response) and test ‘EQ3’ (inelastic response) from the experimental program. Figure 25 a) and b) provides a comparison between the two responses, respectively. Moreover, the inelastic period of the structure was computed through an eigenvalue analysis considering cracked-section properties. The value obtained from the model was 0.85 seconds, which closely matches the 0.88 seconds from the test report. From Figure 25 and the result of the eigenvalue analysis, it is evident that the model provides a good prediction of the dynamic response of bridge piers in the transverse direction under seismic loads. Additionally, it is worth mentioning that the ability of the fibre element-based models of predicting the seismic 54 response of bridge structures has also been widely demonstrated by several researchers through comparisons with experimental results (Casarotti and Pinho 2006; Bianchi et al. 2011; Billah and Alam 2013). Figure 25. Comparison of the experimental and numerical time-history displacement responses: a) EQ1 and b) EQ3 4.4 Cumulative Damage Model In earthquake engineering, the strength and deformation capacities of a structural component/system are strongly influenced by the cumulative damage it experiences during every displacement cycle as it directly contributes to the deterioration in its state of health. (Krawinkler et al., 1983). Therefore, the gradual loss of element resistance due to inelastic deformation makes the cumulative damage concept one of the bases for the development of a cyclic deformation loading protocol. General cumulative damage concept, in which the Coffin-Mason relationship or the Miner’s rule are often employed, has been utilized in most of the previous works on loading protocol developments to represent the damage imposed by the displacement reversals (Krawinkler et al., 1983; Mergos and Beyer 2014). Additionally, another popular approach serving the same purpose for RC structures is the damage index which suggests that the total damage from earthquake motions is composed of the damage caused by the maximum structural deformation and the absorbed hysteretic energy (Park and Ang 1985). However, the previous models contain constants that need to be determined experimentally which can result in undesirable uncertainties in the loading protocol development. For this reason, in this case study, the damage index selected to represent the damage imposed by the displacement reversals was the cumulative displacement ductility (CDD) factor proposed by Park (1989). The CDD is computed utilizing equation 2 (presented in Chapter 2) and represents the sum of the individual displacement ductility (𝜇𝑖) -350-250-150-50501502500 5 10 15 20 25 30Displacement (mm)Time (seconds)Experimental (Schoettler et al. 2015)Numerical-350-250-150-50501502500 10 20 30 40Displacement (mm)Time (seconds)Experimental (Schoettler et al. 2015)Numerical 55 demands undergone by a structure during cycles of loading. Unlike previous models, CDD factor does not contain any parameters that need to be determined experimentally. Two important observations can be made when this cumulative damage factor is used. First, the structure’s capacity decreases as the number of inelastic cycles increases, and second, inelastic cycles with the largest displacement amplitudes are the most damaging (Krawinkler et al. 2001). Caution must be taken while using the CDD factor because two different cyclic loading protocols could generate the same CDD yet impose significantly different levels of damage to a structure. For instance, loading protocol with 10 cycles and 𝜇𝑖 of 1 in each direction imposes significantly less damage than that with 2 cycles and 𝜇𝑖 of 5, despite having the same CDD of 20. Therefore, in this study, CDD was considered in conjunction with the number of inelastic cycles (N), and the magnitude of the plastic deformation range of each inelastic excursion as target demand parameters for the loading protocol development as will be illustrated in subsequent sections of this manuscript. 4.5 Target Response - Incremental Dynamic Analysis The seismic response of a structure largely depends on the configuration, strength, elastic, and inelastic characteristics of its structural components as well as the seismic input to which the structure may be subjected. For example, for bridge piers, the most convenient parameter to quantify the seismic response is the pier top displacement (or the so-called drift ratio if divided by the pier’s height). Such parameter can be easily assessed from dynamic analysis and used to establish the system displacement ductility (𝜇𝑖) demand. Moreover, 𝜇𝑖 demand is a key parameter that is often evaluated when performing seismic design of bridges following the AASHTO Guide Specifications for LRFD Seismic Bridge Design (2017). For those reasons, in this study, 𝜇𝑖 demand was considered as the primary target demand parameter for the loading protocol development. Target 𝜇𝑖 demand levels of two, four, and eight which correspond to low, moderate, and high levels of seismicity, respectively, were considered. In previous studies on loading protocol development, target 𝜇𝑖 demand levels were achieved following a constant ductility design approach in which the yield strength of the system is modified iteratively to provide the desired ductility value (Bazaez and Dusicka 2016). Such an approach might not be feasible for existing structures whose yield strength cannot be changed. Therefore, in this study, the IDA-based approach was introduced and utilized as a more robust 56 alternative to reach the desired 𝜇𝑖 demands. In IDA, the structural model is subjected to ground motion records, each scaled to multiple levels of intensity to provide a continuous picture of the structure from elasticity to yielding and eventually to failure (Vamvatsikos and Cornell 2002). The steps involved in the proposed procedure to develop the loading protocols are depicted in Figure 20 (presented in Chapter 3) using a flowchart diagram. First, ground motion records representative of the seismicity of the region where the specimen would be constructed need to be selected and spectrally matched. After creating a validated numerical model of the structure, the ground motions are scaled as a part of the IDA framework to achieve the target 𝜇𝑖 demand levels. In this study, scaling factors started from 0.5 and were increased in 0.5 increments until all the target 𝜇𝑖 demand levels were achieved. Linear interpolation can be done to determine the required scaling factors if the scaling results in an overestimation of the target 𝜇𝑖 demand level due to the size of the increments. An example of IDA-curves from the discussed procedure is presented in Figure 26 for two arbitrary ground motion records, and the scaling factors obtained for all ground motions are shown in Table 22. After determining the appropriate scaling factors, nonlinear time-history analyses are performed to generate and store the displacement time-history responses corresponding to the target 𝜇𝑖 demand levels. Then, these responses are converted to displacement ductility time-history responses by dividing them with 𝛿𝑦. Finally, the ductility time-history responses are processed using a rain-flow cycle counting algorithm as well as statistical analyses to develop the loading protocols. Details of the cycle counting algorithm and the statistical analyses are provided next. Figure 26. IDA-curve example for the three target ductility scaling factors. 57 Table 22. Scaling factor extracted from the IDA analysis. Ground Motion Type No. 𝐑𝐞𝐜𝐨𝐫𝐝∗𝐚 Scaling Factors 𝝁𝒊 = 𝟐 𝝁𝒊 = 𝟒 𝝁𝒊 = 𝟖 Crustal 1 Old Ridge Route 0.91 2.84 5.36 2 CIT Athenaeum 2.40 6.00 12.13 3 Coyote Lake Dam 1.70 4.82 8.03 4 Gilroy Array #6 1.92 3.66 7.13 5 Burbank - Howard Rd. 2.45 6.40 11.55 6 Castaic - Old Ridge Route 0.97 1.80 4.05 7 Brentwood VA Hospital 1.30 3.25 8.66 8 Sunland - Mt Gleason Ave 2.1 5.20 8.73 9 Hollister & Pine 1.5 2.83 3.85 10 Palo Alto - SLAC Lab 1.4 2.76 3.88 11 Taft Lincoln School 1.9 4.60 8.55 Subcrustal 1 0720a 1.5 2.96 7.28 2 1416a 0.82 2.45 5.48 3 1416c 0.83 1.98 4.22 4 1421a 1.08 2.44 4.79 5 1421c 1.38 2.11 5.74 6 4359a 1.09 2.79 5.00 7 4359c 0.67 1.21 2.21 8 6987c 1.72 3.69 5.52 9 7133c 1.52 3.98 8.16 10 7147c 2.58 4.69 7.02 11 7157c 3.49 7.39 13.56 Subduction 1 AKT017 2.80 5.10 11.70 2 AKT017 2.10 4.98 7.60 3 AKTH191 4.90 8.60 14.00 4 AKTH191 1.80 4.60 17.70 5 AOM008 3.20 6.32 16.50 6 AOM008 3.20 6.18 14.40 7 AOM028 3.60 10.35 16.80 8 IBR008 3.20 5.13 9.50 9 IBR005 2.85 3.25 7.33 10 YMT002 1.80 5.14 10.30 11 YMT009 1.95 5.10 10.40 a = Records matched to Vancouver hazard spectrum (2% in 50 years) 58 4.6 Cycle Counting The response of a structure under seismic loads consists of asymmetric excursions whose pattern varies with the properties of the ground motions. In conventional component tests involving quasi-static loading, loading histories employed are often characterized with stepwise increasing deformation of symmetric excursions. Because of this and the randomness of the seismic demands, a major challenge in the loading protocol development is to account for the effects of the sequence in which the excursions follow each other. To tackle this challenge, researchers have been utilizing the pre-peak excursions concept (Krawinkler et al. 2001; FEMA 2007; Bazaez and Dusicka 2016). Pre-peak excursions are all the excursions occurring before the latter of the maximum positive and negative displacement peaks. Large excursions that widen the response envelope of a structural component cause most of the damage. According to Krawinkler et al. (2001), excursions occurring after the latter of the maximum positive and negative displacement peaks (i.e., post-peak excursions) cause little damage but considering them may lead to an overestimation of cumulative damage. The overestimation of the cumulative damage because of the post-peak excursions becomes more prominent in numerical models accounting for stiffness and strength degradation. After the absolute maximum displacement, the structure undergoes significant degradation in stiffness and strength, consequently, any post-peak excursion, regardless of its amplitude, would contribute noticeably to hysteretic energy dissipation and cumulative damage (Krawinkler et al. 2001). Therefore, the proposed loading protocol development procedure in this study was primarily based on the pre-peak excursions. Nevertheless, in this work, all excursions were also considered in the case study prototype bridge pier for comparison purposes (please see the next sections). The variable amplitude displacement time-history response (whether based on pre-peak excursions or all excursions) need to be converted into equivalent blocks of constant amplitude cycles before they can be further processed. To achieve this, the rain-flow cycle counting algorithm suggested by ASTM E1049-85 (ASTM 2017) can be used. Additional details on how the rain-flow cycle counting algorithm works can be found in ASTM E1049-85 (ASTM 2017). The resulting cycles from the rain-flow cycle counting correspond to fully closed hysteresis loops in the stress/strain history. This makes them suitable for calculating the CDD given in equation 2 which only applies to 'closed' cycles. The cycles resulting from the rain-flow counting are characterized with excursions having a mean displacement, which is a measure of the symmetry with respect to 59 the undeformed confirmation, unequal to zero (i.e., excursions ranges are not centred to the zero-displacement line). However, according to (Krawinkler et al. 2001), for ordinary ground motion records, the mean displacement tends to be close to zero, hence its effect can be ignored. The rain-flow cycle counting procedure is illustrated in Figure 27. On contrary, for pulse-type near-fault ground motions, the structure experiences a small number of large excursions with significant mean displacement, hence the effect of the mean displacement must be considered. In this study, for the example prototype bridge, the pulse type near-fault ground motions were excluded from the ground motion selection, and therefore the displacement mean effect was ignored. Figure 27. Illustrative procedure for the simplified rain-flow counting: (a) inelastic ductility response history; (b) pre-peak inelastic response history; (c) ordered cycles including mean effect; (d) final ordered cycles. a) b) d) c) 60 4.7 Construction of Loading Protocols The rain-flow counting algorithm described earlier was applied to the displacement ductility time-history responses generated from the IDA analysis of the bridge pier. After the rain-flow counting, the number of inelastic cycles, (i.e., cycles with 𝜇𝑖 > 1), and the CDD values corresponding to the counted cycles were determined. The number of inelastic cycles and the CDD values were normally distributed and hence were conservatively represented by the mean plus one standard deviation (84th percentile). The mean plus one standard deviation of the number of inelastic cycles and the CDD for the three ground motion sets and ductility demand levels considering pre-peak and all excursions are given in Table 23. The values in Table 23 constitute the target demand values in terms of the number of inelastic cycles and CDD that the proposed loading protocols need to achieve. It is worth noting that, for all types of earthquakes, the CDD from pre-peak excursions represented approximately 30% to 60% of those from all excursions for all target ductility demands, which indicates, as anticipated, that considering all excursions would overestimate the CDD. Additionally, the number of inelastic cycles and the CDD are plotted against the target displacement ductility 𝜇𝑖 demand for the three ground motion sets in Figure 28 (a) and (b), respectively. As evident from Figure 28, irrespective of the ground motion set, the number of inelastic cycles and CDD exhibited a nearly linear behaviour with respect to the target 𝜇𝑖 demand. This implies that, for the prototype bridge pier, linear interpolation can be used to evaluate the number of inelastic cycles and CDD corresponding to the target 𝜇𝑖 demand of 2, 4, or 8. However, caution must be exercised when the period increases as such an approach could become invalid. According to Bazaez and Dusicka (2016), the relationship between both the number of inelastic cycles and CDD and target 𝜇𝑖 demand does not show a linear behaviour as the period increases. 61 Table 23. Target values for inelastic cycles and CDD for the prototype bridge pier Ground motion set Max 𝝁𝒊 Inelastic Cycles CDD Pre-peak excursions All excursions Pre-peak excursions All excursions Crustal 2 3 6 9 17 4 5 16 23 61 8 8 23 50 131 Subcrustal 2 4 7 11 17 4 5 19 19 63 8 8 35 50 161 Subduction 2 6 12 14 29 4 12 30 45 94 8 23 50 133 244 For this case study, the values from Table 23 show that the number of inelastic cycles, as well as the CDD from subduction earthquakes, are considerably higher than those from the other two types of earthquakes for all ductility demands. However, the values from crustal and subcrustal earthquakes are similar. The difference between all ductility values (𝜇) is, on average, 17% and 11%, for inelastic cycles and CDD, respectively. To confirm this relative similarity, two statistical tests were carried out to compare the two parameters between the two types of earthquakes for all ductility demands. For both tests, a significance level of 5% was selected. First, the equality of the population variances of the number of inelastic cycles and CDD was tested utilizing the Fisher–Snedecor (F) test (Montgomery 2017). When comparing the test statistics (F0) against the critical value of the F distribution, the null hypothesis (i.e., population variances are equal) was not rejected for all the three ductility demands. This conclusion led to testing the equality of the mean values of the same parameters utilizing a two-sample t-test for equal variances (Montgomery 2017). When comparing the test statistics (t0), the null hypothesis (i.e., mean values of the samples are equal) was also not rejected for all ductility demands. Therefore, no statistical evidence was found to conclude that both population variances and sample means of the number of inelastic 62 cycles and CDD differ between crustal and subcrustal ground motion sets. Consequently, the statistical values for subcrustal earthquakes which are slightly higher than those for crustal earthquakes were conservatively considered for the loading protocol development for all the target ductility demand levels. Figure 28. Comparison of target demand parameters: (a) inelastic cycles; (b) CDD To statistically process the resulting cycles from rain-flow counting, they need to be classified depending on their ranges (i.e., displacement ductility demand amplitudes). Here, the range of a cycle is twice its displacement ductility demand amplitudes because, as mentioned earlier, the mean effect was removed. The first group contained cycles with displacement ductility demand amplitudes from 1 to 1.5 since only inelastic cycles were considered. Displacement ductility demand amplitudes of the cycles contained by each of the subsequent groups were increased in 0.5 increments. Note that each cycle may be placed in one group only. This means that the second group contained only cycles with displacement ductility amplitudes from 1.5 to 2 and not those with displacement ductility amplitudes from 1 to 1.5 as they have been already placed in the first group. The grouping procedure is carried out until the target 𝜇𝑖 ductility demand is achieved. Then, the number of cycles contained in each group was statistically analyzed and represented by the mean number of cycles plus one standard deviation (84th percentile). For example, Figures 29 (a) and (b) provide results of grouping the pre-peak cycles generated from subduction earthquakes, and crustal and subcrustal earthquakes, respectively, at target displacement ductility demand level of 4. 01020304050600 2 4 6 8 10Number of Inelastic CyclesDisplacement Ductility (µ)All ExcursionsPre-peak ExcursionsSubcrustalSubductionCrustala)0501001502002503000 2 4 6 8 10CDDDisplacement Ductility (µ)All ExcursionsPre-peak ExcursionsSubcrustalSubductionCrustalb) 63 Figure 29. Grouping of pre-peak inelastic cycles (N) for (a) crustal and subcrustal earthquakes and (b) subduction earthquakes at target displacement ductility level of 4. 4.8 Proposed Testing Protocols After grouping and statistically processing the inelastic cycles from the rain-flow counting, the loading protocols can be established. For the prototype bridge pier, two different loading protocols were proposed at each target displacement ductility demand level (i.e., 𝜇 = 2, 4 and 8). The first represented the crustal and subcrustal ground motion sets whereas, the second represented the subduction ground motion set. Conventional loading protocols often consist of several symmetric cycles alternated in positive and negative amplitudes (arbitrarily started at positive amplitude) and gradually increased until the correspondent target displacement ductility demand level is reached. In addition, for better representation of damage, they need to be divided into three stages, each composed of different types of cycles. The same consideration has been implemented for loading protocol development by several other researchers (Krawinkler et al. 2001; Mergos and Beyer 2014; Bazaez and Dusicka 2016). The first stage is aimed to visualize low damage states and consists of ten elastic cycles applied to all the protocols with the following distribution: 3 cycles with 0.25(𝜇𝑖), 0.5(𝜇𝑖) and 0.75(𝜇𝑖) amplitudes, and one cycle with 1.0(𝜇𝑖) amplitude (FEMA 2007). The second stage, which imposes most of the damage, consists of the inelastic cycles. The number of the cycles and their amplitudes in this stage depends on the results of the grouping of rain-flow counting cycles as well as the target CDD. First, based on engineering judgment, reasonable cycle amplitudes need to be established for the proposed loading protocols. Then, inelastic cycles from the grouping results are assigned to the proposed cycle amplitudes to 00.511.522.533.541 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4Inelastic cycles (N)Displacement Ductility AmplitudeAverageStandard Deviation84th Percentilea)00.511.522.533.541 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4Inelastic cycles (N)Displacement Ductility AmplitudeAverageStandard Deviation84th Percentileb) 64 achieve the target CDD. For the prototype bridge pier, the proposed cycle amplitudes, and their corresponding number of inelastic cycles of the second stage of the loading protocols are presented in Tables 24 and 25 for Crustal and Subcrustal, and Subduction earthquakes, respectively. To illustrate how the number of inelastic cycles in both tables was determined, as an example, the first portion of the proposed loading protocol of the subduction ground motion set at 𝜇𝑖 level of 4 is considered. For this case, the 84th percentile of the number of inelastic cycles contained by the first group, i.e., with displacement ductility amplitude ranging from 1 to 1.5, was round down to 3 (see Fig. 29 (b)). These 3 cycles were assigned to a cycle amplitude of 1.2, which falls within the range of the first group of the proposed loading protocol (Table 24 and 25). The same procedure was repeated for the remaining cycles until the target CDD (see Table 23) was achieved. The CDD from the second stage of the proposed loading protocol for the subduction ground motion set at target 𝜇𝑖 level of 4 was 51.2, which approximately matches the target CDD considering pre-peak excursions (see Table 23). It is worth mentioning that the cycle amplitudes within the range of each group are proposed to closely follow the distribution of cycles from the rain-flow counting (see Fig. 27 (d)). After that, the loading protocols are followed by the third stage, which contains “trailing” cycles that are added to account for the damage caused by the post-peak excursions without overestimating the CDD. The amplitude of the trailing cycles was set as 75% of the target displacement ductility for each correspondent protocol. This consideration was made to avoid either an excessive or insignificant number of trailing cycles (Krawinkler et al. 2001). The number of the trailing cycles is determined by matching the CDD of the proposed loading protocols with the target CDD when all excursions are considered. For example, in the case of loading protocol of the subduction ground motion set and at 𝜇𝑖 level of 4, 7 trailing cycles with an amplitude of 3 were added to increase the CDD of the loading protocol by 42 to match the target CDD when all excursions are considered (Table 23). The number of trailing cycles is given inside parenthesis in Tables 24 and 25. The proposed protocols for the prototype bridge pier are depicted in Figure 30 and Figure 31, for Crustal and Subcrustal, and Subduction earthquakes, respectively. 65 Table 24. Case study loading protocol for crustal and subcrustal earthquakes. Cycle Amplitude 𝝁𝒊 = 𝟐 𝝁𝒊 = 𝟒 𝝁𝒊 = 𝟖 Inelastic Cycles CDD Inelastic Cycles CDD Inelastic Cycles CDD 1.2 2 4.8 2 4.8 2 4.8 1.5 1(2) 3 1 3 1 3 2 1 4 1 4 2 8 2.5 - - - - - - 3 - - (7) - - - 4 - - 1 8 1 8 5 - - - - - - 6 - - - - 1(9) 12 8 - - - - 1 16 Totala 4 11.8 5 19.8 8 51.8 Totalb 6 17.8 12 62.8 17 159.8 a Pre-peak cycles b Pre-peak + trailing cycles Figure 30. Crustal and subcrustal case study loading protocol -8-6-4-2024680 4 8 12 16 20Displacemet DuctilityNumber of CyclesElastic CyclesInelastic CyclesTrailing Cycles 66 Table 25. Case study loading protocol for subduction earthquakes Cycle Amplitude 𝝁𝒊 = 𝟐 𝝁𝒊 = 𝟒 𝝁𝒊 = 𝟖 Inelastic Cycles CDD Inelastic Cycles CDD Inelastic Cycles CDD 1.2 3 7.2 3 7.2 4 9.6 1.5 2(4) 6 2 6 0 9 2 1 4 2 8 2 12 2.5 - - 2 10 2 15 3 - - 2 (7) 12 2 18 4 - - 1 8 2 16 5 - - - - 2 20 6 - - - - 2 (9) 24 8 - - - - 1 16 Total 6 17.2 12 51.2 23 139.6 Totalb 10 29.2 19 93.2 32 247.6 a Pre-peak cycles b Pre-peak + trailing cycles Figure 31. Subduction case study loading protocol -8-6-4-2024680 4 8 12 16 20 24 28 32 36Displacemet DuctilityNumber of CyclesElastic CyclesInelastic CyclesTrailing Cycles 67 CHAPTER 5: SENSITIVITY ANALYSIS The loading protocol procedure proposed in Chapter 3 and demonstrated in Chapter 4 allows experimentalists and researchers to develop component-specific loading protocols that better represent the demands imposed by the seismic hazard considered. In this chapter, to investigate the viability of a general loading protocol for RC bridge piers, a sensitivity analysis is carried out to proposed suitable loading protocols that will be representative of a wide range of RC bridge piers designed with parameters within their practical range. The necessity of a general loading protocol lies in the lack of consistency when sharing experimental data between experimentalists and researchers around the world; but it is worth noting that no unique and “best” loading protocol exists, simply because seismic events are always different and there are many different structural configurations that the component can be part of. However, consensus-based loading protocols are required for correct comparison of results from different laboratories, and to strengthen tasks such as establishing damage states or evaluating the seismic performance of the structural component. 5.1 Preliminary Screening of Bridge Engineering Design Parameters Many parameters define the design (and correspondent numerical model) of RC bridge piers. These parameters may be geometric, such as column diameter or column height or they may relate to the material or component behaviour such as concrete and reinforcement steel. The development of component-specific quasi-static loading protocols is a function of such parameters because they define the elastic and inelastic characteristics of the system that define the response of the structure, as well as a function of parameters such as the seismic input and the configuration of the structural components. Therefore, the question is whether the variation of these parameters significantly affects the response of the structure. The answer to this question will allow quantifying the effect of such parameters and will indicate whether their variation must be explicitly considered or neglected. For that reason, to evaluate the viability of a loading protocol proposed for a wide range of RC bridge pier designs, it is critical to identify and evaluate the most important parameters that influence the seismic response (i.e., inelastic reversed-cyclic behaviour) of the structure, and consequently their effect on the development of their correspondent loading protocol. 68 Several methods are available to assist with the identification of significant parameters or factors of an experiment. These methods are normally called screening or sensitivity studies. A common method implemented in these cases is an approach called “one-factor-at-a-time” (Wu and Hamada, 2011). Here, one factor is changed at a time, while the rest is set as constant to evaluate the apparent impact it has upon the result. However, its iterative nature requires high computational efforts and does not account for the factor’s interaction. Another popular method for this kind of study is the “design of experiments” (DOE) approach. This statistical approach investigates the effect of the factors while reducing the computational effort as compared to other methods. This approach is an often-used tool for screening experiments where there are relatively many factors under consideration. Therefore, from the DOE framework, a 2𝑘 factorial analysis was adopted for the preliminary screening of factors in this study. At the initial stage, 6 important factors in RC bridge pier design are considered: concrete compressive strength (𝑓𝑐′), steel yield strength (𝑓𝑦), longitudinal reinforcement ratio (𝜌𝑙), transverse reinforcement ratio (𝜌𝑠), aspect ratio (H/D), and axial-load ratio (P/Po). H is the height of the structure and is set as a variable parameter. D is the column diameter, which in conjunction with the concrete cover (r), are set as constant parameters with values of 915 mm and 50.8 mm, respectively. Moreover, the aspect load ratio is calculated by dividing the axial load (P) by the product of 𝑓𝑐′ and 𝐴𝑔. The two levels required for each factor were arbitrarily named “low” and “high” and the correspondent values were selected based on their practical range. The values are shown in Table 26. The task at hand is to reduce the number of experiments to be considered for the subsequent loading protocol development by discarding those variables with an insignificant effect on the response. Note that, because there are only two levels for each factor, it is assumed that the response is approximately linear over the range of the factor levels chosen. A significance level of 5% is considered to reject the null hypothesis (i.e., α = 0.05). Table 26. Level of factors Factor Levels Compressive Strength of Concrete Yield strength of steel Longitudinal Reinforcement Ratio Transverse Reinforcement Ratio Aspect Ratio Axial Load Ratio 𝒇𝒄′ (MPa) 𝒇𝒚 (MPa) 𝝆𝒍 (%) 𝝆𝒔 (%) H/D (-) P/Po (%) Low (-) 30 410 4 1 0.7 5 High (+) 60 520 8 4 1.3 20 69 The same numerical modelling technique as presented in Chapter 3 was implemented, but this time with a more refined model utilized in Opensees to create a total of 64 (26) columns (i.e., the total number of factor combinations). The bridge piers were modelled using an inelastic displacement-based element with a circular section. Nonlinearity was captured through the concrete and reinforcement steel material definition which were considered as per the Concrete01 and Reinforcing steel material models from the Opensees library, respectively. The concrete material model corresponds to a uniaxial Kent-Scott-Park concrete material object with degraded linear unloading/reloading stiffness according to the work of Karsan and Jirsa (1969) and no tensile strength. The reinforcement steel material model is also a uniaxial material object and is implemented as the reinforcement for the concrete fibre section. To connect the top of the column to the centroid of the superstructure a rigid link element was utilized. A uniaxial material object was utilized to capture strain penetration effects at the column-to-footing intersection. The entire numerical model scheme is shown in Figure 32. Figure 32. Numerical model (discretization of the bridge columns) Confined concrete Steelreinforcement Unconfinedconcrete Discretized section LpL -LpLocation of applied axial loadNodeColum Height (L)Displacement-based elementStrain penetration sectionZero-length element 70 The preliminary analysis was carried out with five different subduction records were arbitrarily selected to analyze the structure and obtain an average value of the output parameters and study the structure. Three different outputs were derived including the maximum displacement (MD), the total damage (TD) and the number of inelastic cycles (NIC) undergone by the structure. The number of inelastic cycles was obtained using the rain flow cycle counting algorithm (ASTM 2017), and the total damage was computed using equation 2. It should be noted that since the experiment results come from numerical analysis, only a single replicate experiment can be considered; therefore, high-level interactions (i.e., more than 2-way interactions) between the factors were set as an experimental error to provide noise to the statistical model. The statistical analysis was performed using the software Minitab (2020) and the results obtained are the percent contributions, the F-test score, and the correspondent conclusion of the test of each factor as reported in Table 26. From the statistics reported in the table, it can be observed that 𝑓𝑐′, 𝜌𝑠, H/D and P/Po are the most significant factors (i.e., P-value less than 0.05) for the number of inelastic cycles and the maximum displacement of the structure. As observed from Table 27, the factorial analysis suggests that 𝑓𝑦 is not a significant factor, this result may be attributed to the fact that the response studied is the displacement ductility demand, which is derived from the yield displacement (𝛿𝑦) of the structure. Additionally, even though the selected factors are not significant for the total damage, they will be considered for the entire loading protocol procedure. This way, the final factors to be considered are: 𝑓𝑐′, 𝜌𝑙, H/D and P/Po. The parameters that are not significant will be set as constant with a value equal to the average of the low and high levels considered for the preliminary analysis (i.e., 𝑓𝑦 = 465 𝑀𝑃𝑎 and 𝜌𝑣 = 1%). Table 27. Results from the preliminary factorial analysis Factor Contribution (%) F-test P-value MD NIC TD MD NIC TD MD NIC TD 𝑓𝑐′ 1.560 1.860 0.060 13.06 30.00 0.84 << 0.01 << 0.01 0.387 𝑓𝑦 0.043 0.014 0.001 0.36 0.23 0.01 0.565 0.642 0.929 𝜌𝑠 2.560 2.470 0.120 21.39 39.91 1.69 << 0.01 << 0.01 0.223 𝜌𝑣 0.090 0.010 0.050 0.74 0.14 0.70 0.415 0.719 0.429 H/D 18.140 35.530 0.006 151.37 574.4 0.08 << 0.01 << 0.01 0.781 P/Po 10.200 32.880 11.470 85.10 531.58 161.82 << 0.01 << 0.01 << 0.01 71 3.1 Determination of Inelastic Cycles and Total Damage Demands After determining the most influential factors for the ductility displacement of a bridge pier, a revised experiment was designed to study the influence of such factors on the development of loading protocols for all three seismic hazards considered. A matrix of combinations of these factors was created following the same logic of the factorial design, as shown in Table 28. From the table, the positive (+) and negative (-) signs represent the high and low levels of a factor and are equal to the correspondent value from Table 25, respectively. Note that those factors that were set to be constant (i.e., 𝑓𝑦, 𝜌𝑙, 𝐷, 𝑟) are not included in Table 28; however, they are part of each bridge column design; in other words, each design is formed by 4 variable factors and 4 constant factors. For example, combination number 6 from Table 28 represents a column design with 𝐷 =915 𝑚𝑚, 𝑟 = 50.8 𝑚𝑚, 𝑓𝑐′ = 60 𝑀𝑃𝑎, 𝑓𝑦 = 465 𝑀𝑃𝑎, 𝐻/𝐷 = 4, 𝜌𝑙 = 4%, 𝜌𝑠 = 1% and 𝑁 =5%. This way, the total number of columns is 16 and each combination will be used to develop a component-specific loading protocol and then evaluate its sensitivity to each variable as well as its similarity between column designs. Table 28. Final experiment design Combination 𝒇𝒄′ (𝑴𝑷𝒂) 𝑯/𝑫 (-) 𝝆𝒍 (%) 𝑵 (%) 1 - - - - 2 + - - - 3 - + - - 4 + + - - 5 - - + - 6 + - + - 7 - + + - 8 + + + - 9 - - - + 10 + - - + 11 - + - + 12 + + - + 13 - - + + 14 + - + + 15 - + + + 16 + + + + With all the combinations established, the loading protocol development procedure was carried out as presented in Chapter 3. The first step is to analyze the structure and determine the 72 structural properties required to characterize the protocols. The same approach as presented in Chapter 4, was implemented. The displacement ductility capacity of the pier was obtained using a static pushover analysis (SPA). This parameter is the ratio of the displacement at ultimate (𝛿𝑢) to the displacement at yield (𝛿𝑦). The structural properties are shown in Table 29. Table 29. Structural properties of all the factor combinations Combination Yield Displacement (𝜹𝒚) (mm) Ultimate Displacement (𝜹𝒖) (mm) Displacement Ductility Capacity 1 20 297 14.85 2 19 277 14.58 3 78 750 9.62 4 73 588 8.05 5 24 344 14.33 6 23 323 14.04 7 96 1390 14.48 8 91 841 9.24 9 21 166 8.10 10 21 122 5.95 11 83 228 2.75 12 31 191 2.36 13 26 368 14.43 14 24 194 8.08 15 104 422 4.08 16 100 283 2.83 The columns were subjected to a progressive time-history analysis (IDA) to extract the correspondent scaling factors required to reach the target 𝜇𝑖 demands (i.e., target 𝜇𝑖 = 2, 4 𝑎𝑛𝑑 8). After reaching the targets, the correspondent ductility displacement response was obtained, and the rain-flow counting algorithm was utilized to determine the total number of inelastic cycles (i.e., cycles with 𝜇𝑖 >1), and the correspondent total CDD was determined utilizing equation 2. The total inelastic cycle count was further processed by splitting the count into groups of 0.5 step size. This way the first group will be formed by inelastic cycles of 𝜇𝑖 > 1 to 𝜇𝑖 < 1.5 and so on until the target 𝜇𝑖 value has been reached. After all the groups have been created, all the cycles contained in each group are then assigned to an amplitude equal to the upper bound of the correspondent group This 73 consideration was taken to allow better investigating stiffness and strength degradation of the structural component that is tested. As an example, the cycle counting for target 𝜇𝑖 = 4 will be formed by several inelastic cycles assign to ductility displacement amplitudes of 1.5, 2, 2.5, 3 and 4. Moreover, this approach provides conservative values since it indicates that the protocol will comprise larger amplitude cycles, which are more damaging than smaller amplitude cycles. Note that the distribution of cycles depends on the response of the structure; therefore, some ductility displacement amplitudes might not have cycles assigned for some cases; however, the sum of cycles and the CDD must match the total cycle count and the values from the statistics, respectively. Thus, this procedure was repeated for all the 11 records for each type of earthquake presented in Table 18. After the analysis, the number of inelastic cycles of each group, as well as the total CDD were found to be normally distributed between all 11 records of the three seismic hazards, and hence were conservatively represented by the mean plus one standard deviation (84th percentile) of each case, as also recommended by Krawinkler et al (2001). After obtaining all the values for all the target 𝜇𝑖 demands, they were compared between the different bridge columns. The total CDD, as well as the number of inelastic cycles in each counting group, were found to be similar between the combinations. Besides, they were also found to be normally distributed, meaning that the average will be a potential prediction of the values. For those reasons, it was decided to statistically confirm this similarity utilizing the standardized residuals approach proposed by Montgomery (2017). This diagnostic tool suggests the use of standardized residuals to evaluate the existence of outliers in the data. The presence of outliers indicates that the average value is not a good estimation of all the data points. The standardized residuals are calculated as follows: 𝒅𝒊 =𝒆𝒊√𝑺 (10) where S is the standard deviation of the data and 𝑒𝑖 is the residual from a particular data point calculated as the difference between the data point and the average value of all the data. As discussed in Montgomery (2017), if the residuals are normally distributed, the standardized residuals should be approximately normal with mean zero and unit variance. This means that about 68 percent of the standardized residuals should fall within the limits ±1, about 95 percent of them should fall within ±2, and nearly all of them should fall within ±3. A residual bigger than 3 or 4 standard deviations is a potential outlier. After calculating the standardized residuals for all cases, 74 the maximum value found was 2.58, which corresponds to the cycles contained in the 3.5 displacement ductility group for the combinations with crustal records. Therefore, no means were found to conclude the existence of outliers in any cycle counting group nor in the total CDD compared between the bridge columns. This conclusion was obtained for all target 𝜇𝑖 demands and all three earthquake types. Consequently, the average value was utilized to represent the number of inelastic cycles in each cycle counting group and the total CDD of all the bridge columns in the construction of the final loading protocols in the next section. The average number of inelastic cycles is reported in Table 30 for each type of earthquake. It is worth mentioning that for some combinations the ductility capacity of the column is less than the target 𝜇𝑖 demands of four or eight (see Table 29); in such cases, the average value was calculated without their consideration. For example, for target 𝜇𝑖 = 4, only 14 combinations will be used since 2 of them have ductility capacity less than 4. Similarly, for target 𝜇𝑖 = 8, only 11 combinations are considered. Table 30. Average values for the number of inelastic cycles Cycle Amplitude Crustal Subcrustal Subduction 𝝁𝒊 = 𝟐 𝝁𝒊 = 𝟒 𝝁𝒊 = 𝟖 𝝁𝒊 = 𝟐 𝝁𝒊 = 𝟒 𝝁𝒊 = 𝟖 𝝁𝒊 = 𝟐 𝝁𝒊 = 𝟒 𝝁𝒊 = 𝟖 1.5 1.75 2.31 1.82 2.63 3.39 3.73 5.56 5.85 6.10 2 1.25 1.39 1.18 1.69 2.39 2.91 2.50 4.62 5.18 2.5 - 1.23 1.18 - 1.54 2.00 - 2.92 3.46 3 - 1.15 1.18 - 1.00 1.64 - 1.92 3.10 3.5 - 1.00 1.09 - 0.77 1.27 - 1.15 2.27 4 - 1.00 0.73 - 1.00 0.91 - 1.00 2.00 4.5 - - 0.55 - - 0.91 - - 1.63 5 - - 0.55 - - 0.36 - - 1.10 5.5 - - 0.46 - - 0.46 - - 0.91 6 - - 0.09 - - 0.36 - - 0.64 6.5 - - 0.46 - - 0.27 - - 0.64 7 - - 0.09 - - 0 - - 0.46 7.5 - - 0 - - 0 - - 0.09 8 - - 1.00 - - 1.00 - - 1.00 Total cycles 3.00 8.08 10.36 4.31 10.07 15.82 8.06 17.46 28.55 Total CDD 19.44 50.10 94.91 25.31 72.23 167.64 41.19 72.23 284.55 75 3.1 Proposed Testing Protocols Nine different loading protocols were proposed accounting for all the ductility target demands and all the earthquake types considered. The loading protocols consist of several symmetric cycles alternated in positive and negative amplitudes and gradually increased until the correspondent target ductility demand is reached. As implemented in Chapter 4, to better represent the damage that affects the structure, the protocols are divided into three different stages. The first stage is formed by ten elastic cycles with the following distribution, 3 cycles at 0.25(𝜇𝑖), 0.5(𝜇𝑖) and 0.75(𝜇𝑖) amplitudes and one cycle at 1.0(𝜇𝑖). This stage is constant for all the proposed protocols to visualize low damage states. The second stage is meant to replicate the inelastic demands imposed into the structure (i.e., inelastic cycles) and is characterized by the values reported in Table 30. Note that since most of the average values in Table 30 have decimal numbers, the values were balanced between the cycle groups to have integer numbers without altering the total cycle count. For instance, for crustal earthquakes and target 𝜇𝑖 demand of two, 2 cycles are assigned for a displacement ductility amplitude of 1.5, and 1 cycle for displacement ductility amplitude of 2. The peculiar distribution of cycles in the proposed loading protocol was determined having in mind that large damaging cycles are less common than moderate and low damaging cycles (see Table 30 and Figure 29). The proposed number and amplitude of each inelastic cycle of this stage are presented in Tables 30, 31 and 32 for crustal, subcrustal and subduction earthquakes, respectively. However, it is worth mentioning that irrespective of the ground motion set, the number of inelastic cycles and CDD exhibited a nearly linear behaviour with respect to the target demand, which indicates that linear interpolation can be used to evaluate those parameters corresponding other to target demand values. Finally, for the third stage, a series of “trailing” cycles are implemented. As previously discussed, the pre-peak approach is considered to avoid an overestimation of the cumulative damage, consequently giving the post-peak cycles (referred here as the trailing cycles) a secondary consideration. These cycles are added after the completion of stage two and are calculated as the necessary cycles to match the total CDD of the test with the statistics presented in Table 30. To avoid having an excessive number of trailing cycles, an amplitude of 75% of the maximum target ductility demand was set for each correspondent protocol. For example, for target 𝜇𝑖 demand of eight, the trailing cycles will have an amplitude of being 6. The trailing cycles for all ductility demands are shown in parenthesis in Tables 31, 32 and 76 33. The proposed nine protocols are illustrated in Figures 33, 34 and 35, for crustal, subcrustal and subduction earthquakes, respectively. Table 31. Proposed inelastic sequence of loading for crustal earthquakes. Cycle Amplitude 𝝁𝒊 = 𝟐 𝝁𝒊 = 𝟒 𝝁𝒊 = 𝟖 Inelastic Cycles CDD Inelastic Cycles CDD Inelastic Cycles CDD 1.5 2 (4) 18 2 6 2 6 2 1 4 2 8 2 8 3 - - 2 (3) 30 2 12 4 - - 1 8 1 8 5 - - - - 1 10 6 - - - - 2 (2) 48 8 - - - - 1 16 Total 3 22 7 52 11 108 Figure 33. Proposed testing protocols for crustal earthquakes: target ductility of a) 2, b) 4 and c) 8 a) b) c) 77 Table 32. Proposed inelastic sequence of loading for subcrustal earthquakes. Cycle Amplitude 𝝁𝒊 = 𝟐 𝝁𝒊 = 𝟒 𝝁𝒊 = 𝟖 Inelastic Cycles CDD Inelastic Cycles CDD Inelastic Cycles CDD 1.5 3 (3) 18 4 12 4 12 2 2 8 2 8 3 12 2.5 - - 2 10 2 10 3 - - 1 (5) 36 2 12 4 - - 1 8 2 16 5 - - - - 1 10 6 - - - - 2(5) 84 8 - - - - 1 16 Total 5 26 10 74 17 174 Figure 34. Proposed testing protocol for subcrustal earthquakes: target ductility of a) 2, b) 4 and c) 8 a) b) c) 78 Table 33. Proposed inelastic sequence of loading for subduction earthquakes. Cycle Amplitude 𝝁𝒊 = 𝟐 𝝁𝒊 = 𝟒 𝝁𝒊 = 𝟖 Inelastic Cycles CDD Inelastic Cycles CDD Inelastic Cycles CDD 1.5 5 (5) 30 6 18 6 18 2 3 12 4 16 5 20 2.5 - - 3 15 4 20 3 - - 2 (1) 18 4 24 3.5 - - 0 0 3 21 4 - - 2 16 3 24 5 - - - - 2 20 6 - - - - 2 (8) 120 8 - - - - 1 16 Total 8 42 17 75 30 283 Figure 35. Proposed testing protocol for subduction earthquakes: target ductility of a) 2, b) 4 and c) 8 c) a) b) 79 3.1 Comparison of the Proposed Loading Protocols The proposed loading protocols for the prototype bridge pier were compared with those found in the literature being commonly used for seismic performance evaluation of reinforced concrete bridge piers. First, for the proposed loading protocols reflective of crustal and subcrustal earthquakes, ACI 374 (ACI 2013) and the Modified New Zealand (MNZ) (Hines et al. 2002) loading protocols from the University of California San Diego were considered, whereas those reflective of subduction earthquakes, loading protocols proposed by Bazaez and Dusicka (2016) were considered for comparison purposes. The selected proposed loading protocols were the ones developed for target displacement ductility demand level of 8 to have a clear and complete comparison of cycles of distribution and amplitudes progression of the loading protocols. Figure 36 provides a comparison between the proposed loading protocols and those selected from literature for the three types of earthquakes. For clear visual comparisons between the loading protocols, trailing cycles were removed, and all the loading protocols were arranged such that all of them end at the same cycle, i.e., the last cycle having a displacement ductility amplitude of 8. Note that the beginning of shorter protocols has been indicated by a dashed line in Figure. Moreover, comparisons of the total number of cycles (i.e., including elastic, inelastic and trailing cycles), the number of inelastic cycles and the CDD of the loading protocols are provided in Table 34. The ACI 374 and the MNZ protocols are not representative in terms of the number of inelastic and 𝜇𝑖 when compared to the proposed protocols for crustal and subcrustal earthquakes, as shown in Figures 36 (a) and (b), respectively. Even though the demand values obtained for crustal and subcrustal earthquakes are relatively close (see Table 34), the conventional protocols exhibit characteristics that differ differently between both proposed loading protocols. The number of cycles of the ACI 374 protocols is less by 2 and 9 cycles than those from the proposed protocols for crustal and subcrustal earthquakes, respectively. On the other hand, the number of inelastic cycles from the MNZ protocol is higher by 5 cycles for crustal earthquakes and lower by 3 cycles for subcrustal earthquakes. However, despite generating a close CDD value, the ACI 374 and the MNZ Protocols’ inelastic cycles are characterized by noticeably larger 𝜇𝑖 amplitudes than those of both proposed loading protocols. Consequently, this imposes significantly larger damage than the actual damage the bridge pier would undergo during a seismic event generated by the crustal or subcrustal earthquake. Avoiding such a scenario was the main reason to consider the CDD for 80 pre-peak cycles as the main reference for the damage demand and incorporate trailing cycles into the proposed loading protocols. The peculiar distribution of cycles in the proposed loading protocols was determined considering the statistical values and the results from the rain-flow counting, which showed that large damaging cycles are less common than moderate and low damaging cycles. The response of the structure to crustal and subcrustal earthquakes can also be related to their specific characteristics such as the shearing of the transmitting rock with a motion that is transverse to the direction of travel, energy concentrated at the surface (sharp body wave peaks) and energy/moment ratios with high-frequency spectra (Scholz, 2019). Moreover, the unrealistic demands resulting from the conventional protocols can also be explained by the fact that these protocols were developed to represent demands from structural components and/or systems with a short fundamental period (i.e., T = 0.2 – 0.3 seconds) (FEMA 2007; ACI 2013). Clearly, the conventional protocols do not represent the inelastic demands expected for the porotype RC bridge piers neither for crustal nor for subcrustal earthquakes. The aggressive application of inelastic cycles of the conventional loading protocols can cause significant degradation in strength and stiffness of the structure and, consequently, lead to underestimating its inelastic capacity. Therefore, adopting the proposed approach to developing structural component specific-loading protocols, in which the properties of the structural system, as well as the seismic input, are taken into consideration, is highly recommended for an accurate and realistic seismic performance assessment using quasi-static cyclic tests. Similar comparisons were performed between the proposed loading protocols for subduction earthquakes and those developed by Bazaez and Dusicka (2016). Bazaez and Dusicka (2016) developed loading protocols considering 262 ground motions from subduction earthquakes for bridge piers with fundamental periods of 0.5, 1, and 2 seconds, and target ductility demand levels of 2, 4, and 8. For the comparisons presented here, loading protocols from Bazaez and Dusicka (2016) corresponding to a fundamental period of 1 second (i.e. the closest period to that of the prototype bridge pier), and target ductility demand level of 8 was selected and is presented in Figure 36 (c). The main difference between the loading protocols proposed in this study and those proposed by Bazaez and Dusicka (2016) is in the procedure by which the target ductility demand levels are achieved. An IDA-based approach, which largely depends on scaling of ground motions, is considered in the present study, whereas a constant ductility design approach was considered in the latter. As evident in Figure 36 (c), the progression of the proposed protocol for 81 subduction earthquakes is consistent with that of the protocol developed by Bazaez and Dusicka (2016). Also, the proposed loading protocol is characterized by the same number of inelastic cycles. However, the ductility demand amplitudes of the additional inelastic cycles in the loading protocol developed by Bazaez and Dusicka (2016) are small, and hence their contribution to the CDD is minimal. For instance, the percentage difference between the CDD of the proposed loading protocol and that of the loading protocol by Bazaez and Dusicka (2016) at the target ductility demand level of 8 is 20% (Table 34). Besides, considering only 11 ground motion records was sufficient to generate loading protocols comparable to those generated using 262 ground motion records. Finally, it is worth mentioning that the difference between the generalized protocols presented in this chapter and those from the case study is approximately 12% for crustal and subcrustal earthquakes and 8% for subduction earthquakes. This indicates that the proposed protocols are valid for those column designs that have factors that are in between those used in the experiment design and presented in Table 26. The results imply that the IDA-based approach is valid and yields loading protocols consistent with those developed by employing the classical constant ductility design-based approach. Moreover, the differences between the demands from the proposed and the standard loading protocols also indicate that the assessment of drift, strain, or damage limit states, for example, will result in improved design guidelines regarding energy dissipation under cyclic loading. 82 Figure 36. Comparison of protocols considering a) crustal earthquakes: b) subcrustal earthquakes; and c) subduction earthquakes. -8-6-4-2024680 5 10 15 20 25Ductility DisplacementNumber of CyclesProposed ProtocolMNZ ProtocolACI 374 ProtocolStarting points-8-6-4-2024680 3 6 9 12 15 18 21 24 27DuctilityDisplacementNumber of CyclesProposed ProtocolMNZ ProtocolACI 374 ProtocolStarting points-8-6-4-2024680 5 10 15 20 25 30 35 40Ductility DisplacementNumber of CyclesProposed ProtocolBazaez and Dusicka (2016)a) b) c) 83 Table 34. Comparison of conventional and proposed protocols Demand Parameter Ductility Demand ACI 374 Protocol MNZ Protocol Crustal Earthquakes Proposed Protocols Subcrustal Earthquakes Proposed Protocols Bazaez and Dusicka 2016 Protocol Subduction Earthquakes Proposed Protocols Total Cycles 2 6 12 13 15 19 18 4 10 18 17 20 34 27 8 18 24 20 27 51 40 Inelastic Cycles 2 4 9 3 5 9 8 4 8 15 7 10 19 17 8 16 21 11 17 34 30 Ratioa (%) 2 67 75 23 33 47 45 4 80 83 42 50 56 63 8 89 88 55 63 67 75 CDD 2 12 27 22 26 24 42 4 40 69 52 74 89 75 8 144 153 108 174 225 283 a=Inelastic cycles to total cycles ratio 84 CHAPTER 6: CONCLUSIONS 6.1 Summary and Conclusions An Incremental Dynamic Analysis (IDA) based procedure was introduced for the development of suitable quasi-static loading protocols for the cyclic seismic testing of bridge piers. The approach was implemented by utilizing an archetype bridge pier in southwestern British Columbia, Canada. The structure was designed and modelled with force-based fibre elements and subjected to three different types of earthquakes: crustal, subcrustal and subduction earthquakes. The ASTM E1049-85 simplified rain-flow counting procedure was utilized to extract the number of inelastic cycles in the displacement ductility time-history responses of the bridge pier and compute the correspondent Cumulative Displacement Ductility (CDD) damage index. Both parameters were taken as the main reference and the results from statistical analysis on them were implemented to propose suitable loadings protocol for low, moderate, and high ductility demands. Moreover, the sensitivity of the proposed procedure was evaluated with the important bridge engineering design parameters. A factorial analysis was utilized to determine the most influential factors and study their effect on the displacement ductility response of the structure. Suitable loading protocols were proposed that correspond to a realistic yet conservative representation of the cyclic deformation history to which RC bridge piers are likely to be subjected considering various seismic events. A statistical analysis carried out on the similarity of the number of inelastic cycles as well as the total damage (CDD) between the different bridge columns (i.e., each different combination of factors) indicated that there are no means to conclude that the values obtained for each structure are different, and hence it was shown that the mean value is a good prediction for both parameters. Thus, the proposed loading protocols are a representative seismic response of bridge columns with design parameters values that fall within the low and high factor levels considered. Observations made on the inelastic demands between the three different types of earthquakes showed that those from crustal and subcrustal earthquakes are relatively similar, whereas those from subduction earthquakes are considerably higher, whereby nine different loading protocols were proposed to account for all three types of earthquakes. Moreover, the distribution of cycles obtained from the rain-flow counting showed that the number of high damaging cycles is considerably less than moderate to low damaging cycles. Additionally, the number of inelastic cycles and the CDD factor 85 showed a nearly linear behaviour as the achieved ductility demand increases. Therefore, linear interpolation will produce reasonable results for ductility demands other than the target values. The comparison of the proposed protocols with conventional protocols showed that the applications of these protocols have big considerations in terms of cumulative damage when implemented for RC bridge piers. The most important difference is that the proposed protocols were developed considering a pre-peak approach to avoid strength and stiffness degradation, which is not considered in the conventional protocols. The conventional protocols overestimate the inelastic demands from crustal and subcrustal earthquakes and underestimate those from subduction earthquakes. The aggressive and closely linear application of inelastic cycles makes them unrepresentative of the demand expected for RC bridge piers, showing that the proposed protocols are a better representation of the inelastic demands expected in this type of structure. 6.2 Limitations Although the proposed procedure to develop quasi-static loading protocols demonstrated that they better represent the inelastic demands that RC bridge piers with single column are likely to be subjected to in Southwestern BC, several drawbacks should be noted. The first and most obvious limitation to be observed is that the proposed loading protocols are only applicable to RC bridge column designs that fall within the range of the variables considered and cannot be extrapolated to other cases. Second, the procedure is dependent on the correct selection of ground motions that can be carried out with different approaches, which depending on the case, may not accurately reflect the seismicity of the selected site if different types of earthquakes are considered, and therefore, produce unrealistic results. Third, the proposed procedure is based on an IDA approach where the original ground motions are scaled depending on the desired target demands, which can produce unrealistic earthquake magnitudes if the target values are too high. Moreover, the iterative nature of the approach can result in high computational efforts if the correct assumptions are not made from the beginning. In the end, one can argue that the proposed loading histories are not a close fit to any of the cases on which its derivation is based. This cannot be argued but lies in the nature of the problem. There is no single predetermined loading protocol that can accurately represent all structural responses; however, based on the reference parameters accounted for, the match between the proposed protocols and the obtained structural responses is considered to be satisfactory. 86 6.3 Research Contribution After all, it was demonstrated that the proposed loading protocols can be implemented for all the bridge columns with parameters that are in between the low and high factor levels considered. Moreover, they were compared with standard loading protocols and were found to more representative for RC bridge piers of a single column. Consequently, they represent a well-founded option that will be useful for future experimentalists intending to test bridge columns for quasi-static loading and considering crustal, subcrustal and subduction earthquakes. The procedure introduced in this study was aimed to provide a relatively simple development approach of cyclic loading protocols for academic research and practical engineering. The development procedure will permit one to obtain and work with consistent data that can be used to test or establish various seismic performance levels of RC bridge piers. The procedure is also useful to improve analytical modelling, detailing, and retrofitting of this type of structure, and can even be adapted for different structural systems. Avoiding the overestimation of inelastic demands was the biggest motivation for developing and proposing appropriate testing protocols. 6.4 Future Research To help ensure the adoption of the proposed protocols as a common approach, significant future research is still required. This research should focus on developing a better numerical model that can predict with more precision the displacement time-history response of the structure. More refined hysteretic models that consider, for example, low cycle fatigue effects and/or extrinsic actions such as climate damage will improve the results presented in this study and will refine the procedure with a stronger foundation to developed more accurate loading protocols for RC bridge piers. 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John Wiley & Sons. 93 APPENDICES Appendix A: RC bridge pier displacement-based fibre model example for Opensees wipe all; model basic -ndm 2 -ndf 3 ; node 1 0.000 0.000; node 4 37 0; node 5 87 0; node 6 137 0; node 7 187 0; node 8 237 0; node 2 288 0; fix 1 1 1 1; uniaxialMaterial Concrete01 1 -5.2 -0.002 0 -0.004; uniaxialMaterial Concrete01 2 -7.24384891 -0.005930479 -6.604318133 -0.02086128; uniaxialMaterial ReinforcingSteel 3 75.2746 85.9 28400 852 0.011 0.122; section Fibre 1 { patch circ 2 20 10 0.0 0.0 0.0 21.6875 0.0 360.0; patch circ 1 20 5 0.0 0.0 21.6875 24 0.0 360.0; layer circ 3 18 1.561450089 0.0 0.0 20.67 0 340; }; geomTransf PDelta 1; #element forceBeamColumn 1 1 2 5 1 1; element dispBeamColumn 1 1 4 5 1 1; element dispBeamColumn 2 4 5 5 1 1; element dispBeamColumn 3 5 6 5 1 1; element dispBeamColumn 4 6 7 5 1 1; element dispBeamColumn 5 7 8 5 1 1; element dispBeamColumn 6 8 2 5 1 1; mass 2 1.35093 1.35093 9378.23; 94 recorder Node -file ele1global.out -time -node 2 -dof 2 disp; timeSeries Linear 1 pattern Plain 1 1 {load 2 -522 0 0}; constraints Plain numberer Plain system BandGeneral algorithm Linear integrator LoadControl 0.1; analysis Static; analyze 10; loadConst -time 0.0; rayleigh 0.0 0.0 0.00194106 0.0; set patternID 2 set GMdirection 2 set GMfile "EQ1.txt" set accelSeries "Series -dt 0.00391 -filePath $GMfile -factor 386.4" pattern UniformExcitation $patternID $GMdirection -accel $accelSeries wipeAnalysis constraints Plain numberer RCM system UmfPack test NormDispIncr 1.0e-8 1000 algorithm NewtonLineSearch integrator Newmark 0.5 0.25 analysis Transient analyze 300000 0.0001 print ele 1 Completed 95 Appendix B: Rain-flow cycle counting algorithm MATLAB code Three different MATLAB codes were developed to comply with the proposed IDA-based procedure to develop loading protocols. Files work with the imperial system. 1. Factors Interpolation: in this step, several time history displacement responses with their correspondent scaling factors are feed to the code to obtain the appropriate scaling factor for each target demand value (see Figure 26). The name format of the input files is as follows: Xele1globalEQYScaleZ where X is the element number, Y is the record number and Z is the scale number. function []=Factors_Interpolation n_records=11; n_columns=16; filePattern = fullfile('./', '*.out'); files = dir(filePattern); for i=1:length(files) filename = files(i); fn=filename.name; %Matrix of given data. Careful with the format Data(i,1)=str2num(fn(1:strfind(fn,'e')-1));%Column with the combination number Data(i,2)=str2num(fn(strfind(fn,'Q')+1:strfind(fn,'S')-1));%column with the earthquake number Data(i,3)=str2num(fn(strfind(fn,'S')+5:strfind(fn,'.out')-1));%column with the scaling factor end Data=sortrows(Data); Data=sortrows(Data,2); IDA_iterations=zeros(n_columns,n_records);%How many scale factors per case column-EQ group for i=1:n_columns for j=1:n_records f=find(Data(:,1)==i & Data(:,2)==j); if isempty(f)==1 IDA_iterations(i,j)=0; else IDA_iterations(i,j)=size(f,1);%number of factors per column end end end %values for the different factors sfp=zeros(max(max(IDA_iterations)),n_records,n_columns); 96 for i=1:n_columns for j=1:n_records f=find((Data(:,1)==i & Data(:,2)==j)); if isempty(f)==1 else for t=1:size(f,1) sfp(t,j,i)=Data(f(t),3); end end end end %Matrix of zeros to store all the IDA results in a 3D matrix. IDA(:,:,:)=zeros(max(max(IDA_iterations)),n_records,n_columns); %Yield displacements in inches yield_D(:,1)=[0.669; 0.630; 2.638; 2.520; 0.787; 0.768; 3.228; 3.031; 0.709; 0.689; 2.795; 2.756; 0.866; 0.846; 3.465; 3.386]; %Maximum displacement capacities in inches fail_D(:,1)=[11.693 10.906 29.528 23.150 13.780 12.717 54.724 33.110 6.353 4.803 8.976 7.520 14.488 7.638 16.614 11.142]; for i=1:1:n_columns for j=1:n_records for k=1:IDA_iterations(i,j) sf=sfp(k,j,i); decimal=mod(sf,1);%integer or floating if decimal==0 filename=sprintf('%dele1globalEQ%dScale%d.out',i,j,sf); else filename=sprintf('%dele1globalEQ%dScale%3.1f.out',i,j,sf); end Dr=load(filename); D=Dr(10:length(Dr),2);%files start from line 10 maxD=max(abs(D)); if maxD>fail_D(i) maxD=fail_D(i);%avoid extremely large displacements. end maxDD=maxD/(yield_D(i)); IDA(k,j,i)=maxDD;%Max disp. ductility value per scaling factor. clear D clear Dr end end end % Interpolation to obtain factors Df(:,:,:)=zeros(n_columns,n_records,3);%Those values that does not reach the target ductility values will have 0 97 for d=1:1:3% 3 ductility values evaluated for i=1:1:n_columns for j=1:1:n_records for k=1:IDA_iterations(i,j) if k==1 if IDA(k,j,i)>(2^d) Df(i,j,d)=((2^d))/(IDA(k,j,i))*(sfp(k,j,i)); break end else if IDA(k-1,j,i)<(2^d) && IDA(k,j,i)>(2^d) limit=sfp(k,j,i)-sfp(k-1,j,i); if limit>5 %interpolation bounds less than 5 Df(i,j,d)=0; break else Df(i,j,d)=(((sfp(k,j,i)-sfp(k-1,j,i))/(IDA(k,j,i)-IDA(k-1,j,i)))*((2^d)-IDA(k-1,j,i)))+sfp(k-1,j,i); break end end end end end end end Df end 2. Ductility Check: after obtaining the factors, the structure is analyzed with the scaled records and the achieved ductility is checked in this step. The name format of the input files is as follows: Xele1globalEQYDZScaleN where X is the element number, Y is the record number, Z is the target demand value and N is the scale number. function []=Ductility_check filePattern = fullfile('./', '*.out'); files = dir(filePattern); for i=1:length(files) filename = files(i); fn=filename.name; %Matrix of given data. Careful with the format Data(i,1)=str2num(fn(1:strfind(fn,'e')-1));%column with the combination number Data(i,2)=str2num(fn(strfind(fn,'Q')+1:strfind(fn,'D')-1));%column with the earthquake number Data(i,3)=str2num(fn(strfind(fn,'D')+1:strfind(fn,'S')-1));%column with the ductility level Data(i,4)=str2num(fn(strfind(fn,'S')+5:strfind(fn,'.out')-1));%column with the scaling factor end 98 Data=sortrows(Data); Data=sortrows(Data,1); %Yield displacements in inches yield_D(:,1)=[0.669; 0.630; 2.638; 2.520; 0.787; 0.768; 3.228; 3.031; 0.709; 0.689; 2.795; 2.756; 0.866; 0.846; 3.465; 3.386]; %Displacements capacities in inches fail_D(:,1)=[11.693 10.906 29.528 23.150 13.780 12.717 54.724 33.110 6.353 4.803 8.976 7.520 14.488 7.638 16.614 11.142]; %Assuming a complete matrix n_columns x n_records has been provided for r=1:length(files) i=Data(r,1); j=Data(r,2); d=Data(r,3); sf=Data(r,4); if sf==0 Data(r,5)=0; else decimal=mod(sf,1); if decimal==0 filename=sprintf('%dele1globalEQ%dD%dScale%d.out',i,j,d,sf); else filename=sprintf('%dele1globalEQ%dD%dScale%3.4f.out',i,j,d,sf); end Dr=load(filename); D=Dr(11:length(Dr),2); maxD=max(abs(D)); if maxD>fail_D(i) maxD=fail_D(i); end maxDD=maxD/(yield_D(i)); Data(r,5)=maxDD; clear Dr clear D end end Data end 99 3. Rain-flow cycle counting: when the scaling factors and the correspondent ductility has been checked, the time-history displacement responses are utilized to obtain the required reference parameters for further post processing. The name format of the input files is the same as the ductility check step. function []=RainFlowCounting_Subduction %Initial Data n_records=11; n_columns=16; target_d=[2 4 8];% 3 ductility values evaluated d_steps=[0.5];%cycle count step size. Careful to not give decimals when dividing by the ductility %Yield displacements in inches yield_D(:,1)=[0.669; 0.630; 2.638; 2.520; 0.787; 0.768; 3.228; 3.031; 0.709; 0.689; 2.795; 2.756; 0.866; 0.846; 3.465; 3.386]; %counting d_levels=size(target_d,2); %Output file alphabet=char(97:122); varnames1={'Column' 'Record 1' 'Record 2' 'Record 3' 'Record 4' 'Record 5' 'Record 6' 'Record 7' 'Record 8' 'Record 9' 'Record 10''Record 11''Mean' 'Std' 'Mean + Std' 'Round up'}; varnames2={'Lower bound' 'Upper bound' 'Record 1' 'Record 2' 'Record 3' 'Record 4' 'Record 5' 'Record 6' 'Record 7' 'Record 8' 'Record 9' 'Record 10''Record 11' 'Mean' 'Std' 'Mean + Std' 'Round up'}; %reading file names filePattern = fullfile('./', '*.out'); files = dir(filePattern); for i=1:length(files) filename = files(i); fn=filename.name; %Matrix of given data. Data(i,1)=str2num(fn(1:strfind(fn,'e')-1));%Column with the combination number Data(i,2)=str2num(fn(strfind(fn,'Q')+1:strfind(fn,'D')-1));%column with the earthquake number Data(i,3)=str2num(fn(strfind(fn,'D')+1:strfind(fn,'S')-1));%column with the ductility level Data(i,4)=str2num(fn(strfind(fn,'S')+5:strfind(fn,'.out')-1));%column with the scaling factor end Data=sortrows(Data); Data=sortrows(Data,2); sfp=zeros(n_columns,n_records,d_levels); 100 %putting the factors in a 3D array for i=1:n_columns for j=1:n_records f=find(Data(:,1)==i & Data(:,2)==j); if isempty(f)==1 sfp(i,j,:)=0; else for t=1:size(f,1) if Data(f(t),3)==2 sfp(i,j,1)=Data(f(t),4); elseif Data(f(t),3)==4 sfp(i,j,2)=Data(f(t),4); elseif Data(f(t),3)==8 sfp(i,j,3)=Data(f(t),4); end end end end end %Assuming a complete matrix n_columns x n_records has been provided for t=1:size(d_steps,2) %output variables cyclesAP=zeros(n_columns,n_records+5,d_levels);%5 to account for the statistics CDDAP=zeros(n_columns,n_records+5,d_levels); cyclesPP=zeros(n_columns,n_records+5,d_levels); CDDPP=zeros(n_columns,n_records+5,d_levels); IcyclesAP=zeros(n_columns,n_records+5,d_levels);%5 to account for the statistics ICDDAP=zeros(n_columns,n_records+5,d_levels); IcyclesPP=zeros(n_columns,n_records+5,d_levels); ICDDPP=zeros(n_columns,n_records+5,d_levels); Eq_type=sprintf('Subduction%d.xlsx',t); %name depends on each case. for d=1:d_levels CCMAP=zeros((target_d(d)/d_steps(t)-(1/d_steps(t)))+1,n_records+6,n_columns);%6 to account for the columns of statistics CDDMAP=zeros((target_d(d)/d_steps(t)-(1/d_steps(t)))+1,n_records+6,n_columns);%+1 to account for the "total" row CCMPP=zeros((target_d(d)/d_steps(t)-(1/d_steps(t)))+1,n_records+6,n_columns); CDDMPP=zeros((target_d(d)/d_steps(t)-(1/d_steps(t)))+1,n_records+6,n_columns); for i=1:1:n_columns for j=1:n_records sf=sfp(i,j,d); 101 if sf==0 CCMAP(1:end-1,2+j,i)=0;%Raw Cycle counting matrix CDDMAP(1:end-1,2+j,i)=0;%Raw CDD matrix CCMPP(1:end-1,2+j,i)=0; CDDMPP(1:end-1,2+j,i)=0; else decimal=mod(sf,1); if decimal==0 filename=sprintf('%dele1globalEQ%dD%dScale%d.out',i,j,target_d(d),sf); else filename=sprintf('%dele1globalEQ%dD%dScale%3.4f.out',i,j,target_d(d),sf); end Dr=load(filename); D=[Dr(11:length(Dr),1) (Dr(11:length(Dr),2)/yield_D(i))]; nlines=size(D,1); n=0; %Extracting the peaks [~,locs1]=findpeaks(D(:,2)); [~,locs2]=findpeaks(-D(:,2));%to find the negative peaks sizepks=size(locs1,1)+size(locs2,1);% total peaks pks=zeros(sizepks,1); time=zeros(sizepks,1); if locs1(1)<locs2(1) It always go peak-valley locs=interleave2(locs1,locs2); else locs=interleave2(locs2,locs1); end for c=1:sizepks if c==1 n=n+1; pks(n)=0; time(n)=0; elseif c==nlines n=n+1; pks(n)=D(locs(c),2); time(n)=D(locs(c),1); pks(n)=D(c,2); time(n)=D(c,1); else n=n+1; pks(n)=D(locs(c-1),2); time(n)=D(locs(c-1),1); end end %Time History Response Peaks Vector x=pks; tp=time; 102 clear pks D Dr time %identify the maximum or minimum peak [Pmax,p1]=max(x); [Pmin,p2]=min(x); Pmax2=-Pmin;%We get the absolute maximum if Pmax>Pmax2 Pmaxf=Pmax; fp=p1; else Pmaxf=Pmax2; fp=p2; end xpp=x(1:fp);%Prepeakvalues [C,hist,edges]=rainflow(x,'ext'); [Cpp,histpp,edgespp]=rainflow(xpp,'ext'); clear x xpp %sorted cycle counting Cs=sortrows(C,2); Cpps=sortrows(Cpp,2); CDD=Cs(:,1).*Cs(:,2); CDDpp=Cpps(:,1).*Cpps(:,2); s=0; for ps=(2*d_steps(t)):(2*d_steps(t)):(target_d(d)*2) s=s+1; rowsap=find(Cs(:,2)>=(ps-2*d_steps(t)) & Cs(:,2)<ps); if isempty(rowsap)==1%check if vector is empty ccountingAP(s,1)=0; CDDcAP(s,1)=0; else for n=1:1:size(rowsap) cyclesindap(s,n)=Cs(rowsap(n),1); CDDindap(s,n)=CDD(rowsap(n)); end ccountingAP(s,1)=sum(cyclesindap(s,:)); CDDcAP(s,1)=sum(CDDindap(s,:)); clear cyclesindap CDDindap end end cyclesAP(i,j+1,d)=sum(ccountingAP);%cycles per record CDDAP(i,j+1,d)=sum(CDDcAP);%total CDD per record %total inelastic cycles per record IcyclesAP(i,j+1,d)=sum(ccountingAP((1/d_steps(t)+1):end,1)); %vector with total CDD per record ICDDAP(i,j+1,d)=sum(CDDcAP((1/d_steps(t)+1):end,1)); CCMAP(1:end-1,j+2,i)=ccountingAP((1/d_steps(t)+1):end,1); CDDMAP(1:end-1,j+2,i)=CDDcAP((1/d_steps(t)+1):end,1); %PREPEAK EXCURSIONS s=0; for ps=(2*d_steps(t)):(2*d_steps(t)):(target_d(d)*2) s=s+1; 103 rowspp=find(Cpps(:,2)>=(ps-2*d_steps(t)) & Cpps(:,2)<ps); if isempty(rowspp)==1%check if vector is empty ccountingPP(s,1)=0; CDDcPP(s,1)=0; else for n=1:1:size(rowspp) cyclesindpp(s,n)=Cpps(rowspp(n),1); CDDindpp(s,n)=CDDpp(rowspp(n)); end ccountingPP(s,1)=sum(cyclesindpp(s,:)); CDDcPP(s,1)=sum(CDDindpp(s,:)); clear cyclesindpp CDDindpp end end clear Cs Cpps CDD CDDpp cyclesPP(i,j+1,d)=sum(ccountingPP);%cycles per record CDDPP(i,j+1,d)=sum(CDDcPP);%Total CDD per record IcyclesPP(i,j+1,d)=sum(ccountingPP((1/d_steps(t)+1):end,1)); ICDDPP(i,j+1,d)=sum(CDDcPP((1/d_steps(t)+1):end,1)); CCMPP(1:end-1,j+2,i)=ccountingPP((1/d_steps(t)+1):end,1); CDDMPP(1:end-1,j+2,i)=CDDcPP((1/d_steps(t)+1):end,1); end end %CYCLE COUNTING AND CDD MATRIX %All peaks. %Cycle Counting CCMAP(1:end-1,1,i)=(1:d_steps(t):target_d(d)-d_steps(t))'; CCMAP(1:end-1,2,i)=(1+d_steps(t):d_steps(t):target_d(d))'; CCMAP(1:end-1,n_records+3,i)=mean(CCMAP(1:end-1,3:n_records+2,i),2); CCMAP(1:end-1,n_records+4,i)=std(CCMAP(1:end-1,3:n_records+2,i),0,2); CCMAP(1:end-1,n_records+5,i)=CCMAP(1:end-1,n_records+3,i)+CCMAP(1:end-1,n_records+4,i); CCMAP(1:end-1,n_records+6,i)=round(CCMAP(1:end-1,n_records+5,i)); CCMAP(end,3:n_records+6,i)=sum(CCMAP(1:end-1,3:n_records+6,i)); %CDD CDDMAP(1:end-1,1,i)=(1:d_steps(t):target_d(d)-d_steps(t))'; CDDMAP(1:end-1,2,i)=(1+d_steps(t):d_steps(t):target_d(d))'; CDDMAP(1:end-1,n_records+3,i)=mean(CDDMAP(1:end-1,3:n_records+2,i),2); CDDMAP(1:end-1,n_records+4,i)=std(CDDMAP(1:end-1,3:n_records+2,i),0,2); CDDMAP(1:end-1,n_records+5,i)=CDDMAP(1:end-1,n_records+3,i)+CDDMAP(1:end-1,n_records+4,i); CDDMAP(1:end-1,n_records+6,i)=round(CDDMAP(1:end-1,n_records+5,i)); CDDMAP(end,3:n_records+6,i)=sum(CDDMAP(1:end-1,3:n_records+6,i)); %Pre Peaks %Cycles CCMPP(1:end-1,1,i)=(1:d_steps(t):target_d(d)-d_steps(t))'; CCMPP(1:end-1,2,i)=(1+d_steps(t):d_steps(t):target_d(d))'; 104 CCMPP(1:end-1,n_records+3,i)=mean(CCMPP(1:end-1,3:n_records+2,i),2); CCMPP(1:end-1,n_records+4,i)=std(CCMPP(1:end-1,3:n_records+2,i),0,2); CCMPP(1:end-1,n_records+5,i)=CCMPP(1:end-1,n_records+3,i)+CCMPP(1:end-1,n_records+4,i); CCMPP(1:end-1,n_records+6,i)=round(CCMPP(1:end-1,n_records+5,i)); CCMPP(end,3:n_records+6,i)=sum(CCMPP(1:end-1,3:n_records+6,i)); %CDD CDDMPP(1:end-1,1,i)=(1:d_steps(t):target_d(d)-d_steps(t))'; CDDMPP(1:end-1,2,i)=(1+d_steps(t):d_steps(t):target_d(d))'; CDDMPP(1:end-1,n_records+3,i)=mean(CDDMPP(1:end-1,3:n_records+2,i),2); CDDMPP(1:end-1,n_records+4,i)=std(CDDMPP(1:end-1,3:n_records+2,i),0,2); CDDMPP(1:end-1,n_records+5,i)=CDDMPP(1:end-1,n_records+3,i)+CDDMPP(1:end-1,n_records+4,i); CDDMPP(1:end-1,n_records+6,i)=round(CDDMPP(1:end-1,n_records+5,i)); CDDMPP(end,3:n_records+6,i)=sum(CDDMPP(1:end-1,3:n_records+6,i)); %at least one cycle with the target ductility if sf==0 else if CCMAP(end-1,end,i)==0 CCMAP(end-1,end,i)=1; end if CCMPP(end-1,end,i)==0 CCMPP(end-1,end,i)=1; end end end %All peaks %Total Cycles cyclesAP(1:end,1,d)=(1:n_columns); cyclesAP(1:end,n_records+2,d)=mean(cyclesAP(1:end,3:n_records+1,d),2); cyclesAP(1:end,n_records+3,d)=std(cyclesAP(1:end,3:n_records+1,d),0,2); cyclesAP(1:end,n_records+4,d)=cyclesAP(1:end,n_records+2,d)+cyclesAP(1:end,n_records+3,d); cyclesAP(1:end,n_records+5,d)=round(cyclesAP(1:end,n_records+4,d)); %Total CDD CDDAP(1:end,1,d)=(1:n_columns); CDDAP(1:end,n_records+2,d)=mean(CDDAP(1:end,3:n_records+1,d),2); CDDAP(1:end,n_records+3,d)=std(CDDAP(1:end,3:n_records+1,d),0,2); CDDAP(1:end,n_records+4,d)=CDDAP(1:end,n_records+2,d)+CDDAP(1:end,n_records+3,d); CDDAP(1:end,n_records+5,d)=round(CDDAP(1:end,n_records+4,d)); 105 %Pre peaks %Total Cycles cyclesPP(1:end,1,d)=(1:n_columns); cyclesPP(1:end,n_records+2,d)=mean(cyclesPP(1:end,3:n_records+1,d),2); cyclesPP(1:end,n_records+3,d)=std(cyclesPP(1:end,3:n_records+1,d),0,2); cyclesPP(1:end,n_records+4,d)=cyclesPP(1:end,n_records+2,d)+cyclesPP(1:end,n_records+3,d); cyclesPP(1:end,n_records+5,d)=round(cyclesPP(1:end,n_records+4,d)); %Total CDD CDDPP(1:end,1,d)=(1:n_columns); CDDPP(1:end,n_records+2,d)=mean(CDDPP(1:end,3:n_records+1,d),2); CDDPP(1:end,n_records+3,d)=std(CDDPP(1:end,3:n_records+1,d),0,2); CDDPP(1:end,n_records+4,d)=CDDPP(1:end,n_records+2,d)+CDDPP(1:end,n_records+3,d); CDDPP(1:end,n_records+5,d)=round(CDDPP(1:end,n_records+4,d)); %Inelastic Cycles %Total Cycles IcyclesAP(1:end,1,d)=(1:n_columns); IcyclesAP(1:end,n_records+2,d)=mean(IcyclesAP(1:end,3:n_records+1,d),2); IcyclesAP(1:end,n_records+3,d)=std(IcyclesAP(1:end,3:n_records+1,d),0,2); IcyclesAP(1:end,n_records+4,d)=IcyclesAP(1:end,n_records+2,d)+IcyclesAP(1:end,n_records+3,d); IcyclesAP(1:end,n_records+5,d)=round(IcyclesAP(1:end,n_records+4,d)); %Total CDD ICDDAP(1:end,1,d)=(1:n_columns); ICDDAP(1:end,n_records+2,d)=mean(ICDDAP(1:end,3:n_records+1,d),2); ICDDAP(1:end,n_records+3,d)=std(ICDDAP(1:end,3:n_records+1,d),0,2); ICDDAP(1:end,n_records+4,d)=ICDDAP(1:end,n_records+2,d)+ICDDAP(1:end,n_records+3,d); ICDDAP(1:end,n_records+5,d)=round(ICDDAP(1:end,n_records+4,d)); %Pre peaks %Total Cycles IcyclesPP(1:end,1,d)=(1:n_columns); IcyclesPP(1:end,n_records+2,d)=mean(IcyclesPP(1:end,3:n_records+1,d),2); IcyclesPP(1:end,n_records+3,d)=std(IcyclesPP(1:end,3:n_records+1,d),0,2); IcyclesPP(1:end,n_records+4,d)=IcyclesPP(1:end,n_records+2,d)+IcyclesPP(1:end,n_records+3,d); IcyclesPP(1:end,n_records+5,d)=round(IcyclesPP(1:end,n_records+4,d)); %Total CDD ICDDPP(1:end,1,d)=(1:n_columns); ICDDPP(1:end,n_records+2,d)=mean(ICDDPP(1:end,3:n_records+1,d),2); ICDDPP(1:end,n_records+3,d)=std(ICDDPP(1:end,3:n_records+1,d),0,2); ICDDPP(1:end,n_records+4,d)=ICDDPP(1:end,n_records+2,d)+ICDDPP(1:end,n_records+3,d); ICDDPP(1:end,n_records+5,d)=round(ICDDPP(1:end,n_records+4,d)); 106 %Creating Cell Arrays Tables to print %All cycles Tcyclesap=cell2table(num2cell(cyclesAP(:,:,d)),'VariableNames',varnames1); TCDDap=cell2table(num2cell(CDDAP(:,:,d)),'VariableNames',varnames1); Tcyclespp=cell2table(num2cell(cyclesPP(:,:,d)),'VariableNames',varnames1); TCDDpp=cell2table(num2cell(CDDPP(:,:,d)),'VariableNames',varnames1); %Inelastic cycles TIcyclesap=cell2table(num2cell(IcyclesAP(:,:,d)),'VariableNames',varnames1); TICDDap=cell2table(num2cell(ICDDAP(:,:,d)),'VariableNames',varnames1); TIcyclespp=cell2table(num2cell(IcyclesPP(:,:,d)),'VariableNames',varnames1); TICDDpp=cell2table(num2cell(ICDDPP(:,:,d)),'VariableNames',varnames1); if d==1 range1=sprintf('A1'); range2=sprintf('%s%d',alphabet(size(cyclesAP,2)+2),1); else range1=sprintf('A%d',(d-1)*(size(cyclesAP,1)+1)+2*d-1); range2=sprintf('%s%d',alphabet(size(cyclesAP,2)+2),(d-1)*(size(cyclesAP,1)+1)+2*d-1); end %Print the values in the created text files writetable(Tcyclesap,Eq_type,'Sheet','Total Cycles','Range',range1); writetable(TCDDap,Eq_type,'Sheet','Total CDD','Range',range1); writetable(Tcyclespp,Eq_type,'Sheet','Total Cycles','Range',range2); writetable(TCDDpp,Eq_type,'Sheet','Total CDD','Range',range2); writetable(TIcyclesap,Eq_type,'Sheet','Total Inelastic Cycles','Range',range1); writetable(TICDDap,Eq_type,'Sheet','Total Inelastic CDD','Range',range1); writetable(TIcyclespp,Eq_type,'Sheet','Total Inelastic Cycles','Range',range2); writetable(TICDDpp,Eq_type,'Sheet','Total Inelastic CDD','Range',range2); for c=1:1:n_columns CC_ap=cell2table(num2cell(CCMAP(:,:,c)),'VariableNames',varnames2); CDD_ap=cell2table(num2cell(CDDMAP(:,:,c)),'VariableNames',varnames2); CC_pp=cell2table(num2cell(CCMPP(:,:,c)),'VariableNames',varnames2); CDD_pp=cell2table(num2cell(CDDMPP(:,:,c)),'VariableNames',varnames2); 107 if c==1 rangea=sprintf('A1'); rangeb=sprintf('A%d',size(CCMAP,1)+3); else rangea=sprintf('A%d',(2*c-2)*(size(CCMAP,1)+1)+3*c-2); rangeb=sprintf('A%d',(2*c-1)*(size(CCMAP,1)+1)+3*c-1); end sheetname1=sprintf('CycleC_ap_D%d',target_d(d)); sheetname2=sprintf('CycleC_pp_D%d',target_d(d)); writetable(CC_ap,Eq_type,'Sheet',sheetname1,'Range',rangea); writetable(CDD_ap,Eq_type,'Sheet',sheetname1,'Range',rangeb); writetable(CC_pp,Eq_type,'Sheet',sheetname2,'Range',rangea); writetable(CDD_pp,Eq_type,'Sheet',sheetname2,'Range',rangeb); end clear CC_ap CDD_ap CC_pp CDD_pp Tcyclesap Tcyclespp TCDDap TCDDpp TIcyclesap TIcyclespp TICDDap TICDDpp ccountingPP CDDcpp ccountingAP CDDcAP ccountingPP ccountingAP CDDcAP CDDcPP end end end
Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Quasi-static cyclic loading protocol development for the seismic performance assessment of bridge piers
Text