DAMAGE-FREE SEISMIC-RESISTANT SELF-CENTERING FRICTION-DAMPED BRACED FRAMES WITH BUCKLING-RESTRAINED COLUMNS. A Dissertation.

Transcription

1 DAMAGE-FREE SEISMIC-RESISTANT SELF-CENTERING FRICTION-DAMPED BRACED FRAMES WITH BUCKLING-RESTRAINED COLUMNS A Dissertation Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Felix C. Blebo May, 2015

2 DAMAGE-FREE SEISMIC-RESISTANT SELF-CENTERING FRICTION-DAMPED BRACED FRAMES WITH BUCKLING-RESTRAINED COLUMNS Felix C. Blebo Dissertation Approved: Accepted: Dissertation Advisor Dr. David A. Roke Department Chair Dr. Wieslaw Binienda Committee Member Dr. Craig C. Menzemer Dean of the College Dr. George K. Haritos Committee Member Dr. Qindan Huang Interim Dean of the Graduate School Dr. Rex Ramsier Committee Member Dr. Xiaosheng Gao Date Committee Member Dr. Desale Habtzghi ii

3 ABSTRACT Conventional concentrically braced frame (CBF) systems have limited drift capacity prior to brace buckling, and related damage leads to deterioration in strength and stiffness. CBFs are also susceptible to weak story failure. A pin- supported self-centering frictiondamped braced frame system with buckling-restrained columns (FDBF-BRC) is being developed to provide significant drift capacity while limiting damage due to residual drift and soft-story mechanisms. The FDBF-BRC system consists of beams, columns, and braces branching off a central column, with buckling restrained columns (BRCs) incorporated into the system at the first story external column positions. The BRCs and friction generated at lateral-load bearings at each floor level are used to dissipate energy to minimize the overall seismic response of the FDBF-BRC system. Vertically aligned post-tensioning bars provide additional overturning moment resistance and aid in self-centering the system to eliminate residual drift. The pin support condition and the lateral stiffness of the system enable it to exhibit a nearly uniform inter-story drift distribution. In this study, a suite of 44 DBE-level ground motions used in FEMA P695 is numerically applied to several FDBF- BRCs to demonstrate the seismic performance of the system. The results show that the FDBF-BRC system has a nearly uniform inter-story iii

4 drift response, high ductility, and a high energy dissipation capacity, and is an effective seismic-resistant system. iv

5 ACKNOWLEDGEMENTS The research presented in this dissertation was conducted at the University of Akron, Department of Civil Engineering, in Akron, Ohio. During the study, the chairmanship of the department was held by Dr. Wieslaw K. Binienda. The author would like to thank his research advisor and chair of his dissertation committee, Dr. David Roke, for his constant guidance, support, direction, and advice for the past four years. The author also appreciates the time and contributions of Dr. Craig Menzemer, Dr. Xiaosheng Gao, Dr. Qindan Huang, and Dr. Desale Habtzghi who served on my dissertation committee. The author would like to thank the following people for their contributions to his research: the civil engineering department staff, particularly Ms. Kimberly Stone, and fellow researchers, particularly Amir Gandomi, Mojtaba Dyanati and Mehdei Kafaeikivi. Most importantly, the author would like to extend his sincerest thanks to his friends and family who have offered help and inspiration along the way. The author extends a special thanks to his parents their counsel and immense contribution throughout his academic life. For most part, the author is extremely thankful for the unwavering support, patience, and love of his wonderful wife Gifty. This dissertation is dedicated to her. v

6 TABLE OF CONTENTS Page LIST OF TABLES... ix LIST OF FIGURES... xiii CHAPTER I. INTRODUCTION Overview Research Objectives Research Scope Organization of Dissertation...6 II. LITERATURE REVIEW Self-centering concentrically braced frame systems Pin-supported systems and soft-story failure Buckling-Restrained Braces...15 III. FDBF-BRC SYSTEM BEHAVIOR Description of FDBF-BRC System FDBF-BRC Limit States BRC Yielding PT Bar Detensioning and Yielding Member Yielding and Failure...33 vi

7 IV. DESIGN CRITERIA AND PROCEDURE Initial Design Stage Design Parameter Selection Design BRC Yield Strength Design Strength Initial Member Selection Design Strength and PT Steel Area Selection Design Roof Drift Capacity Member Force Design Demands First Mode Design Forces Higher Mode Design Forces Design of the Adjacent Gravity Columns...47 V. PARAMETRIC STUDY Prototype Building Prototype Design Numerical Model Nonlinear Static Analysis Monotonic Pushover Study Cyclic Pushover Study...58 VI. NONLINEAR DYNAMIC ANALYSIS Analytical Model Ground Motion Record Selection...80 vii

8 6.3 Dynamic Response Response to DBE-Level Ground Motions Effective Pseudo-Acceleration and Member Force Demand Estimation...88 VII. COMPARATIVE STUDY OF FDBF-BRC, FDBF, AND SC-CBF SYSTEMS Application of Proposed Design Recommendations Nonlinear Dynamic Analyses VIII. SUMMARY AND CONCLUSIONS Summary Motivation for Present Research Research Objectives Research Scope Observations and Findings FDBF Design Results Nonlinear Static Analysis Results Nonlinear Dynamic Analysis Results Conclusions Recommendations Original Contributions of Research Future work REFERENCES viii

9 LIST OF TABLES Table Page 5.1 Design dead loads at each floor level Design live loads at each floor level Summary of gravity loads on each adjacent-gravity column Summary of gravity loads on the lean-on column and tributary seismic mass for one FDBF-BRC Summary of gravity column sections and lean-on column areas Comparison of design parameters Design Summary of the various frame configurations Summary of 44 DBE-level ground motion characteristics Summary of θdbe and θs,max to DBE-level ground motions for s 4-1, 4-2, and Summary of θdbe and θs,max to DBE-level ground motions for s 5-1, 5-2, and Summary of θdbe and θs,max to DBE-level ground motions for s 6-1, 6-2, and Summary of DCF values for DBE-level ground motions Mean and coefficient of variation of θdbe, θs,max, and DCF to DBE-level ground motions Summary of the probability of θdbe, and θs,max exceeding 2.5% (assuming log normal distribution) Summary of θrr, and θrs to DBE-level ground motions for s 4-1, 4-2, and ix

10 6.9 Summary of θrr, and θrs to DBE-level ground motions for s 5-1, 5-2, and Summary of θrr, and θrs to DBE-level ground motions for s 6-1, 6-2, and Mean, coefficient of variation of θrr, and θrs to DBE-level ground motions Summary of the probability of θrr, and θrs exceeding 0.2% radians after a DBE-level ground motion (assuming log normal distribution) Summary of normalized peak PT force responses to DBE-level ground motions Mean and coefficient of variation of the normalized PT force, and the probability of the peak PT force exceeding PTY (assuming log normal distribution) Summary of fbrc,dbe to DBE-level ground motions Summary of fbrc,t to the DBE-level ground motions Summary of fbrc,c to DBE-level ground motions Mean and coefficient of variation of fbrc,dbe, fbrc,t, fbrc,c under DBE-level ground motions Probability of fbrc,dbe, fbrc,t, fbrc,c exceeding unity (assuming log normal distributions) Summary of fbr1 to DBE-level ground motions Summary of fbr2 responses to DBE-level ground motions Summary of fbr3 to DBE-level ground motions Summary of fbr4 to DBE-level ground motions Mean and coefficient of variation of fbri, and vbn for each frame under the DBE-level ground motions Summary of probability of fbri exceeding brace force design demands for each frame configuration Statistics for αn and probability of exceedance of SAn for s 4-1, 4-2, 4-3, 5-1, 5-2, and Statistics for αn and probability of exceedance of SAn for s 6-1, 6-2, x

11 6.28 Statistics for αn and probability of exceedance of γ n SAn for the selected frames Comparison of design parameters for frames design based on Design II Statistics for αn and probability of exceedance of γ n SAn for the frames designed using BRC yield force (Design II) Mean, coefficient of variation of θdbe and θs,max and DCF for frames designed based on factored design demand under the DBE-level ground motions Summary of the probability of θdbe, and θs,max exceeding 2.5% (assuming log normal distribution) Mean and coefficient of variation θrs, and the probability of θrs exceeding 0.2% (assuming log normal distribution) probability of fbrc,dbe, fbrc,t, fbrc,c exceeding unity for each frame under the DBE-level ground motion (assuming log normal distribution) Summary of probability fbri exceeding unity for each frame under the DBE-level ground motion Comparison of design parameters for SC-CBF, FDBF, and FDBF-BRC Summary of peak roof and θs,max responses to DBE-level ground motions for SC-CBF, FDBF, and FDBF-BRC θdbe and θs,max statistics and probabilities of exceedance for SC-CBF, FDBF, and FDBF-BRC Summary of DCF responses to DBE-level ground motions for SC-CBF, FDBF, and FDBF-BRC DCF statistics for SC-CBF, FDBF, and FDBF-BRC Summary of θrr and θrs to DBE-level ground motions for SC-CBF, FDBF, and FDBF-BRC θrr and θrs statistics and probabilities of exceedance for SC-CBF, FDBF, and FDBF-BRC Summary of peak first and second story external column stresses due to DBE-level ground motions, for SC-CBF, FDBF, and FDBF-BRC Summary of peak third and fourth story external column stresses due to DBE-level ground motions, for SC-CBF, FDBF, and FDBF-BRC xi

12 7.10 Summary of peak stress in first and second story braces due to DBE-level ground motions, for SC-CBF, FDBF, and FDBF-BRC Summary of peak stress in third and fourth story braces due to DBE-level ground motions, for SC-CBF, FDBF, and FDBF-BRC xii

13 LIST OF FIGURES Figure Page 2.1 Unbonded post-tensioned precast wall: (a) elevation; (b) enlarged cross section near base (Kurama et al. 1999a) Moment-resisting connections: (a) conventional; (b) post-tensioned (Ricles et al. 2001) Schematic of an SC-CBF system (Roke 2010) SC-CBF behavior under lateral loading: (a) elastic response prior to column decompression; (b) rigid-body rotation due to rocking (Roke 2010) Schematic of an FDBF system: (a) initial position; (b) rotation about the base of the FDBF central column (Blebo 2013) Typical self-centering behavior of FDBF from: (a) static pushover analysis; (b) dynamic analysis (Blebo 2013) Schematic of Sub-standard Braced and Stiff Rocking Core (Qu et al. 2014) Inter-story Drift Concentration vs. SRC Stiffness Ratio (Pollino et al. 2013) Schematic of pin-supported wall system (Qu et al. 2012) Drift Concentration factor (DCF) before and after retrofit (Qu et al. 2012) Inter-story drift ratio (IDR) of the building before and after retrofit: (a) original structure; (b) retrofitted structure (Qu et al. 2012) Comparison of cyclic behavior of conventional steel braces and buckling-restrained braces (Adam 2013) BRB main components (adapted from Star Seismic 2013) Schematic of an FDBF-BRC system: (a) initial position; (b) rotation about the base of the FDBF-BRC central column xiii

14 3.2 Idealized overturning moment-roof drift response of an FDBF-BRC: (a) PT bar detensioning occurring after BRC yielding in tension; (b) PT bar detensioning occurring between BRC yielding in compression and BRC yielding in tension; (c) PT bar detensioning occurring prior to BRC yielding in compression Free body diagram of the FDBF-BRC at PT bar yielding Rigid body rotation of an FDBF-BRC Design cases for the adjacent gravity column: (a) loading to PT bar yielding; (b) unloading after PT bar yielding Prototype building used for the parametric study: (a) typical floor plan; (b) elevation Member selections for Member selections for Member selections for Member selections for Member selections for Member selections for Member selections for Member selections for Member selections for Typical Hysteretic Behavior of BRB (BRC in this study): (a) Experimental (Merritt et al. (2003)) and (b) OpenSees analytical model Monotonic pushover results for s 4-1, 4-2, and Monotonic pushover results for s 5-1, 5-2, and Monotonic pushover results for s 6-1, 6-2, and Monotonic pushover results for s 4-1, 5-1, and Monotonic pushover results for s 4-2, 5-2, and Monotonic pushover results for s 4-3, 5-3, and xiv

15 5.18 Cyclic pushover results for s 4-1, 4-2, and 4-3: to 1.0 % roof drift Cyclic pushover results for s 5-1, 5-2, and 5-3: to 1.0 % roof drift Cyclic pushover results for s 6-1, 6-2, and 6-3: to 1.0 % roof drift Cyclic pushover results for s 4-1, 5-1, and 6-1: to 1.0 % roof drift Cyclic pushover results for s 4-2, 5-2, and 6-2: to 1.0 % roof drift Cyclic pushover results for s 4-3, 5-3, and 6-3: to 1.0 % roof drift Spectral accelerations of the ground motions used in this study θdbe for 44 DBE-level ground motions: (a) grouped by αy; (b) grouped by RBRC θs,max for each frame under the 44 DBE-level ground motions: (a) grouped by αy; (b) grouped by RBRC θrr for each frame after each of the 44 DBE-level ground motions: (a) grouped by αy; (b) grouped by BRC θrs for each frame after each of the 44 DBE-level ground motions: (a) grouped by αy; (b) grouped by BRC The nine different frame configurations studied, and the FDBF-BRC design trend Recommended design curve for FDBF-BRC systems Schematic of an: (a) SC-CBF system; (b) FDBF (fixed base) system Schematic of an: (a) FDBF (pinned base) system; (b) FDBF-BRC system Roof drift response of FDBF-BRC system to ground motion CLW-TR Comparison of DCF for SC-CBF, FDBFs, and FDBF-BRC systems Inter-story drift distributions for SC-CBF Inter-story drift distributions for FDBF (Fixed) Inter-story drift distributions for FDBF (Pinned) Inter-story drift distributions for FDBF-BRC xv

16 7.10 Response of third story brace under CLW-TR ground motion; (a) Axial force, out-of-plane displacement (b) out-of-plane displacement/deformation time history response xvi

17 CHAPTER I INTRODUCTION 1.1 Overview Buildings with conventional lateral force resisting systems are designed to ensure the safety of the occupants during an earthquake. Structural damage due to an earthquake is allowed (FEMA 450), but collapse is not permitted. The allowable damage is anticipated to be repairable, but the repairs may or may not be economical. Steel concentrically-braced frame (CBF) systems are a commonly used lateral force resisting system. These systems are economical and have significant strength and stiffness. However, CBFs suffer from limited system ductility capacity prior to brace buckling and related structural damage. Under the design basis earthquake (DBE), CBFs are expected to undergo drift demands that will yield or buckle the brace members. This damage may result in residual lateral drift after the earthquake. The system ductility capacity can be increased while maintaining stiffness through the use of bucklingrestrained braces; however, buckling-restrained braced frame (BRBF) systems may exhibit significant residual drift after an earthquake (Fahnestock et al. 2007). Steel moment resisting frames (MRF) are another frequently used seismic resistant system. 1

18 MRFs have high ductility capacity but, like CBFs, they are susceptible to soft-story failure and residual drift after an earthquake. A stiffer pin-supported structural system may have more resistance to soft-story failures (e.g. MacRae et al. 2004, Qu et al. 2012, and Qu et al. 2014). A stiff pinsupported self-centering friction-damped braced frame with buckling resistant columns (FDBF-BRC) system is being developed to provide significant nonlinear drift capacity while limiting damage, residual drift, and soft-story behavior. The purpose of this study is to analytically evaluate the seismic response of the self-centering FDBF-BRC system and to establish it as an effective lateral force resisting system. 1.2 Research Objectives The overall objective of this research program is to determine the effect of design parameters on the behavior and seismic response of the FDBF-BRC system under the design basis earthquake, and to develop a design procedure for the system. The specific objectives necessary to achieve the overall objectives are the following: 1. To design different configurations of FDBF- BRC using three different frame strength factors and three different BRC strength factors; 2. To conduct nonlinear static analyses on each of the designed FDBF- BRC configurations to assess the behavior of the systems; 3. To conduct nonlinear dynamic analyses on each FDBF- BRC configuration to determine the response to a suite of DBE-level ground motions; 2

19 4. To study the results of the designs, static analyses, and dynamic analyses to determine the effect of the frame strength and the BRC strength on the behavior and seismic response of the FDBF-BRC system. 5. To recommend an appropriate frame strength factor and BRC strength factor to use when designing FDBF- BRCs for use in seismic applications. 6. To establish a design procedure for the FDBF- BRC system. 7. To compare the performance of the FDBF- BRC system with the performance of its precursor systems. 1.3 Research Scope are: To achieve the research objectives, eight research tasks were undertaken. These tasks 1. Develop a preliminary procedure to design FDBF-BRCs with different frame strength factors and three different BRC strength factors. Preliminary performance-based seismic design (PBD) criteria were used to design the nine FDBF-BRC systems in this study. Three different frame strength factors were used; for each frame strength factor, three different BRC strength factors were considered. The design criteria are adapted from those for rocking SC-CBFs with friction-based energy dissipation (Roke et al. 2010) and the AISC seismic provisions for BRBFs (AISC 2010a). 3

20 2. Develop finite element models for the FDBF-BRC systems. Detailed nonlinear finite element models were created in OpenSees (Mazzoni et al. 2009) to represent the designed FDBF-BRC systems. The main components of the model are the FDBF-BRC structural members, the BRCs, the PT bars, the adjacent gravity columns, and the lean-on column, which is used to account for the stiffness of the gravity system and the global P-Δ effects on the FDBF- BRC system. 3. Conduct nonlinear static analyses using the finite element models of the designed FDBF-BRC systems. Nonlinear static analyses were performed on the nonlinear finite element models for each FDBF-BRC configuration to determine the system behavior under monotonic and cyclic static pushovers. 4. Conduct nonlinear dynamic analyses using the finite element models of the designed FDBF-BRC systems. Nonlinear dynamic analyses were performed to assess the response of the nine FDBF-BRC systems under a suite of DBElevel ground motions. 5. Assess the overall behavior and performance of the designed FDBF-BRC systems. The design and analysis results for the various FDBF-BRC systems were compared to determine which combinations of frame strength factor and BRC strength factor exhibited the most desirable behavior and seismic performance. 4

21 6. Assess the effectiveness of the proposed design procedures in estimating the member force design demands. Two design procedures were proposed for estimating member force design demands for each of the frame. The member force design demands from each of the proposed design procedure were compared with their corresponding member force responses under the DBE ground motions to evaluate the accuracy of the proposed procedure. 7. Validate the FDBF-BRC design procedure and design trends and offer recommendations for efficient design. An FDBF-BRC system was designed based the design trends from the research program. Nonlinear dynamic analyses were performed to evaluate the response of the designed FDBF-BRC system under a suite of DBE-level ground motions. 8. Compare the performance of the FDBF-BRC system with the performance of its precursor systems. Nonlinear dynamic analyses were performed to compare the performance of the FDBF-BRC system to the performances of its precursor systems under the same suite of DBE-level ground motions to demonstrate improvement in the performance of the FDBF-BRC system over the SC-CBF system and the FDBF system. 5

22 1.4 Organization of Dissertation The remaining chapters of this dissertation are organized as follows: Chapter 2 discusses previous research relevant to this study. Chapter 3 introduces the intended behavior of an FDBF-BRC under lateral loading. Chapter 4 describes the design criteria and preliminary design procedure for the FDBF-BRC system. These include design parameter selection and estimation of member force design demands. Chapter 5 presents a parametric study of the FDBF-BRC system and discusses the prototype building. The nine FDBF-BRC designs and the results of the monotonic and cyclic nonlinear numerical static pushover analyses are compared. Chapter 6 introduces the DBE-level ground motions used in the nonlinear numerical dynamic analyses and discusses the analysis results for the nine FDBF- BRC systems. An alternative design approach is introduced here. This chapter establishes a design trend for the FDBF-BRC system from the dynamic analysis results. Chapter 7 presents an FDBF-BRC system designed based on the design trend established in Chapter 6, and discusses the response of the system to a suite of DBE level ground motions. This chapter also compares the seismic performance of the FDBF-BRC system to the seismic performance of its precursor systems (the SC-CBF system and the FDBF system). 6

23 Chapter 8 summarizes the research, presents the conclusions, and provides recommendations for future research. 7

24 CHAPTER II LITERATURE REVIEW This chapter first presents general information on self-centering (SC) systems. Specific SC systems, as well as other seismic-resistant structural systems that are relative to this research, that have been developed from previous studies are then presented. SC lateral force resisting systems are newly developed earthquake-resistant structural systems. SC systems are designed to maintain the strength and stiffness of conventional structural systems while limiting structural damage and residual drift under seismic loading. Unlike conventional systems, SC structural systems have specially designed connections that decompress and open at a specific level of earthquake loading, initiating rigid-body rocking within the system. The rocking behavior softens the lateral forcelateral drift response of the SC system without damaging the structural members. SC systems typically have high-strength post-tensioning (PT) bars or strands that provide a restoring force to return the connection to its initial closed state after an earthquake (i.e., providing self-centering). Energy dissipation elements that are deformed by the rocking behavior are often incorporated into the system to reduce the peak seismic response. 8

25 SC systems were originally developed for concrete structures (e.g., Kurama et al. 1999, Priestley et al. 1999). Kurama et al. (1999) introduced the design concepts for unbonded post-tensioned precast concrete walls (as shown in Figure 2.1) to limit the earthquake-induced damage that occurs in conventional cast-in-place concrete shear walls and in precast walls designed to emulate cast in-place shear walls. This concept has been adapted to develop seismic-resistant steel braced frame systems (e.g., Roke et al. 2006, Eatherton et al.2010). Priestley et al. (1999) performed experimental test on unbonded prestressed and post-tensioned concrete frames that were being developed to improve the seismic performance of conventional in-situ reinforced concrete frames. The study constructed and tested a 60% full size five-story precast building under simulated seismic loading. The results showed that the building suffered low structural damage and low residual drift as expected. The concept has been adapted to develop SC seismicresistant steel moment-resisting frame systems (e.g., Ricles et al. 2001, 2002; Garlock et al. 2005; Rojas et al. 2005) like that shown in Figure Self-centering concentrically braced frame systems Rocking concentrically-braced frame systems that have large drift capacity and minimal residual drift relative to the conventional CBF under seismic loading have recently been developed and studied (e.g., Roke et al. 2006, Tremblay et al. 2008, Wiebe and Christopoulos 2009, Eatherton et al.2010). One such system is the self-centering concentrically braced frame (SC-CBF) system (Roke et al. 2006), which is shown schematically in Figure 2.3. The SC-CBF is designed to decompress at the base at a specific level of lateral loading, initiating a rigid-body rotation (rocking) on its base as 9

26 shown in Figure 2.4. The SC-CBF system incorporates an extra set of columns so that the frame is separated from the gravity loading, permitting the rocking behavior without damaging the floor slabs. Friction at lateral-load bearings between the gravity system and SC-CBF at each floor level (rather than member yielding) is used to dissipate energy in the system. PT bars that run vertically over the SC-CBF s height provide a restoring force to return the frame to its foundation (i.e., self-centering the system). Roke et al. (2010) developed a performance-based design (PBD) procedure for SC- CBF systems that targets a performance condition of immediate occupancy (IO) under the DBE and collapse prevention (CP) under the MCE. Sause et al. (2010) performed experimental studies on a 0.6-scale test structure to assess the PBD criteria developed for the system. The experimental results from these hybrid earthquake simulations showed that the SC-CBF achieved the objectives of the PBD criteria: no significant structural damage occurred under DBE-level ground motions, and only a small loss of prestress in the PT bars occurred under MCE-level ground motions. In all cases, the SC-CBF system self-centered after the seismic response. Eatherton et al. (2010) introduced a self-centering system that concentrates structural damage in replaceable fuse elements. The system is called a controlled rocking braced frame system and consists of: (1) steel braced frames that remain essentially elastic and are allowed to rock about the column bases; (2) vertical PT strands that provide active self-centering forces; and (3) replaceable energy dissipating elements that act as structural fuses, yielding to effectively limit the forces imposed on the rest of the structure. Quasi-static cyclic tests were conducted on 0.43-scale controlled rocking 10

27 system test structures. The test results showed that the controlled rocking system successfully concentrated structural damage in the replaceable fuse elements and eliminated residual drift when the load is removed. Ma et al. (2010) conducted a shake table test on a 0.68-scale controlled rocking CBF system on the E-Defense shake table facility in Miki, Japan. Results from tests suggest that the controlled rocking CBF system is a viable lateral-load resisting system that can eliminate residual drift after an earthquake and can control damage in the structural members. Results from ongoing studies show that local member yielding may occur at the base of the SC-CBF first story external columns due to the concentrated vertical force acting on a single SC-CBF column during rocking. To address this issue, Blebo (2013) introduced a self-centering friction-damped braced frame (FDBF), which provides significant nonlinear drift capacity (without column uplift and pounding) while limiting structural damage and residual drift. Figure 2.5 shows the schematic of the FDBF system. The FDBF system consists of beams, columns, and braces branching off a central column. Similar to the SC-CBF, friction at lateral-load bearings between the gravity system and the FDBF at each floor level is used to dissipate energy in the system. Vertically oriented PT bars provide additional overturning moment resistance and help to reduce residual drift. Nonlinear static and dynamic analyses were performed to assess the effect of the selected yield strength of the system and the roof drift at PT bar yielding on the behavior 11

28 and seismic response of the FDBF system using nine prototype structures. The results of the study showed increasing yield strength increases the frame member sizes, frame weight, and PT bar area. The initial PT stress decreases with increasing roof drift capacity. Dynamic analyses indicate that the peak roof drift and story drift response of the FDBF is not a function of any of the selected design parameters that were examined. However, increasing the roof drift capacity at PT bar yielding reduces the probability of yielding the PT bars and the probability of the peak brace force response exceeding the brace force design demand during earthquake. Prior to PT bar yielding, the system does not exhibit residual drift, as shown in Figure 2.6. Nonetheless, the mean story drifts under the DBE were quite high and the design procedure (which was adapted from the SC-CBF design procedure developed by Roke et al. (2010)) underestimated the factored design demands in the braces (Blebo 2013). There is, therefore, a need to increase the energy dissipating capacity of the system and to develop a design procedure specifically for the FDBF for better member force design demand estimation. 2.2 Pin-supported systems and soft-story failure Earthquake resistant multi-story buildings are conventionally designed using assumed building mass and stiffness. There may be a substantial difference in the actual seismic demand and capacity over the height of the building due to differences in the building mass and stiffness distribution at the time of the earthquake, among other factors. This may lead to inter-story drift concentration and soft-story failure (Pollino et al. 2013). The inter-story drift concentration is quantified by the drift concentration factor (DCF). MacRae et al. (2004) defined DCF as: 12

29 DCF di max hi r H where di and hi are the inter-story displacement and height of story i, Δr is roof displacement, and H is the overall height of the structure. A DCF equal to 1.0 indicates a perfectly uniform inter-story drift distribution (i.e., the floor displacements are linearly proportional to their heights); while a DCF greater than 1.0 indicates the presence of inter-story drift concentration in the structure. Steel conventional concentric braced frames and steel moment resisting frames are susceptible to soft-story failure during earthquakes (e.g., MacRae et al. 2004, Qu et al. 2012). Many structural collapses have occurred during earthquakes as a result of the softstory mechanism. A study by Villaverde (1991) on the 1985 Mexico City earthquake identified soft-story mechanisms as the cause of the upper floor collapse of many buildings. Studies show that incorporating stiff pin-supported systems into existing multi-story buildings originally designed with braced frames or moment frames can prevent the problem of soft-story failure. MacRae et al. (2004) proposed the use of continuous stiff columns to mitigate large story deformation in CBFs. The columns are modeled as pinsupported and are connected to the braces by hinges. Pushover and dynamic analyses were performed to establish a relationship between the drift concentration and column stiffness. The results show that increasing the flexural stiffness of the column reduces the 13

30 DCF. Pollino et al. (2013) and Qu et al. (2014) developed a similar approach to mitigate soft-story mechanisms in sub-standard buildings through the use of a stiff Rocking Core (RC), which may be a steel truss or prestressed concrete wall, as shown in Figure 2.7. The RC is pin-supported and has high stiffness over the height of the structure to create a more uniform ductility demand and story drift profile. Energy dissipating elements are placed between the RC and the existing frame to minimize the overall system drift. Numerical analysis was performed to investigate the behavior and performance with existing sub-standard moment-resisting frame and braced frame multi-story buildings. As expected, the inter-story drift concentration decreases with increasing stiffness ratio of the RC, as shown in Figure 2.8. Qu et al. (2012) experimentally implemented a pin-supported wall frame system to retrofit an eleven story steel reinforced concrete frame structure that was originally designed with moment resisting frames. Figure 2.9 shows the schematic of the system. A total of six pinned supported walls were used in the study. Each wall was prestressed using PT tendons to avoid cracking (and the resulting wall stiffness degradation) during strong earthquakes. Similar to the stiff RC discussed earlier, the stiffness of the wall is essential to controlling the drift pattern of the structure. The pin-supported walls are connected to the foundation by hinges and connected to the existing frame at each floor by horizontal trusses. Dampers were placed between the pin-supported wall and the adjacent frame columns to dissipate a substantial amount of the seismic induced energy. Linear-elastic static and dynamic analyses were performed to compare the seismic performance of the building before and after incorporating the pin-supported walls. The 14

31 results show that the pin-supported walls redistributed the story drift and reduced the DCFs of the structure (as shown in Figure 2.10), creating a more uniform drift distribution (as shown in Figure 2.11), thus avoiding the soft-story failure of the original building. 2.3 Buckling-Restrained Braces Ordinary braces exhibit buckling deformations when subjected to high compressive force and exhibit unbalanced hysteric behavior under tension yielding and compression yielding. Consequently, they suffer significant strength degradation under monotonic compression and cyclic loading (Xie 2005). Buckling-restrained braces (BRBs) have been developed to prevent buckling, and the consequent strength deterioration, in conventional braces. BRBs exhibit symmetric hysteric behavior in compression yield and tension yielding. Figure 2.12 compares the behavior of conventional braces and BRBs. Unlike conventional braces, BRBs dissipate energy through stable tension-compression hysteric behavior (Clark et al. 1999). As shown in Figure 2.13, BRBs consist of the following components: (i) a steel core, made up of a short middle section called the yielding zone and larger end sections called the unyielding zone, that resists axial force; (ii) the surrounding concrete-filled hollow structural steel (HSS) that prevents buckling in the steel core by providing lateral restraint along the entire length of the brace; (iii) the bond-preventing layer that separates the steel core from the surrounding buckling-restraining unit so that the axial loads are resisted only by the steel core; and (iv) the connection unit that extends beyond the casing and 15

32 connects the brace to the frame (Clark 2000, Xie 2005, Sabelli 2004). BRBs are designed such that the buckling capacity of the encasing member is higher than the yield capacity of the BRB steel core to ensure that the brace does not buckle; Watanabe et al. (1988) proposed that the Euler buckling load of the steel tube should be at least 1.5 times the yield strength of the steel core. Star Seismic, LLC is one of the proprietary BRB manufacturers. Due to the availability of test results on different configurations of Star Seismic LLC s BRBs, the BRBs used in this study were assumed to be manufactured by Star Seismic, LLC. Merritt (2003) performed subassemblage testing of eight full-scale buckling-restrained braces manufactured by Star Seismic. The tests were carried out in accordance to the proposed SAEOC-AISC Recommended Provisions for Buckling-Restrained Braced s (AISC/SEAOC 2001). The results of the testing (Merritt 2003) showed that each BRB specimen demonstrated stable hysteresis behavior prior to fracture and dissipated a substantial amount of energy. Based on the test results, the following expressions were developed for calculating the adjusted tension strength factor, ω, and the adjusted compression strength factor, β: (2.2) by by (2.3) by by 16

33 (2.4) by where Δ = axial deformation of steel core, in. (mm) Δby = value of brace deformation, Δb, at first significant yield of the test specimen, in. (mm) From Equations (2.2), (2.3), and (2.4), the average values of ω and β at a deformation level of 1.5Δbm (i.e., Δ = 7.5Δby) were calculated to be 1.44 and 1.15, respectively. Δbm is defined as the value of brace deformation, Δb, at the design roof drift. The average values of ω and β are used in determining the member force design demands for structural members adjoining the BRBs. The current AISC Seismic Provisions (AISC 2010a) require that the values of ω and β for design should be determined at a deformation level of 2.0Δbm (i.e., Δ = 10.0 Δby). BRBs are incorporated into the FDBF system (culminating in the FDBF-BRC system) to increase the system s energy dissipation capacity, thereby reducing its peak drift response to ground motions as proposed by Blebo (2013). The BRBs are located at the first story external column position and are therefore referred to as bucklingrestrained columns (BRCs) in this study. The FDBF-BRC is a stiff, pinned-base selfcentering system designed to prevent soft-story failure during earthquakes and reduce residual drift after earthquakes. 17

34 Figure 2.1 Unbonded post-tensioned precast wall: (a) elevation; (b) enlarged cross section near base (Kurama et al. 1999a) 18

35 Figure 2.2 Moment-resisting connections: (a) conventional; (b) post-tensioned (Ricles et al. 2001) 19

36 Figure 2.3 Schematic of an SC-CBF system (Roke 2010) 20

37 Δroof (a) (b) Δgap θbase Figure 2.4 SC-CBF behavior under lateral loading: (a) elastic response prior to column decompression; (b) rigid-body rotation due to rocking (Roke 2010) 21

38 (a) (b) Lateral-load bearing (typ) Applied force FDBF central column L PT PT bar (typ) Adjacent gravity column (typ) FDBF external column (typ) b frame b bay Tension PT bar Compression PT bar Figure 2.5 Schematic of an FDBF system: (a) initial position; (b) rotation about the base of the FDBF central column (Blebo 2013) 22

39 (a) 4 x 105 Overturning Moment (kip-ft) st and 2 nd cycles loading 1 st and 2 nd cycles unloading PT bar detensioning (typ) Roof Drift (%) (b) 2.0 Roof Drift (% rad.) Time (s) Figure 2.6 Typical self-centering behavior of FDBF from: (a) static pushover analysis; (b) dynamic analysis (Blebo 2013) 23

40 Figure 2.7 Schematic of Sub-standard Braced and Stiff Rocking Core (Qu et al. 2014) Figure 2.8 Inter-story Drift Concentration vs. SRC Stiffness Ratio (Pollino et al. 2013) 24

41 Figure 2.9 Schematic of pin-supported wall system (Qu et al. 2012) Figure 2.10 Drift Concentration factor (DCF) before and after retrofit (Qu et al. 2012) 25

42 Figure 2.11 Inter-story drift ratio (IDR) of the building before and after retrofit: (a) original structure; (b) retrofitted structure (Qu et al. 2012) Figure 2.12 Comparison of cyclic behavior of conventional steel braces and bucklingrestrained braces (Adam 2013) 26

43 connection unit steel core restraining system Figure 2.13 BRB main components (adapted from Star Seismic 2013) 27

44 CHAPTER III FDBF-BRC SYSTEM BEHAVIOR Conventional concentrically-braced frames (CBFs) are commonly used earthquakeresistant steel frame structural systems because of their stiffness and economy. However, when subjected to design-basis earthquakes, CBFs suffer damage due to the system s limited drift capacity and the inherent possibility of a soft-story mechanism. This damage may lead to residual drift. Self-centering CBF (SC-CBF) systems have been developed to offer an increased drift capacity and eliminate residual drift in CBFs. However, results from ongoing studies have shown that under seismic loading, local yielding may occur at the base of the SC- CBF first story column, which may be due to concentrated vertical force acting on a single SC-CBF column following column uplift. A pin-supported friction-damped braced frame (FDBF-BRC) system, a modification of the SC-CBF system in which there is no column uplift, is being developed to providing significant nonlinear drift capacity while limiting residual drift and soft-story mechanisms and their related damage. 3.1 Description of FDBF-BRC System The FDBF-BRC system is shown schematically in Figure 3.1. The system comprises a single column at the middle of the bracing bay, with beams and columns branching off 28

45 the central column. The arrangement of the structural members is similar to that of a conventional CBF system; however, vertical post-tensioning (PT) bars are located at each end of the FDBF-BRC to provide additional overturning moment resistance and selfcentering behavior. Buckling restrained columns (BRCs) are incorporated into the system at the first story external column position. The FDBF-BRC system is isolated from the floor diaphragms to permit relative vertical displacement between the ends of the FDBF-BRC beams and the adjacent gravity columns. Vertical friction forces are generated at each floor level where lateral-load bearings transfer inertia forces from the floor diaphragms to the FDBF-BRC. The friction generated at the lateral-load bearings is used to supplement the energy dissipation of the BRCs to reduce the overall seismic response of the FDBF- BRC system. This behavior has been adapted from SC-CBF systems with friction bearings (e.g., Sause et al. 2010). The friction at the lateral-load bearings has been shown to provide reliable energy dissipation that is not dependent on member yielding or structural damage (Roke et al. 2010). Overturning moment from applied lateral loads causes rotation about the base of the FDBF-BRC central column. This rotation can be idealized as a rigid-body rotation, as shown in Figure 3.1(b). The behavior is similar to that of an FDBF (Blebo 2013) and concrete Pin-Supported Wall- Systems (Qu et al. 2012); however, the PT tendon in the pin-supported walls is intended to prevent cracking and the resulting degradation of the wall stiffness, not to self-center the system. The rotation about the FDBF-BRC column base elongates the BRC and the PT bar closest to the applied force (on the 29

46 tension side of the frame) and shortens the BRC and the PT bar at the side toward which the frame is rotating (on the compression side of the frame). Therefore, applied lateral force: (i) increases the compressive force in the compression BRC (ultimately leading to BRC compression yielding), (ii) reduces the compressive force and introduces a tensile force to the tension BRC (ultimately leading to BRC tension yielding), (iii) increases the tensile force in the tension PT bar (ultimately leading to PT bar yielding), and (iv) reduces the tensile force in the compression PT bar (ultimately leading to detensioning of the PT bar to an unloaded state). 3.2 FDBF-BRC Limit States The FDBF-BRC has an initial lateral stiffness that is a function of the elastic stiffness of the frame, the cross-sectional area of the BRC yielding core, and the area of the PT bars. Four desirable limit states related to the system behavior soften the lateral force-lateral drift response of the system: (1) yielding of the compression BRC, (2) yielding of the tension BRC, (3) detensioning of the compression PT bars, and (4) yielding of the tension PT bars. Two additional undesirable structural limit states are considered for the performance of the FDBF-BRC system: (5) yielding of the FDBF-BRC structural members (the beams, columns, or braces), and (6) failure of the FDBF-BRC structural members. Limit states (1) and (2) are identical to those identified for BRBFs (Fahnestock 2006), limit state (3) is a distinct characteristic of an FDBF system (Blebo 2013), and limit states (4) to (6) are identical to those identified for SC-CBFs (Roke et al 2006). The 30

47 six limit states are indicated schematically in Figure 3.2, which shows typical overturning moment-roof drift responses of FDBF-BRC systems. The difference in the occurrence of limit states is a function of certain design parameters, as discussed in the following sections BRC Yielding Gravity load from the FDBF-BRC frame, combined with the initial PT bar force, induces a compressive force into the BRCs in the undeformed structure. Rotation about the base of the FDBF central column stretches the BRC closest to the applied force (on the tension side of the frame) and further compresses the BRC at the side toward which the structure is rotating (on the compression side of the frame). Therefore, applied lateral force increases the compressive force in the compression BRC but reduces the compressive force before introducing a tensile force in the tension BRC; consequently, the compression BRC always yields before the tension BRC PT Bar Detensioning and Yielding Detensioning occurs when the compressive stress in the bars from the rotation of the system negates the initial tensile stress in the compression PT bars, reducing the stiffness of the system, as further overturning moment is resisted only by an increase in tension in the tension PT bars (the compression PT bars are assumed to buckle due to their length rather than resist compressive stresses). Detensioning is not expected to cause structural damage. The position of this limit state with respect to the other limit states is not fixed. It often occur after BRC yielding in tension (as shown in Figure 3.2(a)); however, it 31

48 could occur prior to BRC yielding in tension or BRC yielding in compression (as shown in Figures 3.2(b), and 3.2(c), respectively); depending on ratio of the area of BRC steel core, Asc, to the initial stress in the PT bars, σ0,pt. A smaller Asc/σ0,PT value leads to PT bar detensioning occurring after BRC yielding in tension; while a larger Asc/σ0,PT value leads to PT bar detensioning occurring prior to BRC yielding in compression. PT bar detensioning may also occur before or after PT bar yielding, depending on the initial stress in the PT bars. Detensioning precedes PT bar yielding when the initial stress in the PT bars is less than 50% of their yield stress and occurs after PT bar yielding when the initial stress in the PT bars is greater than 50% of their yield stress (Blebo 2013). In this study, detensioning always preceded PT bar yielding because all studied FDBFs were designed for a 3.0% roof drift capacity, which required a very low initial stress in the PT bars. The rotation of the FDBF-BRC system elongates the tension PT bars and increases the strain in the bars. When the strain demand in the bars exceeds their yield strain, the PT bars yield. This is the first structural damage in the system but is repairable; the system can easily be restored to its original state by restressing the PT bars. Yielding of the tension PT bars is intended to limit the forces that can be transmitted into the frame members; therefore, the members are designed to remain linear-elastic when the tension PT bars yield. 32

49 3.2.3 Member Yielding and Failure The FDBF-BRC is expected to offer increased lateral drift capacity prior to member yielding, much like SC-CBF systems (Roke et al. 2010). The FDBF-BRC s rotation behavior softens the lateral force-lateral drift response (as indicated in Figure 3.2), reducing the rate of increasing member force demand in the system. However, even after PT bar yielding, further increase in member force will ultimately yield the structural members, leading to permanent damage and residual drift. Structural member failure is defined as the loss of member force capacity due to excessive deformation (such as member yielding or buckling). Member failure may lead to instability and collapse of the system. If members are properly designed and detailed, deformation beyond member yielding will be required to cause member failure. Member failure is avoided in the FDBF-BRC system by 33

50 (a) L PT Lateral-load bearing (typ) FDBF-BRC central column PT bar (typ) (b) Applied force Tension PT bar Adjacent gravity column (typ) b frame b bay FDBF-BRC external column (typ) Tension BRC Buckling Compression PT bar restrained column (typ) Compression BRC Figure 3.1 Schematic of an FDBF-BRC system: (a) initial position; (b) rotation about the base of the FDBF-BRC central column 34

51 (a) Overturning Moment PT bar yielding Member yielding X PT bar detensioning BRC yielding in tension BRC yielding in compression Member failure Roof Drift (b) Overturning Moment PT bar yielding Member yielding X BRC yielding in tension PT bar detensioning BRC yielding in compression Member failure (c) Roof Drift Overturning Moment PT bar yielding Member yielding X BRC yielding in tension BRC yielding in compression PT bar detensioning Member failure Figure 3.2 Idealized overturning moment-roof drift response of an FDBF-BRC: (a) PT bar detensioning occurring after BRC yielding in tension; (b) PT bar detensioning occurring between BRC yielding in compression and BRC yielding in tension ; (c) PT bar detensioning occurring prior to BRC yielding in compression 35 Roof Drift

52 CHAPTER IV DESIGN CRITERIA AND PROCEDURE Preliminary performance-based seismic design (PBD) criteria were developed for the friction-damped braced frame with buckling-restrained columns (FDBF-BRC) system in this study. The design criteria were adapted from those for rocking self-centering concentrically braced frames (SC-CBFs) with friction-based energy dissipation (Roke et al. 2010), which target immediate occupancy (IO) performance under the design basis earthquake (DBE) and collapse prevention (CP) performance under the maximum considered earthquake (MCE). The proposed FDBF-BRC design procedure comprises four stages: the initial design stage, the BRC design stage, the PT bar design stage, and the member design stage. 4.1 Initial Design Stage This section describes the initial steps of the FDBF-BRC design procedure, which involve developing the design response spectrum and using the equivalent lateral force (ELF) procedure to determine design lateral forces and the design overturning moment for the FDBF-BRC system. The lateral force vector generated from the ELF procedure (ASCE 2010) is the total lateral force acting on the entire building. It must therefore be divided by the number of 36

53 FDBF-BRCs acting in one direction of the building to obtain the ELF force vector for each FDBF-BRC, FELF: F ELF Fi, i (4.1) N where, Fi is the total lateral force acting at floor level i and N is the number of frames acting in one direction of the building. There are four identical FDBF-BRCs oriented in each direction of the prototype building used for this study (as shown in Chapter 5); therefore, each FDBF-BRC will therefore support one quarter of the total lateral force. From the ELF force vector, FELF, the ELF base shear and overturning moment are determined using the following equations: n V b, ELF FELF, i (4.2) 1 OM ELF T h F ELF (4.3) where {h} = vector of heights from the base of the FDBF-BRC to each floor level Design Parameter Selection The BRC strength and coefficient of friction at the lateral-load bearings define the energy dissipation capacity of the FDBF-BRC system. The seismic energy that will be input into the building during an earthquake is directly proportional to the base shear the FDBF-BRC will resist; the higher the base shear, the greater the energy dissipation capacity necessary to prevent structural damage. Therefore, the BRC yield force capacity 37

54 BRCY,d is selected based on the design base shear Vb,ELF, which was derived from the ELF procedure. The BRC strength factor RBRC is used to related the BRC strength (i.e., axial force capacity) to Vb,ELF. The selection of PT bars defines the strength of the FDBF-BRC system at PT bar yielding. A dimensionless frame strength factor αy is used in the design of FDBF-BRC systems, and is defined as the ratio of the overturning moment of the system at PT bar yielding (OMY) to OM,ELF. The choice of the initial PT force defines the drift capacity at PT bar yielding. The designer may choose values of the BRC strength factor RBRC, the frame strength factor αy, and the yield roof drift θy to achieve the desired system behavior Design BRC Yield Strength From the selected value of RBRC, the BRC axial yield strength demand is determined as follows: BRC Y, d RBRC Vb, ELF (4.4) BRC A F Y, d sc ysc (4.5) where Asc = cross sectional area of yielding segment of steel core (in 2 ) Fysc = specified minimum yield stress of the BRC steel core (ksi) ϕ = 0.90 (LRFD) 38

55 The BRCs used in this study are assumed to be manufactured by Star Seismic, LLC. The manufacturer s specified minimum yield stress of the steel core is 38 ksi (Robinson 2009). The required BRC steel core area, Asc,d, is calculated as follows: A sc, d BRCY, d RBRC Vb, 0.9 Fysc 0.9 Fysc ELF (4.6) The BRC must then be selected such its steel core area Asc Asc,d. The actual yield force (kips) of the selected BRC steel core is: BRC Y A sc F ysc (4.7) The actual yield stress of the steel core, Fysc, was determined as 42 ksi from a coupon test performed by Merritt (2003); therefore, in this study, Fysc = 42 ksi. The AISC Seismic Provision (AISC 2010a) requires that the BRC connections and adjoining members should be designed to resist forces calculated based on adjusted BRC strength, which is greater than BRCY. The adjusted BRC strength in tension, BRCAT, and adjusted BRC strength in compression, BRCAC, are defined as follows: BRCAT A sc F ysc (4.8) BRCAC A sc F ysc (4.9) where ω is the strain hardening adjustment factor, and β is the compression strength adjustment factor. As mentioned previously, in this study, β and ω are determined at a deformation level of 2.0Δbm (= 10.0 Δby) from Equations (1.1) and (1.2): 39

56 by by (4.10) by by (4.11) 4.4. Design Strength Determination of the frame design strength begins with arbitrarily selection of member sizes to perform modal analysis, followed by selection of frame strength factor to define the strength of the frame Initial Member Selection As the design of the FDBF-BRC system uses modal analysis, it is necessary to start with arbitrarily selected section sizes for each structural member. Modal analysis of the frame is then carried out using standard linear elastic structural analysis software (e.g., SAP2000 (Computers and Structures, Inc. 2014). The FDBF-BRC mode shapes are then used to calculate modal spatial distributions of mass and lateral force. The modal spatial distribution of mass for mode i, {si}, is calculated as: s i i m i (4.12) where, [m] = seismic mass matrix { i} = mode shape vector for mode i 40

57 Γi = the mode participation factor for mode i, given by: i T m i i M i (4.13) {i} = { } T for a four-story FDBF-BRC M i = modal mass, defined as: M i T m i i (4.14) Design Strength and PT Steel Area Selection From the selected value of αy, the design frame strength is determined: OM OM Y, d Y ELF (4.15) The yield strength of the system is determined from a free-body diagram of the FDBF-BRC at tension PT bar yielding, as shown in Figure 4.2. For a conservative design, at tension PT bar yielding, the compression PT bars are assumed to have already been detensioned, so they provide no further overturning moment resistance. The applied lateral forces (FY,i at each floor i) generate friction forces (FED,i = μ FY,i) at the lateral-load bearings. The overturning moment resistance at tension PT bar yielding (OMY) can be determined as follows: OM Y b frame bbay ( PTY BRCAT BRCAC ) FED, i (4.16)

58 OM Y b frame bbay ( PTY BRCAT BRCAC ) Vb, Y (4.17) 2 2 OMY can be expressed as the base shear Vb,Y multiplied by the effective height h * of the system. Assuming a first mode force distribution for design, Equation (4.17) can be simplified as follows: OM Y b frame ( PTY BRC AT BRC AC) OM Y (4.18) 2 where h * 1 b bay 2 (4.19) The required design PT yield force can then be determined: PT Y, d b frame 2 OMY 1 ( BRC BRC AT AC ) (4.20) The PT bar area (APT) must then be selected such that APT σy,pt PTY,d, where σy,pt is the yield stress of the PT bars (σy,pt = 120 ksi in this study). 4.5 Design Roof Drift Capacity The roof drift capacity of the FDBF-BRC at PT bar yielding is determined based on the assumed rigid-body rotation of the system about the base of the central column and the elongation capacity of the PT bars. The elongation capacity of the PT bars can be determined as follows: 42

59 L PT, Y L PT Y, PT E PT 0, PT (4.21) Where σ0,pt and EPT are the initial stress and the elastic modulus of the PT bars, respectively. Assuming small deflections, the roof drift capacity at PT bar yielding is calculated assuming rigid body rotation of the FBDF, as shown in Figure 4.3. L PT, Y b Y frame 2 (4.22) The value of σ0,pt required to achieve the selected roof drift capacity θy can be determined by combining Equations (4.21) and (4.22) as follows: 0, PT Y, PT Y b 2 L frame PT E PT (4.23) Due to elastic frame deformations, the initial stress derived from Equation (4.23) produces a roof drift capacity larger than the selected roof drift capacity (i.e., the elastic deformation of the FDBF creates additional roof drift that does not cause elongation of the PT bars). A modification factor, λ, is introduced into Equation (4.23) to account for the elastic deformation of the frame. 0, PT Y, PT Y b 2 L frame PT E PT (4.24) λ can be determined from nonlinear pushover analysis and is always less than 1.0 (Blebo and Roke 2015). Determination of λ will be discussed in more detail in Chapter 5. 43

60 4.6 Member Force Design Demands For consistency with the design of rocking SC-CBFs with friction-based energy dissipation, the member force design demands for the FDBF-BRC are determined using a modified response spectrum analysis procedure First Mode Design Forces As previously mentioned, the structural members of the FDBF-BRC are intended to remain linear elastic at PT bar yielding; therefore, it is necessary to account for the effects of PT bar yielding in the member force design demands. The rigid-body rotation of the FDBF-BRC is predominantly a first-mode response; therefore, the first mode design forces include the PT bar yield force. The first mode lateral forces are determined as follows: F s g m g (4.25) where g = acceleration due to gravity. The first mode lateral forces {F1} must then be scaled to be in equilibrium with the PT bar yielding condition: F (4.26) 1, d Y,1 F 1 where αy,1 = a scale factor determined as follows: OM OM Y Y Y,1 (4.27) T OM1 h F 1 where {h} = the vector of floor heights from the base of the structure. 44

61 The applied forces for determination of the first mode member force design demands are indicated in Figure 4.2. The applied forces are the design lateral forces F1,d,i (denoted as FY,i on the figure), the vertical friction forces FED,i = μ F1,d,i, and the PT bar yield force PTY. The member force design demands are calculated from an elastic analysis of the FDBF-BRC Higher Mode Design Forces To maintain consistency with the design of rocking SC-CBFs with friction-based energy dissipation, the number of modes used for design is dependent upon the cumulative effective modal mass. The number of modes required to achieve a cumulative effective modal mass greater than or equal to 95% of the total structural mass will be used to determine the member force design demands. The modal lateral forces for mode i are determined as follows: F m SA g i i i i (4.28) where SAi = the design spectral acceleration value for mode i. The i th mode member force design demands are then determined from elastic analysis. The modal member force design demands are then combined using the square root of the sum of the squares (SRSS) method to estimate the design demands for the main structural members, Fx,fdd. The SRSS method assumes no correlation between the modes (Villaverde 2009). The SRSS method equation is written as: 45

62 F N N F F,,,,, x fdd i x fdd j x fdd i 1 j (4.29) where N is the number of modes. As with the rocking SC-CBF system with friction-based damping, the structural members should be designed to accommodate moment as well as axial load (Roke et al. 2010) due to the high member force design demands and the large connections necessary to transmit those forces. The structural members are therefore designed based on combined axial load and bending moment interaction (AISC 2010b). Pr P c n 8 M 9 bm rx nx M M b ry ny 1.0 Pr for P c n 0.2 (4.30) Pr 2 P c n M bm rx nx M b ry M ny 1.0 Pr for P c n 0.2 (4.31) where, Pr = factored design axial force demand determined from second-order analysis c = compression resistance reduction factor, equal to 0.9 Pn = nominal compressive strength of the member Mrx = factored design strong-axis bending moment demand determined from second-order analysis Mry = factored design weak-axis bending moment, assumed to be zero 46

63 b = flexural bending resistance reduction factor, equal to 0.9 Mny and Mnx = nominal flexural strength about each cross-sectional axis of the member If the interaction equations are not satisfied, the member sizes must be increased and more iterations of the design must be performed. 4.7 Design of the Adjacent Gravity Columns After the PT bars and the FDBF-BRC structural members have been designed, the design of the adjacent gravity columns then follows to conclude the design procedure. The adjacent gravity column is designed to support all the gravity loads from the tributary floor area, in addition to the vertical friction forces from the lateral load bearings. Three main load cases are considered in the design: (1) loading to PT bar yielding, (2) unloading after PT bar yielding, and (3) zero lateral loading. Load cases (1) and (2) are shown in Figure 4.4(a) and 4.4(b), which show free body diagrams of the adjacent gravity column at PT bar yielding and after PT bar yielding, respectively. At PT bar yielding, the friction forces from the lateral load bearings act vertically downward on the FDBF-BRC, resisting the rocking response, so the opposing forces act upward on the adjacent gravity column. This may create tension in the adjacent gravity column. However, during unloading after PT bar yielding, the friction forces act upward on the FDBF-BRC, resisting the downward motion of the system to return to its initial position, and thus act downward on the adjacent gravity column. These forces increase the total compressive force on the adjacent gravity column. 47

64 The zero lateral load case consists of only gravity loads; no friction forces are present in this design case. Unloading after PT bar yielding is the governing load case for design of the adjacent gravity columns, as the friction forces in the loading condition were not sufficient to cause tension in the adjacent gravity columns. The design demands from this case are compared to the capacity of the selected members using Equations (4.30) and (4.31) with the moments in the adjacent gravity columns assumed to be zero (Roke et al. 2010). 48

65 Direction of motion PT Y PT = 0 F Y,4 F ED,4 Centerline of gravity column F Y,3 F ED,3 F Y,2 F ED,2 BRB AT F Y,1 F ED,1 BRB AC b frame b bay Figure 4.1 Free body diagram of the FDBF-BRC at PT bar yielding 49

66 roof θ b frame 2 θ θ Figure 4.2 Rigid body rotation of an FDBF-BRC 50

67 (a) F 4 F 4 F 4 F 3 F 3 F 3 F 2 F 2 F 2 F 1 F 1 F 1 V ED V ED 0 (b) F 4 F 4 F 4 F 3 F 3 F 3 F 2 F 2 F 2 F 1 F 1 F 1 V ED V ED 0 Figure 4.3 Design cases for the adjacent gravity column: (a) loading to PT bar yielding; (b) unloading after PT bar yielding 51

68 CHAPTER V PARAMETRIC STUDY The effects of BRC strength factor RBRC and the frame strength factor αy on the behavior of an FDBF-BRC system subjected to lateral loading are presented in this chapter. This study considered three different BRC strength factors and three different frame strength factors. A total of nine frames were designed. The results of each frame design are presented in this chapter. The results of pushover analyses performed to assess the behavior of the FDBF-BRC system are also presented. 5.1 Prototype Building The prototype structure is a four story office building designed for a site with stiff soil in Los Angeles, CA. Figure 5.1 shows the floor plan of the building and a typical elevation of the prototype structure. The floor plan is a simple, 9-bay by 9-bay configuration with 20 ft. bay widths. Four FDBF-BRCs are located along each axis of the building. Simplifying assumptions were made to estimate the gravity load and the seismic mass. The dead load comprises the weights of the concrete floor slab, steel floor deck, mechanical equipment, floor and ceiling finishes, cladding weight, and an estimated weight per square foot of structural steel. Table 5.1 shows the dead load acting at each 52

69 floor level of the prototype building. The dead loads used for the design were 80 psf, 87 psf, and 87 psf, for the first, second, and third floors, respectively. The total roof dead load is 51.3 psf. The floor live load, roof live load, and partition load were assumed equal to 50 psf, 20 psf, and 15 psf, respectively (ASCE 2010) as shown in Table 5.2. Table 5.3 summarizes the gravity loads acting on the tributary area of each FDBF-BRC adjacent gravity column. Table 5.4 lists the total gravity loads acting on the lean-on column and the seismic mass per each floor. The seismic mass is calculated based on the dead loads and the partition loads that are tributary to one FDBF-BRC (ASCE 2010). The sizes of the gravity columns and the adjacent gravity columns, and the areas of the lean-on columns are shown in Table Prototype Design Three prototype frames were designed using BRC strength factors RBFC = 1.0, 2.0, and 3.0. For each value of RBRC, three frames were designed with different frame strength factors. For simplicity in the discussion of the results, the prototype frames will be referred to using the convention αy-rbrc. For example, s 4.0-1, 4.0-2, and have αy equal to 4.0 and RBRC equal to1.0, 2.0, and 3.0, respectively. Table 5.6 summarizes several design parameters for each frame. The tabulated parameters are the frame strength factor (αy), the BRC strength factor (RBRC), the roof drift capacity at PT bar yielding (θy), the total frame weight (WT), the PT bar area (APT), the cross sectional area of yielding segment of the BRC steel core (Asc), PT bar yield force capacity (PTY), BRC yield force capacity (BRCY), adjusted BRC tension force capacity (BRCAT), adjusted 53

70 BRC compression force capacity (BRCAC), and the ratio of PT bar force capacity to BRC axial force capacity (PTY/BRCY). The coefficient of friction at lateral load bearings, μ, is assumed to be constant (equal to 0.45 (Roke et al. 2010)), and the roof drift at PT bar yielding (θy) was set equal to 3.0% radians for each frame. Table 5.6 shows that, for a constant αy, increasing RBRC increases Asc and BRCY, but reduces APT, WT, and PTY/BRCY. For example, for frames with αy equal to 4.0, 4-1 (which has the lowest RBRC value), has the highest APT, WT, and PTY/BRCY values, but 4-3 (which has the highest RBRC value) has the lowest APT, WT, and PTY/BRCY values. The results also show that, for a constant RBRC, increasing αy increases APT, WT, and PTY/BRCY; but Asc remains constant. For instance, for frames with RBRC equal to 2.0, 4-2 (which has the lowest αy value), has the lowest APT, WT, and PTY/BRCY values among the three frames, and 6-2 (which has the highest αy value of 6.0) has the highest APT, WT, and PTY/BRCY values. These results suggest that Asc, APT, WT, and PTY/BRCY may be functions of αy and RBRC. As expected, BRCY, BRCAC, and BRCAT increase as RBRC increases; however, they are independent of αy. Table 5.7 presents the weight of the various structural elements that constitute WT for each frame. Consistent with the results in Table 5.6, the results show that for a constant RBRC, as αy increases, the weight of the braces, beams, internal and external columns, and the PT bars increase, while the weight of the BRC remain constant as expected. For a constant αy, as RBRC increases, the weight of the braces, beams, internal and external columns, and the PT bars decrease; however, the weight of the BRC increases. The member sizes for each frame configuration are shown on Figures 5.2 through Note 54

71 that varying αy or RBRC has a greater effect on the size of the external columns and the first floor braces than the sizes of the internal columns and the beams, since the external columns and the first floor braces are along the force path from the roof-level PT bar anchorage to the ground level. The sizes of the external columns and the first floor braces are much larger than those of the internal columns and the beams for each frame configuration. As shown in Table 5.6, for a constant RBRC, increasing αy increases APT, thereby increasing WT; however, for a constant αy, increasing RBRC reduces APT, thereby decreasing WT. 5.3 Numerical Model Nonlinear static numerical simulations were conducted using OpenSees (Mazzoni et al. 2009). For each prototype structure, the numerical model represents a single FDBF- BRC and its tributary area, as shown in Figure 5.1. The basic components of the numerical models are the FDBF-BRC, the PT bars, the adjacent gravity columns, the lean-on column, and the lateral load bearings. At this stage of the research, the FDBF- BRC structural members (beams, columns and braces) are modeled as linear elastic elements so that the force demands in these members can be determined. The nodes in the model are located at the working points of the connections between FDBF-BRC members, and these connections are assumed to be rigid (Roke et al. 2010). Nonlinearities in the FDBF-BRC analytical model are included in only the BRCs, the lateral-load bearings, and the PT bars. The BRCs are modeled as nonlinear elements using the Steel02 material to exhibit hysteric behavior and isotropic hardening in tension 55

72 and compression, as shown in Figure The BRCs are modeled with 3% strain hardening. The lateral-load bearings are modeled as contact-friction elements, which are gap elements (i.e., elements that resist only compressive forces) that can develop a friction force perpendicular to the contact force. In this study the lateral-load bearings are modeled with a coefficient of friction μ = 0.45 (Roke et al. 2010). Initial gaps in these elements are set to 0.02 inches at each floor level (i.e., these elements provide no compressive resistance until the compressive deformation exceeds 0.02 inches). The PT bars are modeled as corotational truss elements. The PT bar elements have a postyielding stiffness equal to 2% of their elastic stiffness. These elements are modeled in series with tension-only gap elements at the base, which permit the PT bars to yield in tension, but not to resist any compressive forces (to accommodate PT bar compressive buckling). The analytical models included lean-on columns to account for the stiffness of the gravity-load resisting system as well as the mass of the floor diaphragms. The lean-on column is also used to model the global P-Δ effects. The nodes at the floor levels of the lean-on column and the adjacent gravity columns are modeled to have the same degree of freedom in the lateral direction (i.e., each floor has one horizontal degree of freedom). The lateral loads were applied to the lean-on column. 5.4 Nonlinear Static Analysis Pushover analyses were used to assess the behavior of the FDBF-BRC system under lateral loading. The lateral loads used in the static analyses were proportional to the first mode lateral forces, to be consistent with the design procedure. Two types of static 56

73 analyses were performed on the structure: nonlinear monotonic pushovers and nonlinear cyclic pushovers. The static pushover responses were then compared to study the effect of RBRC and αy on the behavior of the FDBF-BRC system Monotonic Pushover Study Monotonic pushovers were performed to determine the stiffness and strength of each frame configuration and to verify the occurrence of the limit states. Each FDBF-BRC frame was pushed beyond the designed roof drift at PT bar yielding, which is 3.0% radians. The monotonic pushover analysis started with the determination of the modification factor λ (see Equation (4.33)) using the procedure introduced by Blebo et.al (2015). λ is obtained iteratively by running several preliminary analyses using the initial stress in the PT bars, σ0,pt, calculated from Equations (4.33) and (5.1). Equation (5.1) defines the λ value for the i th pushover analysis: 1.0 i Y Y ( i 1) i 1 i 1 i 1 (5.1) where θy(i-1) = the roof drift at PT bar yielding in the (i-1) th pushover analysis. Figures 5.12 through 5.17 show the results of the monotonic pushover analyses. Note that each frame exhibits the four limit states discussed previously: (1) BRC yielding in 57

74 compression, (2) BRC yielding in tension, (3) compression PT bar detensioning, and (4) tension PT bar yielding. As expected, BRC yielding in compression always occurs before BRC yielding in tension, since the analysis begins with both BRCs in compression. Consistent with the behavior of the FDBF system (Blebo 2013), PT bar detensioning occurred before PT bar yielding in each frame because the initial stress is less than 50% of the PT bar yield stress. Figures 5.12, 5.13, and 5.14 compare the monotonic pushover results for frames with the same αy values (αy =4.0, 5.0, and 6.0, respectively) but different RBRC values. Note that the initial stiffness of the frame increases with increasing RBRC; however, after the occurrence of the first three limit states, the stiffness reduces significantly at a rate directly proportional to RBRC. Figures 5.15, 5.16, and 5.17 compare monotonic pushover results for frames with constant RBRC values (RBRC =1.0, 2.0, and 3.0, respectively) but different αy values. The results indicate that increasing αy increases the stiffness of the frame. This is consistent with the behavior of FDBF systems (Blebo 2013) Cyclic Pushover Study Cyclic pushovers were performed for the nine FDBF-BRC systems to quantify the energy dissipation capacity of each frame and to compare the behavior of the nine frames. Figures 5.18, 5.19, and 5.20 compare the cyclic pushover response for frames with the same αy values (αy = 4.0, 5.0, and 6.0, respectively) but different RBRC values pushed to 1.0% roof drift. Note that each frame exhibited a flag-shaped hysteresis loop with an 58

75 area proportional to RBRC. s with RBRC = 3.0 have the widest hysteresis loops and s RBRC = 1.0 have the narrowest hysteresis loops, suggesting that a higher RBRC value may lead to greater energy dissipation. Figures 5.21, 5.22, and 5.23 compare the cyclic pushover response for frames with constant RBRC (RBRC = 1.0, 2.0, and 3.0, respectively) but different αy values pushed to 1.0% roof drift. Note that the sizes of the hysteresis loops are virtually the same regardless of the value of αy, indicating that energy dissipation may not be influenced by αy. Figures 5.18 through 5.23 indicate that residual drift may be a function of RBRC and αy. For a constant αy, residual drift reduces with a reduction in RBRC (as shown in Figures 5.18, 5.19, and 5.20); for a constant RBRC, residual drift reduces with increasing αy (as shown in Figures 5.21, 5.22, and 5.23). These responses suggest that the ratio of the PT bar yield force to BRC yield force (PTY/BRCY) may be the primary parameter determining the residual drift in the system. 59

76 Table 5.1 Design dead loads at each floor level Floor 1 Floor 2 & 3 Roof Dead Loads (psf) (psf) (psf) Floor/roof slab Floor/roof deck Roofing material Mechanical weight Ceiling material Floor finish Steel fireproofing Structural steel Exterior wall (per sq. ft. of floor area) Total Table 5.2 Design live loads at each floor level Floors 1-3 Roof Dead Loads (psf) (psf) Office Partitions Roof live load Table 5.3 Summary of gravity loads on each adjacent-gravity column Floor Dead Load (kip) Live Load (kip)

77 Table 5.4 Summary of gravity loads on the lean-on column and tributary seismic mass for one FDBF-BRC Floor Dead Load (kip) Live Load (kip) Mass (kip-s 2 /in) Table 5.5 Summary of gravity column sections and lean-on column areas. Gravity Column Section Adjacent Gravity Column Section Lean-on Column Area (in 2 ) 1st and 2nd Stories 3rd and 4th Stories 1st and 2nd Stories 3rd and 4th Stories 1st and 2nd Stories 3rd and 4th Stories 4-1 W10x49 W10x49 W10x88 W10x W10x49 W10x49 W10x88 W10x W10x49 W10x49 W10x88 W10x W10x49 W10x49 W10x88 W10x W10x49 W10x49 W10x100 W10x W10x49 W10x49 W10x100 W10x W10x49 W10x49 W10x100 W10x W10x49 W10x49 W10x112 W10x W10x49 W10x49 W10x112 W10x Table 5.6 Comparison of design parameters αy RBRB θy (%) WT (kips) APT (in 2 ) ABRB (in 2 ) PTY (kips) BRBY (kips) BRBAT (kips) BRBAC (kips) PTY/BRBY

78 Table 5.7 Design Summary of the various frame configurations Weight Weight Weight Weight Weight Weight Weight Weight Weight (kip) (kip) (kip) (kip) (kip) (kip) (kip) (kip) (kip) Brace Beam Ext. Column Int. Columns BRC PT bar Total (WT)

79 Figure 5.1 Prototype building used for the parametric study: (a) typical floor plan; (b) elevation 63